SUBBOTOOLS User's Manual
Contents
1. i EE tell zm where A a a 2 2 Parameters Estimation Consider a set of N observations 2 xy and assume that they are independently drawn from an unknown Subbotin distribution We are interested in the estimation of the parameters a b m of this unknown distribution A first approach to this problem commonly known as method of moments is based on the comparison between the theoretical moments of the density see Section 2 1 The Subbotin Fam ilies of Densities page 2 and the sample moments computed starting from the N observations The procedure goes as follows We compute the sample mean My the sample variance M and the sample absolute deviation Mj First we obtain an estimation of the central parameter of the distribution using the sample mean We set m My Second we obtain the estimation for the parameter b as the unique root of the equation D 3 DT 1 b _ M 2 62 Mi Chapter 2 Overview 4 and finally the estimated value for the parameter a is given by MT 1 b T 3 b a b71 A second possible approach consists in the maximization of the empirical likelihood i e in the method generally known as maximum likelihood estimation More specifically instead of maximizing the likelihood of the sample we minimize the negative log likelihood computed taking the logarithm of the likelihood function and chancing it sign Moreover using the first order conditions is easy to derive the expression of2. interval formed by two consecutive observations of the dataset See Section 5 1 3 Minimization on Intervals page 11 A search algorithm is implemented on the local minima found inside these intervals to find the global minimum 5 3 Fitting the asymmetric Laplace density For the asymmetric Laplace a global maximization of the reduced one paramter log likelihood to find the value of the parameter m The parameters a_l and a r are then computed using the formula provided in see Chapter 2 Overview page 2 5 4 subbofit subbofit fits a Subbotin density on a set of data provided as standard input The method used to estimate the density parameters and the ouput format can be set with command line options Usage subbofit options list files list where files list is a list of files containing the observations on which the density is estimated If files list is empty the program reads the data from standard input The possible options in options list are M method choose a fitting method If method 0 uses maximum likelihood estimation if method 1 uses method of moments See Section 2 2 Parameters Estimation page 3 Chapter 5 Fitting the Subbotin Density 13 0 output choose an output format If output 0 print to the standard ouput the estimated b a m and reduced negative log likelihood If output 1 prints each datapoint with the associated distribution function If output 2 prints each data
3. 1 Method of Moments 00 ccc cece eee ee 11 5 1 2 Unconstrained Minimization 00 11 5 1 3 Minimization on Intervals sleeeeeseeeesn 11 5 2 Fitting the Subbotin density in the asymmetric case 12 5 3 Fitting the asymmetric Laplace density 12 5 4 subbOofit ikbnesenenee een ue eae ae ERISd MESE PEG ERE 12 5 5 SUbbOAlit eese Debent ER e n RI e d a dan 13 DG subbolatituraisenrea lella 14 D laplaafili sca lilla gialli 15 Exploring the symmetric Subbotin Log Likelihood PTTT 16 6 1 SubbOSHOW oe RLLEHPLP BOUE LR einer a ee OL As eie 16 Installa ioni ispira X RERO de C ho 17 License and Copyright 18 Concept Index eo2299 99929 e dan 19 Chapter 1 Introduction 1 1 Introduction The SUBBOTOOLS package is intended as an help to the use of the Subbotin family of proba bility densities in a statistical analysis environment The package contains various programs for the maximum likelihood estimation of an unknown distribution and for the generation of pseudo random variables The program subbofit can be used for the fitting estimation of a Subbotin probability density on an user supplied dataset The programs subboafit and subbolafit perform this estimation on two larger classes of probability densities the asymmetric Subbotin densities a 5 parameter family and the so called less asymmetric Subbotin densit
4. 1529 0 1672 0 5969 0 2583 15040113 m 0 003864 0 1503 0 4651 0 4840 0 3193 0 4899 Upper triangle covariances Lower triangle correlation coefficients ssss s SS ee bl br al ar m log like 8 1340e 01 9 7619e 01 8 6487e 01 9 0047e 01 3 8641e 03 1 612e 00 with a convenient heading added to the estimates Chapter 3 Tutorial 8 3 2 Graphic Tutorial Now let s start gnuplot and see how the programs in SUBBOTOOLS can be used inside its graphic environment Simply type gnuplot and you will be greeted by a fairly large amount of informations and finally a prompt As a first exercise let s compare the result of our fitting procedure with the original function Using gnuplot gt set log y gnuplot gt plot lt subbofit 0 2 lt testdata txt exp abs x 2 you can produce a graph similar to the one in Fig 1 The option 0 2 print for each point of the file testdata txt the value of the probability density associated with the fitted parameters gnuplot interprets this output and produces the corresponding graph With our small sample the obtained estimate is quite different from the theoretical Laplace density from which the data were originally generated A similar comparison can be performed on the distribution function using the command gnuplot gt set key top left gnuplot gt unset log y gnuplot gt plot lt subbofit 0 1 lt testdata txt gnuplot gt replot x lt 0 exp abs x 2 1
5. exp abs x 2 where the option 0 1 now prints the value of the estimated distribution function for each input points The result you should obtain is similar to the one reported in Fig 2 The program subboshow is conceived for the visual exploration the Subbotin reduced lok likelihood For a precise definition Chapter 5 Fitting Subbotin Density page 11 The program takes a set of observations and print out the associated value of the Subbotin reduced negative log likelihood for a grid of b and m values Using the previously generated observations try the following gnuplot gt set contour gnuplot gt unset clabel gnuplot gt set grid x y z gnuplot gt splot subboshow b0 5 1 5 m 1 11 lt testdata txt i Ow 1 and you will obtain a 3D plot representing the negative log likelihood of the dataset similar to Fig 3 If one is interested in the behaviour of the reduced negative log likelihood with respect to a specific parameter then it is enough to specify a single point for the other parameter For instance if one is interested in the behavior of the function with respect to the parameter m keeping fixed the value of b one can use gnuplot gt plot subboshow b0 5 B 1 m 05 08 M 500 testdata txt i O w and obtain a 2D plot similar to Fig 4 6 The options b and m set the region for which points are generated while B and M set the number of points See the help of subboshow for m
6. the estimated value of the parameter a as an explicit function of b and m Substituting this expression in the negative log likelihood one obtains an object function that depends only on 2 parameters b and m We call this function reduced megative log likelihood and it reads N L a b log 20 T 1 1 0 11087 Y 1s mp j l In the case of the asymmetric density the method of moments is not so attractive Indeed it is impossible to obtain simple expressions relating the parameters characterizing the density to the various moments of the sample Then the use of numerical methods becomes mandatory and one is naturally led toward maximum likelihood estimation The negative log likelihood per observation in the case of asymmetric density reads u 1 m dj e 1 M iena 5 j Nb 5 zj m ai T yj m ar A abi T 1 1 5 a b T 1 1 b As can be seen both L a b and L b b are not analytic functions of their arguments This is way their minimization is a non trivial task that must be handled with particular care see Chapter 5 Fitting Subbotin Density page 11 The situation is much easier for the case of the asymmetric Laplace density Indeed by substituting b 1 in the previous expression one obtains the reduced log likelihood as a function of the mean m L m 21og VS VS 1 where 1 1 ux2 m z s uom zj lt m x gt m It is easy to see that the function above posses
7. these programs find the symmetric subbofit and asymmetric subboafit Sub botin density or the asymmetric Laplace density laplaafit that better fit a given set of observations The observations are considered independently drawn from the same unknown probability density The values of the density parameters are esti mated depending on the family using maximum likelihood or with the method of moments see Chapter 5 Fitting Subbotin Density page 11 subboshow this program takes a set of observations as input and produces a graphic showing the symmetric Subbotin log likelihood of this set as a function of the density parameters see Chapter 6 Exploring Subbotin Log Likelihood page 16 All these programs deal in a way or another with one of the Subbotin families of densities The symmetric Subbotin densities depend on three parameters usually denoted m a and b The first parameter m represents the central tendency of the density ie the position of its center Indeed it is at the same time the mean the mode and the median of the density The second parameter a is a scale parameter that express the spread of the distribution It is proportional but in general not equal to the density standard deviation The last parameter b is a shape parameter It tells how fat are the tails of the density with respect to its central part The family of asymmetric Subbotin densities extends the previous family by int
8. 1 6 Chapter 2 Overview 3 Due to its symmetry the Subbotin density has all odd central moments equal to 0 The central moment of order 2 reads while the absolute deviation is _ an TA Mai ab TA The asymmetric Subbotin density extends the family described above by considering different values for the parameters a and 6 in the two halves of the density Its functional form depends on five parameters a positioning parameter m two scale parameters a and a respectively for the values below or above m and two shape parameters and 6 characterizing respectively the lower and upper tail of the density Its functional form reads d z m i er lt M f a a b m br r m ar x gt m where A abi T 1 1 5 a b T 1 1 6 It is rather straightforward to obtain the expression of the central moments of the asymmetric Subbotin density They are in general all different from zero the density being skewed Since we are not going to make use of the central moments for the asymmetric case their expressions are not reported here The asymmetric Laplace density restricts the family described above by considering fixed values for the tail parameters b b 1 Its functional form depends on three parameters a positioning parameter m and two scale parameters a and a for the parts of the density below or above m respectively Its functional form reads Le gx m fuma bym
9. 6 E BExamples o292ReRoetRe t e ee REIP UEUE 6 F IUE sfollati 11 Fitting Laplace asymmetric 12 Fitting Subbotin asymmetric 12 Fitting Subbotin symmetric 11 G SOUP LOG iad says E AI e aye hae 5 Graphics rei cided tile hate ae oe hee 5 Glicine Pie Se 1 Input iii ela ile RR 5 Introd ctlom 2 2 4 0 2109 ae lea il 19 L Likelihood satana 3 M Method of Moments llle esee 11 Minimization on intervals 11 Minimization unconstrained Lf ep I sali dai ees alee 5 Overview ici ii ER Ge PIRE TER eI 2 P Paraineters 2o oil eda 2 plotutilT8 254 a orbes Ned eerie 5 Probability density 2 hx EEG E pcd 2 R Reduced log likelihood LL 3 Subbotin central moments 2 Subbotin likelihood 125 12 mr dae RIS 3 Subbotin parameters 0 eee eee eee 2 Subbotin probability density 2 Subbotin reduced log likelihood 3 T Tutorial 2 2 330542 ee e204 ila dada ees 6 U Unconstrained Minimization 11
10. SUBBOTOOLS User s Manual For version 1 2 7 March 2014 Giulio Bottazzi Copyright c 2003 2014 Giulio Bottazzi Permission is granted to make and distribute verbatim copies of this manual provided the copy right notice and this permission notice are preserved on all copies Permission is granted to copy and distribute modified versions of this manual under the condi tions for verbatim copying provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one Permission is granted to copy and distribute translations of this manual into another language under the above conditions for modified versions except that this permission notice may be stated in a translation approved by the Free Software Foundation Table of Contents T 8 IntroducliOns 2222 ipa nie viari 1 OVeVIOW i asia ae be ae 2 2 1 The Subbotin Families of Densities L 0 2 2 2 Parameters Estimation 0 000 cece cece ad erdoi sasi 3 2 3 Programs ipa OUUpUL coeunt riore Ra DER Ed rei eroe vox 5 PUGS nitore ei 6 3 1 Command Line Tutorial 0 00 eee ee eee 6 9 2 Graphie Tutorial eevlee seg Sth cee rre RE ees 8 Generating Subbotin Random Variates 10 Acl SW DO ROM ise al ere 10 4 2 g bboageniocicirweerese yea cw bee gees dern el edad ER ER d 10 Fitting the Subbotin Density 11 5 1 Fitting the Subbotin density in the symmetric case 11 5 1
11. a single global minimum Once the central parameter m has been estimated the value of the other two parameters are directly obtained a and a are determined by the relations a S S Sr ar Sr VS Sr Chapter 2 Overview 5 2 3 Programs Input Output Both subbofit and subboshow require user supplied data The data can be read in ASCII format from files whose name are specified at the end of the command line after any option or from the standard input Different data are separated by white characters spaces or tabs or newlines Lines beginning with a fence symbol are considered comments and ignored The output of subboshow and the output of subbofit subboafit subbolafit and laplaafit whith the options 0 1 or 0 2 are intended to be used to produce pictures They consist in tabulated data of 2 for 2 dimensional plots or 3 for 3 dimensional plots columns where each line correspond to a different point In the case of 3 dimensional plots the triplet associated to different values of the first variable are separated by 2 newlines This format has been chosen to provide an easy interface to the gnuplot plotting program The 2 dimensional plots can be easily displayed also making use of the graph utility of the plotutils package For an example of the use of the graphic output Section 3 2 Graphic Tutorial page 8 The gnuplot program FAQ can be found at http www gnuplot info fag For information about the
12. appear the search is stopped 5 2 Fitting the Subbotin density in the asymmetric case In the asymmetric case it is impossible to implement a simple and direct approach for the density estimation based on the method of moments Even if the method is still valid in principle its application requires the solution of a 5 dimensional system of non linear equations This strongly reduces its attractiveness as a first guess generator Another difference with respect to the symmetric case is the practical inconvenience of re ducing the number of variables in the log likelihood function Indeed in the asymmetric case it is again possible to use suitable first order conditions to remove the explicit dependence of the log likelihood function on some variables namely a and a However this method would lead to an expression for the reduced log likelihood that despite the reduction of the number of independent variables form 5 to 3 appears extremely complicated and computationally more demanding For the above reasons in order to obtain the asymmetric Subbotin fit the program subboafit starts with an unconstrained minimization of the log likelihood function over all its 5 parameters for the expression of the log likelihood function see see Section 2 1 The Subbotin Families of Densities page 2 Then the programs proceeds with an interval constrained minimization similar to the one performed by subbofit whit the parameter m limited inside a compact
13. e log likelihood function L becomes not differentiable when m x for j 1 N i e when the parameter m takes the value of one of the observations The method of Interval Constrained Minimization try to overcome this problem by evaluating the function L only in domains where it is analytical More specifically one searches for the minimum inside any compact interval x x 1 In this way a list of local minima one for each interval is produced The minimization inside each interval is performed on a smoothed Chapter 5 Fitting the Subbotin Density 12 version of L obtained with a change of variables in such a way that the first derivative results well defined but the number and location of minima remains unaffected Once the local minima are computed inside all the intervals the local minimum associated with the smallest value of the function L constitutes the global minima i e the point one was looking for The algorithm actually implemented does not apply this straightforward procedure because the execution of a constrained minimization problem for each interval can become too expensive when the size of the sample increases Instead a search algorithm is implemented on the set of these intervals Initially the minimization problem is solved on a small group of intervals surrounding the point m4 This initial set is enlarged progressively if new global minima are found When for a given number of steps no new global minima
14. e manual after the configuration do cd doc make pdf make html For more detailed installation instruction see the file INSTALL Chapter 8 License and Copyright 18 8 License and Copyright SUBBOTOOLS is copyright c 2002 2014 Giulio Bottazzi SUBBOTOOLS is a collection of free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 2 of the License or at your option any later version This package is distributed in the hope that it will be useful but WITHOUT ANY WAR RANTY without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with this package if not write to the Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA You can also find the GPL on the GNU web site In addition I kindly ask you to acknowledge SUBBOTOOLS and its author in any program or publication in which you use SUBBOTOOLS You are not required to do so it is up to your common sense to decide whether you want to comply with this request or not Concept Index Concept Index A Asymmetric Subbotin probability density 2 B Bugs Report sis eor br ERR REA il C Central moments urico ea ra dat 2 Command Line 2522 2 Imm ber Re Rei
15. ed in the order b a m print a short help Chapter 5 Fitting the Subbotin Density 11 5 Fitting the Subbotin Density The programs subbofit and subboafit try to find the Subbotin density or the asymmetric Subbotin density that better fit a user supplied dataset These programs obtain their estimation via numerical minimization of the reduced negative log likelihood see Section 2 2 Parameters Estimation page 3 The program subbolafit fit the same asymmetric Subbotin density fitted by subboafit but in the particular case in which the two scale parameters are set equal i e a a Due to the non analytic nature of the Subbotin log likelihood function both in the symmetric and asymmetric case a straightforward minimization of the negative likelihood function can be unreliable Instead the programs use a multi steps approach 5 1 Fitting the Subbotin density in the symmetric case In the symmetric case subbofit obtains a first guess of the parameters values using the method of moments Then it refines this first guess performing an unconstrained minimization of the log likelihood function Finally it splits the minimization procedure inside different sub domains whose interior constitute an analyticity region for the log likelihood function and that together constitute a neighborhood covering of the previously found values By comparing the various local minima in these domains the program finds the global minimum Let s analyze
16. ee Section 5 4 subbofit page 12 5 6 subbolafit subbolafit fits a less asymmetric density on a user supplied set of data The less asym metric density is the asymmetric Subbotin density see Section 2 1 The Subbotin Families of Densities page 2 with the two scale parameters equal al ar The ouput format can be set with command line options Usage subbolafit options list files list where files list is a list of files containing the observations on which the density is estimated If files list is empty the program reads the data from standard input The possible options in options list are 0 output choose an output format If output 0 print to the standard ouput the estimated b a m and reduced negative log likelihood If output 1 prints each datapoint with the associated distribution function If output 2 prints each datapoint with the associated density V verbosity choose a verbosity level If verbosity 0 print only the final results More infor mation can be obtained increasing the verbosity level mM set the parameter m equal to M The optimization is performed only on the two remaining parameters G step tol iter eps algo set the parameter for the global numerical optimization I step tol iter eps algo set the parameter for the interval constrained numerical optimization s steps set to steps the step of the discrete optimi
17. ich the log likelihood is computed If files list is empty the program reads the data from standard input The possible options in options list are M num print the reduced negative log likelihood for num equally spaced values of the pa rameter m B num print the reduced negative log likelihood for num equally spaced values of the pa rameter b m min max print the reduced negative log likelihood for values of m in the range min max b min max print the reduced negative log likelihood for values of 6 in the range min max n print a short help and default parameters values Chapter 7 Installation 17 7 Installation You require the GSL Gnu Scientifi Library to be present on your system in order to install SUBBOTOLS Check that these libraries are properly installed on your system before proceeding to the installation of the present package For informations about GSL including installation instructions check http www gnu org software gs1 SUBBOTOLS comes with a configure program in the GNU style Installation can be as simple as tar xvzf subbotools version tar gz cd subbotools version configure make make install where version stands for a number indicating the package version The commands above install the SUBBOTOLS executable in usr local bin and the docu mentation in the form of info files in usr local info To obtain pdf or html version of th
18. ies a 4 parameter family see Chapter 2 Overview page 2 The program laplaafit restrict the estimation to the family of asymmetric Laplace densities a subset of the asymmetric Subbotin In addition the programs subbogen and subboagen can be used to generate pseudo random variates drawn respectively form a user specified Subbotin or asymmetric Subbotin density see Chapter 4 Subbotin Random Variates page 10 while subboshow allows the exploration of the symmetric Subbotin log likelihood function for a definition see Section 2 2 Parameters Esti mation page 3 of an user supplied dataset see Chapter 6 Exploring Subbotin Log Likelihood page 16 All the programs in this package make use of functions from the GNU Scientific Library a collection of numerical routines for scientific computing These libraries must be installed on your system for SUBBOTOOLS to compile and run More information about GSL can be found at the project home page http www gnu org software gs1 Follow the instruction provided there to download and install the GSL library Using appropriate command line options the programs subbofit subboafit subbolafit laplaafit and subboshow can be used to produce pictures In fact they can produce tabu lated data in ASCII format representing 1 dimensional curves or 2 dimensional surfaces The output structure made of newline separated records of two or tree space separated numeric fields is in a format suitab
19. le to be sent to the common plotting utilities ordinarily found on Unix systems For instance I usually use this graphic output options in conjunction with Gnuplot or the graph utility found in the plotutils package For more information on the output format of these programs See Section 2 3 Programs Input Output page 5 SUBBOTOOLS has been developed on Linux systems It is freely distributed under the GNU General Public License and is provided as is without any explicit or implicit warranty see Chapter 8 License and Copyright page 18 In principle it should work in any flavor of Unix with minor or without modifications No porting to different OS s has been undertaken nor planned The last version of SUBBOTOOLS can be found at http www sssup it giulio Comments and bug reports are welcome E write to giulio bottazzi sssup it Please read the INSTALL file provided with the package for installation instructions and the file README for an updated description of the package components Chapter 2 Overview 2 2 Overview The SUBBOTOOLS package contains the following programs subbogen subboagen these programs generate random variable extracted from a symmetric subbogen or asymmetric subboagen Subbotin density The parameters that defines the density can be provided in the program command line or read from standard input see Chapter 4 Subbotin Random Variates page 10 subbofit subboafit subbolafit laplaafit
20. lready installed the SUBBOTOOLS package without problems 3 1 Command Line Tutorial In order to avoid to damage possibly relevant files before trying the following steps create a new directory something like subbotest and move there mkdir subbotest cd subbotest Now that you are in a more or less safe environment let s start We begin by generating a meaningful set of numbers to work with subbogen N 100 m0 al bi R 5 gt testdata txt It is useful to review the meaning of the command line parameters of the previous command Using subbogen we generated 100 random numbers N 100 from a Subbotin distribution with m 0 a 1 and b 1 i e a Laplace symmetric exponential distribution centered in zero and with variance equal to 2 The numbers are generated initializing the RNG random number generator with a seed equal to 5 R 5 and are placed in a file named testdata txt One can generate independent samples by using the option R with different integer numbers the default value for the seed is 0 The file testdata txt now just contains a column of 100 number you can easily inspect it using an editor We now use this file to investigate the properties of the program subbofit Begin with the simplest command subbofit testdata txt that should generate an output similar to 8 8637e 01 8 7919e 01 3 3469e 02 1 6159e 00 The meaning of these four number is as follows the first three number
21. ore options Using this command you can check if the parameters values previously found with subbofit do in fact constitute a minimum Chapter 3 Tutorial 9 1 1 I f f Lp YIN th 7 th 0 8 01L a T 001 L 6 Fa S PL N 0 001 i i i 0 i e I Me 5 jJ 2 Figure 1 Estimated fit and theoretical th Figure 2 Estimated fit and theoretical th probability density probability distribution 674 672 1 67 668 666 664 662 1 66 658 656 654 i VM 0 06 0 04 0 02 0 002 0 04 0 06 0 08 m Figure 3 Negative reduced log likelihood as a Figure 4 Negative reduced log likelihood as function of m and b a function of b Chapter 4 Generating Subbotin Random Variates 10 4 Generating Subbotin Random Variates The programs subbogen and subboagen can be used to generate pseudo random variables drawn respectively from a Subbotin and an asymmetric Subbotin density 4 1 subbogen subbogen generate a sequence of pseudo random variables from a Subbotin density and print them to standard output in ASCII format separated by a newline character The parameters of the density can be specified as option on the command line or can be read from standard input Usage subbogen options where options can be N num generate a n
22. plotutils package check http www gnu org Chapter 3 Tutorial 6 3 Tutorial This tutorial will drive you across some simple examples on the use of the programs provided with this package The tutorial is divided in two parts In the first part the use of the programs from the command line will be illustrated We will learn how to generate pseudo random vari ables drawn from a Subbotin density and how to obtain the best Subbotin fit on a given dataset The second part of the tutorial is more graphical and we will see how to use the programs dis tributed with SUBBOTOOLS to generate plots inside the Gnuplot plotting environment Previous knowledge of Gnuplot is not assumed however I will not explain all the details of the different commands that I will use For more information on the different commands and switches you are referred to the exhaustive help system of Gnuplot itself try help after Gnuplot invocation The choice of gnuplot as the plotting utility is due to my long experience with it and to its proven reliability and easiness of use The output of the SUBBOTOOLS programs has been designed according to the gnuplot requirement as illustrated in Section 2 3 Programs Input Output page 5 Due to the essential looseness of these requirements however it should be relatively easy to adapt it to different utilities In this tutorial Pm going to suppose that you are familiar with the Unix shell command line interface and that you a
23. point with the associated density V verbosity choose a verbosity level If verbosity 0 print only the final results More infor mation can be obtained increasing the verbosity level mM set the parameter m equal to M The optimization is performed only on the two remaining parameters G step tol iter eps algo set the parameter for the global numerical optimization I step tol iter eps algo set the parameter for the interval constrained numerical optimization s steps set to steps the step of the discrete optimization algorithm on the set of local minima obtained with interval constrained optimization n print a short help with the default values for the various parameters The parameters of the numerical optimization are set using a comma separated list of 5 parameters Empty fields leave the default unchanged The meaning of the various parameter is as follows step initial step size of the searching algorithm tol tolerance of the line search algorithm iter maximum number of iterations eps gradient tolerance the search is stopped when gradient lt eps algo choose the optimization method 0 Fletcher Reeves 1 Polak Ribiere 2 Broyden Fletcher Goldf 3 Steepest descent 5 5 subboafit subboafit fits an asymmetric Subbotin density on a user supplied set of data The ouput format can be set with command line options Usage subboafit option
24. represent the esti mated values for b a and m respectively while the last number is the negative log likelihood associated with these three values As can be seen the estimated values for b and a are quite far from the real values The situation is different for different seeds but with only 100 observations an error of about 1096 is not uncommon It could also be the case that the program produces some warning message like status iteration is not making progress towards solution These messages are generated by an intermediate optimization step and in general they can be safely ignored They are more common in the case of small samples Chapter 3 Tutorial 7 In addition to the maximum likelihood estimation of the parameters which is the default the program subbofit also implements a method of moments You can choose different estimation methods with the command line option M Let s opt for the method of moment subbofit M 1 testdata txt and you should obtain something like 9 5699e 01 9 1046e 01 5 2919e 02 1 6174e 00 In this particular case due to the relative small size of the sample the method of moments provides better estimates than the maximization of the likelihood Of course the log likelihood associated to these estimates is larger In order to obtain more details on the procedure of fitting you can increase the verbosity of the program by using the option V For more details Chapter 5 Fitting Subbotin Densi
25. roducing two different values for the parameters a and b at the left and at the right of the density mode m This extended group of densities constitutes a 5 parameters family The parameters are commonly denoted 0 br aj ar and m In this case the positioning parameter m represent the mode of the density but in general it is neither its mean nor its median The asymmetric Laplace family is a subset of the asymmetric Subbotin family obtained putting bj b 1 For a precise definition of the Subbotin asymmetric Subbotin and asymmetric laplace families Section 2 1 The Subbotin Families of Densities page 2 For the estimation of the density that better fits a provided sample different methods can be applied For a short discussion of the problem Section 2 2 Parameters Estimation page 3 The strategy of the estimation procedures implemented in the programs distributed with this package is described in Chapter 5 Fitting Subbotin Density page 11 The program of this package share a common procedure to read in input and print out output Input is read from standard input in ASCII format and output is sent to standard output always in ASCII format For more details see Section 2 3 Programs Input Output page 5 2 1 The Subbotin Families of Densities The functional form of the Subbotin density is characterized by three parameters a position ing parameter m a scale parameter a and a shape parameter b and reads 5 4 5 a GATA
26. s list files list where files list is a list of files containing the observations on which the density is estimated If files list is empty the program reads the data from standard input The possible options in options list are 0 output choose an output format If output 0 print to the standard ouput the estimated b a m and reduced negative log likelihood If output 1 prints each datapoint with the associated distribution function If output 2 prints each datapoint with the associated density V verbosity choose a verbosity level If verbosity 0 print only the final results More infor mation can be obtained increasing the verbosity level mM set the parameter m equal to M The optimization is performed only on the two remaining parameters Chapter 5 Fitting the Subbotin Density 14 G step tol iter eps algo set the parameter for the global numerical optimization I step tol iter eps algo set the parameter for the interval constrained numerical optimization s steps set to steps the step of the discrete optimization algorithm on the set of local minima obtained with interval constrained optimization n print a short help with the default values for the various parameters The parameters of the numerical optimization are set using a comma separated list of 5 parameters The meaning of the parameters and the syntax is the same as for subbofit s
27. the procedure in more details 5 1 1 Method of Moments The first step of the fitting procedure consists in the estimation of the values of the parameters based on the method of moments i e using the sample mean My the sample variance M and the sample absolute deviation Mj and following the procedure outlined in see Section 2 2 Parameters Estimation page 3 Let bo ao mo be the parameters values obtained with the method of moments 5 1 2 Unconstrained Minimization In the second step we run an unconstrained minimization procedure on the reduced negative log likelihood function L see Section 2 2 Parameters Estimation page 3 using as starting point the couple bo mo previously determined with the method of moments In the case in which the value of m is provided on the command line using the option m see Section 5 4 subbofit page 12 the search for the minimum is reduced to a one dimensional problem Moreover in this case the function L is always analytical and the unconstrained minimization solve the problem finding the global minimum In general when the value for m is not provided due to the non analyticity of the object function the unconstrained minimization procedure does not generate reliable results and it is simply used as a first rough estimate of the solution Denote with b a4 m4 the set of values obtained with the present procedure 5 1 3 Minimization on Intervals If b lt 1 the reduced negativ
28. ty page 11 Another interesting feature is the possibility of fitting just the b and a parameters providing the true value of m directly from the command line This is useful when you work with normalized data whose mean has been previously subtracted Since this is a quite common situation subbofit provides the m command line option With the file generated above do subbofit m O lt testdata txt and you should obtain 8 808684e 01 8 777596e 01 0 000000e 00 1 616923e 00 Remember do not confuse the option M which selects the estimation method with the option m which sets the value of the mean Finally let s try with the asymmetric version of the Subbotin distribution Do subboafit lt testdata txt and you should obtain something similar to 8 1340e 01 9 7619e 01 8 6487e 01 9 0047e 01 3 8641e 03 1 6126e 00 The six numbers are the estimate values of bl b_r a l a r and m respectively plus the value of the associated negative log likelihood Increasing the verbosity level you can have a detailed description of the estimates standard errors The command subboafit V 1 lt testdata txt gives you something like FINAL RESULT _____ r 2 00 LL Uu00005 correlation matrix value std err bl br al ar m bl 0 8134 0 186 0 0069 0 0102 0 0048 0 0130 br 0 9762 0 2413 0 1532 0 0083 0 0220 0 0176 al 0 8649 0 1416 0 3884 0 2426 0 0056 0 0068 ar 0 9005 0
29. ty level mM set the parameter m equal to M The optimization is performed only on the two remaining parameters G step tol iter eps algo set the parameter for the global numerical optimization n print a short help with the default values for the various parameters The parameters of the numerical optimization are set using a comma separated list of 5 parameters The meaning of the parameters and the syntax is the same as for subbofit see Section 5 4 subbofit page 12 Chapter 6 Exploring the symmetric Subbotin Log Likelihood 16 6 Exploring the symmetric Subbotin Log Likelihood The possibility of visually explore the reduced negative log likelihood of a given dataset becomes very useful when the fitting program produces some error and you want to directly verify if the found minima is actually the global minima you were looking for The program subboshow provides an easy way of performing such a graphical exploration It is essentially intended to be used inside the gnuplot graphic environment The format of the output has been conceived to be sent to gnuplot s plot or splot commands For an example of use See Section 3 2 Graphic Tutorial page 8 6 1 subboshow The program subboshow allows the visual exploration of the Subbotin reduced negative log likelihood of an user supplied dataset Usage subboshow options list files list where files list is a list of files containing the observations with wh
30. umber num of independent random variables mM set the value of the parameter m equal to M ca A set the value of the parameter a equal to A b B set the value of the parameter b equal to B R seed set the seed of the random number generator to seed i read the density parameter from the standard input They should be provided in the order b a m n print a short help 4 2 subboagen subboagen generate a sequence of pseudo random variables from an asymmetric Subbotin density and print them to standard output in ASCII format separated by a newline character The parameters of the density can be specified as option on the command line or can be read from standard input Usage subbogen options where options can be N num generate a number num of independent random variables mM set the value of the parameter m equal to M a A if A is a single number set the values of both al and ar equal to A If A is a couple of comma separated numbers the first is used to set al the second to set ar b B if B is a single number set the values of both bl and br equal to B If B is a couple of comma separated numbers the first is used to set bl the second to set br R seed set the seed of the random number generator to seed i read the density parameter from the standard input They should be provid
31. zation algorithm on the set of local minima obtained with interval constrained optimization n print a short help with the default values for the various parameters The parameters of the numerical optimization are set using a comma separated list of 5 parameters The meaning of the parameters and the syntax is the same as for subbofit see Section 5 4 subbofit page 12 Chapter 5 Fitting the Subbotin Density 15 5 7 laplaafit laplaafit fits an asymmetric Laplace density on a user supplied set of data The asymmetric Laplace density is a special asymmetric Subbotin density see Section 2 1 The Subbotin Families of Densities page 2 with the two tail parameters equal to one The ouput format can be set with command line options Usage laplaafit options list files list where files list is a list of files containing the observations on which the density is estimated If files list is empty the program reads the data from standard input The possible options in options list are 0 output choose an output format If output 0 print to the standard ouput the estimated b a m and reduced negative log likelihood If output 1 prints each datapoint with the associated distribution function If output 2 prints each datapoint with the associated density V verbosity choose a verbosity level If verbosity 0 print only the final results More infor mation can be obtained increasing the verbosi
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