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1. xo 21 By covering 0 1 by cells in this way we obtain an expected frequency distribution for Pr g f 30 B The Algorithms B 1 The commoner underlying sets The symbols representing the commoner underlying sets are given in Table 9 Symbol Meaning S N The set of all distributions of degree N R N The set of all ranked distributions of degree N fA gt gt F N RR N The set of all reverse ranked distributions of degree N f l lt lt f N M A to B N The set of all unimodal distributions of degree N with mode between A amp B inclusive M m N M m to m N M N M 1 to N N U A to B N The set of all U shaped distributions of degree N with trough between A amp B inclusive U m N U m to m N U N U 1 to N N SD c N For c Xy_1 the set of all step down distributions of degree N with step at c i lt ce lt k gt fli gt f k RSD c N The set of all ranked step down distributions of degree N with step at c i lt j lt e lt k fli gt fG gt FQ Table 9 The commoner underlying sets B 2 Selection of f S N Using the computer s RAND function select N 1 points in 0 1 and use them to partition 0 1 resulting in N subintervals Use the subinterval lengths as the f i randomising them first to reduce any biase in the selection process B 3 Selection of r R N pid i 2 N ae 2 ago y 1 N Then y S N gt R N f r where rT Ay f is a linear bi2. To reduce errors and also to make things easier for you when you are specifying the relative frequencies for H2 you do not have to make them sum to 1 the program will normalise them for you 11 2 Merge Blocks Table 5 gives three examples of Merge blocks 13 4 5 6 7 8 12 3 4 5 68 13 4 5 6 8 9 2 0 3 3 1 2d E S ae Oe rk 9 20 7 a Merge block at left b Merge block at right c Merge blocks at both ends Table 5 Merge blocks Each consists of a number of columns which have been merged together so that the detail has been lost of the entries in individual columns but the total of the entries is still known In practice when data is collected in batches it sometimes happens that some batches contain a merge block but others do not To cater for this the program allows merge blocks to be entered independently of H1 and H2 and to overlap their non zero entries The degree is the number of columns that there would have been had the merging not taken place it is 8 in each of the tables in Table 5 This means that merge blocks cannot be used as an artificial way to reduce the degree The use of Merge Blocks introduces such vagueness into a problem that the resulting algorithms can be highly ill conditioned A significant increase in the number of 16 iterations will be needed but will not necessarily improve convergence to an acceptable extent indeed increasing the number of iterations could make convergence properties w
3. 0 0 0 If f is a distribution and g is an integram both of degree N then the probability of g given f is Pr g f M g f where M g is the multinomial coefficient associated with g see Appendix A and f9 f 1 9 f N 9 The best estimate of something is its mean value In order to have a mean value there must be a set of values to find the mean of This program best estimates Pr g f so there has to be a set of Pr g f to find the mean of We start with a set of distributions called the Underlying Set calculate Pr g f for each f in that set and then find the mean of those The mean used is a weighted mean where the weights depend upon available data in the form of an histogram called the given histogram The actual formula used is eee fEP yee JEP Lp g h M 9 1 where P is the underlying set h is the given histogram and Lp g h is the likeliness see Appendix A N The expression for M g contains the product of factorials g i see Appendix A i i so each g i has to be integer valued so g has to be an integram The weights f started life as Pr h f that is M h f but the M h has cancelled since it appeared in both the numerator and denominator of 1 The cancellation of the M h has taken with it any need for h to be integer valued so h may be an histogram rather than specifically an integram In a few simple cases the main ones of which are listed in Appendix C the integ
4. 2 DIS IND 8 7 29E 01 7 29E 01 1 45E 01 1 26E 01 OBS IND 8 20 7 29E 01 13 2 5 DISGRP Group 6 38E 01 6 38E 01 2 05E 01 1 57E 01 OBSGRP Group 80 6 38E 01 52 16 12 DISCUM Overall 6 47E 01 6 47E 01 2 06E 01 1 47E 01 OBS CUM Overall 160 6 47E 01 110 29 21 R DISIND 9 6 33E 01 6 33E 01 2 79E 01 8 85E 02 OBS IND 9 20 6 33E 01 10 8 2 DIS IND 10 6 94E 01 6 94E 01 2 27E 01 7 85E 02 OBS IND 10 20 6 94E 01 16 3 1 DIS IND 11 6 77E 01 6 77E 01 2 78E 01 4 58E 02 OBS IND 11 20 6 77E 01 12 7 1 DIS IND 12 6 96E 01 6 96E 01 2 33E 01 7 15E 02 OBS IND 12 20 6 96E 01 14 5 1 DISGRP Group 6 75E 01 6 75E O1 2 54E 01 7 11E 02 OBSGRP Group 80 6 75E 01 52 23 5 DISCUM Overall 6 56E 01 6 56E 01 2 22E 01 1 22E 01 OBS CUM Overall 240 6 56E 01 162 52 26 iid Laid a sampling_dis csv b sampling_obs csv Program started on 8 12 2013 at 3 55 14 Degree 3 Groups of 4 Code 20 w g 1 Size Priglf E 1 2 3 RFS IND 1 20 4 59E 01 0 6 0 25 0 15 RFS IND 2 20 6 57E 01 08 0 15 0 05 RFS IND 3 20 6 19E 01 0 65 0 15 0 2 RFS IND 4 20 8 89E 01 0 85 0 1 0 05 RFSGRP Group 80 6 56E 01 0 725 0 1625 0 1125 RFSCUM Overall 80 6 56E 01 0 725 0 1625 0 1125 m RFS IND 5 20 4 59E 01 0 55 0 2 0 25 RFS IND 6 20 8 39E 01 0 85 0 15 o RFS IND 7 20 5 23E 01 0 55 0 35 0 1 RFS IND 8 20 7 29E 01 0 65 0 1 0 25 RFSGRP Group 80 6 33E 01 0 65 0 2 0 15 RFSCUM Overall 160 6 47E 01 0 6875 0 18125 0 13125 i RFS IND 9 20 6 33E 01 0 5 0 4 0 1 RFS IND 10 20 6 94E 01 0 8 0 15 0 05 RFS IND 1 20 6 77E 01 0 6 0 35 0
5. as possible and so are not simply a repetition of your typing The basic idea is that it is easier and less error prone to alter an existing value than it is to insert an omitted value so everything is specifically given and nothing is implied Examples are When specifying an histogram you give a block about which you wish to be asked and the program defaults values outside that block to zero Regardless of which block you specify the block stored in data txt always runs from 1 to N and the defaulted zero values are all specifically given When specifying the subintervals to be used for the calculation of cdfs and pdfs you are asked whether you want to specify them yourself or whether you want to leave that to the program Regardless of how you reply 18 data txt always contains the answer do it myself followed by the intervals the program has chosen or were selected by you Regardless of whether or not you say that you want to use a contraction the program always gives you one albeit one which has no effect because it has a magnitude of 1 So the answer to the question Do you want to use a contraction is always stored as 1 for YES followed by the centre and magnitude of a contraction If you say that you do want a contraction then the stored details will be as specified by you If you say that you do not want a contraction then the magnitude will be 1 and you don t have to worry about where t
6. correctly for the types of problem you are normally involved with save you from having to do so most of the time Just be careful not to use any thousands separators etc If you rarely have an interest in anything apart from basic likelinesses then it should be possible to reduce the default number of iterations to substantially fewer than the Factory Setting 100000 or fewer will often be good enough for example see Figure 12 However if the program runs fast enough for you then you should ask yourself why you are reducing the number On the other hand if you are usually interested in PDFs then you might find that an increase to substantially more than the Factory Setting would be convenient 23 If defaults txt contains a value lt 0 the program uses 999 999 to act as a warning that something is not right whilst still allowing the program to run Factory Setting 750000 SEEDing the random number generator There are two options 1 The program chooses the same seed every time it is run This is always the same and you have no control over its value 2 The program chooses a different seed every time it is run If defaults txt contains any integer other than 1 then the program uses option 2 Factory Setting 2 The maximum number of distributions which may be specified in the next item The next item asks for the number of distributions to be saved in sampling_dis csv There is no natural upper limit to this which makes
7. need to restart your computer before you can use the program again If you are analysing confidential or sensitive data then you should delete scratchl txt and scratch2 csv manually 9 The underlying set 9 1 Components of the underlying set As a general rule anything other than data values of h i and g i which has to be stated in order to define the problem forms part of the definition of the underlying set Components include e fundamental type e degree e contraction e essential domains of modal distributions e range of any mode e whether modal distributions are bell shaped 9 2 Fundamental types of underlying set 9 2 1 Unstructured S N S N is the set of all distributions of degree N There is no relationship between the f i other than the requirement that they sum to 1 The fact that the domain Xy is ordered is irrelevant 0 04 gt T T T T T T T 1 1 3 5 7272 S 2 BSB SB TT 2 a2 3B snu B Figure 2 Unstructured general distributions Ls 29 2 9 2 2 Ranked R N A distribution f is ranked if f 1 gt f 2 gt gt f 0 16 5 0 14 i 0 12 1 0 10 4 01 e a 0 01 4 0 06 o 0 001 0 04 Pe a 0 0001 0 02 000 ee a 0 00 r r l L T 0 00001 S N 300 N 400 N 500 1 3 5 7 9 11 43 45 17 19 21 23 25 27 29 0 000001 i a Dray 2 b Lay i for various N semilog scales Figure 3 Ranked distributions 10 9 2 3 Reverse ranked RR N The mirror im
8. sample to that point Factory Setting 0 How frequently a new distribution is to be chosen There are two options 1 every time a distribution is written to sampling_dis csv 2 only at the start of each group If defaults txt contains any integer other than 2 the program uses option 1 Factory Setting 1 The maximum number of observations generated per distribution which may be specified in the next item The next item asks for the number of observations to be generated per distribution in the sampling files There is no natural upper limit to this but the user may voluntarily impose one here If the code used in the next item is 0 or 1 then no upper limit is imposed If the code used is negative other than 1 then the negative of the same limit is applied Factory Setting 50 Code giving the number of observations generated per distribution Each distribution sent to sampling_dis csv is used as a generating distribution to simulate at least one observation You specify the actual number of observations here by giving an integer n which has the effect given by Table 7 If n is negative a random choice of the number of observations is made every time ie it is not a once and for all decision Factory Setting 0 25 n Number of observations simulated per distribution Di A se n 0 w g 1 chosen at random from 1 w g 2 3 chosen at random from 1 n Table 7 Specifying the number of ob
9. stop on the previous run This makes data txt unusable Take the same action as for the previous problem I get a run time error with the error number M6101 This is an under over flow problem There are various possible causes 1 The given histogram has too large a sample size causing underflow The definition of likelinesses involves the factor f see Appendix A so a large sample size can lead to very small numbers The program has been written to handle numbers down to about 10 but this is sometimes not small enough If this should happen to you then the best you can probably do is reduce the sample size of the given data by inputting it as relative frequencies and then reducing the sample size until you find one which works 2 You have been experimenting with the program and have used a completely unrealistic example which has a large h i associated with a very unlikely i This is equivalent to having a large number of observations of something that is very unlikely to happen 27 3 The required integram has too large a sample size The calculation of M g involves several factorials again see Appendix A which can quickly exceed the limits of double precision in the intermediate calculations even if the final value is within limits The program has been written to minimise this problem but it cannot be completely avoided 28 A Notation and Terminology Let Rt be the non negative reals and Nt be the non
10. that item particularly vulnerable to gross typing mistakes To give some protection the user may specify an upper limit to the number of distributions which may be specified in the next item If a number larger than that upper limit is entered it will be reset to that upper limit If the maximum entered here is 0 or negative then no upper limit is imposed Factory Setting 10000 Number of distributions wanted in sampling_dis csv Each time the program is run a number of distributions meeting the problem definition is sent to the file sampling_dis csv This item specifies how many there should be If defaults txt contains an integer lt 0 the program sends 100 distributions to the sampling files Factory Setting 100 Since computers work to a finite number of digits there is always the remote chance that the program will choose the same seed twice in succession Depending upon the circumstances the chances of this happening could be as high as 1 in 200 000 24 Broken into groups of Every n th distribution is followed by group level figures Insert the value of n here A group size of 0 or less or of more than the number of distributions forces a single group consisting of all the distributions In sampling_dis csv each group is followed by the group mean in sampling_obs csv by the group sum in sampling _rfs csv by the overall relative frequencies These are then followed by similar figures for the whole of the
11. 05 RFS IND 12 20 6 96E 01 0 7 0 25 0 05 RFSGRP Group 80 6 75E 01 0 65 0 2875 0 0625 RFSCUM Overall 240 6 56E 01 0 675 0 21667 0 10833 c sampling_rfs csv Figure 13 Contents of the Sampling files experiment consists of a number of tests The whole file consists of a number of repetitions of an experiment and shows the variation which might be expected Many users will not have any need for the sampling files but for those who do they can be the most valuable part of the program If a sample of distributions is needed just to show what distributions look like eg in a report then it should be remembered that results csv contains a sample of 25 distributions intended for that purpose Each line starts with a status field which says what that line is all about By sorting on this a file can be split into its various components of individual results group sums averages and grand total In sampling dis csv the Cumulative Sums show the means over all the completed groups in sampling_obs csv the sums in sampling_rfs csv the overall relative frequencies Details from any incomplete group at the end of the file are not included 22 The program does not place an upper limit on the number of distributions which may be produced However a spreadsheet might an older spreadsheet might have a maximum of about 65 000 rows A more modern spreadsheet will normally cater for more than 1 million rows 17 defaults txt defaults txt i
12. 19 15 results csv results csv is in the form of various tables as below Table 1 Input Data This Table summarises the data as input by you If you are not using any Merge blocks then each will be shown as not used For any Merge block that you are using the column size will show the number of observations you have specified for that block and the extent of the block will be shown by the fields that have been asterisked out If you are not using a contraction then the row contraction will show a contraction of size 1 and centre 1 If you are using a contraction then its size and centre will be as specified by you Table 2 Random selection of 25 distributions For convenience either for your own interest or for use when writing a presentation this section shows a random selection of 25 distributions Also shown is Pr g f Table 3 Convergence of Likelinesses This table shows convergence of the calculated likelinesses as the iterations proceed Plotted points are concentrated towards the beginning and end of the iterative process 0 6114 0 6112 0 6110 0 6108 0 6106 4 0 6104 0 6102 0 6100 0 100 000 200 000 300 000 400 000 500 000 600 000 700 000 800 000 Number of iterations Figure 12 Example of the convergence of calculations This table gives a very good indication of whether or not enough iterations had been used it should always be checked as a m
13. LOVEGROVE MATHEMATICALS GREAT LIKELINESSES Versions BO7 amp BO7C USER MANUAL R Lovegrove GREAT LIKELINESSES is a program for calculating likelinesses that is for finding best estimates of probabilities This is the manual for versions BO7 and BO7C The recommended citation for this manual is Lovegrove R 2014 Great Likelinesses versions B07 amp BO7C User Manual Lovegrove Mathematicals London November 2014 London United Kingdom November 2014 www lovegrovemaths co uk loveroger1024 gmail com Contents 1 Introduction 3 2 There is no warranty 3 3 Versions 3 3 1 Changes implemented in this version 2 004 3 3 2 Known Issues 6 cas Oo dees al be e a a ge ea a ee oe See ae 3 4 Basics 4 5 Using Spreadsheets 7 6 What you should do now 7 7 Running the program 8 8 Stopping 9 9 The underlying set 9 9 1 Components of the underlying Seti 4 5 we ee le wl ee Gee ae 9 9 2 Fundamental types of underlying set 4 10 9 2 1 Unstructured S N 22 04 8S 2S oa a 10 9 227 Ranked AR ON cee aise as Se a es BORER a EN Beier eee Ek 10 9 2 3 Reverse ranked RR N 04 so Oke Ew OR ew ae ES 11 9 2 4 Step down SD c N bogs ti ths oA oe BoA ee oe Ace ek BO 11 9 2 5 Ranked step down RSD c N 000050204 s 11 9 2 6 Unimodal M A to B N ig is ea F aetate BAO le Gena Se 11 9 2 7 U shaped distributions U A to B N 12 9 2 8 IMO GA o desg a e gen che Se
14. The sampling files How many distributions there are in the sample and how many simulated observations there are per distribution are controlled by altering the appropriate values in the file defaults txt The distributions may also be split into groups Figure 13 the size of which is also set in defaults txt The sampling files are useful in modelling and in the design of experiments Think of each row as representing the results of a test where each test consists of a number of observations Each group represents the results of an experiment where each 21 Program started on 8 12 2013 at 3 55 14 Program started on 8 12 2013 at 3 55 14 Degree 3 Groups of 4 Degree 3 Groups of 4 Code 20 w g 1 Code 20 w g 1 Pr g f i 1 2 3 Size Pr g f i 1 2 3 DISIND 1 4 59E 01 4 59E 01 3 16E 01 2 24E 01 OBS IND 1 20 4 59E 01 12 5 3 DIS IND 2 6 57E 01 6 57E 01 1 72E 01 1 71E 01 OBS IND 2 20 6 57E 01 16 3 1 DIS IND 3 6 19E 01 6 19E 01 2 83E 01 9 83E 02 OBS IND 3 20 6 19E 01 13 3 4 DIS IND 4 8 89E 01 8 89E 01 5 59E 02 5 47E 02 OBS IND 4 20 8 89E 01 17 2 1 DISGRP Group 6 56E 01 6 56E 01 2 07E 01 1 37E 01 OBSGRP Group 80 6 56E 01 58 13 9 DISCUM Overall 6 56E 01 6 56E 01 2 07E 01 1 37E 01 OBS CUM Overall 80 6 56E 01 58 13 9 HR HR DISIND 5 4 59E 01 4 59E 01 2 76E 01 2 64E 01 OBS IND 5 20 4 59E 01 11 4 5 DIS IND 6 8 39E 01 8 39E 01 1 25E 01 3 54E 02 OBS IND 6 20 8 39E 01 17 3 o DISIND 7 5 23E 01 5 23E 01 2 74E 01 2 03E 01 OBS IND 7 20 5 23E 01 11 7
15. aai e 0 00 cP ees 0 00 42S ee AE 42 5S 7 9 44 48 15 47 49 21 20 28 27 29 i i a Lm 6 29 i b Lagcagy 2 Figure 6 Unimodal distributions 1 To use a specific mode m say Figure 6a has m 6 choose A B m We then write M m N rather than M m to m N 2 If A 1 and B N then M A to B N becomes M 1 to N N which is written as M N and is the set of all unimodal distributions of degree N Figure 6b Despite the impression given by some mathematical texts unimodal distributions are not usually bell shaped see Figure 6a any bell shape is due to vagueness in the mode Figure 6b You are offered the option to use only bell shaped distributions but you should not usually accept this offer 9 2 7 U shaped distributions U A to B N 0 14 0 14 J 0 12 0 12 J 0 10 4 0 10 0 08 0 08 0 06 Peas 0 06 o e e e 0 04 e 0 04 4 o G e a o a o o 0 02 e 0 02 Pes eo 0 00 op ogee 0 00 ee eeeeee 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 1 3 5 7 9 11 13 15 17 19 2 28 25 27 29 i i a Lu 6 29 4 b Lu as i Figure 7 U shaped distributions The logical dual of unimodal distributions U shaped distributions have a single trough rather than a single mode Otherwise there is no difference between the two 9 2 8 Multi modal The domain Xy is covered by several maximum 9 mini unimodal distributions which may overlap Each mini unimodal distribution ha
16. ages of ranked distributions these increase to the right FQ lt JE lt FCN 9 2 4 Step down SD c N There is a c such that the function values below c are all greater than the function values above c that is i lt c lt k gt f i gt f k 0 16 0 14 0 12 0 10 4 oog 0 06 4 0 04 0 02 eeeeeeeeeeeeeeeeeeeees 0 00 T T r 1 1 T 1 i i 1T3s 5S FTs lt nrvBvBEVvpvAasaags 1 Figure 4 Step down distributions Lsp 6 29 7 9 2 5 Ranked step down RSD c N The same as stepdown except that in addition the functions values below c are ranked that isi lt j lt c lt k f i gt f j gt f k c can be interpreted as a limit of discrimination for a ranked distribution 0 16 5 0 14 4 0 12 0 10 4 0 08 0 06 dl 0 04 0 02 eeeeeeeeeeeseeeeeeeees 0 00 T T T T T T Ls Ss 7FSEPUBsBA VDA Sa 2 nD Figure 5 Ranked step down distributions Lpgpi6 29 7 9 2 6 Unimodal M A to B N Each distribution has precisely one mode The mode does not have to be the same for every distribution used in the analysis although it can be if so required The program asks for a range of i which the mode may take the smallest value is A and the largest is B You will usually use one of two specific cases here 11 0 14 a 0 14 0 12 0 12 0 10 0 10 0 08 4 0 08 PE e G e 0 06 0 06 e e 0 04 e 0 04 lt i 0 02 s Teee 0 02 e 02 4 o tees T
17. asing the chances of a typing mistake The program therefore also gives you the choice of several standard cases dee 3 sga g gt gt 2 oe Da gu g b Contraction centre Cen c Contraction centre Figure 10 Two contractions of M 3 A contraction might cause the underlying set to fail to be of the fundamental type originally chosen for example the contraction of an unimodal distribution might not be unimodal Figure 10 shows M 3 together with two contractions of magnitude 0 6 In Figures 10b amp 10c the shaded area outside the boundary of M 3 shows the location of those distributions which are not unimodal The contraction in 10b causes a significant number of distributions to move into this area and so fail to be unimodal By contrast every distribution in 10c remains in M 3 and so is still unimodal 15 11 The given histogram 11 1 Basics of the specification The given histogram is specified in three parts H1 H2 H3 1 H1 is specified as an histogram 2 H2 is specified as an ordered N tuple of relative frequencies which may be 0s together with a sample size The program finds H2 by multiplying the two together 3 H3 is due to merge blocks and will be discussed separately Because an histogram will usually consist mostly of zeroes when giving H1 also the relative frequencies for H2 you specify a block about which you want to be asked and the program defaults values outside that block to zero
18. atter of routine 20 Table 4 Likelinesses This will be the main table of interest it gives the likelinesses of the standard integrams the required integram g and each of the integrams i It also shows the Multinomial Consistency which is an indication of how well the likelinesses obey the Multinomial Theorem This will usually be just for the purposes of reporting since the program itself makes use of the Multinomial Theorem unnecessary Tables 5 amp 6 Frequency and Cumulative distributions The program partitions each range of interest see Section 13 into 20 subintervals and finds the likeliness that Pr g f or Pr z f as appropriate is in each of those subintervals The results are output as Table 5 which consists of a number of small tables one for each of the standard integrams The centres of the cells have been included to make graph plotting easier The cumulative sums of the likelinesses in each of the sub tables of Table 5 are then formed to give the best estimated CDF for Pr g f and for each Pr i f These are output as Table 6 16 The Sampling files Every time the program is run it forms three sampling files as in Table 6 Filename Content sampling_dis csv Sample of distributions sampling _obs csv Observations simulated by using the distributions in sampling_dis csv as generating distributions sampling_rfs csv Relative frequencies for the observations in sam pling_obs csv Table 6
19. e L unfilled places to the left of the block and R unfilled places to the right Then of the C possible ways in which the remaining L R values may be placed the number which have the next value to the left is C _1 so the proportion which have the next value to the left is the ratio of C _ to L RkO which is a So when deciding where to place the next value use RANDOM to select L Q 0 1 and then place the value to the left if Q lt IFR 32 B 8 Selection of f M A to B N Select r R N Since we know the number of unimodal permutations with a given mode we can count the total number which have a mode of at least A and at most B and then find the proportions of that total which have each of the permissible modes in the range AtoB We can then set those proportions as subintervals of 0 1 and use RANDOM to select one of them as the mode m Having selected that mode we can then proceed as with M m N B 9 Selection of f U m N and f U A to B N The algorithms for the selection of U shaped distributions are identical to those for Unimodal distributions with one exception The construction of an Unimodal distribution starts with the selection of an element of R V but that of a U shaped distribution starts with the selection of an element of RR N B 10 Merge Blocks Say the merge block covers columns M N so that we know h 1 h M 1 We reconstruct h M h N from the m
20. ed Bk hele od Gok oh ee Tera gah 12 9 2 9 Reverse ranked with unimodal slope 13 9 2 10 Ranked with ranked slope 2 04 13 9 2 11 Ranked with unimodal slope ooa aaa 14 10 Contractions 15 11 The given histogram 16 11 1 Basics of the specification o oaoa aaa 44644444 264444 16 Pie Merge Blocks sak ioy a di ah oo A aan a te ek thay es e a a e E 16 12 The required integram 17 13 Frequency distributions and CDFs 17 14 data txt 18 15 results csv 20 16 The Sampling files 17 defaults txt 18 Odds and Ends 19 Troubleshooting A Notation and Terminology B The Algorithms B 1 The commoner underlying sets B 2 Selection of f S N cao base woe eae B 3 Selection of r ANd eet th ee B 4 Selection of f RR N B 5 Selection of f RSD ce N 2464 4 44 B 6 Selection of f SD ce N 2 e poe eae bY B 7 Selection of f M m N B 8 Selection of f M AtoB N B 9 Selection of f U m N and f U A to B N B 10 Merge Blocks 2 2 14 vk ee wR es B 11 Using a distribution to generate a simulated observation B 12 Expected frequency distributions and CDFs C Standard analytically solvable problems 21 23 26 27 29 31 31 31 31 32 32 32 32 33 33 33 33 33 34 1 Introduction GREAT LIKELINESSES the name is a pun on that of Charles Dickens s novel Great Expectations is a program for calc
21. erge block Select f from the underlying set of degree N normalise f M f V so that they become a distribution of degree N M 1 Use that distribution to generate the appropriate number of observations Append those generated observations to h 1 a M 1 to form the given histogram and then continue with the analysis Do all of the above with every selection of a distribution B 11 Using a distribution to generate a simulated observation Having selected f from the underlying set of degree N use the points FD fA f 2 fC f N 1 to partition 0 1 into N subintervals Label those subintervals 1 N from left to right and use RAND to select one of them B 12 Expected frequency distributions and CDFs The program reads the upper and lower limits of the interval of interest from data txt and partitions it into 20 subintervals by using 21 equally spaced points Those 21 points partition 0 1 into usually 22 subintervals including the two not of the same length outside the interval of interest Having found Pr g h the program looks to see which of those 22 intervals it falls into and increases an accumulation register for that interval by f At the end of the iterations it normalises the contents of the accumulation registers to produce the 33 expected frequency distribution The expected CDF is the CDF of that expected frequency distribution C Standard analytically solvable p
22. et onto any centre It can also handle merged data blocks Files used The files used are given in Table 4 Filename Purpose data txt Keeps details of problem for future editing and or use results csv Results for viewing in spreadsheet defaults txt Various defaults to personalise the program sampling_dis csv Sample of distributions sampling_obs csv Observations simulated by using the distri butions in sampling_dis csv sampling _rfs csv Relative frequencies for the observations in sampling _obs csv errlog txt Keeps track of run time errors scratch1 txt scratch2 csv Scratchpads for the program s own use Table 4 Files produced by program data txt and results csv When you start a new problem you type it in from the keyboard Details are saved in the file data txt so that you do not have to type them in again on subsequent runs Results are sent to the file results csv for viewing in a spreadsheet If you want to keep data txt or results csv then make a copy under a different name in the usual way Countdown While the program is carrying out an analysis a countdown to completion is sent to the screen so that you can see progress The analysis is in two parts a fast initial pass during which various items are roughly estimated so as to improve the efficiency of the program followed by a slower second pass during which likelinesses are found During the second pass the countdown includes details of the estima
23. he centre is because the contraction will have no effect When specifying the centre of a contraction you are given the choice between various standard cases and specifying all the co ordinates yourself Regardless of how you reply data txt contains the answer specify them myself followed by all the co ordinates The two things you will most often want to alter are a the number of iterations and b the subintervals used for the calculation of distributions For convenience these have been placed together and are preceded by a line of asterisks across the screen terminating with the words FREQUENT CHANGES HERE Figure 11 VALUES OF g gC 1 gC 2 o gC 3 o g 4 o gC 5 o gC 6 o whether the number of iterations is to be changed a C RRR RR RRR ERR RARER RRR RR RRR REE EREQUENT CHANGES HERE FACTOR BY WHICH THE NUMBER OF ITERATIONS IS TO BE CHANGED 1 0000 method for deciding the interval to be used to find the distributions 2 LIMITS OF INTEREST FOR Pr g f 0 1000E 01 0 0000E 00 LIMITS OF INTEREST FOR Pr 1 f 0 1000E 01 0 0000E 00 LIMITS OF INTEREST FOR Pr 2 f 0 7000E 00 0 3000E 00 LIMITS OF INTEREST FOR Pr 3 f 0 6000E 00 0 1000E 00 LIMITS OF INTEREST FOR Pr 4 f 0 8000E 00 0 4000E 00 LIMITS OF INTEREST FOR Pr 5 f 0 7000E 00 0 6000E 00 LIMITS OF INTEREST FOR Pr 6 f 0 9000E 00 0 7000E 00 Figure 11 Position of commonest changes
24. jection 31 Select f S N and then starting with r N 4 f N and working upwards construct r on f B 4 Selection of f RR N Select r R N and then set f z r N 1 1 B 5 Selection of f RSD c N Select r R N and randomise r c 1 r NV B 6 Selection of f SD c N Select f RSD c N and randomise f 1 f c B 7 Selection of f M m N The set of all injective unimodal distributions of degree N which have a mode of m is denoted by M m N The practical underlying set M A to B N is formed as the union of M A N M B N The set of non injective unimodal distributions has measure zero The basic procedure for forming f M m N is to select r R N and then permute the r i to produce an unimodal distribution with mode m The algorithm needs to determine how that permutation is to be carried out By considering the selection of the m 1 values to the left of m out of the N 1 available since r 1 m it follows that there are C _ unimodal permutations of r which have a mode at m Place r 1 then r 2 then r 3 etc as follows 1 Since it must be that r 1 gt m place r 1 at m 2 Since the distribution is to be unimodal r 2 must be placed at either m 1 or m 1 In general at each stage the already placed r i s must form a contiguous block with the next value being placed at either end we need to choose which end Let there b
25. l the frequency distribution has been constructed but the frequency distribution cannot be constructed until the range of interest has been chosen To get round this the program carries out a quick first pass through the iterations during which it collects sufficient information to enable it to make a reasonable first estimate of the interval which it then stores in data txt The intention is not to get the interval right but rather to be able to present the user with enough information to make a better choice to suit his her own needs modifying data csv accordingly To begin with when choosing the intervals of interest you will not have the slightest idea which ranges to specify so you will leave it to the program to make a choice Sometimes it will get the range right sometimes it will be totally wrong Nonetheless the program will usually provide you with enough information to make a better second 17 guess which you should use to edit data csv before running the program again With experiece you will rarely need to run the program 3 times 14 data txt You may edit data txt with any simple text editor use of a wordprocessor is not recommended since it may leave unwanted and possibly invisible codes behind You would edit data csv if for example you wanted to change the values of some data without having to retype the whole of the problem specification There are two types of item in data txt structural and non st
26. ndom number generator you can now have the program always choose the same seed or always choose a different one Section 17 W Additional protection against typing mistakes has been provided for entries in defaults txt Section17 A 3 2 Known Issues none 4 Basics Basic concepts A coin is of degree 2 a die is of degree 6 a pack of cards is of degree 52 The degree is the number of possibilities that something tossing a coin rolling a die drawing a card may take It is the number of classes in a classification system The possibilities classes are labelled 1 2 N where N is the degree Given a degree we can always define a distribution with that degree as in Table 1 For N a distribution f each f i gt 0 and gt gt f z 1 i l i 1 2 3 4 5 6 f i 0 2 0 1 0 35 0 05 0 25 0 5 Table 1 Distribution of degree 6 Likewise we can define an histogram of degree N as in Table 2 Each h i gt 0 and the h i are not required to be integer valued If each h i is an integer then h is called an integram Table 3 an integram is just a special type of histogram There are circumstances where an integram rather than just an histogram is required io ee g a h i 3 0 42 0 07 18 0 Table 2 Histogram of degree 6 i o 1 2 hG 3 4 3456 0 0 1 0 Table 3 Integram of degree 6 Note that an histogram may take values of 0 but a distribution may not The most important histogram is the zero integram 0
27. negative integers For N N let Xy 1 N N is called the degree Let G N glg Xn gt Nt H N h h Xy gt R so G N c H N The elements of H N are called histograms on Xy and those of G N integer valued histograms shortened to integrams on Xy The histogram h is identified with the point h 1 A N N For h H N the sample size of h is w h X h i i 1 For n Nt Qa n g G N w g n This is the set of all integrams of degree N and sample size n In particular Qy w g is the set of all integrams with the same sample size as g For g G N the Multinomial coefficient associated with g is N Let f Xn 0 1 be such that f i 1 Then f is called a distribution on Xy i 1 S N is the set of all such distributions S N c H N For geG N he H N and P C S N where P 4 Q we define se i feP eG JEP Lp g h M 9 where f is the Daniell integral Lp g h is called the likeliness over P of g given h Since P g or h will usually be clear from the context this terminology is normally shortened by omitting appropriate terms h is called the given histogram g the required integram and P the underlying set More generally any non empty subset of S N is called the underlying set in S N The integram of degree N and sample size 0 is 0 0 which is denoted by 0 or if greater clarity is needed by Oy We have Lp O h 1 fo
28. not choose to merge successive delimiters Select detect special numbers Lotus 1 2 3 Choose Parse as CSV When the spreadsheet opens it may seem that some of the fields have been asterisked out they have not it s just that the default column widths are too small In all spreadsheets when viewing results csv you might find it helpful to widen Columns A and B 6 What you should do now 1 Form a new folder to contain the files used by this program 2 Transfer the exe file and these notes into that folder 3 Run the exe file and select item 999 from the opening menu This will set up various defaults You can now experiment with the program to get a feel for what s going on 7 Running the program To start the program run the exe file The opening screen gives you various options If this is the first time that you have ever run this version then you must choose option 999 in order to set up various defaults Welcome to GREAT LIKELINESSES VERSION B 7C Copyright Roger Lovegrove modified November 2614 g Which do you want 1 Keyboard entry 2 Input from file 5 STOP 999 Restore Factory Settings Please choose 1 2 5 or 999 as appropriate Figure 1 Opening screen Otherwise your choice will normally be between Option 1 if you are starting a completely new problem or Option 2 if you are repeating a previous problem or running a modification of one To start a new problem cho
29. orse rather than better For this reason Merge Blocks should be used with caution If there is no Merge Block then H1 H2 is the given histogram If there is a Merge Block however the situation is more complicated since the program uses the information about the Merge Block to recreate the third histogram H3 H3 however is created every time that a new distribution is selected from the underlying set and will change as that distribution changes so it is not possible to specify it This continually changing nature of H3 is a significant component of the vagueness which is introduced by the use of merge blocks 12 The required integram The required integram g is specified in the same way as is H1 by giving a block about which you want to be asked with values outside that block defaulting to zero The calculated Likeliness of g is shown on screen as part of the countdown display This means that convergence can be monitored whilst the calculations are proceeding 13 Frequency distributions and CDFs The expected frequency distribution of Pr g f can be useful as a predictor of the distribution of relative frequencies This will normally be concentrated in a fairly small region around Lp g f so it would be inefficient to construct it over the whole of 0 1 instead we use a smaller interval to cover the range of interest and then partition that interval into cells The difficulty is that that range of interest cannot be chosen unti
30. ose Option 1 You will then be asked a number of questions the answers to all of which will be numerical For most of these you will develop standard answers which you will soon get used to giving very quickly with practice your fingers will type most of the answers faster than you think of them If you choose Option 2 then the computer will just take over and run the problem giving you an on screen progress report which for simpler problems might flash past so quickly that you are unable to read it 8 Stopping The program will normally come to a stop of its own accord There are times though when you might want to stop it prematurely You can choose Stop from the opening menu This will stop the program before it has really started During data input when asked to answer 1 for YES or 2 for NO give one of the emergency numbers 911 or 999 The program will interpret this as a request to stop However data txt will be partially written and will be unuseable the next time you run the program you will have to input the whole of the problem from the keyboard Once data input has finished all you can really do is force a hard stop You could switch the computer off and remove its batteries but an easier way would be to press CTRL C The program will be forced to stop immediately so it will not have the opportunity to close any open files As a result some files might not be closed properly which could mean that you will
31. r all h P Lp g Q is written as Lp g If we roll a die and throw the number 2 then we have not only thrown a 2 once but have also thrown 1 3 4 5 and 6 zero times each So we can think of ourselves as having 29 thrown the integram 0 1 0 0 0 0 Also we have not actually thrown the number 2 but have rather thrown the face labelled 2 It is very convenient to adopt notation which associates the symbol 2 with 0 1 0 0 0 0 We define i to be that integram zx1 y for which x 1 but x 0 otherwise for example 2 0 1 0 0 0 0 It is usually possible to write i rather than ih without introducing ambiguity Importantly f f i and M 7 1 If P is a singleton set P f then Lp g h M g f9 which is denoted by Pr g f h since this is independent of h the notation may be simplified to Pr g f however the presence of the h although technically unnecessary can sometimes add clarity Now let V C S N Then the likeliness of V over P and given h is h Lp V h hoe P For x 0 1 let Vy f S N Pr g f lt x Then Lp V h is the likeliness over P and given h of the set of those f P for which Pr g f lt x We denote this by Lp Pr glf lt ah The function 0 1 gt 0 1 x Lp Pr g f lt x h is the expected CDF of Pr g f Likewise if 0 lt o lt z lt 1 then we define Lp Pr g f xo x1 h to be Lp V h where V f P Pr g f
32. rals on the RHS can be evaluated theoretically Usually however a numerical approach is needed that is what this program does The process is very simple we replace the integrals by summations and the underlying set P by a sample of points ie distributions selected at random from P That sample can be surprisingly large about a million points are often needed the program defaults to 750 000 but 100 million or more can at times be necessary It is only recently that improvements in computer technology have made it possible for such large problems to be tackled on home computers What the program does The program e Finds Lp g h and other standard likelinesses by sampling from P e Keeps track of convergence as the sampling process proceeds e Produces a separate sample of distributions from P and uses each as a generating distribution to generate simulated observations e Calculates the best estimate of the probability ie the likeliness that Pr g f lt x for a selection of x equally spaced across some interval specified by the user Likewise for Pr 1 f Pr N f This is the likeliness equivalent of building up a CDF e For each of the subintervals x calculates the Likeliness that Pr g f lies in that subinterval This is the likeliness equivalent of building up a PDF It does this for an underlying set P chosen by the user from those listed in 9 2 In addition the program can contract the underlying s
33. ree is no more than 77 then a visual aide de memoire will appear on the screen during data input Figure 9 to help you keep track of where your mini unimodal distributions are This is switched off if the degree is more than 77 since a normal window would not be wide enough to contain it 9 2 9 Reverse ranked with unimodal slope As the name suggests 9 2 10 Ranked with ranked slope As the name suggests 13 Where is the left hand end of piece between 1 and 77 gt Set BARE Eee the right hand end of between 20 and 77 gt For piece 1 of 6 what is the st value that the mode can take Must be between 2 and 32 gt For piece 1 of 6 what is the Must be between 25 and 32 gt Is this mode to be bell shaped Please reply 1 for YES or 2 for NO Where is the left hand end of piece between 1 and 77 gt 1111111111111 ee Ce ere foe Where is the right hand end of piece 2 of 6 between 62 and 77 gt Figure 9 Multi modal distributions aide de memoire 9 2 11 Ranked with unimodal slope As the name suggests 14 10 Contractions A contraction is a mapping of the form f q a f q These are useful for reducing the size of the underlying set a is the magnitude and q the centre of the contraction Although the program gives you the opportunity just to type in the co ordinates of the centre if the degree is large then this can involve a lot of typing which will be boring as well as incre
34. roblems a The Multinomial Theorem gives the likeliness of g given h when the underlying set is singleton b The Law of Succession gives the likeliness over S N of g given h when w g 1 c The Combination Theorem gives the likeliness over S N of g given h when h 0 d The Integram Theorem gives the likeliness over S N of g given h when g amp h are both integrams e The Ranked Law of Succession gives the likeliness over R N of g given h when w g 1 For details of a d see 1 for details of e see 2 References 1 Lovegrove R 2013 The Fundamentals of Likelinesses Lovegrove Mathematical Services Research Report 2013 02 London December 2013 2 Lovegrove R 2013 Ranked Distributions on Finite Domains Lovegrove Mathematical Services Research Report 2013 02 London December 2013 34
35. ructural Structural items affect the layout of the remainder of data txt non structural items do not You are strongly advised not to edit structural items because of the knock on effects for the rest of the file which are usually not as easy to predict as might be thought Each item is preceded by a brief description Descriptions of non structural items are in CAPITALS and are enclosed in square brackets To help you find your way around items in a block of similar items are usually preceded by an indication of where you are eg h 11 These are not descriptions so the lower case and the round brackets should not be taken as indicating a structural item the description is at the start of the block If you make a syntactical mistake whilst typing the details of a new problem then the program can and will ask you to re enter the information If you make a syntactical mistake when editing data txt however then the program cannot ask you to re enter the information because the program will not be running The first you will know of the mistake is when you subsequently try running the program and a run time error occurs details of this will be sent to the screen and to the file errlog txt Behaviour is similar to that of a compiler the error might not be picked up immediately and the reported form of the error might not be the actual form The contents of data txt are in a standard layout chosen to make subsequent editing as easy
36. s intended to contain the answers to questions which most people either would not be interested in or would not usually want to alter These questions could rapidly become annoying if asked every time the program was run Because defaults txt is much simpler than data txt error reporting is minimal either something works or it produces a run time error If the latter then the cause of the problem is easily spotted usually either a text string has not been enclosed in quotes or a non integer numerical value has been used Some errors usually involving nonsensical integer values are easily detectable as errors the program will use standard values if one of these is detected Some items could at least in theory take unlimited integer values For these there is no nonsensical integer value which could be specified as part of the program The user however could voluntarily place limits on these and so to give protection against gross typing etc mistakes is given the ability to give maximum acceptable values Each item has a Factory Setting which is hardwired into the program All items can be reset to their factory settings by selecting item 999 from the opening menu Number of iterations to be used Each iteration corresponds to one distribution selected at random from the underlying set Specify the number here It is possible to alter the number of iterations within the program but the number you put here should if chosen
37. s its own essential domain which is extended to cover the whole of Xy by using values of zero elsewhere 12 The Multi modal distribution is then built up from those mini unimodal distributions by taking a linear combination of them A linear combination needs weights these are not specified precisely but are rather specified by giving a set from which they are to be selected the program then selects them every time it is forming a distribution 0 08 5 0 08 1 0 06 5 0 065 o 2 ood 1 4 5 2 3 6 0 04 e Eee 3 ee iy oe 7 u sft 4 0 02 5 0 02 4 5 0 00 taaa a O E 0 00 een ee 0 10 20 30 40 50 60 0 10 20 30 40 50 60 a Weights taken from S 6 b Weights taken from R 6 0 08 5 0 08 5 1 0 06 4 2 3 0 06 LJ ee kid 2 0 045 ee o ee ee 0 04 e oe 3 wd 4 5 6 024 4 5 Sf 6 0 00 a ee e T a 0 00 a ea e i KJ 0 10 20 30 40 50 60 0 10 20 30 40 50 60 c Weights taken from SD 3 6 d Weights taken from RSD 3 6 Figure 8 Multi modal distributions effects of different types of weights The program labels the modes in the order in which their details are typed into the computer which need not be left to right and so gives you flexibility when defining the problem Figure 8 shows examples with weights selected from different sets where the details of the modes were typed into the program in the order 1 2 3 4 5 6 If the deg
38. servations to be simulated 18 Odds and Ends An unordered list of things to remember and things which do not easily fit in elsewhere e For basic problems no merging contractions or Relative Frequencies no given data required integram 1 750 000 iterations run times using a 64 bit laptop were as given in Table 8 Fundamental Degree N type 2 5 10 25 50 75 100 125 S N 2 3 5 13 32 60 115 170 R N 2 3 5 13 33 61 117 174 M N 2 3 7 18 47 89 172 232 U N 2 3 7 18 49 90 174 250 Table 8 Typical run times seconds e If you have been looking at any file but have forgotten to close it down before running the program again then you will receive a run time error or be thrown back into Windows Close the file and if your system offers you the choice choose Retry If it does not offer you this choice then you may need to restart your computer e When using a spreadsheet to plot results pay careful attention to the scale of the vertical axis Spreadsheets usually choose the scale so as to maximise the vertical spread of the plotted points this can cause the results to seem highly scattered when they are in fact in agreement to several sig figs See Figure 12 for an example e To investigate the effects of the sample size of the given data take advantage of the fact that input relative frequencies are normalised before use so they do not actually have to be relative frequencies provided they are not negative Do not gi
39. ted run time these estimates will be thrown out if the computer is used for any other purpose while the program is running Running from Windows It is recommended that you run the program from within Windows rather than by switching firstly to DOS This is because the program can be so fast with simpler analyses that the screen buffer cannot keep up and so forces the program to slow down significantly Windows has improved screen buffering which largely overcomes this 5 Using Spreadsheets The csv files have been designed to be viewed within a spreadsheet Open your spreadsheet by right clicking on the icon for the file you want to open and selecting open with You should then be offered the choice of programs to use to open the file Choose your favourite spreadsheet you should then be offered a choice of options defining how the spreadsheet is to interpret the file Microsoft Excel As the separator choose a comma As the text delimiter choose a double quote Do not choose to merge successive delimiters OpenOffice Apache OpenOffice Calc As the separator choose a comma As the text delimiter choose a double quote Do not choose to merge successive delimiters For versions 3 3 and later of Calc select detect special numbers you will not be offered this option in earlier versions LibreOffice Calc As the separator choose a comma As the text delimiter choose a double quote Do
40. ulating likelinesses that is for finding best estimates of probabilities This is not a textbook about the theory of likelinesses It is a guide to the use of the program Great Likelinesses A summary of the notation and terminology is given in Appendix A the algorithms are outlined in Appendix B 2 There is no warranty This program is currently under development and is not intended to be used in any situation where there are or might be deleterious consequences arising from that use There is no warranty of any form For example there is no warranty against failure of the program or against failure of the program to produce the correct result By using the program you accept full responsibility for all the consequences of that use If you are not willing to accept that responsibility then do not use the program 3 Versions Version B07 comes in two sub versions BO7 which is the full version and BO7C which is the classical version The full version might not be available yet In the full version the degree is now limited to a maximum of 127 increased from the previous 77 In the classical version the degree is restricted to values which have some significance in the classical problems to do with coins dice and cards these are 2 3 4 5 6 8 13 16 26 32 36 39 52 and 64 3 1 Changes implemented in this version U shaped distributions added Section 9 2 7 W Maximum degree increased from 77 to 127 AW e When seeding the ra
41. ve any data as the input histogram but give it instead as input relative frequencies varying the sample size then does just that e In defaults csv if you choose to have a single group by eg selecting a group size of 0 and also choose to have a new distribution only at the start of a group then 26 every entry in the sampling files will use the same distribution However that distribution will be selected at random and you will not have any say in its choice To have just a single distribution specified by you throughout the whole of the sampling files when running the program specify a contraction of magnitude zero centred on the required distribution 19 Troubleshooting The program freezes immediately after I have chosen Item 2 from the opening menu leaving the menu on the screen You probably stopped the program on the previous run by using one of the emergency numbers 911 or 999 These make data txt unusable see 8 page 9 so the program has frozen while trying to read from it You will firstly need to clear the computer by restarting it Then open data txt if it starts with a message saying that it was formed when the previous run was finished early then that is the problem You will need to run the program from the keyboard The program freezes soon after I have selected Item 2 from the opening menu leaving the message downloading of samples now completed on the screen You probably forced the program to
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