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IEAST User Manual IEAST - An Integrated Environment for

1. 0 5 F 0 s T lag Rho I k 0 5 H i i i 4 1 i i 10 5 0 5 10 Temporal lag Figure 6 3 Extended cross correlations ExtSTXCF between two space time variables Z and Y STR Model Y Z Function Y Z Random disturbances e Figure 6 4 The relationship between space time variable Z and independent variable Y under the assumption of space time regression model 6 4 Plotting correlations versus T Lag S Lag for Z Y The plotting function for correlations is under the menu Plot STACF STPACF STPXCF ExtSTXCF vs T S Lag There are two things need to be specified before plotting Correlation to be plotted The correlations can be plotted are STACF STPACF STXCF STPXCF and ExtSTXCF User need select one of them to plot Plot versus T Lag or S Lag The X axis in the plot can be chosen to be either T Lag or S Lag If T Lag is on X axis then S Lags are represented in different colored curves and vice versa User need to decide which to be plotted on X axis i e T Lag or S Lag As an example we continue the example in section 6 2 and plot its result as shown in Figure 6 5 Correlation Analyses 1 gt AutoCorrelation STACF 58 CHAPTER 6 Partial AutoCorrelation STPACF Cross Correlation STXCF Partial Cross Correlation STPXCF Extended Cross Correlation ExtSTXCF Plot Correlations versus T Lag S Lag Return Selection or lt Enter gt to exit 6 Which corre
2. Load spatial weighting structure Select Load Spatial Weighting From Files and choose a file from current directory Setup 1 gt Space time dataset 2 gt Spatial correlation structure 3 gt Information of datasets 4 gt Return Selection or lt Enter gt to exit 2 Spatial correlation structure 1 gt gt Load from file wet 2 gt gt Define by users 3 gt gt Return Selection or lt Enter gt to exit 1 Spatial correlation structure files in current directory 1 UF S10 10x10NBPP wet 2 UF S10 16x16NBPP wet 3 UF S10 20x20NBPP wet 4 UF S6 14x14NBPP wet Which Spatial correlation structure file to load Select a number 1 4 1 Loading spatial definition file UF S10 10x10NBPP wet done The spatial dimension is 10 x 10 Spatial Order Definition 1 2 QUICK START A SIMPLE EXAMPLE 7 0 0010 9 9 910 0 0 0 010 9 8 7 7 7 8 910 0 09 8 65 5 5 6 8 9 0 10 8 6 5 43 4 5 6 8 10 975 42 12 465 7 9 9 75 3 1013 5 79 975 42 12 465 7 9 10 8 6 5 43 4 5 6 8 10 09 8 65 55 6 8 9 0 010 9 8 7 7 7 8 910 0 0 0010 9 9 910 0 0 0 Spatial Relation Matrix SOD_MAX_S 10 000 000 000 083 050 050 050 083 000 000 000 000 083 050 083 083 083 083 083 050 083 000 000 050 083 125 063 063 063 125 083 050 000 083 083 125 063 125 250 125 063 125 083 083 050 083 063 125 250 250 250 125 063 083
3. 0 3 x e t 1 0 07 x WW e t 1 e Random noise e t N p 0 071 0 001251 e Weight matrices Uniform filename uniform wet Data set 100 points in time 64 sites in two dimensional space filename demo dat The dataset were generated by picking the last 100 generations from 5000 simulated generations 10 2 2 Modeling Procedures As mentioned model types and orders of some models can be simply recognized by investigating their STACF and STPACF especially for pure STAR and pure STMA models In such cases after STACF STPACF have been calculated the model type and orders of the model can be decided immediately However in reality there exist many models especially mixed models i e STARMA which are not easily recognized by investigating their STACF STPACF behaviors Hence a feasible way to discover the model type is to test AICC and BIC for every possible model type with maximum possible orders iteratively and choose the best one Then by evaluating significance of every parameter in the candidate and dropping the insignificant parameters an appropriate and parsimonious model can be obtained In the identification stage the STACF and STPACF of the dataset were computed If their behaviors are obvious as expected the model type and orders of the candidate can be decided immediately However in this STARMA example it was not the case see the autocorrelation analysis in Figure 10 1 All the mode
4. verify pgm load program verify pgm plotacf Synopsis plotacf lt acf_matrix gt lt max_t_lag gt lt max_s_lag gt Description Plot the space time autocorrelation matrices Example 10 3 COMMAND REFERENCES 91 plotacf STACF 16 3 plot2d Synopsis plot2d lt 2d data gt lt var caption gt lt xlabel gt lt ylabel gt Description Plot the array lt 2d data gt in a two dimensional graph with caption and X Y labels X is the the index of the array Y is the values of the corresponding elements Example plot2d TTrendZ Temporal Trend plot3d Synopsis plot3d lt 3d data gt lt mode gt lt var caption gt Description Plot the matrix lt 2d data gt in a three dimensional graph with caption lt mode gt is the style of the 3D plot and can be S M or B S for surface only M for mosaic map style B for Both Example plot3d STrendZ M Spatial Trend pre_estimate Synopsis pre_estimate lt init_phi gt lt init_theta gt lt model gt lt AR_max_t gt lt MA_max_t gt lt max_s_order gt Description This instruction find the initial values of parameters using different algorithms according to their model types Example pre_estimate INIT_PHI INIT_THETA MODEL 2 2 1 print Synopsis print lt model var setup string acf gt lt type gt lt PHI THE PHI THE gt lt variable1l gt lt variable2 gt Var String lt Var ACF gt lt Max T Lag gt Max S Lag Description Print model equation va
5. Correlation Analyses 6 1 The Role of STACF STPACF in STARMA Modeling STACF and STPACF calculated as shown in Eq 2 5 2 7 are quite useful in identifying space time models There are significant differences of the STACF and STPACF among pure STAR pure STMA and mixed STARMA models A tentative model can be decided by determining whether the curves of STACF or STPACF drop abruptly or both of them decay in an exponential manner along the time or spatial axis We call the abruptly dropping behavior cut off and the exponentially decaying behavior tail off By investigating at which lag in space or in time the STACF or STPACF cuts off the spatial and temporal order of the model can be decided In a manner analogous to that of the univariate ARMA subclasses of STARMA family are each characterized by distinct STACF and STPACF behaviors A pure STAR model exhibits an STACF that tails off both in space and in time and STPACF that cuts off after some lags in space and in time In contrast a pure STMA model is characterized by the STACF that cuts off after some temporal and spatial lags and the STPACF that tails off spatially and temporally As to the mixed model STARMA it exhibits STACF and STPACF that both tail off As examples to illustrate the cut off tail off behaviors of STACF STPACF of the dataset we illustrate the space time autocorrelation functions for the following two typical simulated datasets These datasets were ge
6. There is also an algorithm designed to estimate these two together but alternatively In section 8 3 these algorithm are further discussed and shown 8 3 Estimation Algorithms in IEAST There are several estimation algorithms for different purposes The first four algorithms are pre estimations i e estimating initial values The fifth is for estimating model parameters only The sixth is only for estimating spatial weighting structure in the form of SRM that is to say the model parameters are given and fixed The last one is to estimate both of them but alternatively that is in every iteration the estimations of model parameters and SRM will run once 1 Pre estimate Model Param Linear STAR Estimate initial values for linear processes STAR The best linear unbiased estimator BLUE 12 is used to find the initial values 2 Pre estimate Model Param Non linear STMA STARMA Estimate initial values for non linear process STMA and STARMA A space time extension of Hannan Rissanen algo rithm 4 was implemented for this purpose 3 Pre estimate Model Param From STACF STPACF No matter what model type it is once the type is determined the initial values of model parameters can be retrieved from the coefficients in the first few spatial temporal lags in STACF or STPACF depending on model type This is a quicker way to get the initial values and thereafter the algorithm in Algorithm 5 can optimize the estimates 4 Pre e
7. estimate EST_PHI EST_THETA EST_EE INIT_PHI INIT_THETA AR_MASK MA_MASK estimate_hr Synopsis estimate_hr lt result_phi gt lt result_theta gt lt AR_max_t gt lt MA_max_t gt lt max_s_order gt Description Find the initial values for the coming estimation process using Hannan Rissanen algorithm It is good for non linear processes such as STMA or STARMA models In ar guments the maximum temporal order in AR terms is AR_max_t the maximum temporal order in MA terms is MA_max_t and the maximum spatial order is max_s_order The final estimated initial values are stored into the variables result_phi result_theta Example estimate_hr INIT_PHI INIT_THETA 2 2 1 estimate_yw Synopsis estimate_yw lt result_phi gt lt AR_max_t gt lt max_s_order gt Description Find the initial values for the coming estimation process using Yule Walker equations It is good for linear processes such as STAR models In arguments the maximum temporal order in AR terms is AR_max_t and the maximum spatial order is assigned in max_s_order The final estimated initial value are stored into the variables result_phi Example estimate_yw INIT_PHI 2 1 exit Same as the instruction quit for endfor Synopsis for lt var gt lt init gt lt step gt lt final gt endfor Description This is a for loop instruction The variable var will start from init and decrease or increase according to the value in step for each iteration until reaching final Examp
8. or lt Enter gt to exit 2 Model Type 1 STAR 2 STMA 3 STARMA STMA 1 3 1 Max temporal order 1 20 3 1 20 2 Max spatial order 0 20 2 0 20 2 Model type is assigned to be STAR with Max Temporal Order 2 Max Spatial Order 2 7 3 Parameter Masking If there are some parameters need to be temporarily removed or put more attention to users can select Parameter Masking to eliminate the effects in estimation stage of those unwanted parameters The following is an example Model Identification 1 gt Automatic Identification Type Orders 2 gt Artificial Identification Type Orders 3 gt Parameter Masking 4 gt 5 gt Return Selection or lt Enter gt to exit 1 Model type is suggested to be STAR with Max Temporal Order 3 Max Spatial Order 2 Press any key to continue Model Identification 1 gt Automatic Identification Type Orders 2 gt Artificial Identification Type Orders 3 gt Parameter Masking Pad s 5 gt Return Selection or lt Enter gt to exit 3 Set masks for AR terms Y N N y 7 3 PARAMETER MASKING Setting AR AR AR AR AR AR AR AR AR term term term term term term term term term masks for AR terms Enable Disable E t 1 s 0 Enable Disable t 1 s 1 Enable Disable t 1 s 2 Enable Disable t
9. 050 050 083 063 250 250 000 250 250 063 083 050 050 083 063 125 250 250 250 125 063 083 050 083 083 125 063 125 250 125 063 125 083 083 000 050 083 125 063 063 063 125 083 050 000 000 083 050 083 083 083 083 083 050 083 000 000 000 000 083 050 050 050 083 000 000 000 4 Space time autocorrelation analyses Go back to main menu by pressing lt Enter gt in menu 1 Setup select 3 AutoCorrelation Analysis and then select AutoCorrelation ACF and wait results for few minutes if your computer faster enough Do the similar steps to calculate PACF by selecting Partial AutoCorrelation PACF IEAST v1 30 01 STARMA Modeling amp Analysis Main Menu 1 Setup 2 Data Preprocessing 3 Correlation Analyses 4 Model Identification 5 Parameter Estimation 6 Diagnostic Analysis 7 Forecasting 8 Preference 9 Interpreter 10 Exit Selection or lt Enter gt to exit 3 Correlation Analyses gt AutoCorrelation STACF gt Partial AutoCorrelation STPACF E gt Cross Correlation STXCF 4 gt Partial Cross Correlation STPXCF gt Extended Cross Correlation ExtSTXCF gt Plot Correlations versus T Lag S Lag gt Return Selection or lt Enter gt to exit 1 Remove mean of the variable Z 0 0005106068 for autocorrelation Y N Y Mean has been removed f
10. 1 Dataset files in current directory 54 CHAPTER 6 CORRELATION ANALYSES 1 WNV_XCrow_Detroit_10x10R45 dat 2 WNV_XHuman_Detroit_10x10R45 dat Which Dataset file to load Select a number 1 2 2 WNV_XHuman_Detroit_10x10R45 dat is being loaded done File name WNV_XHuman_Detroit_10x10R45 dat J Data size Space 10x10 Time 28 J Maximum 9 000000 J Minimum 0 000000 J Sample Mean 0 137500 J Sample Variance 0 351836 J Value for data missing 99 900000 of Missing Data 0 Press any key to continue Space time dataset 1 gt gt Load major varialbe Z from dataset file dat 2 gt gt Load second varialbe Y from dataset file dat 3 gt gt Import major variable Z from XY T T XY XYTD 4 gt gt Simulation dataset 5 gt gt Save dataset from major variable Z to file dat 6 gt gt Return Selection or lt Enter gt to exit 2 Dataset files in current directory 1 WNV_XCrow_Detroit_10x10R45 dat 2 WNV_XHuman_Detroit_10x10R45 dat Which Dataset file to load Select a number 1 2 1 WNV_XCrow_Detroit_10x10R45 dat is being loaded done File name WNV_XCrow_Detroit_10x10R45 dat Data size Space 10x10 Time 28 J Maximum 27 000000 J Minimum 0 000000 J Sample Mean 0 692143 J Sample Variance 1 545331 J Value for data missing 99 900000 of Missing Data 0 Spatial correlation st
11. 20 unchanged 20 1 99 1 2 QUICK START A SIMPLE EXAMPLE 9 Calculating STPACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Partial Autocorrelation of Z Phi s 1 2 206 073 063 028 003 3 009 006 033 001 053 4 014 031 009 001 017 5 020 015 018 013 025 6 009 008 015 013 017 7 017 026 011 015 013 8 015 015 021 024 003 9 006 013 003 012 051 10 001 025 020 002 027 11 020 002 002 015 024 12 010 002 002 004 025 13 011 016 030 011 019 14 008 027 006 044 032 15 001 041 042 031 040 16 008 004 038 020 012 17 005 029 002 008 005 18 002 018 003 042 019 19 010 018 003 016 008 20 003 005 031 034 011 5 Plotting ACF PACF The results of ACF and PACF can also be plotted versus tempo ral spatial lags as shown in Figure 1 2 by selecting Plot Correlations versus T Lag S Lag Correlation Analyses gt AutoCorrelation STACF gt Partial AutoCorrelation STPACF gt Cross Correlation STXCF 4 gt Partial Cross Correlation STPXCF gt Extended Cross Correlation ExtSTXCF gt Plot Correlations versus T Lag S Lag gt Return Selection or lt Enter gt to exit 6 Which correlation to plot 1 STACF of Z 2 ST
12. STAR 2 for non linear processes STMA mixed model 3 from STACF STPACF values 4 given by users All of them are discussed in section 8 3 8 2 Estimation of Spatial Weighting Structure We found that in some cases spatial correlation structure is more important than the estimates of model parameters Especially spatial correlation structure can provide directional information anisotropy and future tendency of a process Thus IEAST provides several methods to discover the spatial correlation structure by estimating the spatial weighting matrices for a given dataset There are three important matrices related to spatial correlation structure in IEAST i e SOD SRM and SWM SWM is in a form of computation friendly but big and awkward In fact the compact form SOD SRM has one by one mapping to SWM and concise Therefore in estimation of spatial weighting structure SOD SRM are the target to estimate Actually only the elements in matrix SRM are needed to be estimated Because the model parameters and SRM are in a same model equation in some sense at least it is not wise to estimate all of them 65 66 CHAPTER 8 PARAMETER ESTIMATION at the same time Fortunately we found that reasonable small deviations in SRM will not affect the estimation of the model parameters much Thus while implementing these algorithms we intentionally separated these two objectives estimations of spatial correlation structure and model parameters
13. gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Autocorrelation of Z Rho s 1 k 0 S lag 0 1 2 3 4 5 6 7 8 9 10 Tea gs SSS Sas aa CRT Sas Seana Sas Se SSS anaes sr ac asa eStart Sa 1 669 651 552 558 522 475 383 371 291 280 205 2 493 473 399 390 361 333 272 252 194 178 127 3 357 347 291 269 245 228 179 160 117 105 059 4 159 136 105 084 057 051 023 008 013 024 044 5 035 002 019 034 053 070 077 087 091 102 106 6 041 069 083 102 118 133 128 139 140 153 143 7 093 119 132 148 159 177 169 178 175 189 177 8 109 151 154 171 182 204 192 202 195 212 194 9 114 153 160 171 185 201 190 204 196 214 198 10 110 147 152 163 178 196 177 198 186 208 188 Partial Autocorrelation STPACF Similar to AutoCorrelation STACF Continue on last example the following is the result for STPACF Correlation Analyses gt AutoCorrelation STACF gt Partial AutoCorrelation STPACF gt Cross Correlation STXCF 4 gt Partial Cross Correlation STPXCF gt Extended Cross Correlation ExtSTXCF gt Plot Correlations versus T Lag S Lag E gt Return Selection or lt Enter gt to exit 2 Remove mean of the variable Z 0 6921428571 for autocorrelation Y N
14. gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Calculating STACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Autocorrelation of Z Rho s 1 k 0 1 002 001 010 012 2 006 001 002 O11 3 007 011 008 013 4 008 006 008 001 85 86 5 013 6 011 7 007 8 012 9 001 10 015 11 002 12 016 13 016 14 002 15 001 16 005 000 002 010 008 009 023 003 011 010 013 006 007 011 005 009 007 010 010 005 015 001 001 007 006 Calculating STPACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Partial Autocorrelation of Z Phi s 1 10 3 Command References 003 001 004 013 006 000 003 006 022 007 014 004 CHAPTER 10 INTERPRETER In this section a comprehensive description of all instructions in the interpreter is given The command keywords are in bold face In Syntax the arguments in lt gt are required arguments in are optional symbol means OR For example commandi lt argi arg2 gt means command has one required argument which is either a
15. if the estimated model appropri ately represents the dataset the variance of sample STACF of residuals should satisfy Eq 9 4 Chapter 10 Interpreter In TEAST the programming environment provided is an interpreter In general an interpreter parses and executes requested actions in every statement line by line in a program This environ ment provides a method using scripts to compose a modeling job 10 1 The Programming Environment From the point of view of functionalities provided interpreter is parallel to menu driven environ ment Each of them can separately work well However the interpreter gives an even more powerful environment for modeling STARMA processes than menu driven environment does Combining with the suggested STARMA modeling procedure Figure 2 4 under the interpreter users can construct a complete and efficient modeling flow with maximum flexibility The interactive re sponse and attention of users are not required For the different datasets with the same setup and modeling flow the only thing different in the program is the instructions for retrieving dataset from different files Users need not to program it again 10 1 1 Invoke and Quit To enter the interpreter select Interpreter under the main menu As we expected there is a prompt after entering That means it is ready to execute commands or accept program input To leave the interpreter type command quit and press lt Enter gt key as shown
16. 0 40 Z t 1 0 30 W1 Z t 1 C 0 20 Z t 2 0 10 W1 Z t 2 e t Simulating STAR model 100 Done 3 4 DATASET FROM EXCEL FILES AND OTHER SOURCES 29 3 4 Dataset from Excel files and other sources If your space time dataset is in an Excel file you need follow the procedures below to transform it into IEAST dat format 1 Make sure your data in Excel is arranged in one of the three ways XY T T XY or XYTD as described in section 3 2 2 In Excel save your data as an Tab separated file txt or Comma separated file csv Copy the file your_file txt for example to IEAST working directory Following the steps as in section 3 2 to import it into the major variable Z NN 0 Save the dataset in Z to an IEAST dataset file dat TEAST can only accept text data files not binary So if the data files are not in text users need to transform them into ASCII data files in advance 3 5 Important Global Variables Information stored in the following global variables is the important characteristics of your STARMA model They are essential for all the analyses in TEAST Z Matrix This is the major variable storing the main dataset It has N_SITEx N_SITE rows and T columns Y Matrix This is the secondary variable storing the auxiliary dataset It has N_SITE x N_SITE rows and T columns It often is used while cross analyzing between variables Z and Y N_TIME Integer Total num
17. 2 Box Jenkins Modeling The Box Jenkins modeling procedure involves identifying an appropriate ARMA model fitting it to the dataset and using the fitted model for forecasting One of the attractive features of the Box Jenkins method to forecasting is that ARMA processes are a very rich class of possible models and it is usually possible to find a model which provides an adequate description for the data The Box Jenkins modeling procedure involves iterative steps of model identification pa rameter estimation model diagnosis and forecasting Figure 2 2 outlines a modified Box Jenkins procedure for searching an adequate STARMA model Before estimation a step pre estimation is suggested to be done in advance Each of these steps are discussed and illustrated in the sections that follow 23 STARMA Model Definition The STARMA model class is characterized by linear dependence lagged both in space and time Here we follow primarily the notation of Pfeifer s 10 STARMA expresses Z t the observation of the random variable Z at site i i 1 2 N and time t as a weighted linear combination of past observations and random noise inputs which may be lagged both in space and time 2 3 STARMA MODEL DEFINITION OOOOOO 0000000 0000000 0000000 0000000 OODODOS OJUO Ist order OOODOdDO 0000000 0000000 0000000 0000000 OOOOOOO OOOO 2nd order O9OOOD OS 0000000 ODO 0000000 0000000 0000000 JJJ 3rd order 4t
18. 2 s 0 Enable Disable t 2 s 1 Enable Disable t 2 s 2 Enable Disable t 3 s 0 Enable Disable t 3 s 1 Enable Disable t 3 s 2 Enable Disable Set masks for MA terms Y N N E D E E D E E D E E D E E D E E D E E D E E D E E D E 64 CHAPTER 7 MODEL IDENTIFICATION Chapter 8 Parameter Estimation Estimation is the most time consuming stage in some cases In estimation the parameters in the candidate model with the selected type and orders are estimated We implemented this stage with the maximum likelihood estimates MLE In IEAST we have implemented not only the algorithms for estimating model parameters but also the algorithms for spatial weighting structure In other words IEAST can retrieve the spatial weighting information from the dataset 8 1 Estimation of Model Parameters Because of nonlinear nature of STMA and STARMA we have to use a nonlinear optimization algorithm to find the parameters for STARMA family Marquardt s algorithm 8 was chosen to implement the nonlinear optimization of model parameters During the optimization process to avoid the likelihood function reaching a local maxima and to reduce the iterations needed in the optimization we added an extra stage pre estimation to calculate the appropriate initial values for the optimization algorithm In IEAST there are four pre estimation methods implemented 1 for linear processes
19. 29 3 6 Visualize Your Dataset sariri 24 a dd ad 30 Spatial Correlation Structure 33 4 1 Spatial Dimension N SITE and NxN o e e 33 4 2 Spatial Order Definition SOD o o 33 4 3 Spatial Relation Matrix SRM e 34 4 4 Spatial Weighting Matrices SWM eee ee eee 35 4 5 Configure Your SysteM 2 2 EO E ee 35 4 6 Load Spatial Correlation Structure or Weight Matrices Directly from Files 37 Data Preprocessing 39 91 Remove Mean i225 IEEE AA OE I EB eS 39 5 2 DeSeasonalize ee 39 5 3 Difference by One 5 GG ee eS WP OEE a Ge Ee A 41 Did De Trenid e f4 Je ane Peo AE EE GI RE Ee ee WEAN S Al 5 5 Box Cox Transforma 44 5 6 Subsequencing Resampling 0 00 00 000002000048 45 ber pmoothing 4 4 eK nS ak hae A i ee eS SB A 45 0 6 Missing Data terror PRR RR PE a h 46 5 9 Filter with a given STARMA model o 0200022 eee 47 Correlation Analyses 49 6 1 The Role of STACF STPACF in STARMA Modeling 49 6 2 Autocorrelations STACF and STPACF oo 50 6 3 Cross Correlations STXCF STPXCF and ExtSTXCF 53 6 4 Plotting correlations versus T Lag S Lag for Z Y lt o 57 Model Identification 61 7 1 Automatic Identification sooo e ee ee 61 7 2 Artificial Identification 2 ee 62 7 3 Parameter Masking 2 ee 62 Parameter Estimation 65 8 1 Estimation of Model Parame
20. An extended STARMA modeling procedure The area enclosed by dotted line is for the case that model type cannot be decided simply from autocorrelations AICC BIC of every possible model type is evaluated and compared to select the best model type seta g ii e OA Wane A OR ie 22 An example of 3D plotting of dataset o o 31 3D plots of the original dataset o e ee ee 42 5th order polynomial spatial trend and temporal trend average over space 44 3D plots of the dataset after de trending o 44 Dataset before left and after right IDM d smoothing with 3x3 window 46 Autocorrelation functions of a dataset before left and after right filtering 48 STACF and STPACF for STAR 2 1 o 50 STACF and STPACF for STMA LD o 50 Extended cross correlations ExtSTXCF between two space time variables Z and Y 57 The relationship between space time variable Z and independent variable Y under the assumption of space time regression model o o 57 STACF plots versus T Lag left and S Lag right 59 Residuals STACF and STPACF for the example in Quick Start 72 vi 10 1 STACF and STPACF for the simulated dataset STARMA 1 1 LIST OF FIGURES List of Tables 1 1 3 1 3 2 3 3 Accuracies of model type selection using variances AICC BIC and AICC BIC base
21. DATASETS FROM OTHER FORMATS 25 Table 3 2 An example for format T XY of SPACE 10x10 and TIME 1 28 DX LY 1T 1 DX LY 1T DX 1Y 1T 2 D X 2 Y 1 T 1 DX 2 Y 17 2 DX 2 Y 1T 28 D X 10 Y 10 T 1 D X 10 Y 10 T 2 D X 10 Y 10 7 28 Table 3 3 An example for format XYTD of SPACE 10x10 and TIME 1 28 1 3 convenient remember to save your imported dataset to a dat file for next time use by Save dataset from major variable Z to file dat Following the procedure below Space time dataset gt gt Load major varialbe Z from dataset file dat gt gt Load second varialbe Y from dataset file dat gt gt Import major variable Z from XY T T XY XYTD gt gt Simulation dataset gt gt Save dataset from major variable Z to file dat gt gt Return Selection or lt Enter gt to exit 3 The file to be imported must be tab separated txt or comma separated csv and in one of the formats XY T T XY or XYTD Select file format 1 Tab separated txt 2 Comma separated csv Selection 1 2 1 Tab separated files in current directory 1 CCF varCrow XYTD txt 2 CCF varHuman XYTD txt 3 DeerTB 10x10 XYTD txt 4 NOAA_Temperature XYT txt Which Tab separated file to load Select a number 1 4 1 Select data format 1 XY T 2 T XY 26 CHAPTER 3 DATA PREPARATION 3 XYTD Selec
22. Fri Sep 19 18 02 42 2003 patial Relation Matrix SOD_MAX_S 6 O tae Hu H 1i 0 30 0 20 0 11 0 20 0 12 0 07 Advancement of Average Parameter SSE 0 000021 Time elapsed 1 43 50 98 CHAPTER 8 PARAMETER ESTIMATION 8 3 ESTIMATION ALGORITHMS IN IEAST EEEE Iter 6 Fri Sep 19 19 46 33 2003 Spatial Relation Matrix SOD_MAX_S 6 SRM 0 34 0 25 0 21 0 38 0 00 0 33 0 21 0 32 0 49 ePHI 0 30 0 20 0 11 0 20 0 12 0 07 Advancement of Average Parameter SSE 0 000010 Time elapsed 1 47 56 93 Z t C 0 30 Z t 1 0 20 W1 Z t 1 0 11 W2 Z t 1 0 20 Z t 2 0 12 W1 Z t 2 0 07 W2 Z t 2 e t 70 CHAPTER 8 PARAMETER ESTIMATION Chapter 9 Diagnostic Analysis Once the model type order and parameters have been identified and estimated the acquired model can be used to forecast or to analyze the future behavior of the system of interest To prevent unacceptable forecasting errors we have to check if the fitted model is appropriate If there is abundant data on hand it would be beneficial to identify and estimate the model on the first half and check adequacy of the model on the remainder If the amount of data is limited models can be identified estimated and diagnostically checked using the same dataset These checking methods are called diagnostic checks There are various methods to check adequacy of STARMA models e g autocorrelation checking randomne
23. The 27 order neighbors are farther away from than the 1 order neighbors but closer than the 37 order neighbors The model is specified as a STARMA p q Ak my That is autoregressive terms have the maximum lag p in time and Az in space and moving average terms have the maximum lag q in time and m in space There are two important subclasses of STARMA models A stochastic process Z t is said to be a Space Time AutoRegressive process of temporal order p and spatial order A ST AR p Ax if q 0 and a Space Time Moving Average process of temporal order q and spatial order mg ST MA q my if p 0 The STAR process is described as P Ak Z t D gt Y ba WO Z t k e t k 1 1 G 2 2 20 CHAPTER 2 STARMA MODELS AND MODELING and the STMA process is described as q Mk 2 3 Z t e t YY Ow WWe t k k 1 1 64 These two subclasses are essential Not only can many practical stochastic processes be simply attributed to STAR or STMA but also there exist primary statistical differences between them We discuss these issues in detail later 2 4 Elementary Statistical Tools for Analyzing STARMA Models The following are the definitions of the elementary tools for analyzing STARMA models These tools provide a systematic way for revealing the types and orders of the underlying model By the definition in 10 the space time covariance function between I and kt order neighbors at time lag s can be expre
24. Time elapsed ES Iter 3 eTHETA 0 60 0 11 Advancement of Time elapsed ee tae al Iter 4 eTHETA 0 61 0 10 Advancement of Time elapsed Z t e t Mon Aug 09 22 18 Average Parameter 0 0 5 39 Mon Aug 09 22 18 Average Parameter 0 0 6 11 Mon Aug 09 22 18 Average Parameter 0 0 4 54 Mon Aug 09 22 19 Average Parameter 0 0 4 27 48 2004 SSE 0 164246 53 2004 SSE 0 001766 59 2004 SSE 0 000081 04 2004 SSE 0 000010 0 62 e t 1 0 10 W1 re t 1 AICC BIC Variance AICC 6 5541e 04 BIC 2 7763e 04 eVARIANCE 0 0014238 CHAPTER 10 INTERPRETER 10 2 A SAMPLE PROGRAM 0 50 0 10 eTHETA 0 29 0 08 Advancement of Average Parameter SSE 0 000015 Time elapsed 0 0 18 33 Z t 0 50 Z t 1 0 10 W 1 Z t 1 e t 0 29 e t 1 0 08 W 1 e t 1 AICC BIC Variance AICC 6 6759e 04 BIC 2 8363e 04 eVARIANCE 0 0012603 ak ak ak k ak k ak k ak k k ok ok Final Model FEO I IR IK MODEL ARMA fAICC 6 6759e 04 fBIC 2 8363e 04 fVARIANCE 0 0012603 Z t 0 50 Z t 1 0 10 W 1 Z t 1 e t 0 29 e t 1 0 08 W 1 e t 1 Significance sPHI 0 0010000 0 0010000 sTHETA 0 0010000 0 0100000 Residual s STACF STPACF analysis Initializing Covariance Look up Table CovarTB for major varialbe Z gt gt gt gt gt
25. W 1963 An algorithm for least squares estimation of nonlinear parameters Journal of the Society of Industrial And Applied Mathematics 11 431 441 Martin R L and Oeppen J E 1975 The identification of regional forecasting models using space time correlation functions Transactions of the Institute of British Geographers 66 95 118 Pfeifer P E 1979 Spatial Dynamic Modeling Unpublished Ph D dissertation Georgia Institute of Technology Atlanta Georgia Pfeifer P E Deutsch S J 1980 Stationarity and invertibility regions for low order STARMA models Communications in Statistics Simulation and Computation 9 5 551 562 Pfeifer P E Deutsch S J 1980 A three stage iterative procedure for space time modeling Technometrics 22 1 35 47 Pfeifer P E Deutsch S J 1981 Variance of the sample space time autocorrelation function Journal of the Royal Statistical Society Series B 43 1 28 33 Schwarz G 1978 Estimating the dimension of a model The Annals of Statistics 6 2 461 464 95 96 BIBLIOGRAPHY 15 Shibata R 1976 Selection of the order of an autoregressive model by Akaike s information criterion Biometrika 63 117 126 16 Stoffer D S 1986 Modeling risk from a disease in time and space Journal of the American Statistical Association 81 395 762 772
26. Y Mean has been removed for calculating autocorrelations Max spatial lag 4 0 10 10 Max temporal lag 20 1 27 10 Initializing Covariance Look up Table CovarTB for major varialbe Z gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Calculating STPACF 6 3 CROSS CORRELATIONS STXCF STPXCF AND EXTSTXCF 53 gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Partial Autocorrelation of Z Phi s 1 2 084 228 127 147 141 056 072 035 001 020 015 3 002 025 007 058 046 056 028 058 066 020 039 4 183 323 108 097 118 001 035 050 024 024 030 5 056 023 081 072 122 050 030 010 009 031 011 6 018 005 036 060 047 118 000 058 023 073 000 7 006 075 013 007 066 049 015 055 010 046 055 8 005 119 061 070 102 202 025 128 055 107 038 9 021 030 001 020 032 193 025 178 092 261 132 10 021 040 044 064 140 379 068 207 386 405 209 6 3 Cross Correlations STXCF STPXCF and ExtSTXCF IEAST provides not only single variable s correlation analysis but also the cross correlation be tween two space time variables i e Zand Y The model based on for t
27. below 1 Setup 2 Data Preprocessing 3 AutoCorrelation Analysis 4 Model Identification 5 Parameter Estimation 6 Diagnostic Analysis 7 Forecasting 8 Simulation 9 Preference 10 Interpreter 11 Exit 77 78 CHAPTER 10 Selection or lt Enter gt to exit 10 See you IEAST v1 30 01 STARMA Modeling amp Analysis 1 Setup 2 Data Preprocessing 3 AutoCorrelation Analysis 4 Model Identification 5 Parameter Estimation 6 Diagnostic Analysis 7 Forecasting 8 Simulation 9 Preference 10 Interpreter 11 Exit Selection or lt Enter gt to exit 10 1 2 Program Mode and Interactive Mode INTERPRETER There are two modes in the interpreter i e program mode and interactive mode Under the prompt if the first word of the inputs is a command listed in section 10 3 this line of statement will be treated in an interactive way and executed right away If the first word is a number a line number separated by tailing space this line of input will be put into system memory as a part of a program and not be executed in the mean time Simply speaking in the interpreter prompt input line if the leading is a integer number from 1 to 10000 then the entire line will be put into system memory as part of a stand by program Otherwise the command entered will be executed right after the Enter key pressed Take a look at the example below The input list after prompt means show
28. current program But after that the input 45 list means store the statement list into line 45 of the program When we list the program again there is an extra line 45 list added list 10 load data verification dat 20 load weight uniform wet 30 stacf ST_ACF Z 16 3 40 plot ST_ACF 16 3 ACF 50 end 45 list list 10 load data verification dat 10 2 A SAMPLE PROGRAM 79 20 load weight uniform wet 30 stacf ST_ACF Z 16 3 40 plot ST_ACF 16 3 ACF 45 list 50 end 10 1 3 Notices for the Interpreter All data types data structures variables expressions functions and operators are inherited from of GNU Octave http www octave org Line number is for program sequence reference and flow control Line number can be a integer from 1 to 10000 Line number 0 is reserved for system use In program the line number can be arbitrarily chosen however the line numbers are the running sequence of the program Users need to maintain its logical order in the sense of line numbers The executing sequence will follow the order of the line numbers from small to big in a program The newly entered line of program will overwrite the existing one with the same line number in memory All programs need to be either loaded or entered into system memory first then type com mand run to run the program Spaces are only used for separation among commands and arguments Any line of a program shoul
29. model will result in a small estimated noise variance but it is not necessarily desirable In fact it is not advantageous from a forecasting point of view to choose orders arbitrarily 9 4 AICC BIC ANALYSIS 75 large Hence we need to introduce a penalty into the criterion to discourage the fitting of models with too many parameters We extended and implemented a space time extension of bias corrected AIC 1 referred to as the AICC suggested by Hurvich and Tsai 6 to be the criterion in determining the model type orders and checking adequacy of parameters The space time extension of AICC criterion for STARMA can be defined as follows 7 bs TN K K 0 9 2 AICCsTARMA TNlogl FN KIKO K K 6 where the maximum likelihood estimate of the residual variance G is 6 s TN the residuals sum of square errors is 2 T Pp Ax q Mk s zo Y Y ba WOZ t k 0 040 WWe t v t 1 k 1 I 0 k 1 I 0 T is the number of points in time of the observations N is the number of sites in each observation is the estimated parameters for autoregressive terms is the estimated parameters for moving average terms K B is the number of elements in K O is the number of elements in O In the definition of AICC Eq 9 2 the second term is the penalty for high order models which makes the value of AICC higher than that of lower order model Thus in determining an adequate model we only
30. need to find a model with some type orders and parameters which has a minimum value of AICC However because AICC has tendency of overfitting autoregressions as shown in 7 and 15 another criterion i e BIC Bayesian Information Criterion is suggested to correct this tendency BIC is suggested by 2 14 and extended to space time case as follows S B 9 3 BICsrarma TNlog 6 ll K amp K log TN Both AICC and BIC are relative measures They are useful when select the best model out of many candidates In IEAST they are especially useful when using the IEAST programming environment in which iterative testing of AICC BIC on different models is a powerful method to pick a good model for the dataset out of lots of candidates The following is an analysis of AICC BIC to the example in Quick Start section 1 2 Diagnostic Analysis Statistical Significance AICC BIC Analysis STACF of Residuals STPACF of Residuals 76 CHAPTER 9 DIAGNOSTIC ANALYSIS Selection or lt Enter gt to exit 2 The residual sum of square errors 12 70624 Estimated Mean 0 00001 Estimated Noise Variance 0 00127 AICC 66675 467 BIC 28313 603 9 5 Residuals Variance With regard to variance checking it has been shown by 10 that if a process the residuals in our case is pure white noise then 1 9 4 variance pig s NTa where fio s is the sample STACF as defined in Eq 2 5 That is
31. oc0o00000000000000000000 oc0o000000000000000000000 oc0o00000000000000000000 oc0o00000000000000000000 oc0o000000000000000000000 4 3 Spatial Relation Matrix SRM After the SOD defined we need to specify the relative importance of a cell in a given spatial order This information is stored in SRM which has the same size as that of SOD In the last example of SOD there are eight cells in first spatial order We can give the following relative importance to the cells in first spatial order with anisotropic weights to emphasize the importance of the northern point of the central cell weight 0 3 All weights in the same spatial order should sum to one In TEAST SRM can be given either by users or by system if isotropic 4 2 SRM 1 o0 1 4 4 SPATIAL WEIGHTING MATRICES SWM 35 4 4 Spatial Weighting Matrices SWM These are the actual matrices used by IEAST for calculations Given SOD SRM and N_SITE SWM can be generated by system automatically SWM is normally a huge matrix with SOD_MAX_S see section 3 5 rows and N_SITE x N_SITE columns 4 5 Configure Your System To configure your two dimensional spact time system under the menu 1 Setup choose Configure System and answer the first few questions if necessary Spatial dimension If you already have your dataset you don t have to enter this number It will be overwritten by the file loading thereafter I
32. space The 3D plots of the original dataset are shown in Figure 5 1 File name Combined_WAM_20x20full dat Data size in space 20x20 J in time 28 of Missing Data 0 Data Preprocessing 1 gt Remove Mean 2 gt De seasonalize 1 B7dd Z t 3 gt Diference by one 1 B Z t 4 gt De trend 5 gt 6 gt Subsequencing Resampling 7 gt Smoothing 8 gt Missing Data 9 gt Filter with a given STARMA model 10 gt Undo previous action 11 gt Return Selection or lt Enter gt to exit 4 De trending J 1 gt gt Trend in space Trend Surface Analysis 5 4 DE TREND 43 gt gt Trend in time gt gt Remove trend in space time Selection or lt Enter gt to exit 1 Trend in space 1 gt gt gt Polynomial Regression OLS 2 gt gt gt Fourier Analysis 3 gt gt gt Average over time 4 gt gt gt Remove the Trend Surface 5 gt gt gt Return Selection or lt Enter gt to exit 1 Enter the order of the trend surface 1 1st 2 2nd 3 3rd 1 10 5 Spatial trend Poly Regression gt ZTS 1 gt gt Trend in space Trend Surface Analysis 2 gt gt Trend in time 3 gt gt Remove trend in space time 4 gt gt 5 gt gt Return Selection or lt Enter gt to exit 2 s
33. t 2 e t Again remember to save the dataset to a dat file after simulation is done S a Space time dataset gt gt Load major varialbe Z from dataset file dat gt gt Load second varialbe Y from dataset file dat gt gt Import major variable Z from XY T T XY XYTD gt gt Simulation dataset gt gt Save dataset from major variable Z to file dat gt gt Return Selection or lt Enter gt to exit 4 Weight matrices are not yet specified Press any key to load wet file Spatial correlation structure files in current directory 1 UF S6 10x10NBPP wet 2 UF S6 14x14NBPP wet 3 UF S6 20x20NBPP wet 4 UF S8 16x16NBPP wet Which Spatial correlation structure file to load Select a number 1 4 1 Loading spatial definition file UF S6 10x10NBPP wet done The spatial dimension is 10 x 10 Spatial Order Definition oonan o ankr wk OD OPNEN POD aOwrorw io OPNEN FOI anh wD oonan o Spatial Relation Matrix S0D_MAX_S 6 000 125 063 063 063 125 000 125 063 125 250 125 063 125 063 125 250 250 250 125 063 063 250 250 000 250 250 063 063 125 250 250 250 125 063 125 063 125 250 125 063 125 000 125 063 063 063 125 000 Spatial dimension NxN N 10 4 100 Model type 1 STAR 2 STMA 3 STARMA N A 1 3 1 Max temporal order N A 1 20 2 28 CHAPTER 3 DATA PREPARATION Max spatial ord
34. unix system 3 Give commands under prompt as shown below mkdir lt Your_Working_Directory gt cd lt Your_Working_ Directory gt gunzip ieast v1 xx xx tar gz tar xvf ieast vi xx xx tar 4 To start give the command octave ieast m 1 6 Performance The modeling performance of IEAST using various model type selection criteria in 150 Monte Carlo simulated datasets of STAR STMA and mixed processes 50 datasets for each is shown in Table 1 1 We used four different criteria variance of residuals AICC only BIC only and AICC BIC respectively The parameters for generating datasets were randomly chosen with higher weights on the parameters having lower spatial or temporal orders As expected using AICC or variance has a bias toward overparameterized mixed models BIC has much better performance than AICC All 50 simulated STAR datasets were correctly classified as STAR The accuracies of type determination are 86 and 66 for datasets based on STMA and mixed models respectively Using the product of AICC and BIC as the criterion can give even better performance where the accuracies can reach at least 78 for all types These criteria provide reliable quantitative measures for determining model choice whereas previous researches relied on subjective qualitative behaviors of STACF and STPACF 1 7 Menu System Current version version 1 30 01 of IEAST is designed as a menu driven selection system if not in the interpr
35. with the following STARMA model and parameters Z t 0 30Z 1 0 20W Z t 1 0 10W Z 1 0 20Z t 2 0 10W Z t 2 0 07W Z t 2 e t where Ele t 0 Varle t 0 001251 The following are the steps needed for STARMA analysis and modeling of the given dataset 1 2 QUICK START A SIMPLE EXAMPLE 5 1 Invoking IEAST On most systems the way to invoke IEAST is with the command octave ieast m Under UNIX system you can simply type ieast m under Octave prompt IEAST displays a short message and then the main menu If you get into trouble you usually can interrupt IEAST by type Control C Doing this will normally return you to your operating system To exit IEAST select Exit under the main menu or press several lt Enter gt till exit Now run GNU Octave under your OS Unix Windows Mac then enter the main program ieast be sure you are in the same directory as IEAST system octave GNU Octave version 2 1 36 i686 pc cygwin Copyright C 1996 1997 1998 1999 2000 2001 2002 John W Eaton This is free software see the source code for copying conditions There is ABSOLUTELY NO WARRANTY not even for MERCHANTIBILITY or FITNESS FOR A PARTICULAR PURPOSE For details type warranty Report bugs to lt bug octave bevo che wisc edu gt gt gt dir ieast v1 30 01 gt gt cd ieast v1 30 01 gt gt ieast IEAST v1 30 01 STARMA Modeling amp Ana
36. 0 083 000 083 000 000 000 4 100 2004 0 000001 0 13 W2 Z t 1 0 06 W2 Z t 2 STARMA modeling is pretty close to the one used to simulate It is a good result actually This is only a simple example for demonstration The later chapters will introduce these procedures in more details 1 3 FEATURES 13 1 3 Features TEAST version 1 30 0lis mainly for STARMA modeling and analysis and features A true spatial and temporal analysis software Full configurability The spatial and temporal correlation structures can be specifically de signed by users Interactive programming environment A script interpreter is provided Highly integrated instructions are provided for simplifying programming efforts and improving efficiency of analysis and modeling Estimation of spatial weighting structure The spatial weighting structure can be retrieved and estimated from given datasets Flexibility of defining spatial structure A three step definition for spatial structure makes IEAST more flexible in the definition of spatial structure i e SOD gt SRM gt SWM Improved estimation algorithm The introduction of space time extension of Hannan Rissanen algorithm can greatly reduce the estimation time and possibility of converging to wrong re sults Improved diagnostic measures The space time AIC BIC provide a good criteria for the selection of model type and orders especially useful in programmi
37. 083 050 083 083 083 083 083 050 083 000 000 050 083 125 063 063 063 125 083 050 000 083 083 125 063 125 250 125 063 125 083 083 050 083 063 125 250 250 250 125 063 083 050 050 083 063 250 250 000 250 250 063 083 050 050 083 063 125 250 250 250 125 063 083 050 083 083 125 063 125 250 125 063 125 083 083 000 050 083 125 063 063 063 125 083 050 000 000 083 050 083 083 083 083 083 050 083 000 000 000 000 083 050 050 050 083 000 000 000 Correlation Analyses 1 gt AutoCorrelation STACF 2 gt Partial AutoCorrelation STPACF 3 gt Cross Correlation STXCF 4 gt Partial Cross Correlation STPXCF 5 gt Extended Cross Correlation ExtSTXCF 6 gt Plot Correlations versus T Lag S Lag 52 CHAPTER 6 CORRELATION ANALYSES 7 gt Return Selection or lt Enter gt to exit 1 Remove mean of the variable Z 0 6921428571 for autocorrelation Y N Y Mean has been removed for calculating autocorrelations Max spatial lag 4 0 10 10 Max temporal lag 20 1 27 10 Initializing Covariance Look up Table CovarTB for major varialbe Z gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Calculating STACF gt gt gt gt gt gt gt gt gt gt gt
38. 50 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 load data demo dat load weight uniform wet stac ST_ACF Z 16 3 plotacf ST_ACF 16 3 ACF stpacf ST_PACF Z 16 3 plotacf ST_PACF 16 3 PACF let fAICC 0 MODEL_MAX_T 1 MODEL_MAX_S 1 MASK ones MODEL_MAX_T MODEL_MAX_S 1 let types STAR STMA ARMA for1113 let MODEL types I print string in print var MODEL print string let iTHETA zeros size MASK iPHl zeros size MASK pre_estimate iPHI iTHETA MODEL MODEL_MAX_T MODEL_MAX_T MODEL_MAX_S estimate ePHI eTHETA eRESIDUALS iPHI iTHETA MASK print model MODEL ePHI eTHETA aicc AICC ePHI eTHETA eRESIDUALS bic BIC ePHI eTHETA eRESIDUALS variance eVARIANCE eRESIDUALS print string AICC BIC Variance n print var AICC BIC eVARIANCE if fAICC lt AICC 250 let fAICC AICC fBIC B1C fTYPE MODEL fPHI ePHI fTHETA eTHETA fRESIDUALS eRESIDUALS fVARIANCE eVARIANCE endfor print string r k k k kk kkk nk Final Model gt ote kkk kk kk kkk n let MODEL fTYPE print var MODEL fAICC fBIC fVARIANCE print model MODEL fPHI fTHETA print string n Significance n significance sPHI sTHETA ePHI eTHETA MASK print var sPHI sTHETA print string n Residual s STACF STPACF analysis n stacf ACF_RESIDUALS eRESIDUALS 16 3 plotacf ACF_RESIDUALS 16 3 ACF stpacf PACF_RESIDUALS eRESIDUALS 16 3 plotacf PACF_RESIDUALS 16 3 PACF e
39. Carlo simulated datasets Using variances Simulated datasets Datasets identified as based on STAR STMA Mixed STAR 4 0 96 STMA 4 6 90 Mixed 8 2 90 Using AICC Simulated datasets Datasets identified as based on STAR STMA Mixed STAR 16 0 84 STMA 4 6 90 Mixed 8 2 90 Using BIC Simulated datasets Datasets identified as based on STAR STMA Mixed STAR 10 0 0 STMA 4 86 10 Mixed 18 16 66 Using AICC BIC Simulated datasets Datasets identified as based on STAR STMA Mixed STAR 100 0 0 STMA 4 78 18 Mixed 16 4 80 1 8 Conventions The following should be kept in mind 1 Because of being developed on GNU Octave the IEAST s formats of many outputs inputs from to IEAST are the same as that of Octave 2 Currently the only user interface supported is text mode menu driven 3 The spatial dimension has to be N by N e g 10x10 array of cells that is the arrangement of cells for analysis have to be square The support for N by M is still under construction 4 TEAST system your data files and weight matrix files are all located in the same working directory Chapter 2 STARMA Models and Modeling 2 1 Statistical Modeling Statistical modeling is an important procedure for analyzing stochastic systems because it allows us to connect the dataset in reality to theoretical processes Furthermore these theoretical models can be used to forecast or to control the future behavior of a re
40. IEAST User Manual IEAST An Integrated Environment for Analyzing STARMA Models by Cheng Yu Lee Department of Forestry Michigan State University Copyright 2004 February 14 2005 Contents Contents Preface 1 Introduction 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 Wehatyis TEAST oe Roe a ee ees rere ee ead ata Me AAA Quick Start A Simple Example e e o Features A E NR Requirements roras e gaara Wee Ee a a PaaS Installation Sal hk cele GE EEE AAA OK an Performance opera a e Gaui townie got sok So als MenusSystem Sct be AA Dee ee Ree ees Conventions a ew be eit a ee ete Dee 2 STARMA Models and Modeling 2 1 2 2 2 3 2 4 2 5 2 6 StatisticalsModeling 27 kl dette o nae ee ee Ghats Box Jenkins Modeling e STARMA Model Definition vga pr ee a Re ee a Elementary Statistical Tools for Analyzing STARMA Models Assumptions tt st tle de ie thy eet Be he ad Gee ot cs ech al ho a he ad General STARMA Modeling Procedure for using IEAST 3 Data Preparation 3 1 3 2 3 3 3 4 Load space time dataset from dat files o Import space time datasets from other formats o o Simulation Data exonera a ad A E as Dataset from Excel files and other sources 13 13 14 14 14 16 17 17 18 18 20 21 21 CONTENTS 3 5 Important Global Variables es
41. PACF of Z 3 STACF of Y 4 STPACF of Y 5 STXCF of Z and Y 6 STPXCF of Z and Y 7 Extended STXCF Selection 1 7 1 Plotting from spatial lag 0 0 4 Plotting up to spatial lag 4 0 4 Plotting from temporal lag 1 1 20 Plotting up to temporal lag 20 1 20 y Plotting versus 1 Temporal lags 10 Space Time Autocorrelation STACF Space Time Partial Autocorrelation STPACF T i Spatial lag 0 1 T F T Spatial lag 0 Spatial lag 1 Spatial lag Spatial lag 2 Spatial lag 2 Spatial lag 3 a Spatial lag 3 Spatial lag 4 m Spatial lag 4 05 Le a i A i f i i J os L 4 3 S E gt A a T cd a a o e od Ha pn ie ep gone eee ON ee ce aa Cea eee ae ee ee ee eee ee eee ed 3 OA E Ve 8 i 3 a 05h f A f i J 05 b i i i 4 4 H i i i i i i H i A H i i i i i i i i 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Temporal lag Temporal lag Figure 1 2 Space time ACF and PACF for STAR 2 2 2 Spatial lags Selection 1 2 1 Plotting ACF of Z versus T Lag The plot is also saved as an eps file ACF_PLOT_T eps orrelation Analyses gt AutoCorrelation STACF gt Partial AutoCorrelation STPACF gt Cross Correlation STXCF 4 gt Partial Cross Correlation STPXCF gt Extended Cross Correlation ExtSTXCF gt Plot Correlations versus
42. PROCESSING Spatial Trend average over time Spatial Trend average over time 3 1 6 1 4 1 2 15 08 1 0 6 0 4 0 2 Figure 5 4 Dataset before left and after right IDM d smoothing with 3x3 window 11 gt Return Selection or lt Enter gt to exit 7 Smoothing 1 gt gt Moving Averages in space 2 gt gt 4 gt gt Selection or lt Enter gt to exit 1 Moving Averages 1 gt gt gt Uniform 2 gt gt gt Inverse Distance Method 1 d 3 gt gt gt Inverse Distance Method 1 d72 4 gt gt gt Inverse Distance Method 1 d73 5 gt gt gt Return Selection or lt Enter gt to exit 3 Window size in X odd number 3 10 3 Window size in Y odd number 3 10 3 done 5 8 Missing Data A critical assumption in all space time series models is that the observations are sampled with the same frequency or distance Unfortunately it is often the case that some values of the variable of interest are either missing or unavailable for certain dates or some locations in the sample period There are several ways of dealing with this problem including aggregation and interpolation In IEAST the method Moving Averages in space is used to provide recovering of 5 9 FILTER WITH A GIVEN STARMA MODEL 47 missing values 5 9 Filter with a given STARMA model Sometimes a STARMA model found is used as a fi
43. T Lag S Lag gt Return Selection or lt Enter gt to exit 6 Which correlation to plot 1 STACF of Z 2 STPACF of Z 3 STACF of Y 4 STPACF of Y 5 STXCF of Z and Y 6 STPXCF of Z and Y 7 Extended STXCF Selection 1 7 2 Plotting from spatial lag 0 0 4 Plotting up to spatial lag 4 0 4 Plotting from temporal lag 1 1 20 Plotting up to temporal lag 20 1 20 dy Plotting versus 1 Temporal lags 2 Spatial lags Selection 1 2 1 Plotting PACF of Z versus T Lag The plot is also saved as an eps file PACF_PLOT_T eps 6 Select model type orders According to the autocorrelation results we can decide the 1 2 QUICK START A SIMPLE EXAMPLE 11 tentative model type and orders either automatically or artificially In this case the dataset was given automatically the model STAR S 2 T 2 by IEAST by choosing 4 Model Identification and then Automatic Identification IEAST v1 30 01 STARMA Modeling amp Analysis Main Menu 1 Setup 2 Data Preprocessing 3 Correlation Analyses 4 Model Identification 5 Parameter Estimation 6 Diagnostic Analysis 7 Forecasting 8 Preference 9 Interpreter 10 Exit Selection or lt Enter gt to exit 4 Model Identification 1 gt Automatic Identification Type Orders 2 gt Artificial Identification Type Orders 3 gt Par
44. TRACE HIDEACF PLOTEPS Description Clear the environmental settings 88 CHAPTER 10 INTERPRETER dec Synopsis dec lt var_name gt Description Decrease the value of the scalar variable var_name by one demean Synopsis demean Description Remove the mean of current series in the global variable Z detrend Synopsis detrend lt result gt lt Z gt lt Strend gt lt Ttrend gt lt method gt arg Description According to the spatial and temporal trends found there are several ways to de trend them from the dataset lt Z gt They are 1 detrend spatial trend only method ALL S 2 detrend temporal trend only method ALL_T 3 detrend the best fit to the dataset spatial trend method BF_SS 4 detrend the dataset by the product of temporal trend and spatial trend method SMULT 5 detrend the best fit Strend Ttrend from the dataset method BF_ST Result will be stored in the variable specified in lt result gt lt Z gt is the dataset to be detrended lt Strend gt and lt Ttrend gt are the trends have been found arg is for additional argument not used yet Example detrend both spatial and temporal trends from the dataset Z by the method BF_ST The spatial trend and temporal trend have been found and save in the variables STREND and TTREND respectively The result will be saved into the variable DTZ detrend DTZ Z STREND TTREND BF_ST diff Synopsis diff Z T lt Z gt lt n gt Description Differencing the current serie
45. Trend in time J gt gt gt Polynomial Regression OLS J gt gt gt Fourier Analysis gt gt gt Average over space gt gt gt Remove the trend gt gt gt Return Selection or lt Enter gt to exit 3 Temporal trend Average over space gt ZTC 1 gt gt Trend in space Trend Surface Analysis 2 gt gt Trend in time 3 gt gt Remove trend in space time 4 gt gt 5 gt gt Return Selection or lt Enter gt to exit 3 Best Fit Space Time trend was removed from dataset Z t The fifth order polynomial spatial trend and temporal trend average over space are shown in Figure 5 2 Finally the dataset after de trended both in space and in time is in Figure 5 3 CHAPTER 5 DATA PREPROCESSING Temporal Trend of Z Average over space Trend Surface using PloyRegr 5th order 12 i Temporal Trend Z xy t se 1 N 0 4 23 08 1 0 2 PA 0 1 boot o e 1 ost o1 N Y A 041 A 0 24 ss J o L pri f 1 at aa 0 5 10 15 20 25 30 time Figure 5 2 5th order polynomial spatial trend and temporal trend average over space Z in colored map Spatial Trend average over time Spatial Trend average over time Z x y t ooso90999999 Bee ES Figure 5 3 3D plots of the dataset after de trending 5 5 Box Cox Transform Non normality is common to all natural processes One strategy to make n
46. Under the menu AutoCorrelation Analysis the two selections below provide the calcula tions of autocorrelations 6 2 AUTOCORRELATIONS STACF AND STPACF 5l Autocorrelation STACF px Before computing STACF or STPACF the Spatial Weighting Structure must be specified The spatial dimension of the weighting structure has to coincide with that of the dataset The numbers of maximum temporal and spatial lags will be asked before calculation The maximum temporal lag MAX_T_LAG cannot exceed the number of time periods T in the dataset The maximum spatial lag MAX_S_LAG cannot exceed the highest spatial order SOD_MAX_S in the definition of Spatial Weighting Structure WNV_XCrow_Detroit_10x10R45 dat is being loaded done File name WNV_XCrow_Detroit_10x10R45 dat Data size Space 10x10 Time 28 j Maximum 27 000000 Minimum 0 000000 Sample Mean 0 692143 Sample Variance 1 545331 Value for data missing 99 900000 of Missing Data 0 Loading spatial definition file UF S10 10x10NBPP wet done The spatial dimension is 10 x 10 Spatial Order Definition 0 0010 9 9 910 0 0 0 010 9 8 7 7 7 8 910 O 09 8 6 5 5 5 68 9 0 10 8 6 5 43 4 5 6 8 10 975 42 12 45 7 9 975 3 1013 5 79 9 75 42 12 465 7 9 10 8 6 5 43 4 5 6 8 10 09 8 65 5 5 68 9 0 010 9 8 7 7 7 8 910 O 0 0 010 9 9 910 0 0 0 Spatial Relation Matrix SOD_MAX_S 10 000 000 000 083 050 050 050 083 000 000 000 000
47. _Theta which are the same size as the estimated parameters final_Phi and final_Theta The elements in sig_Phi and sig_Theta represent the importance probability from 0 01 0 The bigger the value is the higher the importance AR_Mask and MA_Mask has the same effect as in the instruction estimate Example significance sig PHI sig_THETA EST_PHI EST_THETA simulate Synopsis simulate lt result gt lt sim_phi gt lt sim_theta gt lt iteration gt lt generation gt lt noise_var gt lt noise_mean gt Description This instruction is doing simulation according to the assigned specifications which are described as below sim phi is the coefficients of AR terms in matrix form sim_theta is the coefficients of MA terms in matrix form The simulated results are picked up after the number of generations assigned in iteration The number of generation simulated is specified in generation noise_var and noise_mean represent the statistical characteristics of the random disturbance Example simulate STAR T251N100 sPHI sTHETA 5000 100 0 00125 0 0 stacf Synopsis stacf lt st_acf gt lt Z gt lt max_t_lag gt lt max_s_lag gt Description Calculate space time autocorrelation according to the series data in series Z the maximum time lag max_t_lag and the maximum spatial lag max_s_lag The final result is stored into the matrix variable stacf Example 94 CHAPTER 10 INTERPRETER stacf ST_ACF Z 16 3 stop Synopsis stop Descri
48. ag 1 Spatial lag 1 Temporal lag 2 gt Spatial lag 2 Temporal lag 3 x Spatial lag 3 0 Temporal lag 4 0 Spatial lag 4 m Temporal lag 5 w Spatial lag 5 e Temporal lag 6 0 Spatial lag 6 Temporal lag 7 e Spatial lag 7 s 7 Ae Temporal lag 8 7 Spatial lag 8 Temporallag9 Spatial lag 9 Femporal lag 10 Spatial lag 10 v A 1 o S S k E E a a E a y i pas 2 2 3 gt a 2 E a a 05 i A i i i 4 0 5 i i 4 1 i i i i i i i 4 i i 1 2 3 4 5 6 7 8 9 10 o 2 4 6 8 10 Temporal lag Spatial lag Figure 6 5 STACF plots versus T Lag left and S Lag right 2 Spatial lags Selection 1 2 2 Plotting STACF of Z versus S Lag The plot is also saved as an eps file ACF_PLOT_S eps 60 CHAPTER 6 CORRELATION ANALYSES Chapter 7 Model Identification Before estimation stage we have to let TEAST know what the model type and orders would be appropriate for the underlying process so that the number of parameters to be estimated or even the estimation algorithm to be used can be decided There are many methods can be used to determine type orders e g using statistical measures for model fitness to search appropriate model iteratively using STACF STPACF In the automatic identification IEAST uses the behavior of STACF STPACF to decide type orders 7 1 Automatic Identification Based on the cut off tail off behavior
49. alistic ecosystem Figure 2 1 shows how a model is used to represent the data generating process of interest Data and knowledge of the underlying process are used to identify a statistical model While modeling a dataset is used to fit this hypothetical model right after the model type and orders are determined The parameters of the model are estimated during the fitting process Then adequacy of this statistical model can be evaluated Finally this statistical model can be used for forecasting analysis control or other purposes The objective is to find a parsimonious model with the smallest number of parameters needed to fit adequately the patterns in the dataset Box and Jenkins 3 examined the use of the ARMA model for analyzing univariate time series data and generating short term forecasts Pfeifer and Deutsch extended the Box Jenkins modeling procedure to the space time ARMA case i e STARMA models 10 12 11 13 Generating Outcomes Z t Stochastic Process Real World A E E EEE peat ent Ate ea E met he e e dada Statistical Modeling Fitting Data Y Forecasting Statistical Modeling Analyzing Model Procedures Control Figure 2 1 Relationship between data generating and statistical modeling procedures 17 18 CHAPTER 2 STARMA MODELS AND MODELING Identification Pre estimation Estimation Yes Figure 2 2 Box Jenkins Modeling 2
50. ameter Masking Selection or lt Enter gt to exit 1 Model type is suggested to be STAR with Max Temporal Order 2 Max Spatial Order 2 7 Estimate model parameters Under the menu 5 Parameter Estimation the model parameters can be estimated as shown below IEAST v1 30 01 STARMA Modeling Analysis Main Menu 1 Setup 2 Data Preprocessing 3 Correlation Analyses 4 Model Identification 5 Parameter Estimation 6 Diagnostic Analysis 7 Forecasting 8 Preference 9 Interpreter 10 Exit Selection or lt Enter gt to exit 5 Parameter Estimation 1 gt Pre estimate Model Param Linear STAR 2 gt Pre estimate Model Param Non linear STMA STARMA 3 gt Pre estimate Model Param From STACF STPACF 4 gt Pre estimate Model Param Specified by users 5 gt Estimate Model Param Fixed SRM 6 gt Estimate SRM Fixed Model Param 7 gt Estimate SRM amp Model Param Alternatively 8 gt Return 12 CHAPTER 1 INTRODUCTION Selection or lt Enter gt to exit 1 Initializing Covariance Look up Tabl e CovarTB for major varialbe Z gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Phi_Now 0 31 0 22 0 13 0 22 0 08 0 06 Press any key to cont
51. asets from other formats IEAST supports three commonly used formats in remote sensing datasets These files have to be in text ASCII and conform with one of the following formats Importing can only work on the major variable Z Notice The X Y and Time must be normalized to integers from 1 Max D X x Y y T t are the corresponding data values for location x y at time t XY T xyt Each row is for different T In each row X changes first then Y For example for a space NxN the first row of your table can be 1 1 2 1 3 1 N 1 1 2 2 2 A A N N See an example in Table 3 1 Table 3 1 An example for format XY T of SPACE 10x10 and TIME 1 28 D XA LY LT7 D DX 2 Y 1T DA 10 Y 10 T 1 D X 1 Y 1T 2 DX 2Y 1T 2 D X 10 Y 10 T 2 DX 1 Y 1 7 28 D X 2 Y 1 T 28 D X 10 Y 10 T 28 T XY txy Each column is for different T In each column X changes first then Y See an example in Table 3 2 XYTD xyd Each row is a record for some location at some time Columns are for X Y Time and Data values The first row can be names for each field See an example in Table 3 3 As an example under the menu 1 Setup select Space time dataset and then Import major variable Z from XY T T XY XYTD In the example the imported dataset is in XYTD format and N_SITE 100 NxN 10x10 and N_TIME 13 Because format dat is 3 2 IMPORT SPACE TIME
52. ats and 3 from simulation 3 1 Load space time dataset from dat files Under the menu 1 Setup select Space time dataset and then Load major variable Z from dataset file dat Follow the procedure as shown below To load secondary variable Y has a similar procedure Setup gt Space time dataset gt Spatial correlation structure gt Information of datasets gt Return Selection or lt Enter gt to exit 1 Space time dataset gt gt Load major varialbe Z from dataset file dat gt gt Load second varialbe Y from dataset file dat J gt gt Import major variable Z from XY T T XY XYTD 4 gt gt Simulation dataset gt gt Save dataset from major variable Z to file dat J gt gt Return Selection or lt Enter gt to exit 1 23 24 CHAPTER 3 DATA PREPARATION Dataset files in current directory 1 NA_Precipitation dat 2 NOAA_Temperature dat 3 US_Precipitation month dat 4 WNV_Crow_WAM_10x10full dat Which Dataset file to load Select a number 1 4 1 US_Precipitation month dat is being loaded done File name US_Precipitation month dat Data size Space 20x20 Time 120 J Maximum 28 145200 Minimum 0 000000 Sample Mean 3 213700 Sample Variance 2 662913 J Value for data missing 99 900000 J of Missing Data 0 3 2 Import space time dat
53. ber of points in time i e T Ncolumns Z N_SITE Integer Total number of cells in Z in two dimensional space i e N_SITE y Nrows Z MODEL String Model type There are three subclasses or types in general STARMA models i e STAR STMA and STARMA mixed model The model type is stored in the global variable MODEL In interpreter you should assign MODEL a string which can be one of STAR STMA and ARMA MODEL_MAX S Integer This is the maximum spatial order in your model It is different from SOD_MAX_S Normally SOD_MAX_S is greater or equal to MODEL_MAX_S MODEL_MAX_T Integer This is the maximum temporal order in your model MAX_T_LAG Integer This is the largest lag in time while calculating autocorrelation functions It can be changed anytime MAX_S_LAG Integer This is the largest lag in space while calculating autocorrelation functions It can be changed anytime SYS_MAX_S Integer This defines the maximum spatial order supported by IEAST Default value 20 30 CHAPTER 3 DATA PREPARATION SYS_MAX_T Integer This defines the maximum temporal order supported by IEAST Default value 20 SOD_MAX_S Integer The maximum spatial order defined in Spatial Order Definition matrix Automatically generated by IEAST for references SOD Matrix Spatial Order Definition SRM Matrix Spatial Relation Matrix SWM Matrix Spatial Weighting Matrices or simply Weight Matrices PHI Matrix The parameters of auto regress
54. d begin with a line number then followed by command and arguments as shown below list 10 load data verification dat 20 load weight uniform wet 30 stacf ST_ACF Z 16 3 40 plot ST_ACF 16 3 ACF 50 end All global variables are shared between menu driven and interpreter If one is changed under interpreter the one in menu driven is changed as well 10 2 A Sample Program In this section to illustrate how to use the interpreter a dataset which was generated by simula tion according to a given known model was analyzed and modeled Using IEAST we can find an appropriate model type and orders to represent the dataset Furthermore some diagnoses were provided for model adequacy evaluation Finally the acquired model and parameters were compared with the given model In this example we found that it is efficient and flexible to analyze a given dataset with the programming environment of IEAST 80 CHAPTER 10 INTERPRETER 10 2 1 Dataset Generation First we needed to generate a dataset as the input by simulating a given STARMA model with known parameters The dataset for this example was generated based on the following specifica tions Spatial dimensions two dimensional space 8 x 8 sites Model type and orders STARMA mixed model with both maximum temporal and spatial orders set equal to one that is STARMA 1 1 e Mathematical expression of the model Z t 0 5 Z t 1 0 1 WY Z t 1 e t
55. d on 150 Monte Carlo simulated datasets eee 16 An example for format XY T of SPACE 10x10 and TIME 1 28 24 An example for format T XY of SPACE 10x10 and TIME 1 28 25 An example for format XYTD of SPACE 10x10 and TIME 1 28 25 vii viii LIST OF TABLES Preface Since the theoretical basis of space time model established in 1970 there have never implemented any general purpose software for space time modeling We began the development of IEAST a true space time analyzing environment in the early 2000 by a grant supported by Michigan Agricultural Experiment Station MAES Michigan State University All work been done since then is in an MatLab like language called GNU Octave As users may notice the environment is a text based menu selection Combining with GNU Plot IEAST provides a complete set of functionalities and sufficient 2D 3D graphical abilities for most space time modeling scenarios Several space time modeling and analyses on spreading of various emerging infectious diseases among animals or human have been done successfully by using IEAST The results were encouraging and promising and we d like to make IEAST be applicable to even more diverse disciplines We keep improving IEAST both on its functionalities supported and applicability to various applications Hopefully in the future there is enough funding or time to let us port the entire system to Window based env
56. data WNV_XCrow_CR45_10x10 dat Y stxcf_ext ExtSTXCF Z Y 20 4 variance Synopsis variance lt variance gt lt residuals gt Description The instruction variance is to calculate the variance of the residuals There are one input variable residuals and one output variance in its format The final result variance will be stored in the variable variance Example variance EST_VARIANCE EST_EE Bibliography 1 Akaike H 1973 Information theory and an extension of the maximum likelihood principle 10 11 12 13 14 2 In Petrov B F C Eds 2nd International Symposium on Information Theory Akademiai Kiado Budapest 267 281 Akaike H 1978 Time series analysis and control through parametric models In Findley D F Ed Applied Time Series Analysis Academic Press New York 1 23 Box G E P Jenkins G M 1970 Time Series Analysis Forecasting and Control Holden Day San Francisco Hannan E J Rissanen J 1982 Recursive estimation of mixed auto regressive moving average order Biometrika 69 81 94 Hordijk L Nijkamp P 1977 Dynamic models of spatial autocorrelation Environment and Planning A 9 505 519 Hurvich C M Tsai C L 1989 Regression and time series model selection in small samples Biometrika 76 297 307 Jones R H 1975 Fitting autoregressions Journal of the American Statistical Association 70 351 590 592 Marquardt D
57. dimension NxN N N A 4 100 10 done Save spatial correlation structure SOD SRM and SWM to file wet Y N N y Available wet files UF S10 10x10NBPP wet UF S10 16x16NBPP wet UF S10 20x20NBPP wet UF S6 14x14NBPP wet Save them as file_name wet demo wet 4 6 LOAD SPATIAL CORRELATION STRUCTURE OR WEIGHT MATRICES DIRECTLY FROM FILES37 4 6 Load Spatial Correlation Structure or Weight Matrices Directly from Files It is easy to load an existing spatial weighting definitions from file The three matrices SOD SRM and SWM will all be loaded and current values of them in computer memory are overwritten The new SOD and SRM will be shown after loading There are some pre existing definition files for com mon spatial correlation structures All of them are named in a style like UF S10 10x10NBPP wet UF means the spatial correlation is uniform or directionless 10 means the structure in this file can support up to spatial order or lag 10 10x10 is the spatial dimension in number of cells supported NB means spatial boundary effect is reduced Finally PP is a common option used for the configuration of correlation structure e g using bishops and rooks moves as spatial order 1 and 2 like the one used in Section 1 2 The other option for this is RN which means the spatial correlation structure is defined as rings outward while the spatial order increasing an RN correla tion structure is used in the following examp
58. e analysis 5 1 Remove Mean It says everything 5 2 De Seasonalize In weekly or monthly data the seasonal component often referred to as seasonality is a component of variation in a space time series which is dependent on the time of year It describes any regular fluctuations with a period of less than one year For example unemployment figures and average daily rainfall they all show marked seasonal variations We are interested in comparing the seasonal effects within the years from year to year removing seasonal effects so that the series is easier to cope with and also interested in adjusting a series for seasonal effects using various models Especially the seasonal components make the datasets nonstationary thereby violate the statistical assumptions Hence these components needs to be removed before analysis In IEAST two methods for de seasonalization are provided Further a spectrum analysis is provided i e the strongest components or fundamental harmonics in frequency spectrum are calculated for assisting user to find the period of seasonal components The two methods and example are shown below 1 B Z t The de seasonalized dataset Zq t Z t Z t dd dd is the period of the seasonal component Z t Z t Average of Z t over space is subtracted from Z t to remove seasonal component The following is an example of NOAA precipitation data with space 14x14 and time 120 months We de seasonalize it with th
59. e first method File name NA_PRCP dat Data size in space 14x14 39 40 CHAPTER 5 DATA PREPROCESSING in time 120 of Missing Data 0 Data Preprocessing 1 gt Remove Mean 2 gt De seasonalize 1 B dd Z t 3 gt Diference by one 1 B Z t 4 gt De trend 5 gt 6 gt Subsequencing Resampling 7 gt Smoothing 8 gt Missing Data 9 gt Filter with a given STARMA model 10 gt Undo previous action 11 gt Return Selection or lt Enter gt to exit 1 Mean of Z has been removed Data Preprocessing 1 gt Remove Mean 2 gt De seasonalize 1 B7dd Z t 3 gt Diference by one 1 B Z t 4 gt De trend 5 gt 6 gt Subsequencing Resampling 7 gt Smoothing 8 gt Missing Data 9 gt Filter with a given STARMA model 10 gt Undo previous action 11 gt Return Selection or lt Enter gt to exit 2 De Seasonalization 1 gt gt 1 B dd Z t 2 gt gt Z t Average Z t 3 gt gt Return Selection or lt Enter gt to exit 1 Based on spectrum of Z the first 10 significant harmonics are at in 1 freq ans 12 00 1 09 6 00 1 20 24 00 1 04 120 00 1 01 1 14 8 00 Deseasonalization 1 B dd Z t Please enter the seasonal factor to be removed dd 1 100 12 The dataset was de seasonalized by 1 B 12 Z The of samples
60. echniques are used to reduce irregularities random fluctuations in space time series They provide a clearer view of the true underlying behavior of a series For example in some series seasonal variation is so strong it obscures any trends or cycles which are very important for the understanding of the process being observed Smoothing can remove seasonality and makes long term fluctuations in the series stand out more clearly The most common type of smoothing technique is moving average smoothing Since the type of irregularities varies from series to series so does the type of smoothing In IEAST both temporal and spatial smoothing are provided Moving Averages in space The are two different methods in moving average They are uni form and IDM Inverse Distance Method In IDM the importance of each cell is inversely proportional three options L 4 4 to the distance between the center and the cell An example with d IDM smoothing with 3x3 window is shown below Moving Averages in time Not implemented yet Splines in space Not implemented yet Krigging in space Not implemented yet Data Preprocessing 1 gt Remove Mean 2 gt De seasonalize 1 B dd Z t 3 gt Diference by one 1 B Z t 4 gt De trend ea S 6 gt Subsequencing Resampling 7 gt Smoothing 8 gt Missing Data 9 gt Filter with a given STARMA model 10 gt Undo previous action 46 CHAPTER 5 DATA PRE
61. er N A 0 20 1 of periods in time of the dataset N A 1 1000 80 a Current Configurations Spatial dimension N_SPACE 10 x 10 Number of samples in time N_TIME 80 J Model type MODEL STAR Model Max T order MODEL_MAX_T 2 J Model Max S order MODEL_MAX_S 1 J Spatial Order Definition oonan o aAankr wk OPNENE O aowrorwda OPNENE O anrkr whan oonan o Spatial Relation Matrix S0D_MAX_S 6 000 125 063 063 063 125 000 125 063 125 250 125 063 125 063 125 250 250 250 125 063 063 250 250 000 250 250 063 063 125 250 250 250 125 063 125 063 125 250 125 063 125 000 125 063 063 063 125 000 Spatial Weighting Matrices SWM Ready Are these configurations good Y N Y Mean of random noise 0 000000 100 100 Variance of random noise 0 00125 100 100 Upper limit of values of Z 1e107 1e10 No Limit 100000000 100000000 Lower limit of values of Z 1e107 1e10 No Limit 100000000 100000000 How many iterations to simulate 1000 1 100000000 dy Please specify the parameters for this STAR model Z t Cee RRR EZ t 1 Cr AWL Z t 1 Cee RRR EZ 6 2 Cree AWLEZ t 2 e t Phi T 1 S 0 10 4 Phi T 1 S 1 10 10 3 I A I ey o Phi T 2 S 0 10 10 2 Phi T 2 S 1 10 10 1 The following model is being simulated Z t C
62. eter The functionalities for analysis are distributed in this menu system Users need to step by step setup and analyze dataset through the menus following a modeling procedure The first two levels of the menu system is shown in Figure 1 3 In all menu selection if nothing entered but the key lt Enter gt the default values quoted in e g 3 will be used or actions canceled or return to previous level of the menu 15 1 7 MENU SYSTEM Main Menu qe J Data Correlation Model Parameter Diagnostic ra J Preprocess _ _ a Identification Estimation J Analysis 8 Preference Interpreter A ae ee ace time utomatic atistic nter dataset 2 Cac gt STACF 2 SCHON J 7 Linear significance Save Env interpreter Spatial Artificial lioi De season STPACF Identification eee DRES Load Env structure Difference Parameter From Info of by one STXCF J masking ACF PACF R STACF Octave Cmd datasets gt lt gt lt gt lt gt De trend STPXCF User defined R STPACF Dir Contents a Subseq amp N tesari piE A Est Param Smoothin Plot Est SRM 8 Correlations J Missing Est SRM amp data Param Filter w STARMA A Undo NS 4 Figure 1 3 The first two levels of the menu system in TEAST 16 CHAPTER 1 INTRODUCTION Table 1 1 Accuracies of model type selection using variances AICC BIC and AICC BIC based on 150 Monte
63. f you are going to simulate a given model specify N here Model type Max temporal order Max spatial order You don t have to give these three information if you are not going to simulate a dataset of samples in time of the dataset This means how many time periods in your dataset You don t have to give this if the dataset is going to be loaded Thereafter it is followed by Spatial Structure Definition Here you need to specify SOD and SRM orderly you cannot assign values to SRM without specifying SOD in advance and then SWM can be generated by IEAST There are two options while specifying SOD Euclidean Distance The system semi automatically decides the SOD based on Euclidean dis tance between the cell and the center Users need to specify the value regions of the spa tial order The minimum Euclidean distance is 1 Note Euclidean distance is equal to Horizontal_Distance to_Center Vertical_Distanceto_Center User defined User assigns the spatial orders in matrix SOD one by one as needed While defining SRM there are also two options Uniform The system automatically decides the SRM so that all cells in each spatial order has same values and sum up to 1 0 User defined User assigns the values in matrix SRM one by one Remember to keep the sum of all cells in each spatial order be 1 0 The following is an example of specifying SOD SRM and then generating SWM At last the weighting definitions are saved to a
64. file wet Setup gt Space time dataset gt Spatial correlation structure gt Information of datasets gt Return 36 CHAPTER 4 SPATIAL CORRELATION STRUCTURE Selection or lt Enter gt to exit 2 Spatial correlation structure 1 gt gt Load from file wet 2 gt gt Define by users 3 gt gt Return Selection or lt Enter gt to exit 2 gt gt gt Define Spatial Order Definition S0D Y N Y gt gt gt Max spatial order in Order Definition SOD_MAX_S N A 1 20 3 gt gt gt Specify SOD by 1 Euclidean Distance 2 User defined 1 2 1 Spatial order 1 Euclidean distance 1 00 7 1 441 4 Spatial order 2 Euclidean distance 4 00 7 1 441 9 Spatial order 3 Euclidean distance 9 00 7 1 441 16 Are they good Y N Y Spatial Order Definition O WWW www Oo WNNNNDN W WNRRRNOO WNRPORN W WNRRRNOO WNNNNDN W OWWWWWo gt gt gt Define Spatial Relation Matrix SRM Y N Y gt gt gt Spatial Relation Matrix 1 Isotropic 2 User defined 1 2 1 Het Isotropic structure are chosen Spatial Relation Matrix SOD_MAX_S 3 000 050 050 050 050 050 000 050 063 063 063 063 063 050 050 063 125 125 125 063 050 050 063 125 000 125 063 050 050 063 125 125 125 063 050 050 063 063 063 063 063 050 000 050 050 050 050 050 000 Translating SRM gt SWM Spatial
65. g Figure 10 1 STACF and STPACF for the simulated dataset STARMA 1 1 10 2 3 Program Description To iteratively test all three possible models STAR STMA STARMA there is a for loop from line 100 to 250 which was used to control the program flow Dataset and uniform weight matrices were loaded to Z W in line 10 and 20 respectively Line 30 60 calculated and plotted STACF and STPACF Line 80 assigned the highest possible temporal and spatial orders Line 150 was for pre estimating the initial values of the parameters using either space time extension of Yule Walker or Hannan Rissanen algorithm according to the model type If the model is STAR the Yule Walker equation is used Otherwise space time extension of Hannan Rissanen algorithm is used Line 160 was for estimating parameters based on the initial values acquired in last steps Lines 180 and 190 calculated the AICC and BIC values for every candidate respectively Line 200 calculated the variance of the residuals After the for loop the candidate with lowest AICC BIC was chosen to be the final model type Then in line 310 the significance of parameters was calculated and the necessary parameters decided In this program the words in lowercase were commands and all variables were in capital The meaning of variables are described below ST_ACF and ST_PACF were used to store the space time autocorrelation results PHI and THETA were matrices which stored the coefficients of autoregres
66. gt Smoothing 8 gt Missing Data 48 CHAPTER 5 DATA PREPROCESSING Space Time Autocorrelation STACF Space Time Autocorrelation STACF 1 T T 3 T T y T 7 1 T z T T 7 T T r Spatial lag 0 Spatial lag 0 Spatial lag 1 Spatial lag 1 x Spatial lag 2 Spatial lag 2 x Spatial lag 3 a Spatial lag 3 5 Spatial lag 4 Spatial lag 4 m Spatial lag 5 0 Spatial lag 5 0 Spatial lag 6 asil i Spatial lag 6 e Do So 8 3 E e E E a A S of e e ee ee E S S A 3 3 2 2 a a 05 ho f A A i i i 4 0 5 L i i i 4 A i i i i i i i i i 4 i i i i i H i i H 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Temporal lag Temporal lag Figure 5 5 Autocorrelation functions of a dataset before left and after right filtering 9 gt Filter with a given STARMA model 10 gt Undo previous action 11 gt Return Selection or lt Enter gt to exit 9 Give a STARMA model to filter the major variable Z Select model type or lt Enter gt for STAR 1 STAR 2 STMA 3 STARMA Selection 1 3 1 Max spatial order 0 9 1 Max temporal order 1 10 2 Assign parameters for AR terms Param for AR term t 1 s 0 1 1 26 Param for AR term t 1 s 1 1 1 36 Param for AR term t 2 s 0 1 1 04 Param for AR term t 2 s 1 1 1 18 The residuals of filtered Z is stored back to itself Chapter 6
67. h order Figure 2 3 An example of definition of spatial orders in 2D systems Let Z t be the N x 1 vector of observed values at time t i e Z t Z t Za t e Zn t where N is the total number of sites in two dimensional space To formulate the spatial relations among sites weight matrices W are introduced as in 10 Assume weight matrix W has elements wi that are the weighting contributions of site 7 to site i and which are nonzero if an only if site and j are l th order neighbors in space Then the general STARMA model can be expressed in the following form P AR q Mx 2 1 Z t Y Y du WZ k Y Y 01 Welt k elt k 1 I 0 k 1 I 0 where p is the maximum autoregressive temporal order q is the maximum moving average temporal order Ax is the spatial order of the kt autoregressive term mx is the spatial order of the kt moving average term dj is the autoregressive parameter at temporal lag k and spatial lag 1 011 is the moving average parameter at temporal lag k and spatial lag 1 WU is the N x N matrix of weights for spatial order l and e t e1 t ea t ev t is the random noise vector at time t The weights wi should reflect an ordering of spatial neighbors Figure 2 3 shows an example of a spatial order definition The first order neighbors corresponding to the 1 order weight matrix are those that are closest to a given site i e the small black dot in the center of Figure 2 3
68. his purpose is a Space Time Regression model 9 and shown below c d 6 1 Zt 5 y WskL Y y Et s 0 k 1 where L is a spatial lag operator and L Z W Z The cross correlation of Z and Y is defined in 9 and shown below gt _El Zie MPa Hy 6 2 PZY s k El Zit wz ELL Vit mA i In cross correlation XCF analysis these two variables are ordered that is XCF Z Y is not the same as XCF Y Z XCF gives the cross correlation of the major variable Z and the secondary variable Y The temporal lag in the correlation results means how many time periods Z lagging behind Y There are three cross correlation provided in IEAST i e Cross Correlation STXCF Partial Cross Correlation STPXCF and Extended Cross Correlation ExtSTXCF The first two are easy to understand without further explanations As to ExtSTXCF it calculates correlations not only on positive temporal lags but also on negative temporal lags at the same time Let s see the following example The graphical result of the extended cross correlation is shown in Figure 6 3 Space time dataset gt gt Load major varialbe Z from dataset file dat gt gt Load second varialbe Y from dataset file dat gt gt Import major variable Z from XY T T XY XYTD gt gt Simulation dataset gt gt Save dataset from major variable Z to file dat gt gt Return Selection or lt Enter gt to exit
69. ial lag 3 8 Spatial lag 3 a Spatial lag 4 m Spatial lag 4 m 0 5 j 4 0 5 4 2 E E a 2 7 a as a eee ee ee 3 iva e a is Luis i i a E J 05 L i 4 al i i i i i i ai i i i i i i f i 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Temporal lag Temporal lag Figure 9 1 Residuals STACF and STPACF for the example in Quick Start gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Calculating STACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Autocorrelation Rho s 1 k 0 0 003 0 005 0 013 0 014 10 0 012 0 010 0 007 0 004 0 002 11 0 009 0 014 0 009 0 005 0 010 12 0 019 0 028 0 003 0 023 0 003 13 0 013 0 006 0 013 0 004 0 030 14 0 019 0 015 0 002 0 004 0 005 15 0 005 0 008 0 002 0 004 0 019 16 0 011 0 009 0 004 0 013 0 015 17 0 012 0 007 0 002 0 007 0 008 18 0 002 0 009 0 004 0 013 0 027 19 0 007 0 000 0 004 0 012 0 002 20 0 014 0 005 0 005 0 019 0 005 9 2 STPACEF of Residuals Same as STACF of Residuals Continue on the last example the residuals STPACF is shown below The results of STACF and STPACF show that there is no significant correlation in space and in time That implies the selected candidate may be a go
70. ial lags Selection 1 2 1 Plotting ExtXCF of Z and Y versus T Lag The plot is also saved as an eps file ExtXCF_PLOT_T eps The cross correlation is based on an assumption of space time regression model Eq 6 1 The linear relationship between variable Z and Y can be shown as in Figure 6 4 It is evident that the autocorrelation of Y also contributes to the cross correlation between Z and Y The cross correlation is spurious When cross correlating the two space time variables this will inflate the correlation and further make it hard to achieve a parsimonious model One solution to this problem is call pre whitening We first fit an STARMA model for the input data sufficient to reduce the residuals to white noise then filter the input data with this model to get the white noise residuals Then filter the response data Z with the same model and cross correlate the filtered response with the filtered input data The function described in section 5 9 can be used for pre whitening a dataset for a given STARMA model with know type maximum orders and parameters 6 4 PLOTTING CORRELATIONS VERSUS T LAG S LAG FOR Z Y 57 Extended Space Time Cross Correlation ExtSTXCF 1 T T T T T Spatial lag 0 Spatial lag 1 x Spatial lag 2 x Spatial lag 3 1 Spatial lag 4 Spatial lag 5 o Spatial lag 6 e Spatial lag 7 4 Spatial lag 8 4 Spatial lag 9 v Spatial lag 10 w
71. in time has been decreased to T 108 5 3 DIFFERENCE BY ONE 41 5 3 Difference by One Differencing is a popular and effective method of removing polynomial trends or gradients from a space time series Za t 1 B Z t Z t Z t 1 where B is a temporal backward operator 5 4 De Trend Trend is a long term or large area movement in a space time series It is the underlying direction an upward or downward tendency and rate of change in a series when allowance has been made for the other components For example in most geographical datasets population clustering is a major trend in space and population increase is a major trend in time for some analysis we need to remove these trends with some methods before analysis In menu De Trend both spatial trend trend surface and temporal trend trend curve can be estimated in three different methods The methods are shown below Polynomial Regression In space 2D or in time the trend surface or curve is approximated by a polynomial with give degree using ordinary least square For example the spatial polynomial regression trend of degree m can be expressed as follow m i 5 1 Strend x y 5 y ayxiy 4 i 0 j 0 where a are the best fit coefficients Fourier Analysis In space 2D or in time the trend surface or curve is approximated by passing the dataset through a band passed filter Average Over Space Time Use average over space time as the temporal spatial
72. inue Parameter Estimation 1 gt Pre estimate Model Param 2 gt Pre estimate Model Param 3 gt Pre estimate Model Param 4 gt Pre estimate Model Param 5 gt Estimate Model Param 6 gt Estimate SRM 7 gt Estimate SRM amp Model Param 8 gt Return Linear STAR Non linear STMA STARMA From STACF STPACF Specified by users Fixed SRM Fixed Model Param Alternatively Selection or lt Enter gt to exit 5 Spatial Relation Matrix SOD_MAX 000 000 000 083 050 050 050 000 083 050 083 083 083 083 000 050 083 125 063 063 063 083 083 125 063 125 250 125 050 083 063 125 250 250 250 050 083 063 250 250 000 250 050 083 063 125 250 250 250 083 083 125 063 125 250 125 000 050 083 125 063 063 063 000 083 050 083 083 083 083 000 000 000 083 050 050 050 Spatial dimension NxN N 10 Aaa Iter 1 Fri Aug 06 13 01 49 0 31 0 22 0 13 0 22 0 08 0 06 Advancement of Average Parameter SSE Time elapsed 0 1 9 91 Z t 0 31 Z t 1 0 22 W1 Z t 1 0 22 Z t 2 0 08 W1 Z t 2 e t The model equation in the result from _S 10 083 000 000 000 083 050 083 000 125 083 050 000 063 125 083 083 125 063 083 050 250 063 083 050 125 063 083 050 063 125 083 083 125 083 050 000 083 05
73. ion Pre Estimation I Y alcule Calculate Dec or inc Statistical Significance space time AICC BIC max S T orders analysis of parameters Y All model types Need to dec inc tested max S T orders Choose the best model type based on AICC BIC analysis Residual s STACF STPACF Exist significant S T correlations End of modeling Figure 2 4 An extended STARMA modeling procedure The area enclosed by dotted line is for the case that model type cannot be decided simply from autocorrelations AICC BIC of every possible model type is evaluated and compared to select the best model type Iteratively testing AICC BIC to decide model type Chapter 3 Data Preparation The targets for STARMA analysis are space time datasets There are two global variables which can be loaded with space time datasets in IEAST i e the major variable Z and the secondary variable Y The major variable is involved in all space time analyses The secondary variable is needed only while doing cross analyses between two variables The only file format from which IEAST can directly retrieve dataset is the octave formated data file dat which is a standard format for storing variables in GNU Octave Besides IEAST also provides some transformations from other data formats to dat for further uses There are three ways to obtain space time datasets which are 1 from dat 2 imported from other form
74. ion of spatial correlation structure consists three steps i e SOD gt SRM SWM These three matrices are stored in the same weight file wet as matrix variables SOD SRM and SWM They together provide a sufficient information for defining a complete spatial correlation structure These three components are described individually below 4 2 Spatial Order Definition SOD It is a 21x21 matrix and defines the spatial orders with respect to the central cell 11 11 in neighborhood In other words the maximum neighborhood coverage can be defined and supported by IEAST is up to 21 by 21 cells For example the matrix shown below gives the spatial order lag definition up to fifth order of every cell Center cell 11 11 is always zero The cells with no relation to the center in the 21x21 matrix are 0s SOD does not give any information about how important some specific cells are but only designate a spatial order for every cell 33 34 CHAPTER 4 SPATIAL CORRELATION STRUCTURE 4 1 SOD oqoooooocoocococoeocococo o oc0o00000000000000000000 oc0o00000000000000000000 oc0o00000000000000000000 oc0o00000000000000000000 oc0o00000000000000000000 oc0o00000000000000000000 2000000 UA YO 00 Oo oooooo SOC COCOCORANNNwWHROTDOCC COCO SFSOCCCOCOWNFRPRFRFNwWCOOTCCCCSO SCOOCCCOWNFOF NWoOOOCCCCSO SCOCCCOCOWNFRFRFRFNwWCOTTCOCCSO SCOOCCCORWNNNWKHhROTOOCCOCCO ooocoocooocoomhWaAwhOIoOoDccCccCcoCoce oc0o000000000000000000000 oc0o000000000000000000000
75. ironment to further provide flexibility and being user friendly The manual gives comprehensive instructions examples of how to use IEAST and some techniques in space time modeling Comments concerning errors or omissions in this edition will be greatly appreciated so that they may be rectified in future editions LIST OF TABLES Chapter 1 Introduction 1 1 What is IEAST Real world datasets in many disciplines are often organized by units of time as well as by ge ographic locations The underlying processes of these datasets are correlated not only in time but also in space It is not always reasonable to analyze these stochastic processes by consider ing space and time separately or by using the well developed univariate AutoRegressive Moving Average ARMA theory Statistical modeling of the datasets produced by stochastic processes that are correlated in both space and time results in the space time extension of univariate ARMA time series models i e Space Time AutoRegressive Moving Average STARMA models Many discrete time discrete space spatio temporal processes can be analyzed by using STARMA theo rems The integrated statistical computing environment IEAST Integrated Environment for Ana lyzing STARMA models was designed for analyzing and modeling the stochastic processes based on general Space Time AutoRegressive Moving Average STARMA models in two dimensional gridded space Current version only support spatial dimen
76. irst datasets are assumed to be zero mean and stationary If they are not some preprocessing or transformations such as differencing de seasonalizing de trending or logarithmic transformation must be done before modeling The processes to be modeled are the stochastic processes whose properties such correlative structures do not vary with location and time To be stationary and having a unique solution just as in the univariate case the causality and invertibility conditions of the process must be satisfied If p 0 in Eq 2 1 the model is a pure STMA model as shown in Eq 2 3 and Z t is always stationary When a STARMA model has p autoregressive terms p gt 0 Z t is stationary if and only if all roots of P Ax det 1 Y Y ba Wa k 1 1 0 are inside the unit circle that is z lt 1 If q 0 in Eq 2 1 the model is a pure STAR model as shown in Eq 2 3 and Z t is always invertible When a STARMA model has q moving average terms q gt 0 Z t is invertible if and only if all roots of 0 q Mk det 1 Y Y Wa k 1 1 0 0 are inside the unit circle that is x lt 1 12 It is assumed that each observation of the space time series has the same expectation func tion standard deviation and probability distribution function We assumed a noise error compo nent which is a sequence of uncorrelated random vectors with a constant distribution Gaussian constant variance and zero mean The random n
77. ive terms It is a matrix with MODEL_MAX_T rows and MODEL_MAX_S columns THE Matrix The parameters of moving average terms It is a matrix with MODEL_MAX_T rows and MODEL_MAX_S columns CovarTB Matrix This is a space time covariance look up table for accelerating the calculations of STACF and STPACF It is a internal global variable E_VARIANCE Real number Estimated variance of random noise 3 6 Visualize Your Dataset IEAST provides 2D 3D graphing abilities to explore datasets and analysis results When dataset is ready under the menu 1 Setup select Information It gives a menu as the following View current system configuration It gives the essential parameters and characteristics of the dataset in major variable 3D Data Plotting The dataset is displayed in 3D While user changes the time the 3D data graph for a given time cross section changes accordingly See an example in Figure 3 1 Average over space temporal trend The dataset is averaged over space It turns out to be the average temporal trend a curve Average over space frequency domain The dataset is Fourier transformed after averaged over space That is the temporal trend is shown in frequency domain Average over time spatial trend The dataset is averaged over time It turns out to be the average spatial trend a surface Average over time frequency domain The dataset is two dimensional Fourier transformed after average over time That is the spa
78. l significance to the example in Quick START section 1 2 Note The analysis of statistical significance will work only after Estimation has been done Diagnostic Analysis gt Statistical Significance gt AICC BIC Analysis gt STACF of Residuals gt STPACF of Residuals Selection or lt Enter gt to exit 1 Analysis of Statistical Significance STAR Param t 1 s 0 0 30 F 1021 97 P 0 001 STAR Param t 1 s 1 0 23 F 181 17 P 0 001 STAR Param t 1 s 2 0 12 F 56 39 P 0 001 STAR Param t 2 s 0 0 20 F 460 46 P 0 001 STAR Param t 2 s 1 0 13 F 58 41 P 0 001 STAR Param t 2 s 2 0 07 F 20 93 P 0 001 Estimates amp P values for AR terms S 0 S 1 S 2 T 1 30 23 12 001 001 001 T 2 02 13 07 001 001 001 9 4 AICC BIC Analysis In addition to the analyses of residuals autocorrelations and parameter statistical significance some other tools can also be helpful in model determination In many cases the STACF and STPACF do not obviously cut off or tail off as they did in pure STAR or pure STMA models In such situations it is not easy to determine the type and orders by observing the behaviors of STACF and STPACF A STAR model with invertibility can be represented as a high order SIMA model On the other hand a SIMA model with causality can be represented as a high order STAR model Fitting a very high order
79. lation to plot 1 STACF of Z 2 STPACF of Z 3 STACF of Y 4 STPACF of Y 5 STXCF of Z and Y 6 STPXCF of Z and Y 7 Extended STXCF Selection 1 7 1 Ploting from spatial lag 0 0 10 Ploting up to spatial lag 10 0 10 Ploting from temporal lag 1 1 10 Ploting up to temporal lag 10 1 10 dy Plotting versus 1 Temporal lags 2 Spatial lags Selection 1 2 1 Plotting STACF of Z versus T Lag The plot is also saved as an eps file ACF_PLOT_T eps Correlation Analyses gt AutoCorrelation STACF gt Partial AutoCorrelation STPACF gt Cross Correlation STXCF 4 gt Partial Cross Correlation STPXCF gt Extended Cross Correlation ExtSTXCF gt Plot Correlations versus T Lag S Lag E gt Return Selection or lt Enter gt to exit 6 Which correlation to plot 1 STACF of Z 2 STPACF of Z 3 STACF of Y 4 STPACF of Y 5 STXCF of Z and Y 6 STPXCF of Z and Y 7 Extended STXCF Selection 1 7 1 Ploting from spatial lag 0 0 10 Ploting up to spatial lag 10 0 10 Ploting from temporal lag 1 1 10 Ploting up to temporal lag 10 1 10 NN ON ND Plotting versus 1 Temporal lags CORRELATION ANALYSES 6 4 PLOTTING CORRELATIONS VERSUS T LAG S LAG FOR Z Y 59 Space Time Autocorrelation STACF Space Time Autocorrelation STACF E E F F E Spatial lag 0 E Temporal l
80. le Here is an example for loading spatial correlation structures Setup gt Space time dataset gt Spatial correlation structure gt Information of datasets gt Return Selection or lt Enter gt to exit 2 Spatial correlation structure 1 gt gt Load from file wet 2 gt gt Define by users 3 gt gt Return Selection or lt Enter gt to exit 1 Spatial correlation structure files in current directory 1 demo wet 2 UF S10 10x10NBPP wet 3 UF S10 16x16NBPP wet 4 UF S10 20x20NBPP wet 5 UF S6 14x14NBPP wet Which Spatial correlation structure file to load Select a number 1 5 1 Loading spatial definition file demo wet done The spatial dimension is 10 x 10 Spatial Order Definition OWWWWWO WNNNNN W WNRPRPEN WwW WNRPORN W WNrRPRPEREN WwW WNNNNN W OWWWWBWO 38 CHAPTER 4 SPATIAL CORRELATION STRUCTURE Spatial Relation Matrix SOD_MAX_S 3 000 050 050 050 050 050 000 050 063 063 063 063 063 050 050 063 125 125 125 063 050 050 063 125 000 125 063 050 050 063 125 125 125 063 050 050 063 063 063 063 063 050 000 050 050 050 050 050 000 Chapter 5 Data Preprocessing To generalize STARMA modeling or to make datasets to satisfy stationarity conditions there are various methods used for preprocessing dataset befor
81. le Let the variable i count down from 10 to 10 with step 1 40 for i 10 1 10 80 endfor goto Synopsis goto lt 1line_no gt Description Change the flow of the program to line_no Example 90 CHAPTER 10 INTERPRETER 80 let A A 1 100 goto 80 help Synopsis help Description Show help menu for commands if Synopsis if lt condition gt lt line_no gt Description Conditionally changig the program flow If the value of the condition is not zero then goto line line_no Otherwise do nothing and continue on next line Condition must be a valid arithmetic statement Example 60 if fAICC lt AICC 90 70 let TYPE STAR 80 fAICC AICC 90 let TYPE STMA inc Synopsis inc lt var_name gt Description Increase the value of the scalar variable var_name by one let Synopsis let lt assignment gt assignment assignment Description Assign one or multiple variable values lt assignment gt must be a valid Octave assignment statement Example let MATRIX_A 1 3 2 1 B 3 MATRIX_C inv MATRIX_A B list Synopsis list begin_line end_line Description Show all or part of a program in memory from begin_line to end_line Example list 10 100 load Synopsis load lt data program weight gt lt filename gt Description Load data program or weight matrices into memory The valid file name for data is dat program is pgm and weight matrices is wet Example load in the program file
82. lgorithm 1 to estimate model parameters this is the best estimate can be found for a linear process and there is no need to optimize it with other algorithms If users also like to find the spatial correlation structure then fix the model parameters and estimate SRM with Algorithm 6 e If the model type is STMA then use Algorithm 2 to estimate initial values of model pa rameters If users also like to find the spatial correlation structure then use Algorithm 7 to estimate both model parameters and SRM e If the model type is STARMA then use Algorithm 2 to estimate initial values of model param eters If users also like to find the spatial correlation structure then fixe model parameters and estimate SRM with Algorithm 6 The following is the estimation stage of the example in Quick Start section 1 2 In Quick Start we only estimated the model parameters Here we also estimate the spatial weighting structure SRM It takes long time to get the results Parameter Estimation 1 gt Pre estimate Model Param Linear STAR 2 gt Pre estimate Model Param Non linear STMA STARMA 3 gt Pre estimate Model Param From STACF STPACF 4 gt Pre estimate Model Param Specified by users 5 gt Estimate Model Param Fixed SRM 6 gt Estimate SRM Fixed Model Param 7 gt Estimate SRM Model Param Alternatively 8 gt Return Selection or lt Enter gt to exit 1 Phi_Now 0 30 0 23 0 12 0 20 0 13 0 07 P
83. ls of three different types i e STAR STMA STARMA with the largest possible orders in time and space were chosen to be candidates According to the model types of the candidates selected in identification stage in the pre estimation stage initial values for the estimation stage were estimated Then these initial values were fed into the estimation stage During the optimization process of estimation stage these candidate models were fitted to the dataset iteratively and the estimates of parameters were found When the model fitting was finished the program entered a diagnosis stage There were diagnostic criteria generated AICC BIC residual variance and parameter significance Based on the values of these criteria the appropriate model type and orders can be decided by user inspection or program selection If necessary the whole process can be re iterated and the type and orders adjusted for the next run according to the criteria obtained 10 2 A SAMPLE PROGRAM 81 Space Time Autocorrelation STACF Space Time Partial Autocorrelation STPACF T r r F T r r Spatial lag 0 Spatial lag 0 Spatial lag 1 Spatial lag 1 gt c Spatial lag 2 x Spatial lag 2 x Spatial lag 3 a Spatial lag 3 a Rho 1 k 0 s T lag o 7 k e i w hal a Lea 1 4 e 4 Phi s T lag i i i i i i i lt i i i i i i i 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Temporal lag Temporal la
84. lter to eliminate the deterministic component contribution from the STARMA process in a dataset The objective is to find the residuals after elimination of a given STARMA process In this function the deterministic component can be removed from the major variable Z for a STARMA model with given type maximum spatial temporal orders and parameters The following is an example of filtering the input dataset with the model Z t 0 26Z t 1 0 36W Z t 1 0 04Z t 2 0 18W Z t 2 e t and the autocorrelation functions before and after removing WNV_XCrow_Detroit_10x10R45 dat is being loaded done File name WNV_XCrow_Detroit_10x10R45 dat Data size Space 10x10 Time 28 J Maximum 27 000000 Minimum 0 000000 Sample Mean 0 692143 J Sample Variance 1 545331 J Value for data missing 99 900000 of Missing Data 0 Loading spatial definition file UF S10 10x10NBPP wet done The spatial dimension is 10 x 10 Spatial Order Definition 00010 9 9 910 0 0 0 010 9 8 7 7 7 8 910 0 09 8 65 5 568 9 0 10 8 6 5 4 3 4 5 6 8 10 975 421245 7 9 9753101357 9 975 4212 45 7 9 10 8 6 5 4 3 4 5 6 8 10 09 8 65 5 5 68 9 0 010 9 8 7 7 7 8 910 0 0 0 010 9 9 910 0 0 0 Data Preprocessing 1 gt Remove Mean 2 gt De seasonalize 1 B dd Z t 3 gt Diference by one 1 B Z t 4 gt De trend 5 gt 6 gt Subsequencing Resampling 7
85. lysis 1 Setup 2 Data Preprocessing 3 AutoCorrelation Analysis 4 Model Identification 5 Parameter Estimation 6 Diagnostic Analysis 7 Forecasting 8 Preference 9 Interpreter 10 Exit Selection or lt Enter gt to exit 1 2 Load dataset Under menu 1 Setup load the dataset for analysis by selecting Load IEAST Dataset Files Setup 1 gt Space time dataset 2 gt Spatial correlation structure 3 gt Information of datasets 4 gt Return Selection or lt Enter gt to exit 1 a Space time dataset 1 gt gt Load major varialbe Z from dataset file dat 2 gt gt Load second varialbe Y from dataset file dat 3 gt gt Import major variable Z from XY T T XY XYTD 4 gt gt Simulation dataset 5 gt gt Save dataset from major variable Z to file dat CHAPTER 1 INTRODUCTION 6 gt gt Return Selection or lt Enter gt to exit 1 Dataset files in current directory 1 demo 1 dat 2 NA_Precipitation dat 3 NOAA_Temperature dat 4 US_Precipitation month dat Which Dataset file to load Select a number 1 4 1 demo 1 dat is being loaded done File name demo 1 dat Data size Space 10x10 Time 100 J Maximum 0 139821 Minimum 0 131962 J Sample Mean 0 000511 Sample Variance 0 037541 Value for data missing 99 900000 of Missing Data 0
86. nce for each estimated parameters can be calculated as below We test the hypothesis that a parameter of interest Br is zero keeping other parameters unrestricted That is let B 61 62 Br 0 Br 1 gt The appropriate test for this hypothesis is based on the statistic TN K S K IS B S B _ y 9 1 S 1 TN K K where T is the number of points in time in the dataset N is the total number of sites in the dataset S 8 is the sum of square residuals for parameter with the restriction 0 S 8 is the sum of square residuals for parameter B without restriction is the estimated parameters for autoregressive terms in column vector form is the estimated parameters for moving average 74 CHAPTER 9 DIAGNOSTIC ANALYSIS terms in column vector form K is the number of elements in K is the number of elements in Significance for the parameter Bj can be found by comparing with standard tables of the F distribution using 1 and TN K K O degrees of freedom During the modeling process any estimated parameter proved being statistically insignif icant should be removed from the model and sometimes space time orders should be reduced to obtain a simpler model which is then considered as the next candidate model Parameters with significance probability lt 0 8 can be tentatively regarded as insignificant from a practical standpoint The following is an analysis of statistica
87. nd run Initializing Covariance Look up Table CovarTB for major varialbe Z gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Calculating STACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Autocorrelation of Z Rho s 1 k 0 10 2 A SAMPLE PROGRAM 334 159 068 022 004 001 005 003 011 010 006 008 009 004 001 110 070 048 033 026 023 017 005 008 006 001 014 023 020 011 Calculating STPACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt 025 026 023 010 015 003 013 001 004 002 001 000 007 005 007 000 003 003 009 005 013 009 007 018 002 010 001 009 001 002 Space Time Partial Autocorrelation of Z Phi s 1 0 65 0 10 Advancement of Average Parameter SSE 0 000003 Time elapsed 0 0 3 95 Z t 0 65 Z t 1 0 10 W1 Z t 1 e t 83 84 AICC BIC Variance AICC 6 6371e 04 BIC 2 8178e 04 eVARIANCE 0 0013105 eTHETA 0 29 0 08 Advancement of Time elapsed ee bee Iter 2 eTHETA 0 55 0 13 Advancement of
88. nerated by picking the last 100 generations from a total of 5000 simulated generations The two dimensional space was assumed to be eight by eight i e 64 sites in total Uniform weight matrices defined in 10 and 16 were used for all spatial orders from zero to third The random noise vector was e t N 0 a with variance o 0 001251 Example 1 The dataset in this example was generated from a pure STAR model with a maximum temporal order two and maximum spatial order one i e STAR 2 1 as the following equation Z t 0 5Z t 1 0 3W Z t 1 0 15Z t 2 e t As the autocorrelation analysis shown in Figure 6 1 the STACF decays slowly both in space or in time in an essentially exponential manner In contrast the STPACF drops abruptly at temporal lag two and spatial lag one After these lags the STPACF fluctuates around zero with small random disturbances Example 2 This dataset was generated from a pure STMA model with the maximum temporal 49 Space Time Autocorrelation STACF Space Time Partial Autocorrelation STPACF 1 y r r r 1 r r Spatial lag 0 Spatial lag 0 Spatial 1291 e Spatial 1331 e Spatial lag 2 Spatial lag 2 Spatial lag 3 0 Spatial lag 3 2 E EN i 0 5 0 5 x ion i LA 3 a A 23 5 Ll A a a SN o i B i R A OB st O eer a emer ag eee cee ES a 0 5 0 5 4 i i s 2 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Temporal lag Tem
89. ng environment The statistical significance can also be evaluated for each parameter Cross correlation analysis provided for analyses of two space time variables 2D 3D plotting abilities 1 4 Requirements Software JEAST version 1 30 01 or later GNU Octave v2 0 16 or later download at http www octave org GNUPLOT v3 8 or later see http www gnuplot info Hardware IEAST has very low hardware dependency It can be ported to any kind of platforms which can run GNU Octave However this is a computing intensive software For most datasets with spatial dimension higher than 20x20 the following specification is suggested Processor AMD K7 1GHz equivalent or higher Memory 128MB or bigger Disk Space 100MB 14 CHAPTER 1 INTRODUCTION 1 5 Installation The system is highly suggested to be installed and run under a unix operating system or vari ants in which IEAST is well tested If you do not have unix you can either install a Linux operating system free download at http www mandrake org or install the Linux Win32 sim ulator CYGWin it simulates a Linux environment under all Windows systems free install can be found at http www cygwin org The instructions for install Octave on Windows can be found at http octave sourceforge net Octave_Windows htm For unix installations follow the instructions below 1 Install GNU Octave into your unix system 2 Install GNUPLOT into your
90. od model to represent the dataset 9 3 STATISTICAL SIGNIFICANCE 73 Diagnostic Analysis Statistical Significance AICC BIC Analysis STACF of Residuals STPACF of Residuals Selection or lt Enter gt to exit 4 Calculating STPACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Partial Autocorrelation Phi s 1 S lag 0 1 2 3 4 Talag O SH Ss sess as 1 0 003 0 002 0 001 0 003 0 010 2 0 001 0 008 0 007 0 021 0 015 3 0 010 0 004 0 024 0 032 0 014 4 0 018 0 008 0 032 0 014 0 000 5 0 008 0 016 0 012 0 008 0 024 6 0 005 0 008 0 030 0 013 0 012 7 0 000 0 013 0 016 0 002 0 040 8 0 014 0 002 0 000 0 015 0 024 9 0 006 0 006 0 010 0 022 0 032 10 0 012 0 018 0 014 0 008 0 001 11 0 009 0 026 0 018 0 008 0 025 12 0 019 0 053 0 004 0 040 0 011 13 0 013 0 011 0 024 0 006 0 064 14 0 018 0 029 0 001 0 009 0 007 15 0 005 0 014 0 003 0 007 0 045 16 0 011 0 019 0 011 0 019 0 031 17 0 011 0 015 0 010 0 009 0 013 18 0 002 0 015 0 007 0 021 0 060 19 0 007 0 001 0 009 0 021 0 005 20 0 014 0 010 0 005 0 030 0 007 9 3 Statistical Significance A tentative model need to be checked to see if it is unduly complex i e having statistically insignificant terms The statistical significance of model parameters is evaluated for this purpose Statistical significa
91. of the correlation results we can tentatively decide the type orders of the dataset as described in section 6 1 For most of pure STAR and pure STMA processes the type orders can be easily identified by users However for the mixed STARMA and some STAR STMA models it is hard to distinguish them by eyes Therefore IEAST provides an algorithm select Automatic Identification Type Orders to replace human inspection and to assist users making decision This identification algorithm is also based on correlation anal yses Notice that this functionality of automatic identification is not necessary correct especially for some practical datasets which have ambiguous behaviors on STACF STPACF Model Identification 1 gt Automatic Identification Type Orders 2 gt Artificial Identification Type Orders 3 gt Parameter Masking D4 gt ssis 5 gt Return Selection or lt Enter gt to exit 1 Model type is suggested to be STMA with Max Temporal Order 3 Max Spatial Order 2 61 62 CHAPTER 7 MODEL IDENTIFICATION 7 2 Artificial Identification If users want to specify the model type orders according to their knowledge to this dataset select Artificial Identification Type Orders Model Identification 1 gt Automatic Identification Type Orders 2 gt Artificial Identification Type Orders 3 gt Parameter Masking Selection
92. oise vector e t is normally distributed at time t and satisfies the following characteristics e t N 0 0 Ly o7In s 0 Ele t e t s H otherwise ElZ t e t s 0 ifs gt 0 2 6 General STARMA Modeling Procedure for using IEAST Based on the statistical tools implemented in IEAST an extended STARMA modeling procedure from Box Jenkins modeling is suggested while using IEAST This is a general model procedure for all possible situations Users do not have to follow all steps but only need to extract the portion needed For example if the model type is known from some prior knowledge of the dataset the area enclosed by dotted line is not necessary for the modeling This modeling procedure is especially useful while programming using IEAST interpreter see Chapter 10 IEAST interpreter provides necessary instruction set to construct the modeling flow This programmability gives users best efficiency and flexibility 22 CHAPTER 2 STARMA MODELS AND MODELING Dataset input Pre processing deseason detrend subseq l Space time autocorrelations STACF STPACF model type based on N TACF STPACE _ gt _ Decide model type Iteratively set model type as based on STACF STPACF o STAR SIMA A Ed ds 1 Decide S T orders Set to max possible S T orders based on STACF STPACF Pre Estimat
93. on normal data resemble normal data is by using a transformation In menu Box Cox Transform this ability is offered Box Cox transform is a comprehensive name of a class of variance stabilizing transforms The transforms are given by z 1 0 A A a ln x A 0 Given a vector of observation data x 2 2 Ln one way to select the power A is to use the A that can maximize the following likelihood function n n n ril A A y f x A ini ie A 1 In lt i 1 a i 1 where n i 1 5 6 SUBSEQUENCING RESAMPLING 45 5 6 Subsequencing Resampling Sometimes we need the dataset to be trimmed into smaller piece either in time or in space or resam ple the dataset to downscale or upscale in time or in space In the menu Subsequencing Resampling IEAST provides the following Subsequencing Resampling abilities Subsequencing in time A subset of dataset in any period of time can be picked from the original dataset Subsequencing in space A subset of dataset in any square area in space can be picked from the original dataset Resampling in time merge There are six methods for resampling cell merging the dataset in time i e sum average number of non zero cells maximum minimum and variance Resampling in space merge Same as in Resampling in time merge Limiting the values of in Z This is used to limit the cell values of dataset to certain extent 5 7 Smoothing Smoothing t
94. or calculating autocorrelations 7 Max spatial lag 4 0 10 MAX_S_LAG 4 unchanged Max temporal lag MAX_T_LAG 20 unchanged 20 1 99 CHAPTER 1 INTRODUCTION Initializing Covariance Look up Table CovarTB for major varialbe Z gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Calculating STACF gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Space Time Autocorrelation of Z Rho s 1 k 0 16 17 18 19 20 259 112 125 012 100 048 020 013 029 019 029 013 020 014 016 018 006 003 004 018 023 013 020 001 008 009 006 004 000 011 006 015 002 015 006 010 012 006 002 003 007 004 008 019 001 027 000 010 006 001 019 Press any key to continue Selection Correlation Analyses AutoCorrelation STACF Partial AutoCorrelation STPACF Cross Correlation STXCF Partial Cross Correlation STPXCF Extended Cross Correlation ExtSTXCF Plot Correlations versus T Lag S Lag Return or lt Enter gt to exit 2 Remove mean of the variable Z 0 0000000000 for autocorrelation Y N N 7 Max spatial lag 4 0 10 MAX_S_LAG 4 unchanged 7 Max temporal lag MAX_T_LAG
95. pacf PACF Z 16 4 autoiden TYPE MAXS MAXT ACF PACF bic Synopsis bic lt result gt lt final_Phi gt lt final_Theta gt lt final_EE gt Description This instruction is used to calculate Bayesian information criterion BIC which is used to correct some tendency in AICC testing The result is a real number Three arguments are used as inputs which are estimated AR coefficients final_Phi estimated MA coefficients final_Theta and final residuals fian1_EE Example bic BIC EST_PHI EST_THETA EST_EE calctrend Synopsis calctrend lt result gt lt Z gt lt S T gt lt method gt arg Description Calculate spatial or temporal trends of dataset lt Z gt Result the trend will be stored in the variable specified in lt result gt lt Z gt is the dataset to be calculated Third argu ment lt S T gt specifies either spatial S or temporal T trend is going to be found jmethod specifies the method used to find the trends there are three different options POLYREG for polynomial regression FOURIER for Fourier transformation and AVERAGE for averaging over time The last argument arg is used by POLYREG and FOURIER In POLYREG it means the maximum order of the polynomial used to fit the dataset In FOURIER means the highest harmonic in spectrum for the trend Example calculate third order polynomial regression spatial trend surface and save to the variable STREND calctrend STREND Z S POLYREG 3 clearmode Synopsis clearmode
96. poral lag Figure 6 1 STACF and STPACF for STAR 2 1 Space Time Autocorrelation STACF Space Time Partial Autocorrelation STPACF 4 1 i Spatial lag 0 E Spatial lag O Spatial lag 1 Spatial lag 1 x Spatial lag 2 x Spatial lag 2 Spatial lag 3 0 Spatial lag 3 a 0 5 0 5 EN A Y F 5 S j A x i j 4 gt ee ae O OELE ee ae fg A lowes ee oe 3 a a a A a oe Al RA OS a lie alone camel f E E x a 0 5 0 5 4 i i 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Temporal lag Temporal lag Figure 6 2 STACF and STPACF for STMA 1 1 order one and maximum spatial order one i e STMA 1 1 as the following equation Z t e t 0 6 e t 1 0 4 W e t 1 In the results shown in Figure 6 2 instead of STACF being decaying it drops abruptly or cuts off at both temporal lag 1 and spatial lag 1 After those lags STACF is always near zero with small random disturbances As to STPACF the envelope of it decays slowly though fluctuating markedly In summary for STAR 2 1 the STPACF cuts off at temporal lag two and at spatial lag one STPACF tails off On the other hand for STMA 1 1 the STACF cuts off at temporal lag one and at spatial lag one and the envelope for STPACF tails off These two examples illustrated how the model type is selected and how the temporal spatial orders are decided by observing the behaviors of STACF and STPACF 6 2 Autocorrelations STACF and STPACF
97. ption Temporarily pause the program for program diagnosis or peeking analysis re sults The program will continue when an empty line is entered press Enter right after the prompt Every command GNU Octave can accept can be used in stop stpacf Synopsis stpacf lt st_pacf gt lt Z gt lt max_t_lag gt lt max_s_lag gt Description Calculate partial space time autocorrelation according to the data in series Z the maximum time lag max_t_lag and the maximum spatial lag max_s_lag The final result is stored into the matrix variable stpacf Example stpacf ST_PACF Z 16 3 stxcf Synopsis stxcf lt result gt lt Vari gt lt Var2 gt lt max_t_lag gt lt max_s_lag gt Description Calculate space time cross correlation between two space time variables Varl and jVar2 with the maximum time lag max_t_lag and the maximum spatial lag max_s_lag The final result is stored into the matrix variable lt result gt Example load data WNV_XHuman_CR45_10x10 dat load data WNV_XCrow_CR45_10x10 dat Y stxcf STXCF Z Y 20 4 stxcf_ext Synopsis stxcf_ext lt result gt lt Var1 gt lt Var2 gt lt max_t_lag gt lt max_s_lag gt Description Calculate space time cross correlation between two space time variables lt Var1 gt and lt Var2 gt with maximum spatial lag max_s_lag from temporal lag max_t_lag to max_t_lag The final result is stored into the matrix variable lt result gt Example load data WNV_XHuman_CR45_10x10 dat load
98. ress any key to continue Parameter Estimation 1 gt Pre estimate Model Param Linear STAR 2 gt Pre estimate Model Param Non linear STMA STARMA 3 gt Pre estimate Model Param From STACF STPACF 4 gt Pre estimate Model Param Specified by users 5 gt Estimate Model Param Fixed SRM 6 gt Estimate SRM Fixed Model Param 7 gt Estimate SRM Model Param Alternatively 8 gt Return Selection or lt Enter gt to exit 7 Estimating Model Parameters amp SRM alternatively Iter 1 Fri Sep 19 10 57 56 2003 Spatial Relation Matrix SOD_MAX_S 6 SRM 0 25 0 25 0 25 ePHI 68 0 30 0 23 0 12 0 20 0 13 0 07 Advancement of Average Parameter SSE 0 074186 Time elapsed 1 46 43 94 ERAN Iter 2 Fri Sep 19 12 44 40 2003 patial Relation Matrix SOD_MAX_S 6 ePHI 0 30 0 22 0 12 0 20 0 13 0 07 Advancement of Average Parameter SSE 0 000422 Time elapsed 1 43 45 69 SEAS Iter 3 Fri Sep 19 14 28 26 2003 patial Relation Matrix SOD_MAX_S 6 O ia Hu H 1 0 30 0 21 0 11 0 20 0 12 0 07 Advancement of Average Parameter SSE 0 000105 Time elapsed 1 48 5 26 ARAN Iter 4 Fri Sep 19 16 16 31 2003 patial Relation Matrix SOD_MAX_S 6 32 0 37 0 00 0 32 2 o o w pue o HS ePHI 0 30 0 20 0 11 0 20 0 12 0 07 Advancement of Average Parameter SSE 0 000045 Time elapsed 1 46 11 13 Aa Iter 5
99. rg or arg2 On other hand commandi argi arg2 means command can have none or one argument which is either argl or arg2 All the instructions IEAST version 1 30 01 and their usage are listed below aicc Synopsis aicc lt result gt lt final_Phi gt lt final_Theta gt lt final_EE gt Description This instruction is used to calculate bias corrected Akaie s information criterion AICC The result is a real number Three arguments are used as inputs which are esti mated AR coefficients final_Phi estimated MA coefficients final_Theta and final residuals fianl EE 10 3 COMMAND REFERENCES 87 Example aicc AICC EST_PHI EST_THETA EST_EE autoiden Synopsis autoiden lt result_type gt lt result_maxs gt lt result_maxt gt lt stacf gt lt stpacf gt Description This instruction is to automatically identify the model type and maximum spatial temporal orders based on the analyses of space time autocorrelations The input ar guments are lt stacf gt and lt stpacf gt which are the matrices of the space time autocorrelation and partial autocorrelation The output will be put into three variables lt result_type gt lt result_maxs gt and lt result_maxt gt The model type is in lt result_type gt in string type STAR STMA or ARMA The maximum spatial order is in lt result_maxs gt The max imum temporal order is in lt result_maxt gt Example load data testdata dat stacf ACF Z 16 4 st
100. riable values system cinfigurations a string enclosed by or autocorrelations Example To print a model equation with type in the variable TYPE model AR param eters in MY_PHI and model MA parameters in MY_THE Then to print the variable VARIANCE and the autocorrelation result in the variable STACF print model TYPE MY_PHI MY_THE z t 0 26 Z t 1 0 37 W1 Z t 1 0 10 W2 Z t 1 0 09 W3x Z t 1 0 04 Z t 2 0 18 W1 Z t 2 0 09 W2 Z t 2 0 06 W3 Z t 2 0 02 Z t 3 0 10 W1 Z t 3 0 00 W2 Z t 3 0 04 W3 Z t 3 e t print var VARIANCE VARIANCE 0 0012665 print acf STACF 16 4 92 CHAPTER 10 INTERPRETER S lag 0 1 2 3 4 We lag TESORO ae ee ee ARA 1 0 332 0 280 0 163 0 006 0 064 9 0 000 0 003 0 003 0 003 0 004 10 0 008 0 013 0 006 0 013 0 004 11 0 003 0 007 0 003 0 006 0 002 12 0 007 0 006 0 008 0 007 0 002 13 0 004 0 005 0 006 0 004 0 001 14 0 011 0 011 0 008 0 001 0 001 15 0 004 0 006 0 004 0 005 0 001 16 0 002 0 001 0 002 0 002 0 002 quit Synopsis quit Description Leave the interpreter and back to menu driven environment rem Synopsis rem lt any string for remark gt Description Comment lines any line with the rem as its beginning whill be skipped while running Example rem the following calculate STPACF renumber Synopsis renumber Description Rearrange all line numbers start f
101. rom 10 increase by 10 The line numbers used by flow control commands are adjusted according to their relative locations In other words the line number may change accordingly run Synopsis run Description Execute the program loaded from the first line of the program save Synopsis save lt data program weight gt lt filename gt Description Save data IEAST program or weight matrices from memory to files The valid file name for data is dat for program is pgm and for weight matrices is wet Example save the data to the data file demo dat save data demo pgm 10 3 COMMAND REFERENCES 93 setmode Synopsis setmode TRACE HIDEACF PLOTEPS Description Set the environmental settings Example print the line number while running for debugging purpose setmode TRACE Example don t print the results of stacf or stpacf while running only calculating setmode HIDEACF Example not only output the graphics with GNUPLOT but also generate a corresponding eps file setmode PLOTEPS shell Synopsis shell lt command gt Description Execute a shell command under IEAST interpreter significance Synopsis significance lt sig Phi gt lt sig_Theta gt lt final_Phi gt lt final_Theta gt AR Mask MA Mask Description Statistical significance is to evaluate the importance of the parameters obtained from estimation stage The outputs of this instruction are stored in sig Phi and sig
102. ructure 1 gt gt Load from file wet 2 gt gt Define by users 3 gt gt Return Selection or lt Enter gt to exit 1 Spatial correlation structure files in current directory 6 3 CROSS CORRELATIONS STXCF STPXCF AND EXTSTXCF 1 UF S10 10x10NBPP wet 2 UF S10 16x16NBPP wet 3 UF S10 20x20NBPP wet Which Spatial correlation structure file to load Select a number 1 15 1 Loading spatial definition file UF S10 10x10NBPP wet done The spatial dimension is 10 x 10 Spatial Order Definition 0 0 010 9 9 910 0 0 0 010 9 8 7 7 7 8 910 0 00865556890 10 8 6 5 43 4 5 6 810 975 4212 45 7 9 975 3 1013 5 7 9 975 4212 45 7 9 10 8 6 5 43 4 5 6 8 10 09 8 655 568 9 0 010 9 8 7 7 7 8 910 O 0 0 010 9 9 910 0 0 0 Correlation Analyses E gt AutoCorrelation STACF E gt Partial AutoCorrelation STPACF E gt Cross Correlation STXCF 4 gt Partial Cross Correlation STPXCF gt Extended Cross Correlation ExtSTXCF gt gt Plot Correlations versus T Lag S Lag Return Selection or lt Enter gt to exit 5 Do you want to remove the mean of Z 0 1375000000 Y N Y n Do you want to remove the mean of Y 0 6921428571 Y N Y n Max spatial lag 10 0 10 XMAX_S_LAG 10 unchanged Max temporal lag 27 1 27 12 Initializing Covariance Look up Table CovarTB for major varialbe Z gt gt gt gt gt gt gt g
103. s by n i e 1 B n Z Example De seasonalize the current series diff Z T lt Z gt 12 dir Synopsis dir data weight program Description List the files in current directory If the optional argument is data then list dataset files only If weight then weighting files only If program then program files only If no argument list them all Example list all the files in current directory dir Example list all weight matrix files in current directory dir weight end Synopsis end Description Terminate the program and back to prompt estimate Synopsis estimate lt est_phi gt lt est_theta gt lt final_EE gt lt init_phi gt lt init_theta gt AR Mask MA_Mask 10 3 COMMAND REFERENCES 89 Description Using the initial values as start the optimization process iteratively approaches to the final estimated parameters This optimization process is based on Marquardt s method In arguments the initial values as used as inputs in init_phi and init_theta The final es timated parameters are stored into the variables est_phi and est_theta The final residuals are in final_EE AR Mask and MA_Mask are used to decide which parameters should be taking into account during optimization process Users can disable or enable any single parameter in the parameter matrices est_phi or est_theta To enable disable the specific parameter simply set the corresponding element in matrix AR_Mask or MA_Mask to one zero Example
104. sion N x N that is the numbers of cells in X and Y direction are equal IEAST takes data files dat spatial weight files wet program files pgm and spatial temporal configurations in setup menu as inputs It outputs in plots text numerical results and equations see Figure 1 1 This system was developed on RedHat Linux 9 0 using the GNU OCTAVE v2 1 40 which is a high level language primarily intended for numerical matrix computations OCTAVE provides a convenient command line interface for numerically solving linear and nonlinear problems and for performing other numerical operations using a language that is largely compatible with MatLab Because of source code compatibility TEAST can be easily run under Windows Unix MacOS without modifications To provide flexibility there are two user interfaces provided in IEAST menu driven mode and interpreter mode programming mode In menu driven mode users can do the modeling or analyzing procedure by selecting a hierarchical menu commands Users need to control the flow of the procedure by themselves In programming mode there is a simple STARMA programming language provided Highly integrated instructions are provided for users to compose STARMA modeling analyzing procedures Users can design an efficient and autonomous modeling procedure for specific applications by simply combining instructions and flow controls It is common to 3 4 CHAPTER 1 INTRODUCTION Dataset file Weigh
105. sive terms and moving average terms respectively Z W MODEL MAX_T_ORDER and MAX_S_ORDER are system reserved global variables which represent the dataset Z t the weight matrices W the model type and the maximum orders in time or space respectively The leading f of some variables meant final results the leading e meant estimated results the leading i meant initial values the leading s meant statistical significance The variable MASK was a matrix used to disable some unimportant or not interesting parameters in PHI or THETA by simply setting to zero to disable or setting to one to enable on the corresponding positions of matrices In this example all parameters in either PHI or THETA were enabled 10 2 4 Results The program and outputs of this example modeling procedure are shown below After entering the interpreter in TEAST by choosing selection 10 in the main menu of menu driven mode there came an interactive prompt for commands Use load command to retrieve the modeling procedure program in file demo pgm The first part of the following results are the program listing The two tables following the program are the sample STACF and STPACF of the dataset The results 82 CHAPTER 10 INTERPRETER followed the tables are analysis output of pre estimation estimation and diagnostic checking Welcome t load program demo pgm list 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1
106. ss testing variance check ing 12 13 statistical significance testing and statistical measures like AICC BIC In IEAST autocorrelation checking variance checking statistical significance testing and AICC BIC are provided 9 1 STACEF of Residuals The most commonly used method for diagnostic checking is to investigate the behaviors of the STACF and STPACF of the residuals Inadequacy of a candidate model will be unveiled in the form of significant space time correlations among the residuals If the candidate model adequately represents the data residuals must be white uncorrelated noise that is all autocorrelations in space or time at nonzero lags are equal to zero or approximately equal to zero in practice If we compute the residuals STACF and STPACF of the example in Quick Start section 1 2 we have the results below The plots of residuals STACF STPACF are shown in Figure 9 1 Diagnostic Analysis gt Statistical Significance gt AICC BIC Analysis gt STACF of Residuals gt STPACF of Residuals Selection or lt Enter gt to exit 3 Initializing Covariance Look up Table CovarTB 71 72 CHAPTER 9 DIAGNOSTIC ANALYSIS Residuals Space Time Autocorrelation R STACF Residuals Space Time Partial Autocorrelation R STPACF F Spatial lag O 1 i i T Spatial lag 0 Spatial lag 1 gt Spatial lag 1 00 Spatial lag 2 x Spatial lag 2 Spat
107. ssed as 2 Iv k 2 4 vu s Es DOE Besa Based on the definition of space time covariance yj s as in Eq 2 4 the following two essential functions can be defined Space Time AutoCorrelation Function STACF and Space Time Partial AutoCorrelation Function STPACF are elementary in determining the preliminary model type and orders from the observed dataset Their calculations are described in 9 5 and 12 The STACF between 1 and kt order neighbors at time lag s can be expressed as A 2 5 pur s Mu O Vex 0 Similar to the univariate ARMA case the STPACF can be found as follows Given s and h which are respectively the maximum temporal and spatial orders the BLP Best Linear Predictor of Z t is s h 2 6 Z tlsn X X ou WOZ k k 1 1 0 For every s 1 p and h 0 A a set of coefficients zx k 1 s 1 0 h can be found for some BLP Z t s Thus the STPACF is the set 2 7 snlPsn is the last element of the set Esn s 1 p h 0 r where 0 lk 1 s 1 0 h is the set of coefficients making VAIA a BLP of Z t with the limitations of maximum temporal order s and maximum spatial order h As shown in 9 and 12 Esn is the solution of the space time extension of Yule Walker equations with maximum temporal order s and maximum spatial order h 2 5 ASSUMPTIONS 21 2 5 Assumptions For a dataset to be meaningful to STARMA analysis there are some assumptions to be met F
108. stimate Model Param Specified by users Users assign initial values based on their knowledge to the dataset 5 Estimate Model Param Fixed SRM If we know the spatial correlation structure basi cally SRM before estimating model parameters this algorithm is used to find the estimates of model parameters Sometimes even spatial correlation structure is unknown we still can use this algorithm with the assumption of isotropic SRM The values of elements in SRM will not affect too much to the model parameters if we use a reasonable spatial correlation structure such as a uniform or isotropic SRM 6 Estimate SRM Fixed Model Param If the model parameters are known then we can estimate the SRM with fixed model parameters For example the parameters in STAR model can simply be estimated by Algorithm 1 which is quick Thereafter the SRM can be estimated using the initial estimates of model parameters 7 Estimate SRM amp Model Param Alternatively With an isotropic SRM as the initial SRM the SRM and model parameters can be estimated alternatively When converged we can obtain a set of model parameter estimates and a SRM estimate However because of so many variables in SRM to be estimated in general this algorithm is very time consuming In summary considering trade offs on computing time and accuracy we have the following sugges tions 8 3 ESTIMATION ALGORITHMS IN IEAST 67 e If the model type is STAR then use A
109. t gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Initializing Covariance Look up Table YCovarTB for 2nd variable Y gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Initializing Cross Covariance Look up Table XCovarTB gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Calculating Extended STXCF 56 CHAPTER 6 CORRELATION ANALYSES Initializing Cross Covariance Look up Table XCovarTB gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt Extended Space Time Cross Correlation of Z and Y Rho s 1 k 0 Ko o o N o o w Q o pur o o N Q o pur 5 o pur o o N o o pur o o N Q o pur o o pur 10 000 001 001 002 001 001 000 000 001 000 001 11 000 000 000 000 000 000 000 000 000 000 000 12 000 000 000 000 000 000 000 000 000 000 000 Plotting from spatial lag 0 0 10 Plotting up to spatial lag 10 0 10 Plotting from temporal lag 12 12 12 Plotting up to temporal lag 12 11 12 eNO ON ND Plotting versus 1 Temporal lags 2 Spat
110. t Matrices Spatial Temporal Programs dat file wet configurations pgm Text numerical results Plots Model equation Figure 1 1 Inputs and outputs of IEAST implement a STARMA modeling procedure in 20 lines of codes This programmability is especially useful when iterative procedure is necessary during analysis This system is designed for two dimensional STARMA analysis However by carefully designing the spatial weight matrices this system can be adapted to one dimensional or even three dimensional systems If the computer memory and computing power are sufficient there is no limitations to the spatial dimensions of the dataset explored But if the spatial dimension is too small the marginal effect at the boundaries would be significant and should be taken into account In univariate ARMA modeling a well known modeling procedure is the three stage itera tive modeling which is referred to as the Box Jenkins method The space time extension of the three stage iterative modeling procedure was first presented in 1980 by Pfeifer and Deutsch The modeling procedure used in IEAST is a refinement of that of Pfeifer and Deutsch see Figure 2 4 This document corresponds to IEAST version 1 30 01 1 2 Quick Start A Simple Example In this section a simple example is used to demonstrate how to use IEAST to analyze and model a dataset The dataset demo 1 dat used for this demonstration was generated by simulation
111. ters 2 0 2 0000 e 65 8 2 Estimation of Spatial Weighting Structure o 65 CONTENTS 8 3 Estimation Algorithms in IEAST 9 Diagnostic Analysis 9 1 STACF of Residuals 9 2 STPACF of Residuals 9 3 Statistical Significance 9 4 AICC BIC Analysis 9 5 Residuals Variance 10 Interpreter 10 1 The Programming Environment e e 10 1 1 Invoke and Quit 10 1 2 Program Mode and Interactive Mode o o 10 1 3 Notices for the Interpreter o o ee ee 10 2 A Sample Program 10 2 1 Dataset Generation e eee e 10 2 2 Modeling Procedures ee 10 2 3 Program Description e e 10 2 4 Results 10 3 Command References Bibliography iii 66 71 71 72 73 74 76 77 77 77 78 79 79 80 80 81 81 86 95 CONTENTS List of Figures 1 1 1 2 1 3 2 1 2 2 2 3 2 4 3 1 5 1 5 2 5 3 5 4 5 5 6 1 6 2 6 3 6 4 6 5 9 1 Inputs and outputs of TEAST o a a a a p a ee 4 Space time ACF and PACF for STAR 2 2 onas 10 The first two levels of the menu system in IEAST 15 Relationship between data generating and statistical modeling procedures 17 Box Jenkins Modeling e 18 An example of definition of spatial orders in 2D systems 19
112. tial trend is shown in 2D frequency domain Histogram Display the histograms of the dataset in space or in time 3 6 VISUALIZE YOUR DATASET 31 Trend Surface using PloyRegr 10th order Z x y t Figure 3 1 An example of 3D plotting of dataset 32 CHAPTER 3 DATA PREPARATION Chapter 4 Spatial Correlation Structure Beside space time datasets another essential step before analysis is to specify a spatial correlation structure for a dataset in practice it is a definition of set of weight matrices The specification describes the relative spatial relations among cells including the definition of spatial orders weight ing distributions and spatial dimension In most of practical cases spatial correlation structures are isotropic directionless Thus once the definition of spatial orders or lags is given IEAST can generate an appropriate set of weight matrices for an isotropic structure 4 1 Spatial Dimension N_SITE and N x N Current version version 1 30 01 of IEAST works only on two dimensional gridded space which have to be an array of N x N The spatial dimension needs to be specified and stored in the system as the global variable N_SITE N_SITE is the total number of sites or cells in the dataset to be analyzed i e N_SITE N The spatial correlation structure can be uniquely defined by a set of weight matrices SRM To make it clear in IEAST the specification of weight matrices SRM or the specificat
113. tion 1 3 3 Example XYTD If the spatial The following is the first line in the file X Y N T Are they the field names Y N Y 1 2 3 4 Hae which field is for coordinate X7 1 4 1 which field is for coordinate Y 1 4 2 which field is for Time 1 4 4 which field is for Data 1 4 3 Reading records from file done Dimension of the gridded space NxN N 10 10 100 Numer of time points T 28 28 10000 How to deal with repeated records at the same location and time To 1 discard the latter 2 replaced with the latter 3 sum up them Selection 1 3 3 There is some cell s in space time not being assigned values Fill them with 1 zeros 2 Default missing marks MISSING 99 3 1 4 User defined values Selection 1 4 1 Empty cells are filled with O File name CCF varCrow XYTD txt J Data size Space 10x10 Time 28 J Maximum 27 000000 J Minimum 0 000000 J Sample Mean 0 692143 Sample Variance 1 545331 Value for data missing 99 900000 J of Missing Data 0 3 3 SIMULATION DATA 27 3 3 Simulation Data IEAST can also generate Monte Carlo simulated STARMA datasets The following example gives a simulation for a dataset based on the facts Space 10x10 cells Time 80 periods Max Temporal Order 2 Max Spatial Order 1 and the model equation Z t 0 40 Z t 1 0 30 W Z t 1 0 20 Z t 2 0 10 W Z
114. trend respec tively Remove Trends Remove either spatial or temporal trend from dataset with various methods Five methods are for removing spatial temporal trends and described below e Only Spatial Trend The same spatial trend is removed from all time period The detrended dataset can be express as Zz y t Za y t Strenal y e Only Temporal Trend The same temporal trend is removed from all spatial sites The detrended dataset can be express as Zi ylt Ze y t gt Ttrena t e S T Trend A proportioned using temporal trend spatial trend is removed from the dataset The detrended dataset can be express as Zo y t Zz y t a Strend y Ttrenalt 42 CHAPTER 5 DATA PREPROCESSING Spatial Trend average over time Zin colored map Spatial Trend average over time Z x y t Figure 5 1 3D plots of the original dataset e Best Fit S Trend A best fit to the dataset factor h is found by OLS then subtract the product of h and the spatial trend from the dataset The detrended dataset can be express as Zz yt Za y t es Strena 2 y e Best Fit S T Trend A best fit to the dataset and proportioned using temporal trend spatial trend is removed from the dataset The detrended dataset can be express as Zz y t Zz y t h Strena y Ttrena t where h is a constant so that h Sirena x y Trrena t best fit to Z t The following is an example of de trending a dataset both in time and in

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