Abstract
This chapter introduces an alternative approach to time-series prediction for stock index data using Interval Type-2 Fuzzy Sets. The work differs from the existing research on time-series prediction by the following counts. First, partitions of the time-series, obtained by fragmenting its valuation space over disjoint equal sized intervals, are represented by Interval Type-2 Fuzzy Sets (or Type-1 fuzzy sets in absence of sufficient data points in the partitions). Second, an interval type-2 (or type-1) fuzzy reasoning is performed using prediction rules, extracted from the (main factor) time-series. Third, a type-2 (or type-1) centroidal defuzzification is undertaken to determine crisp measure of inferences obtained from the fired rules, and lastly a weighted averaging of the defuzzified outcomes of the fired rules is performed to predict the time-series at the next time point from its current value. Besides the above three main prediction steps, the other issues considered in this chapter include: (i) employing a new strategy to induce the main factor time-series prediction by its secondary factors (other reference time-series), and (ii) self-adaptation of membership functions to properly tune them to capture the sudden changes in the main-factor time-series. Performance analysis undertaken reveals that the proposed prediction algorithm outperforms existing algorithms with respect to root mean-square error by a large margin (≥23%). A statistical analysis undertaken with paired t-test confirms that the proposed method is superior in performance at 95% confidence level to most of the existing techniques with root mean square error as the key metric.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Abbreviations
- CSV:
-
Composite secondary variation
- CSVS:
-
Composite secondary variation series
- FOU:
-
Footprint of uncertainty
- IT2:
-
Interval type-2
- IT2FS:
-
Interval type-2 fuzzy set
- IT2MF:
-
Interval type-2 membership function
- MFCP:
-
Main factor close price
- MFTS:
-
Main factor time-series
- MFVS:
-
Main factor variation series
- RMSE:
-
Root mean square error
- SFTS:
-
Secondary factor time-series
- SFVS:
-
Secondary factor variation series
- SFVTS:
-
Secondary factor variation time-series
- T1:
-
Type-1
- T1FS:
-
Type-1 fuzzy set
- VTS:
-
Variation time-series
- \( A_{i,j} \) :
-
Type-1 fuzzy set for partition \( P_{i} \) of MFTS
- \( \tilde{A}_{i} \) :
-
Interval type-2 fuzzy set for partition \( P_{i} \) in MFTS
- \( B_{i} \) :
-
Classical set for partition \( Q_{i} \) of MFVS
- \( B^{\prime}_{i} \) :
-
Classical set for partition \( Q_{i} \) for SFVS/CSVS
- \( c(t) \) :
-
Close Price on \( t{\text{th}} \) day
- \( c_{l} \) :
-
Left end point centroid of IT2FS
- \( c_{r} \) :
-
Right end point centroid of IT2FS
- \( c \) :
-
Centroid of an IT2FS
- c′:
-
Measured value of \( c\left( t \right) \) in centroid calculation
- \( c^{x} \) :
-
Type-1 centroid of \( \mu_{{A_{x} }} (c(t)) \)
- \( m_{A} \) :
-
Mean values of the distributions of RMSE obtained by algorithms A
- \( P_{i} \) :
-
Ith partition for close price time series (of MFTS)
- \( Q_{i} \) :
-
Ith partition for variation series (of MFVS/SFVS/CSVS)
- \( s_{A} \) :
-
Standard deviation of the respective samples obtained by algorithms A
- \( V_{M}^{d} \left( t \right) \) :
-
Main factor variation series with delay \( d \)
- \( V_{{S^{i} }}^{d} \) :
-
Ith secondary factor variation series with delay \( d \)
- \( V_{S}^{d} \left( t \right) \) :
-
Composite secondary variation series with delay \( d \)
- \( W_{{S^{i} }} \) :
-
Weight for ith secondary factor
- \( \mu_{A} (x) \) :
-
Type-1 membership function of linguistic variable \( x \) in fuzzy set \( A \)
- \( \bar{\mu }_{{\tilde{A}}} (x) \) :
-
Upper membership function of IT2FS \( \tilde{A} \)
- \( \underline{\mu }_{{\tilde{A}}} (x) \) :
-
Lower membership function of IT2FS \( \tilde{A} \)
- \( \Delta_{{S^{i} }} \) :
-
Total difference variation for CSVS of ith secondary factor
- \( \hat{\Delta }_{{S^{i} }} \) :
-
Normalized value of \( \Delta_{{S^{i} }} \) for CSVS of ith secondary factor
References
Engineering Statistic and book [Online]. http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc41.htm
Chen, S. M., & Chen, C. D. (2011). Handling forecasting problems based on high-order fuzzy logical relationships. Expert Systems with Applications, 38(4), 3857–3864.
Chen, S. M., & Tanuwijaya, K. (2011). Fuzzy forecasting based on high-order fuzzy logical relationships and automatic clustering techniques. Expert Systems with Applications, 38(12), 15425–15437.
Wu, C. L., & Chau, K. W. (2013). Prediction of rainfall time series using modular soft computing methods. Elsevier, Engineering Applications of Artificial Intelligence, 26, 997–1007.
Wu, C. L., Chau, K. W., & Fan, C. (2010). Prediction of rainfall time series using modular artificial neural networks coupled with data-preprocessing techniques. Elsevier, Journal of Hydrology, 389(1–2), 146–167.
Barnea, O., Solow, A. R., & Stonea, L. (2006). On fitting a model to a population time series with missing values. Israel Journal Of Ecology & Evolution, 52, 1–10.
Chen, S. M., & Hwang, J. R. (2000). Temperature prediction using fuzzy time series. IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics, 30(2), 263–275.
Song, Q., & Chissom, B. S. (1993). Fuzzy time series and its model. Fuzzy Sets and Systems, 54(3), 269–277.
Song, Q., & Chissom, B. S. (1993). Forecasting enrollments with fuzzy time series—Part I. Fuzzy Sets and Systems, 54(1), 1–9.
Song, Q., & Chissom, B. S. (1994). Forecasting enrollments with fuzzy time series—Part II. Fuzzy Sets and Systems, 62(1), 1–8.
Song, Q. (2003). A note on fuzzy time series model selection with sample autocorrelation functions. Cybernetics &Systems, 34(2), 93–107.
Jalil, A., & Idrees, M. (2013). Modeling the impact of education on the economic growth: Evidence from aggregated and disaggregated time series data of Pakistan. Elsevier Economic Modelling, 31, 383–388.
Edwards, R. P., Magee, J., & Bassetti, W. H. C. (2007). Technical analysis of Stock Trends, 9th Edition, p. 10. New York, USA: Amacom.
Raynor, W. J., Jr. (1999) The international dictionary of artificial intelligence. IL: Glenlake Publishing.
Mallat, S. (1997). A wavelet tour of signal processing. San diego, CA, USA: Academic Press.
Bhowmik, P., Das, S., & Konar, A., Nandi, D., & Chakraborty, A. (2010) Emotion clustering from stimulated encephalographic signals using a Duffing Oscillator. International Journal of Computers in Healthcare, 1(1).
Gupta, S. C., & Kapoor, V. K. (2002). Fundamental of mathematical statistics. New Delhi, India: S. Chand & Sons.
Jansen,B. H., Bourne, J. R., Ward, J. W. (1981). Autoregressive estimation of short segment spectra for computerized EEG analysis. IEEE Transactions on Biomedical Engineering, 28(9).
Hjorth, B. (1970). EEG analysis based on time domain properties. Electroencephalography and Clinical Neurophysiology, 29, 306–310.
Sanei, S. (2013) Adaptive processing of brain signals. USA: Wiley.
Chen, S. M. (1996). Forecasting enrollments based on fuzzy time series. Fuzzy Sets and Systems, 81(3), 311–319.
Hwang, J. R., Chen, S. M., & Lee, C. H. (1998). Handling forecasting problems using fuzzy time series. Fuzzy Sets and Systems, 100(1–3), 217–228.
Lee, L. W., Wang, L. H., & Chen, S. M. (2007). Temperature prediction and TAIFEX forecasting based on fuzzy logical relationships and genetic algorithms. Expert Systems with Applications, 33(3), 539–550.
Wong, W. K., Bai, E., Chu, A. W. C. (2010). Adaptive time-variant models for fuzzy-time-series forecasting. IEEE Transaction on Systems, Man, Cybernetics-Part B: Cybernetics, 40(6).
Cheng, C. H., Chen, T. L., Teoh, H. J., & Chiang, C. H. (2008). Fuzzy time-series based on adaptive expectation model for TAIEX forecasting. Elsevier, Expert Systems with Applications, 34, 1126–1132.
Chen, T. L., Cheng, C. H., & Teoh, H. J. (2007). Fuzzy time-series based on Fibonacci sequence for stock price forecasting. Elsevier, Physica A, 380, 377–390.
Huarng, K., & Yu, T. H. K. (2006). The application of neural networks to forecast fuzzy time series. Physica A, 363(2), 481–491.
Yu, T. H. K., & Huarng, K. H. (2010). A neural network-based fuzzy time series model to improve forecasting. Expert Systems with Applications, 37(4), 3366–3372.
Kuo, I. H., Horng, S. J., Kao, T. W., Lin, T. L., Lee, C. L., & Pan, Y. (2009). An improved method for forecasting enrollments based on fuzzy time series and particle swarm optimization. Expert Systems with Applications, 36(3), 6108–6117.
Lu, W., Yang, J., Liu, X., & Pedrycz, W. (2014). The modeling and prediction of time-series based on synergy of high order fuzzy cognitive map and fuzzy c-means clustering. Elsevier, Knowledge Based Systems, 70, 242–255.
Yu, T. H. K., & Huarng, K. H. (2008). A bivariate fuzzy time series model to forecast the TAIEX. Expert Systems with Applications, 34(4), 2945–2952.
Yu, T. H. K., & Huarng, K. H. (2010). Corrigendum to “A bivariate fuzzy time series model to forecast the TAIEX”. Expert Systems with Applications, 37(7), 5529.
Chen, S. M., & Chang, Y. C. (2010). Multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques. Information Sciences, 180(24), 4772–4783.
Wong,H. L., Tu, Y. H., & Wang, C. C. (2009). An evaluation comparison between multivariate fuzzy time series with traditional time series model for forecasting Taiwan export. In Proceedings of World Congress on Computer Science Engineering, pp. 462–467.
Chen, S. M., & Tanuwijaya, K. (2011). Multivariate fuzzy forecasting based on fuzzy time series and automatic clustering clustering techniques. Expert Systems with Applications, 38(8), 10594–10605.
Huarng, K., Yu, H. K., & Hsu, Y. W. (2007). A multivariate heuristic model for fuzzy time-series forecasting. IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics, 37(4), 836–846.
Sun, B. Q., Guo, H., Karimi, H. R., Ge, Y., Xiong, S. (2015). Prediction of stock index futures prices based on fuzzy sets and multivariate fuzzy time series. Neurocomputing, Elsevier, 151, 1528–1536.
Chen, T. L., Cheng, C. H., & Teoh, H. J. (2008). High-order fuzzy time-series based on multi-period adaptation model for forecasting stock markets. Elsevier, Physica A, 387, 876–888.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.
Zhang, W. (1992). Applications engineer, Aptronics Incorporated, Copyright © by Aptronix Inc.
Virkhareg, N., & Jasutkar, R. W. (2014) Neuro-fuzzy controller based washing machine. International Journal of Engineering Science Invention, 3, 48–51.
Konstantinidis, K., Gasteratos, A., & Andreadis, I. (2005). Image retrieval based on fuzzy color histogram processing. Elsevier, Optics Communications, 248, 375–386.
Halder, A., Konar, A., Mandal, R., Chakraborty, A., Bhowmik, P., Pal, N. R., et al., (2013) General and interval type-2 fuzzy face-space approach to emotion recognition. IEEE Transactions On Systems, Man, And Cybernetics: Systems, 43(3).
Karnik, N. N., & Mendel, J. M. (1999). Applications of type-2 fuzzy logic systems to forecasting of time-series. Elsevier, Information Sciences, 120, 89–111.
Huarng, K., & Yu, H. K. (2005). A type 2 fuzzy time series model for stock index forecasting. Physica A, 353, 445–462.
Bajestani, N. S., & Zare, A. (2011). Forecasting TAIEX using improved type 2 fuzzy time series. Expert Systems with Applications, 38(5), 5816–5821.
Chen, S. M., & Chen, C. D. (2011). TAIEX forecasting based on fuzzy time series and fuzzy variation groups. IEEE Transactions on Fuzzy Systems, 19(1), 1–12.
Mendel, J. M., & Wu, D. (2010). Perceptual computing: Aiding people in making subjective judgements. Hoboken, NJ: IEEE-Wiley press.
Chakraborty, S., Konar, A., Ralescu, A., & Pal, N. R. (2015). A fast algorithm to compute precise type-2 centroids for real time control applications. IEEE Transaction on Cybernetics, 45(2), 340–353.
Chen,S. M., Chu, H. P., Sheu, T. W. (2012). TAIEX forecasting using fuzzy time series and automatically generated weights of multiple factors. IEEE Transaction on Systems, Man, and Cybernetics-Part A: Systems and Humans, 42(6), 1485–1495.
Mendel,J. M., Hagras, H., Tan, W. W., Melek, W. W., & Ying, H. (2014) Introduction to type-2 fuzzy logic control: Theory and applications. USA: IEEE-Wiley Press.
Nilsson, N. J. (1980). Principles ofartificial intelligence. CA: Morgan Kaufmann.
Web Link: http://www.computationalintelligence.net/ieee_cyb/index_prediction_appendix1.pdf
Garibaldi,J. M. (2010). Alternative forms of non-standard fuzzy sets: A discussion chapter. In Proceedings of Type-2 Fuzzy Logic: State of the Art and Future Directions, London.
Wu, D. (2011). A constraint representation theorem for interval type-2 fuzzy sets using convex and normal embedded type-1 fuzzy sets, and its application to centroid computation. In Proceedings of World Conference on Soft Computing, San Francisco, CA.
Web Link for historical Data TAIEX. [Online]. Available: http://www.twse.com.tw/en/products/indices/tsec/taiex.php
Storn, R., & Price, K. G. (1997). Differential Evolution- a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4), 341–359.
Das, S., & Suganthan, P. N. (2011) Differential evolution: A survey of the state-of-the-art. IEEE Transaction on Evolutionary Computation, 15(1), 4–31.
Chen, S. M., Kao, P.-Y. (2013). TAIEX forecasting based on time series, particle swarm optimization techniques and support vector machines. Information Science, Elsevier, 247, 62–71.
Cai, Q., Zhang, D., Zheng, W., & Leung, S. C. H. (2015). A new fuzzy time series forecasting model combined with ant colony optimization and auto-regression. Knowledge-Based Systems, 74, 61–68.
Chen, Mu-Yen. (2014). A high-order fuzzy time series forecasting model for internet stock trading. Future Generation Computer Systems Elsevier, 37, 461–467.
Gardner, E., & McKenzie, E. (1989). Seasonal exponential smoothing with damped trends. Management Science, 35(3), 372–376.
Kuremoto, T., Kimura, S., Kobayashi, K., & Obayashi, M. (2014). Time series forecasting using a deep beliefnetwork with restricted boltzmann machines (online open access). Neurocomputing, 137(5), 47–56.
Yildiz,O. T. (2013) Omnivariate rule induction using a novel pairwise statistical test. IEEE Transactions On Knowledge And Data Engineering, 25(9).
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 2.1
See Tables 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12 and 2.13.
Appendix 2.2: Source Codes of the Programs
-
% MATLAB Source Code of the Main Program and Other Functions for Time
-
% Series Prediction by IT2FS REasoning
-
% Developed by Monalisa Pal
-
-
% Under the guidance of Amit Konar and Diptendu Bhattacharya
% Main Program %______________________________________________________________ clear all; clc; str=input(‘Data file having Main Factor:’); load(str); closeMF=close; n=input(‘Number of secondary factors:’); n=str2double(n); SF=zeros(length(closeMF),n); for i=1:n strSF=input([‘Data file having ’,num2str(i),’-th Secondary Factor:’]); load(strSF); SF(:,i)=close; end clear close; tic; %% Training [A,B,VarMF,Au,Al]=partitioning(closeMF); plotpartitions(closeMF(s:e1),A(s:e1),Al,Au); FLRG=tableVI(B(2:e1),A(1:e1-1),A(2:e1)); VarSF=overallvar(VarMF,SF,e1,n); BSF=fuzzyvarSF(VarSF); FVG=tableIX(B(2:e1),BSF(2:e1)); WBS=tableX(FVG); %% Validation (Differential Evolution) sd=extractSD(A(s:e1),closeMF(s:e1)); [UMF,LMF,typeMF,rmse]=MFusingDE(A(s:e1),closeMF(s:e1), Al:Au-1,sd,closeMF(e1+1:e2),A(e1:e2),FLRG,WBS,BSF(e1:e2)); plotDE(rmse); plotFOU(UMF,LMF,Al:Au-1); %% Inference forecasted=predict(closeMF(e2+1:f),A(e2:f),Al,Au,UMF,LMF,typeMF, FLRG,WBS,BSF(e2:f)); [CFE,ME,MSE,RMSE,SD,MAD,MAPE]= errormetrics(forecasted,closeMF(e2+1:f)); disp(‘CFE’);disp(CFE); disp(‘ME’);disp(ME); disp(‘MSE’);disp(MSE); disp(‘RMSE’);disp(RMSE); disp(‘SD’);disp(SD); disp(‘MAD’);disp(MAD); disp(‘MAPE’);disp(MAPE); plotforecasted(forecasted,closeMF(e2+1:f),RMSE); comp_time=toc; disp(‘Execution Time’);disp(comp_time); %End of Main Program
% Function KMmethod to compute IT2FS Centroid %____________________________________________
function centroid=KMmethod(LMF,UMF,x)
diff=ones(1,length(x))*5000; for i=1:length(x) if UMF(i)>0 theta(i,:)=[UMF(1:i) LMF(i+1:length(LMF))]; centroid(i)=defuzz(x,theta(i,:),’centroid’); diff(i)=abs(x(i)−centroid(i)); end end [mindiff sw_index]=min(diff); cl=x(sw_index); theta_l=[UMF(1:sw_index) LMF(sw_index+1:length(LMF))];
diff=ones(1,length(x))*5000; for i=1:length(x); if i<length(x) if UMF(i+1)>0 theta(i,:)=[LMF(1:i) UMF(i+1:length(LMF))]; centroid(i)=defuzz(x,theta(i,:),’centroid’); diff(i)=abs(x(i)−centroid(i)); end end end; [mindiff sw_index]=min(diff); cr=x(sw_index); theta_r=[LMF(1:sw_index) UMF(sw_index+1:length(LMF))];
centroid=(cl+cr)/2; end % End of functiom KMmethod
%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Error Metric Calculation
function [CFE,ME,MSE,RMSE,SD,MAD,MAPE]=errormetrics(predicted,TestCP) % a=isnan(predicted1); % p=size(TestCP1); % z=1; % for i=1:p % if a(i)~=1 % predicted(z)=predicted1(i); % TestCP(z)=TestCP1(i);z=z+1; % end % end CFE=sum(TestCP−predicted); % disp(‘CFE=‘); % disp(CFE); ME=mean(TestCP−predicted); % disp(‘ME=‘); % disp(ME); MSE=mean((TestCP−predicted).^2); % disp(‘MSE=‘); % disp(MSE); RMSE=sqrt(MSE); % disp(‘RMSE=‘); % disp(RMSE); SD=std(TestCP−predicted); % disp(‘SD=‘); % disp(SD); MAD=mean(abs(TestCP−predicted)); % disp(‘MAD=‘); % disp(MAD); MAPE=mean(abs(TestCP−predicted)./TestCP)*100; % disp(‘MAPE=‘); % disp(MAPE); End
%%%%%%%%%%%%%%%%% % RMSE calculation
function rmse=evalfit(x,TestCP,TestA,UMF,LMF,typeMF,FLRG,WBS,TestB) Al=x(1); Au=x(end)+1; forecasted=predict(TestCP,TestA,Al,Au,UMF,LMF,typeMF,FLRG,WBS,TestB); [~,~,~,rmse,~,~,~]=errormetrics(forecasted,TestCP); End
%%%%%%%%%%%%%%%%%%
% Find Standard Deviations(SD)
function sd=extractSD(A,close) numFS=unique(A); % converting the close series into a matrix FS=zeros(length(numFS),length(close)); for i=1:length(A) FS(A(i),i)=close(i); end b=1; for i=1:size(FS,1) temp=find(FS(i,:)); %Case2: Partition has one point if length(temp)==1 sd(b)=0.001; % flag(b)=2; b=b+1; %Case3: Partition has two points elseif length(temp)==2 sd(b)=std(close(temp)); % flag(b)=3; b=b+1; %Case 1 and 4: More than 2 contiguous or discrete points else indx=zeros(length(temp),1); l=1; for j=2:length(temp) %contiguous points have been labelled sequentially if (temp(j)−temp(j−1))==1 indx(j−1)=l; indx(j)=l; elseif j>2 && (temp(j−1)−temp(j−2))==1 && (temp(j)−temp(j−1))~=1 l=l+1; end end if max(indx)==0 sd(b)=std(close(temp)); % flag(b)=4; b=b+1; else for j=1:max(indx) temp1=temp(indx==j); % selecting days where the contiguous points occur tempsd=std(close(temp1)); % if tempsd>=1 sd(b)=tempsd; % flag(b)=1; b=b+1; % end end end end end end %%%%%%%%%%%%%%%%% % Forming Membership Functions(MFs)
function [UMF,LMF,typeMF]=formingMF(A,close,x,sd) numFS=unique(A); % converting the close series into a matrix FS=zeros(length(numFS),length(close)); for i=1:length(A) FS(A(i),i)=close(i); end b=1; UMF=zeros(length(numFS),length(x)); LMF=ones(length(numFS),length(x)); typeMF=zeros(length(numFS),1); for i=1:size(FS,1) temp=find(FS(i,:)); %Case2: Partition has one point if length(temp)==1 m=round(close(temp)); typeMF(i)=1; UMF(i,:)=max(UMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); LMF(i,:)=min(LMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); b=b+1;
%Case3: Partition has two points elseif length(temp)==2 m=round(mean(close(temp))); typeMF(i)=1; UMF(i,:)=max(UMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); LMF(i,:)=min(LMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); b=b+1; %Case 1 and 4: More than 2 contiguous or discrete points else indx=zeros(length(temp),1); l=1; for j=2:length(temp) %contiguous points have been labelled sequentially if (temp(j)−temp(j−1))==1 indx(j−1)=l; indx(j)=l; elseif j>2 && (temp(j−1)−temp(j−2))==1 && (temp(j)−temp(j−1))~=1 l=l+1; end end if max(indx)==0 m=round(mean(close(temp))); typeMF(i)=1; UMF(i,:)=max(UMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); LMF(i,:)=min(LMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); b=b+1; else % c=0; for j=1:max(indx) temp1=temp(indx==j); % selecting days where the contiguous points occur m=round(mean(close(temp1))); % if sd>=1 UMF(i,:)=max(UMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); LMF(i,:)=min(LMF(i,:),trimf(x,[m−3*sd(b) m m+3*sd(b)])); b=b+1; % c=c+1; % end end % if c==1 % typeMF(i)=1; % else typeMF(i)=2; % end end end end %Ensuring flat top loc1=0;loc2=0; for i=1:size(UMF,1) for j=1:1:size(UMF,2) if UMF(i,j)>0.999 loc1=j; break; end end if j~=size(UMF,2) for j=size(UMF,2):−1:1 if UMF(i,j)>0.999 loc2=j; break; end end end if loc1~=0 UMF(i,loc1:loc2)=1; end end end %%%%%%%%%%%%%%%%%%%%%%% % Gaussian MF Creation
function [UMF,LMF,typeMF]=formingMF(A,close,x,sd) numFS=unique(A); % converting the close series into a matrix FS=zeros(length(numFS),length(close)); for i=1:length(A) FS(A(i),i)=close(i); end b=1; UMF=zeros(length(numFS),length(x)); LMF=ones(length(numFS),length(x)); typeMF=zeros(length(numFS),1); for i=1:size(FS,1) temp=find(FS(i,:)); %Case2: Partition has one point if length(temp)==1 m=round(close(temp)); typeMF(i)=1; UMF(i,:)=max(UMF(i,:),gaussmf(x,[sd(b) m])); LMF(i,:)=min(LMF(i,:),gaussmf(x,[sd(b) m])); b=b+1;
%Case3: Partition has two points elseif length(temp)==2 m=round(mean(close(temp))); typeMF(i)=1; UMF(i,:)=max(UMF(i,:),gaussmf(x,[sd(b) m])); LMF(i,:)=min(LMF(i,:),gaussmf(x,[sd(b) m])); b=b+1; %Case 1 and 4: More than 2 contiguous or discrete points else indx=zeros(length(temp),1); l=1; for j=2:length(temp) %contiguous points have been labelled sequentially if (temp(j)−temp(j−1))==1 indx(j−1)=l; indx(j)=l; elseif j>2 && (temp(j−1)−temp(j−2))==1 && (temp(j)−temp(j−1))~=1 l=l+1; end end if max(indx)==0 m=round(mean(close(temp))); typeMF(i)=1; UMF(i,:)=max(UMF(i,:),gaussmf(x,[sd(b) m])); LMF(i,:)=min(LMF(i,:),gaussmf(x,[sd(b) m])); b=b+1; else % c=0; for j=1:max(indx) temp1=temp(indx==j); % selecting days where the contiguous points occur m=round(mean(close(temp1))); % if sd>=1 UMF(i,:)=max(UMF(i,:),gaussmf(x,[sd(b) m])); LMF(i,:)=min(LMF(i,:),gaussmf(x,[sd(b) m])); b=b+1; % c=c+1; % end end % if c==1 % typeMF(i)=1; % else typeMF(i)=2; % end end end end %Ensuring flat top loc1=0;loc2=0; for i=1:size(UMF,1) for j=1:1:size(UMF,2) if UMF(i,j)>0.999 loc1=j; break; end end if j~=size(UMF,2) for j=size(UMF,2):−1:1 if UMF(i,j)>0.999 loc2=j; break; end end end if loc1~=0 UMF(i,loc1:loc2)=1; end end end %%%%%%%%%%%%%%% % FUZZY Secondary Factor Variation
function BSF=fuzzyvarSF(VarSF) BSF=zeros(length(VarSF),1); for i=2:length(VarSF) for j=1:14 if VarSF(i)<−6 BSF(i)=1; elseif VarSF(i)>=6 BSF(i)=14; elseif VarSF(i)>=(j−1)−6 && VarSF(i)<j−6 BSF(i)=j+1; end end end end
%%%%%%%%%%%%%%%%%%%%%%%%%%% %% MF Using DE
function [UMF,LMF,typeMF,rmse]=MFusingDE(A,close,x,sd,TestCP,TestA,FLRG,WBS,TestB) genmax=50; F=0.2; % Scale Factor Cr=0.9; % Cross−over probability NP=20; % no. of population members gen=1; %% Initialization Zmin=ones(1,length(sd))*0.1; Zmax=sd; Z=zeros(NP,length(sd)); for i=1:NP Z(i,:)=Zmin+rand*(Zmax−Zmin); end %% rmse=zeros(genmax,NP); while(gen<=genmax) disp(‘Gen=‘); disp(gen); %% Mutation V=zeros(NP,length(sd)); for i=1:NP j=datasample(1:NP,1); while j==i j=datasample(1:NP,1); end k=datasample(1:NP,1); while k==i || k==j k=datasample(1:NP,1); end l=datasample(1:NP,1); while l==i || l==j || l==k l=datasample(1:NP,1); end V(i,:)=Z(j,:)+F.*(Z(k,:)−Z(l,:)); for j=1:length(sd) % Ensuring V(i,j) is within Zmax and Zmin if V(i,j)<Zmin(j) V(i,j)=Zmin(j); elseif V(i,j)>Zmax(j) V(i,j)=Zmax(j); end end end %% Crossover U=zeros(NP,length(sd)); for i=1:NP for j=1:length(sd) if rand<=Cr U(i,j)=V(i,j); else U(i,j)=Z(i,j); end end end %% Selection for i=1:NP [UMFu,LMFu,typeMFu]=formingMF(A,close,x,U(i,:)); [UMFz,LMFz,typeMFz]=formingMF(A,close,x,Z(i,:)); if evalfit(x,TestCP,TestA,UMFu,LMFu,typeMFu,FLRG,WBS… ,TestB)<evalfit(x,TestCP,TestA,UMFz,LMFz,typeMFz… ,FLRG,WBS,TestB) Z(i,:)=U(i,:); end end %% Storing fitness over generations for i=1:NP [UMFz,LMFz,typeMFz]=formingMF(A,close,x,Z(i,:)); rmse(gen,i)=evalfit(x,TestCP,TestA,UMFz,LMFz,typeMFz… ,FLRG,WBS,TestB); end %% gen=gen+1; end [~,indx]=min(rmse(genmax,:)); [UMF,LMF,typeMF]=formingMF(A,close,x,Z(indx,:)); End %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Over All Variation
function VarSF=overallvar(VarMF,SF,e,n) % VarSF1=zeros(size(SF,1),n); for i=1:n for j=2:size(SF,1) VarSF1(j,i)=(SF(j,i)−SF(j−1,i))*100/SF(j−1,i); end end % % tempDiffer=zeros(e,n); for i=1:n for j=3:e tempDiffer(j,i)=abs(VarSF1(j−1,i)−VarMF(j)); end end DifferSF=sum(tempDiffer); % % WVSF=zeros(1,n); for i=1:n WVSF(i)=sum(DifferSF)/DifferSF(i); end % % WSF=zeros(1,n); for i=1:n WSF(i)=WVSF(i)/sum(WVSF); end % % VarSF=zeros(1,size(SF,1)); for i=1:size(SF,1) VarSF(i)=sum(VarSF1(i,:).*WSF); end % End %%%%%%%%%%%%%%%%%%%%%%%%%% %% Partitioning The Universe of Discourse(UOD)
function [A,B,Var,Au,Al]=partitioning(CP) %% A=zeros(length(CP),1); B=zeros(length(CP),1); Var=zeros(length(CP),1); %% templ=min(CP); tempu=max(CP);
if (roundn(templ,2)−templ)>0 Al=roundn(templ,2)−100; else Al=roundn(templ,2); end
% if (roundn(tempu,2)−tempu)>0 % Au=roundn(tempu,2); % else % Au=roundn(tempu,2)+200; % end Au=Al; while(1) Au=Au+200; if Au>tempu break; end end
nFS=(Au–Al)/200;
for i=1:length(CP) %partioning the close series of main factor for j=1:nFS if CP(i)>=(j−1)*200+Al && CP(i)<j*200+Al A(i,1)=j; break; end end end %% for i=2:length(CP) % finding the var series Var(i)=(CP(i)−CP(i−1))*100/CP(i−1); end
for i=2:length(Var) % partioning the var series for j=1:14 if Var(i)<−6 B(i)=1; elseif Var(i)>=6 B(i)=14; elseif Var(i)>=(j−1)−6 && Var(i)<j−6 B(i)=j+1; end end end end %%%%%%%%%%%%%%%%%%%%%% %% Plot RMSE with adaptation
function plotDE(rmse) figure,plot(1:size(rmse,1),min(rmse,[],2),’kx−’,’MarkerSize’,5); xlabel(‘Generations −−>‘); ylabel(‘RMSE −−>‘); title(‘Evolving parameters to minimize RMSE’); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Plot Forecasted Price
function plotforecasted(predicted,TestCP,RMSE) figure, subplot(3,2,[1 2 3 4]); plot(TestCP,’k*:’); hold on plot(predicted,’ko−’); ylabel(‘Close Price’); axis([0 length(TestCP)+5 min(min(predicted),min(TestCP))−1000 max(max(predicted),max(TestCP))+1000]); hold off legend(‘Actual’,’Predicted’,’location’,’SouthEast’); subplot(3,2,[5 6]); stem(TestCP−predicted,’ko−’); text(27,max(TestCP−predicted)+50,{‘RMSE=‘,RMSE}); axis([0 length(TestCP) min(TestCP−predicted) max(TestCP−predicted)+150]); xlabel(‘Testing days’); ylabel(‘Error’); end %%%%%%%%%%%%%%%%%%%%%% %% Plot FOU(Foot Print of Uncertainity)
function plotFOU(UMF,LMF,x) for i=1:size(UMF,1) figure,shadedplot(x,LMF(i,:),UMF(i,:),[0.8 0.8 0.8]); xlabel(‘Close’); ylabel(‘Membership values’); title([‘FOU for fuzzy set A’, num2str(i)]); end end %%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Plot Partitions
function plotpartitions(close,A,Al,Au) numFS=unique(A); % converting the close series into a matrix FS=zeros(length(numFS),length(close)); for i=1:length(A) FS(A(i),i)=close(i); end
% Plot Input Close plot(close,’k*−’); hold on; for i=1:length(numFS) part=Al+(i−1)*200; plot([1 length(close)],[part part],’k:’); hold on; end hold off; axis([1 length(close) Al Au]); xlabel(‘Training days’); ylabel(‘Close’); title(‘Fuzzifying training data’); % End %%%%%%%%%%%%%%%%%%% %% PredictionFunction
function forecasted=predict(TestCP,A,Al,Au,UMF,LMF,typeMF,FLRG,WBS,B x=Al:Au−1; AB=horzcat(A,B); forecasted=zeros(length(TestCP),1); for i=2:size(AB,1) a=AB(i−1,1); b=AB(i−1,2); consequent=find(FLRG(a,:,b)); centroid=zeros(1,size(FLRG,1)); if isempty(consequent) if typeMF(a)==1 forecasted(i−1)=sum(x.*UMF(a,:))/sum(UMF(a,:)); elseif typeMF(a)==2 avgMF=(LMF(a,:)+UMF(a,:))/2; forecasted(i−1)=sum(x.*avgMF)/sum(avgMF); % forecasted(i−1)=KMmethod(LMF(a,:),UMF(a,:),x); end else %SF1=0;SF2=0;SF3=0; NF=0; for j=1:length(consequent) %Case1: a=T1FS, consequent(j)=T1FS if typeMF(a)==1 && typeMF(consequent(j))==1 predy=interp1(x,UMF(a,:),TestCP(i−1),’linear’,’extrap’); temp=ones(1,length(x))*predy; projMF=min(temp,UMF(consequent(j),:)); centroid(consequent(j))=sum(x.*projMF)/sum(projMF); %Case2: a=T1FS, consequent(j)=IT2FS elseif typeMF(a)==1 && typeMF(consequent(j))==2 predy=interp1(x,UMF(a,:),TestCP(i−1),’linear’,’extrap’); temp=ones(1,length(x))*predy; projUMF=min(temp,UMF(consequent(j),:)); projLMF=min(temp,LMF(consequent(j),:)); avgMF=(projLMF+projUMF)/2; centroid(consequent(j))=sum(x.*avgMF)/sum(avgMF); % centroid(consequent(j))=KMmethod(projLMF,projUMF,x); %Case3: a=IT2FS, consequent(j)=T1FS elseif typeMF(a)==2 && typeMF(consequent(j))==1 predU=interp1(x,UMF(a,:),TestCP(i−1),’linear’,’extrap’); % predL=interp1(x,LMF(a,:),TestCP(i−1),’linear’,’extrap’); temp=ones(1,length(x))*predU; projMF=min(temp,UMF(consequent(j),:)); centroid(consequent(j))=sum(x.*projMF)/sum(projMF); %Case4: a=IT2FS, consequent(j)=IT2FS elseif typeMF(a)==2 && typeMF(consequent(j))==2 predU=interp1(x,UMF(a,:),TestCP(i−1),’linear’,’extrap’); predL=interp1(x,LMF(a,:),TestCP(i−1),’linear’,’extrap’); tempU=ones(1,length(x))*predU; tempL=ones(1,length(x))*predL; projUMF=min(tempU,UMF(consequent(j),:)); projLMF=min(tempL,LMF(consequent(j),:)); avgMF=(projLMF+projUMF)/2; centroid(consequent(j))=sum(x.*avgMF)/sum(avgMF); % centroid(consequent(j))=KMmethod(projLMF,projUMF,x); end if j<a forecasted(i−1)=forecasted(i−1)+centroid(consequent(j))*FLRG(a,consequent(j),b)*WBS(b,1); elseif j==a forecasted(i−1)=forecasted(i−1)+centroid(consequent(j))*FLRG(a,consequent(j),b)*WBS(b,2); else forecasted(i−1)=forecasted(i−1)+centroid(consequent(j))*FLRG(a,consequent(j),b)*WBS(b,3); end end for j=1:length(consequent) if j<a NF=NF+FLRG(a,consequent(j),b)*WBS(b,1); elseif j==a NF=NF+FLRG(a,consequent(j),b)*WBS(b,2); else NF=NF+FLRG(a,consequent(j),b)*WBS(b,3); end end forecasted(i−1)=forecasted(i−1)/NF; end end end %%%%%%%%%%%%%%%%%%%%%%%%%%% %% Shade Plot
function [ha hb hc] = shadedplot(x, y1, y2, varargin) y = [y1; (y2−y1)]’; ha = area(x, y); set(ha(1), ’FaceColor’, ’none’) % this makes the bottom area invisible set(ha, ’LineStyle’, ’none’)
% plot the line edges hold on hb = plot(x, y1, ’k’, ’LineWidth’, 2); hc = plot(x, y2, ’k’, ’LineWidth’, 2); hold off
% set the line and area colors if they are specified switch length(varargin) case 0 case 1 set(ha(2), ’FaceColor’, varargin{1}) case 2 set(ha(2), ’FaceColor’, varargin{1}) set(hb, ’Color’, varargin{2}) set(hc, ’Color’, varargin{2}) otherwise end
% put the grid on top of the colored area set(gca, ’Layer’, ’top’) %%%%%%%%%%%%%%%%%%%%%%%%%%% %% Table IX
function FVG=tableIX(B_MF,B_SF) FVG=zeros(14,14); temp=zeros(14,14); for t=2:length(B_MF) Bx=B_MF(t); Bz=B_SF(t−1); temp(Bz,Bx)=1; FVG=FVG+temp; temp=zeros(14,14); end end %%%%%%%%%%%%%%%%%%%%%%% %% TableVI
function FLRG=tableVI(B,fromA,toA) endA=max([max(fromA),max(toA)]); FLRG=zeros(endA,endA,14);
temp=zeros(endA,endA,14); for i=1:length(B) temp(fromA(i),toA(i),B(i))=1; FLRG=FLRG+temp; temp=zeros(endA,endA,14); end end %%%%%%%%%%%%%%%%%%%%%% %% Table X
function BS=tableX(FVG) BS=zeros(14,3);
for i=1:14 if sum(FVG(i,:))~=0 BS(i,1)=sum(FVG(i,1:i−1))/sum(FVG(i,:)); BS(i,2)=FVG(i,i)/sum(FVG(i,:)); BS(i,3)=sum(FVG(i,i+1:14))/sum(FVG(i,:)); end end end %%%%%%%%%%%%%%%%%%
How to run the program with workspace construction in MATLAB is available in the url: http://computationalintelligence.net/fuzzytimeseries/howtorun.pdf.
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Konar, A., Bhattacharya, D. (2017). Self-adaptive Interval Type-2 Fuzzy Set Induced Stock Index Prediction. In: Time-Series Prediction and Applications. Intelligent Systems Reference Library, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-319-54597-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-54597-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-54596-7
Online ISBN: 978-3-319-54597-4
eBook Packages: EngineeringEngineering (R0)