Linguistic Fuzzy Logic Forecaster 1 The software
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1. lt lt Cn be fixed nodes within a b such that c a cn b and n gt 2 We say that fuzzy sets A1 An E F a b are basic functions forming a fuzzy partition of a b if they fulfill the following conditions for i 1 n 2 A x 0 for x ci 1 i41 where for uniformity of notation we put Co C Gand Cn41 Cn D 3 A is continuous 4 A strictly increases on c 1 ci and strictly decreases on ci Ci 1 5 for all x a b i l Let a fuzzy partition of a b be given by basic functions A Ay n gt 2 and let f a b R bea function that is known on a set x1 x7 of points The n tuple of real numbers F Fn given by Re Year f te Ail i T Z 4il is a fuzzy transform of f with respect to the given fuzzy partition The numbers F Fn are called the components of the fuzzy transform of f E e ET 11 Let F f be the fuzzy transform of f with respect to A1 An E F a b Then the function fr given on a b by frla X FAi a i 1 is called the inverse fuzzy transform of f 2 2 Linguistic description fuzzy IF THEN rules Fuzzy IF THEN rules can be understood as a specific conditional sentence of natural language of the form IF X is A AND AND X is A THEN Y is B where A1 An and B are evaluative expressions very small roughly big etc An example fuzzy IF THEN rule is IF the number of cars sold in the current year is more2. or less small and the half year sales increment is medium THEN the upcoming half year increment will be medium The part of the rule before THEN is called the antecedent and the part after it is called the consequent Fuzzy IF THEN rules are usually gathered in a linguistic description Ry IF X is A AND AND Xn is Ai THEN Y is Bj Rm i IF X is Am AND AND Xn is Amn THEN Y is Bm 2 3 Perception based logical deduction Perception based logical deduction PbLD is a special method of deducing con clusions on the basis of a linguistic description This method can be described as follows if a linguistic description consisting of fuzzy IF THEN rules together with an observation of some value of the variable X are given then the PbLD chooses the most specific fuzzy rule s among the most fired ones and derives a conclusion based on such preselected fuzzy rule s More details can be found in 2 or in 1 2 4 Implementation of these techniques Let time series x t 1 7 is viewed as a discrete function x on a time axis t Then Fafe X1 Xn is the fuzzy transform of the function x with respect to a given fuzzy partition The inverse fuzzy transform then serve us as a model of the trend cycle of a given time series By subtracting the trend cycle inverse fuzzy transform values from the time series lags we get pure seasonal 12 components This is how the fuzzy transform helps us to model and decompose a given
3. time series Logical dependencies between components X1 Xn of the fuzzy transform may be described by the fuzzy rules These rules are generated automatically from the given data and are used for forecasting the next components Fuzzy transform components as well as their first and second order differences are used as antecedent variables For forecasting future fuzzy transform components based on the generated fuzzy rules a special inference method perception based logical deduction 2 is used This is how fuzzy rules and the PbLD are implemented in the software The seasonal components are forecasted autoregressively Finally both fore casted components trend cycle and seasonal are composed together to obtain the forecast of time series lags References 1 V Nov k M t pni ka A Dvo k I Perfilieva V Pavliska and L Vav kov 2010 Analysis of seasonal time series using fuzzy approach International Journal of General Systems 39 305 328 2 Nov k V and Perfilieva I 2004 On the Semantics of Perception Based Fuzzy Logic Deduction International Journal of Intelligent Systems 19 1007 1031 3 Perfilieva I 2006 Fuzzy Transforms theory and applications Fuzzy Sets and Systems 157 993 1023 4 M t pni ka A Dvo k V Pavliska and L Vav kov 2010 Linguistic approach to time series analysis and forecasts Proc FUZZ IEEE 2149 2157 13
4. 20128 17816 12268 642 7962 13932 15936 12628 12267 12470 18944 21259 22015 18581 15175 10306 10792 14752 13754 11738 Predictor selection Features Description stats Best SMAPE valid error 0 10005 test error 0 08707 Trend cycle type Inverse FT Partition period 12 Predictor type Steps ahead linguistic Variables S t amp dS t gt d5 t 1 Season type LMS linear combination Seasonal periodicity 12 Season dependency depth 3 Decomposition Multiplicative Time Os of Os est Predictors finished L Figure 8 LFL Forecaster There is a windows with three tab pages in the right bottom area of the interface These tab pages particularly e Features e Description e Stats display distinct information related to the results used models measured errors etc 1 2 1 Features This tab page presents the main features of a winning model see Figure 9 It contains the following features e Trend cycle type Trend cycle type determines the method that was used for modelling the trend cycle Features Description Stats Trend cycle type Inverse FT Partition period 12 Predictor type Steps ahead linguistic Variables S t amp d5ft gt dS t 1 Season type LMS linear combination Seasonal periodicity 12 Season dependency depth 3 Decomposition Multiplicative Figure 9 Tab page Features Remark So far the inverse fuzzy transform denoted
5. Linguistic Fuzzy Logic Forecaster Software documentation user s manual LFL Forecaster is a specialized software tool for an analysis and forecasting time series developed by the Institute for Research and Applications of Fuzzy Modeling IRAFM University of Ostrava Czech Republic It is based on two methods originally developed by members of IRAFM The first method is the fuzzy transform and the second one is the perception based logical deduction 1 The software Figure 1 displays the interface of the software JS LFLForecaster v 2 9 6 0 build 4597 2010 jrafm osu cz e B amp A Open Save ZoomFit Compute Global view Trend Cycle Season Linguistic variables Data frequency In sample Out sample Name Comment Monthly LearningSet N A E Cut off Testing PR validation Set 18 Forecasting horizon Learning set Validation set Forecast Input Data Predictor selection Features Description Stats Best SMAPE Predictors Figure 1 LFL Forecaster There are the following four icons in the main menu e Open icon that is used to select a time series file upload e Save icon that is used to save the forecasts to files e ZoomFit icon that is used to zoom out the graph if it has been zoomed in manually by mouse befo
6. able 1 Every single fuzzy rule can be taken as a sentence of natural language For instance the very first fuzzy rule appearing in the generated linguistic descrip tion displayed on Figure 10 IF X is ml sm AND AX is qr sm THEN AX 41 is me may be read as follows If the number of cars sold in the current year is more or less small and the half year sales increment is quite roughly small then the upcoming half year increment will be negative medium 1 2 3 Stats Stats tab page provides users with distinct forecasting error Basically there is only one accuracy measure the well known SMAPE Symmetric Mean Ab solute Percentage Error implemented in the software so far Note that further Abbreviation Ev expression sm small me medium bi big ve very si significantly ex extremely ml more or less ro roughly qR quite roughly vR very roughly Table 1 Evaluative linguistic expressions and their abbreviations Features Description valid error 0 10005 trend cycle valid error 0 07081 test error 0 08707 Figure 11 Tab page Stats criterions and accuracy measures are under the consideration The following in formation is at disposal e Validation error Validation error is the forecasting error computed on the validation set Trend cycle validation error Trend cycle validation error is the forecasting error of the trend cycle val ues computed on the validation set Thi
7. an entire seasonal period to be a linear combination of previous periods The LFLF software is ready to be enriched by other methods e Seasonal periodicity Seasonal periodicity denotes the detected periodicity that was used for the forecasting the seasonal component e Season dependency depth Seasonal dependency depth determines number of whole seasonal periods used for forecasting next seasonal periods Recall that users specifies the minimal and the maximal number of such periods see Figure 6 This value already denotes the optimal number of periods within the range defined by a user e Decomposition This feature specifies whether the chose decomposition was either Additive or Multiplicative 1 2 2 Description Features Stats Steps ahead linguistic Linguistic Rules count 14 a Signature S t amp d5 t gt dS t 1 H 1 amp ml sm amp qr sm amp gt amp me 2 amp ze amp me amp gt amp qr bi 3 amp ro sm amp qr bi amp gt amp si sm 4 amp qsm amp si sm amp gt amp wr bi 5 amp mime amp vr bi amp gt amp ml me 6 amp ve SM amp me amp gt amp ro bi v Figure 10 Tab page Description This tab page presents the linguistic description containing fuzzy rules gen erated from fuzzy transform components of a given time times series Abbre viations of evaluative linguistic expressions that are used in the tab page are introduced in T
8. by inverse FT is the only method that is at disposal The LFLF is ready to be enriched by other methods Partition period Partition period displays the partition period of the winning model i e the number of time series values that is covered by any basic function of the model that is used to describe and forecast a given time series Predictor type Predictor types describes whether Simple denoted by the word Linguis tic or More steps ahead denoted by Steps ahead linguistic trend cycle prediction was used Remark The LFLF is ready to be enriched by other methods hence the word linguistic appear in the trend cycle predictor name Variables This feature displays antecedent and consequent variables that appear in fuzzy rules describing the trend cycle model Particularly S denotes the trend cycle components dS their differences and d2S their second order differences The argument t t 1 etc denotes the time lag of the component For example taking S t amp dS t dS t 1 from Figure 9 denotes the fact that X and AX are the antecedent variables and AX is the consequent variable of the winning model and hence we deal with rules of the form IF X is A AND AX is Aa THEN AX441 is Apettas Season type This feature denotes the model and consequently the method that is used to forecast seasonal components Remark So far only the LMS linear combination method that assumes
9. eason depth Season depth determines minimal and maximal number of whole seasonal period that may be used for forecasting next seasonal period e g for monthly time series the season depth is one year and the next seasonal values are forecasted using seasonal values from 1 to 4 years see Figure 6 e Decomposition Models that are searched for are given as compositions of trend cycle and seasonal components Hence particular type of de composition has to be chosen Check box Additive By ticking this check box models using additive decomposition will be searched for Check box Multiplicative By ticking this check box models using multiplicative decomposition will be searched for If both are ticked the software selects better one e Periodicity Periodicity is the length of whole seasonal period Radio button auto By choosing this radio button periodicity is automatically determined by the software Radio button user By ticking this radio button a user sets up the periodicity manually 1 1 5 Linguistic Variables Global view Trend Cycle Season Linguistic Variables Value Difference 2nd Difference Total From 1 from 1 from 1 from 2 to 3 to 3 to 3 to 3 gt Figure 7 Tab page Linguistic Variables This tab page allows to set up minimal and maximal numbers of the input variables that may occ
10. es in Quebec 1960 1968 a Validation Set 12 Forecasting horizon 12 Figure 3 Tab page Global Global tab page serves users to set up e Data frequency This option allows to choose frequency of the data e g monthly daily etc e Validation set A user sets up a length of this set here e Forecasting horizon Forecasting horizon is the number of values to be forecasted e Check box Cut off Testing It can be ticked if a testing set is available In this case values of the testing set are not used for computation These data are purely used for evaluation of prediction error 1 1 2 View Global View Trend Cycle Season Linguistic Variables 7 Trend Cycle F Partition Figure 4 Tab page View e Check box Trend Cycle Trend cycle is displayed on the graph if this option is ticked e Check box Partition Fuzzy partition is displayed on the graph if this option is ticked 1 1 3 Trend Cycle Global view Trend Cycle Season Linguistic Yariables Linguistic predictor type Partition period Linguistic description V Simple auto min rules V More steps ahead user Figure 5 Tab page Trend Cycle e Linguistic predictor type There are two ways how to forecast the future components of the fuzzy transform Check box Simple By ticking this check box the next component of the fuzzy transform will be forecasted from previous n com
11. ponents and their first and second differences It means that we forecast from forecasted values one step ahead Check box More steps ahead By ticking this check box one may avoid the problem of forecasting from forecasted values This is due to the construction of several independent models one model forecasts one step ahead another one forecasts two step ahead etc up to the desired number of models steps ahead to be forecasted If both are ticked the software selects the better one e Partition period Partition period determines width of basic functions By this we mean the number of time series values covered by one basic function Radio button auto By choosing this radio button the software automatically determines the partition period Radio button user By ticking this radio button a user determines the partition period manually e Linguistic description Here a user sets up the minimal number of fuzzy rules that should occur in a winning linguistic description This parameter prevents an extremely small linguistic description consisting of number of fuzzy rules that is below a critical number to win to be selected 1 1 4 Season Global View J Trend Cycle Season Linguistic Variables E Season depth Decomposition Periodicity from 1 7 Additive auto to F Multiplicative user 12 Figure 6 Tab page Season Season tab page allows to set up e S
12. re e Compute icon that is used to run the implemented forecast methods HH h Open Save ZoomFit Compute Figure 2 Main menu icons LFLF software package allows to open the text files of the following formats 1 Comment Title 66 70 74 68 i e the data is in a row See Example 1 or 2 Comment Title 1 66 70 74 63 AUN i e the data is in the column where each value is preceded by its index See Example 2 Title is a name of a time series that is followed by an unlimited number of real numbers delimited by a blank space TAB space etc Comment is a description of a time series that is preceded symbol delimited by a space Comment and Title are not obligatory Example 1 Monthly car sales in Quebec 1960 1968 Car sales 8728 12026 14395 14587 18791 9498 8251 7049 9545 712397 9874 11887 13784 15926 18821 11143 7975 7610 Example 2 Monthly car sales in Quebec 1960 1968 Car sales 1 8728 2 12026 14395 14587 13791 9498 8251 7049 DRAA Lo 1 1 Main menu The interface of the software contains five tab pages which serve a user to set up details for a prediction process e Global e View Trend Cycle e Season Linguistic Variables 1 1 1 Global Global yiew Trend Cycle Season Linguistic Variables Data frequency In sample Out sample Name Comment Monthly LearningSet 84 V Cut off Testing Car_sales Monthly car sal
13. s serves for the determination of the winning model that is to be used for the trend cycle forecast Testing error Testing error is the error computed on the testing set if a testing set was available and used This error is never used to determine the winning model The LFLF software is equipped with the ability to compute the testing error in order to provide users with a high users comfort Without this functionality users would have to export their forecasts and measure the precision manually 1 3 Saving outputs Generated linguistic description model features and time series forecasts can be saved using the Save icon see Figure 2 10 Exported MS Excel file contains forecasted values values of the trend cycle and some features of the winning model such as a validation error partition period etc Exported text file contains the forecasted values of a time series the forecasted trend cycle components trend cycle predictor linguistic description etc 2 Appendixes 2 1 The fuzzy transform The idea of the fuzzy transform is to transform a given function defined in one space into another usually simpler space and then to transform it back The simpler space consist of a finite vector of numbers The reverse transform then leads to a function which approximates the original one More details can be found in 3 The fuzzy transform is defined with respect to a fuzzy partition which con sists of basic functions Let c
14. ur among antecedents of automatically generated fuzzy rules e Value By values we directly mean the components of the fuzzy transform e Difference In this block we set up a number of first order differences of fuzzy transform components that are given as follows differences between components AX Xi Xi 1 e 2nd Difference These are values of second order differences of components of the fuzzy transform that are given as follows A X AX AXi e Total This block is used to set up a total number of input variables 1 2 Outputs Figure 2 displays the interface of the LFLF after a time series is forecasted E LFLForecaster v 2 9 6 0 build 4597 2010 irafm osu cz a Carsalesitt I i 7 _ E N Eeo File Series Help amp B Open Save ZoomFit Compute iew Trend Cycle Season Linguistic Variables Data frequency In sample Out sample Name Comment Monthly v Learning Set T Cut off Testing Car_sales Monthly car sales in Quebec 1960 1968 Validation Set i Forecasting horizon 1 2 3 4 5 f Learning set esso 8725 12026 14395 14587 13791 9498 8251 7049 9545 9364 zi E paidagonk 7237 9374 11837 13784 15926 13821 11143 7975 7610 10015 12759 8816 Forecast 10677 10947 15200 17010 20900 116205 12143 8997 5568 11474 12256 10583 Input Data njawi ne 10862 10965 14405 20379
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