FINANCIAL FORECASTING: Comparison Of ARIMA, FFNN And SVR

FINANCIAL 

FORECASTING: 

Comparison Of 

ARIMA, FFNN 

And SVR 

Introduction 

Experiments And 

Results For 

ARIMA 


Results For FFNN 


Results For SVR 

Comparison 

Conclusion And 

Future Work 

APPENDIX 


FFNN 

SVR 

References 

FINANCIAL FORECASTING: 

Comparison Of ARIMA, FFNN And SVR 

Ashish Gajanan Lahane (05329R01) 

Under the guidance of 

Prof. Bernard L. Menezes 

July 8, 2008

Outline 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

1 Introduction 

2 Experiments And Results For ARIMA 

3 Experiments And Results For FFNN 

4 Experiments And Results For SVR 

5 Comparison 

6 Conclusion And Future Work 

7 APPENDIX 




8 References






Introduction 


Results For 






INTRODUCTION 



Future Work 

APPENDIX 




References

Abbreviations 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Feed Forward Neural Networks 

AutoRegressive Integrated Moving Average 

Support Vector Regression 

SVR using Polynomial Kernel 

SVR using Radial Basis Function Kernel 

AutoCorrelation Function 

Partial AutoCorrelation function 




SVR POLY 

SVR RBF 

ACF 

PACF

Literature Survey 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Financial time series: nonlinear, one of noisiest and most 

difficult signals to forecast 

For financial forecasting: 

FFNN better ARIMA[16, 7, 12, 13, 2, 11] 

SVR better than ARIMA[6] 

FFNN better than SVR[17, 6] 

SVR better than FFNN[4, 5]

Problem Definition 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

To compare three models: 




for one-day-ahead forecasting performance on three important indices 

in Indian stock market: 

BSE Sensex 

CNX IT 

S&P CNX Nifty

Error Measures 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Mean Absolute Percent Error (MAPE) 

MAPE = 100 × 

Mean Absolute Error (MAE) 

N∑ 

|E t | 

t=1 

MAE = 

N 

where Y t ,E t represent desired outputs and corresponding errors 

at t=1,2,...,N respectively. 

Directional Symmetry (DS) 

DS = d correct 

N × 100 

where d correct = number of times the forecaster predicted the 

direction of the series right 

N∑ 

t=1 

N 

|E t| 

Y t

Background 






Introduction 

ARIMA go 


Results For 






FFNN go 



Future Work 

APPENDIX 




References 

SVR go






Introduction 


Results For 








Future Work 

EXPERIMENTS AND 

RESULTS FOR ARIMA 

APPENDIX 




References

ACF PACF with d = 0 For BSE Sensex 






Introduction 


Results For 








Future Work 

APPENDIX 




References







Introduction 


Results For 








Future Work 

APPENDIX 




References







Introduction 


Results For 








Future Work 

APPENDIX 




References

Model Identification 






More than one differencing not needed 

Introduction 


Results For 








Future Work 

APPENDIX 




References 

Summary 

Suitable models are 

AR(1) with φ 1 = 1 (i.e. ARIMA(0,1,0)) 

AR(2) 

ARIMA(p, 1, q) with 0 ≤ p, q ≤ 2 

MA(q) (i.e. ARIMA(0,0,q)) models unsuitable 

0 ≤ p, d, q ≤ 2 should be tried

Results For ARIMA 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

BSE Sensex 

p d q MAPE DS 

1 1 1 1.351 79.49 

1 0 1 1.362 74.87 

1 1 0 1.406 77.95 

2 0 1 1.413 74.36 

2 0 0 1.424 77.95 

2 1 0 1.425 78.46 

1 0 0 1.428 76.41 

0 1 2 1.429 74.36 

1 0 2 1.430 76.41 

0 1 0 1.432 75.38 

0 1 1 1.435 76.41 

1 1 2 1.442 76.92 

2 1 1 1.468 74.87 

2 0 2 1.472 77.44 

2 1 2 1.661 74.36 

0 2 0 2.100 64.10 

1 2 0 2.754 65.13 

2 2 0 3.188 68.21 

2 2 2 20.078 58.46 

1 2 2 20.094 58.46 

0 2 2 21.819 55.38 

2 2 1 22.268 52.82 

1 2 1 22.776 55.38 

0 2 1 23.246 52.31 

0 0 2 24.551 50.26 

0 0 1 53.822 46.15 

0 0 0 65.824 41.03 

CNX IT 

p d q MAPE DS 

1 1 1 1.479 79.77 

1 0 1 1.480 79.10 

2 0 1 1.481 78.32 

1 0 0 1.482 77.00 

2 0 0 1.483 76.58 

2 1 0 1.500 75.89 

0 1 2 1.532 74.98 

1 1 2 1.544 76.85 

1 0 2 1.560 73.07 

0 1 0 1.585 75.00 

0 1 1 1.588 72.67 

1 1 0 1.598 72.83 

2 1 1 1.606 72.92 

2 0 2 1.612 71.11 

2 1 2 1.623 71.91 

0 2 0 2.166 71.12 

1 2 0 3.015 69.56 

2 2 0 3.861 68.77 

2 2 2 25.613 55.65 

1 2 2 42.084 50.62 

0 2 2 64.256 50.62 

2 2 1 77.953 50.62 

1 2 1 93.480 50.62 

0 2 1 93.493 50.62 

0 0 2 175.95 50.62 

0 0 1 183.54 50.62 

0 0 0 243.51 50.62 

S&P CNX Nifty 

p d q MAPE DS 

1 1 1 1.755 79.24 

2 0 2 1.813 77.08 

1 1 2 1.827 78.13 

1 1 0 1.833 79.17 

1 0 0 1.842 78.13 

2 0 0 1.856 77.08 

2 1 1 1.856 75.00 

2 1 2 1.862 77.08 

0 1 1 1.871 76.04 

0 1 2 1.873 75.00 

2 1 0 1.873 76.04 

1 0 2 1.888 77.08 

1 0 1 1.890 72.92 

0 1 0 1.901 77.08 

2 0 1 1.908 77.08 

0 2 0 2.477 71.88 

1 2 0 3.292 67.71 

2 2 0 3.695 66.67 

0 0 1 42.688 43.75 

1 2 2 43.182 52.08 

2 2 2 44.405 52.08 

0 2 1 45.573 52.08 

1 2 1 47.657 52.08 

2 2 1 53.251 52.08 

0 0 0 72.477 50.00 

0 0 2 78.234 47.42 

0 2 2 99.433 47.90 

Conclusion 

ARIMA(1,1,1) outperformed other ARIMA models






Introduction 


Results For 








Future Work 


RESULTS FOR FFNN 

APPENDIX 




References

FFNN Parameters 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

FFNN Parameter 

Values Tried 

Hidden layers 1,2 

Trans func in hidden layer 

tanh, sigmoid 

Trans func in output layer 

tanh, sigmoid, linear 

[2 1],[3 1],[5 1],[7 1],[10 1],[15 1],[20 1], 

[40 1],[3 2 1],[5 2 1],[5 3 1],[7 2 1], 

Neurons per hidden layer [7 3 1],[7 5 1],[10 2 1],[10 3 1], 

[10 5 1],[10 7 1],[15 2 1],[15 3 1], 

[15 5 1],[15 7 1],[15 10 1] 

Window size 5,7,10,20,40,60,80 

Neurons IN output Layer 1 

6440 FFNN models per series







Introduction 


Results For 








Future Work 

APPENDIX 




References 

Summary 

BSE 

S&P CNX 

Sensex CNX IT Nifty 

Min MAPE 1.1180 1.1863 1.0834 

Avg MAPE 13.0838 9.4293 10.8195 

BSE 

S&P CNX 

Sensex CNX IT Nifty 

Max DS 66.6667 62.1429 66.9524 

Avg DS 44.9146 47.8497 50.0907 

Very good performance for value forecasting 

Visible effect of random weight initialisation

Best FFNN Models 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Observation 

Number Of 

Transfer 

Min Win Neurons Per function 

MAPE MAE DS Size Layer Per Layer 

BSE Sensex 1.1180 182.78 53.85 5 [7 2 1] [tanh tanh lin] 

CNX IT 1.1863 56.48 58.13 40 [10 1] [sigmoid lin] 

S&P CNX Nifty 1.0834 62.53 66.67 80 [15 2 1] [sigmoid sigmoid lin] 

Neurons Transfer 

Max Window Per function Per 

DS MAPE MAE Size Layer Layer 

BSE Sensex 66.6667 3.3060 606.3522 80 [20 1] [tanh tanh] 

CNX IT 62.1429 1.2795 61.0987 60 [3 1] [sigmoid lin] 

S&P CNX Nifty 66.9524 1.5371 88.4242 80 [2 1] [tanh lin] 

linear transfer function in last layer

Transfer Function In Last Layer 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

# of models with MAPE60 

and linear trans funct in 

Series # of models with DS>60 last layer % 

BSE Sensex 42 21 50.00 

CNX IT 9 7 77.78 

S&P CNX Nifty 134 74 55.22 

Conclusion 

linear transfer function in last layer gave good results






Introduction 


Results For 








Future Work 


RESULTS FOR SVR 

APPENDIX 




References

SVR POLY Vs SVR RBF Min MAPE 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Avg MAPE 

SVR POLY SVR RBF SVR POLY SVR RBF 

BSE Sensex 1.4089 3.3324 2.5730 5.5208 

CNX IT 1.4695 3.8992 2.4327 6.9512 

S&P CNX Nifty 1.7230 1.7726 2.5094 3.5316 

Max DS 

Avg DS 

SVR POLY SVR RBF SVR POLY SVR RBF 

BSE Sensex 58.5492 50.1233 49.9497 40.2311 

CNX IT 55.2343 48.2390 49.9724 41.0412 

S&P CNX Nifty 61.9048 51.6624 54.1753 43.7640 

Conclusion 

SVR POLY outperforms SVR RBF

SVR POLY Parameter Analysis 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

BSE Sensex 

MAPE MAE DS C ε d γ r 

1.4089 234.1536 58.5492 249748.22 0.0213 2 1.2470 4.5355 

1.4880 257.8835 57.1429 792599.26 0.0106 2 2.5047 3.9518 

1.7269 294.6667 56.6667 402023.38 0.0184 2 2.2115 4.4671 

2.8427 499.2695 51.2500 296975.62 0.0119 3 0.5706 2.4815 

2.9433 489.9205 40.5128 936002.69 0.0099 2 2.8285 4.3638 

5.8682 992.6019 40.5263 575563.83 0.0075 4 3.7899 3.1718 

CNX IT 


1.4695 70.6894 55.2343 882549.15 0.0099 2 0.4234 1.9312 

1.4872 67.1975 52.4432 400000.00 0.0134 2 2.1210 3.3123 

1.5261 68.9893 54.9010 498213.28 0.0124 2 0.2328 2.7086 

3.7557 184.6057 43.4358 991231.61 0.0091 3 3.9946 2.4384 

3.8041 178.3055 44.1263 435563.83 0.0155 3 2.6472 4.2321 

S&P CXT Nifty 


1.7230 98.6894 61.9048 325315.10 0.0153 2 4.7430 0.9691 

1.7883 99.6453 57.1429 474668.02 0.0212 2 1.5805 1.4250 

2.2224 123.0545 58.3333 162817.77 0.0208 4 3.2448 3.2247 

2.4657 140.2473 45.6790 524259.16 0.0187 4 0.1475 3.9702 

3.5384 202.6957 57.4468 333194.98 0.0201 4 1.4261 1.3642 

3.6755 212.2441 56.0976 606613.45 0.0094 2 2.8840 4.4192 

Observations 

Best fit: 

polynomial 

kernel of 

degree 

d = 2 

Higher 

values of C 

Very small 

values of ε






Introduction 


Results For 






COMPARISON 



Future Work 

APPENDIX 




References

Comparison On Value Forecasting 






Introduction 

MAPE ARIMA FFNN SVR 

BSE Sensex 1.3513 1.1180 1.4089 

CNX IT 1.4788 1.1863 1.4695 

S&P CNX Nifty 1.7549 1.0834 1.7230 


Results For 








Future Work 

APPENDIX 




References 

Observation 

FFNN outperforms ARIMA and SVR

Comparison On Directional Forecasting 






Introduction 

DS Naive Heuristic ARIMA FFNN SVM 

BSE Sensex 54.5455 79.4872 66.6667 58.5492 

CNX IT 52.5253 79.7718 62.1429 55.2343 

S&P CNX Nifty 51.0204 79.2417 66.9524 61.9048 


Results For 








Future Work 

APPENDIX 




References 

Observation 

ARIMA outperforms FFNN and SVR






Introduction 


Results For 








Future Work 

CONCLUSION AND 

FUTURE WORK 

APPENDIX 




References

Conclusion 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

ARIMA: ARIMA(1,1,1) is better fit than other ARIMA models 

FFNN: Models with linear transfer function in the last layer 

perform better 

SVR: SVR models with polynomial kernels of degree 2 are 

better fit than other SVR models 

Value Forecasting: FFNN models perform better than ARIMA 

and SVR for these three series 

Directional Forecasting: ARIMA models perform better than 

FFNN and SVR for these three series

Future Work 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Addition of technical indicators such as Moving Average (MA), 

Moving Average Convergence/Divergence (MACD), On Balance 

Volume (OBV) etc[14, 18, 8] 

Addition of fundamental factors such as Price Earnings ratio 

(PE ratio), rate of change of company sales, Price Dividend 

ratio (PD ratio), rate of change of profit margin etc[1] 

N-days-ahead forecasting 

Forecasting on several other series 

Use of combining methods such as TopK, Best, DTopK, AFTER 

etc[19, 15]






Introduction 


Results For 






THANKS! 



Future Work 

APPENDIX 




References






Introduction 


Results For 






APPENDIX 



Future Work 

APPENDIX 




References






Introduction 


Results For 








Future Work 


[appendix] 

APPENDIX 




References

ARIMA Equation 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

ARIMA(p, d, q) model is given by: 

ϕ∇ d X t = θε t 

p∑ 

where:- AR part: ϕ = 1 − ϕ i L i , 

MA part: θ = 1 + 

i=1 

q∑ 

θ j L j 

j=1 

I(difference) part: ∇ d = (1 − L 1 ) d 

Here X t−1 , X t−2 , ......X 2 , X 1 is the time series data 

L is lag operator, i.e. L i X t = X t−i , 

ϕ i and θ j are the model parameters, 

ε t is a white noise process with zero mean and variance σ 2 .

ARIMA Methodology 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

1 Model Identification using ACF and PACF plots or other 

methods 

2 Model Estimation using methods such as likelihood or Bayesian 

3 Forecasting 

For further reading, please refer [3]. 

go back






Introduction 


Results For 








Future Work 


[appendix] 

APPENDIX 




References

A Simple Neuron 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

n∑ 

y = φ( w i x i − θ) 

i=1

Transfer Functions 






Introduction 


Results For 








Future Work 

Linear 

Equation 

a = n 

range = range of 

n 

graphical 

representation: 

Sigmoid 

Equation 

a = 

1 

1 + e −n 

range = [0,+1] 

graphical 


Tanh 

Equation 

a = tanh(n) 

range = [-1,+1] 

graphical 


APPENDIX 




References







Introduction 


Results For 








Future Work 

APPENDIX 




References 

go back






Introduction 


Results For 








Future Work 


[appendix] 

APPENDIX 




References

Basic Principle 






Example 1: Classification 

Introduction 


Results For 








Future Work 

APPENDIX 




References 

Example 2: Regression 

f (m 1 , m 2 , r) = C m 1m 2 

r 2 

log ⇓ 

g(x, y, z) = ln(f (m 1 , m 2 , r)) = ln C + ln m 1 + ln m 2 − 2 ln r = c + x + y − 2z

SVM:Maximal Separating Hyperplane Classifier 






Introduction 


Results For 








Future Work 

APPENDIX 




References

SVM(1) 






Primal Form 

Introduction 


Results For 








Future Work 

APPENDIX 




References 

Dual Form 

1 

minimize: 

2 ||w||2 , 

subject to: c i (w · x i − b) ≥ 1, ∀ : 1 ≤ i ≤ n 

maximize: 

n∑ 

α i − 1 ∑ 

α i α j c i c j 〈x i · x j 〉 

2 

i=1 

i,j 

subject to: α i ≥ 0 and 

n∑ 

α i c i = 0 

i=1

SVM(2) 






Solution To Dual Problem 

From the Solution we get 

Introduction 


Results For 








Future Work 

APPENDIX 




References 

Soft Margin 

f = 

w = ∑ i 

α i c i x i 

n∑ 

α i c i 〈x i · x j 〉 − b = ∑ 

α i c i 〈x i · x j 〉 − b 

i=1 

i∈SV 

1 

minimize: 

2 ||w||2 + C ∑ ξ i , 

i 

subject to: c i (w · x i − b) ≥ 1 − ξ i , ∀ : 1 ≤ i ≤ n.







Introduction 


Results For 








Future Work 

APPENDIX 




References 

subject to: 

minimize: 

⎧ 

⎨ 

⎩ 

1 

2 ||w||2 + C 

l∑ 

(ξ i + ξi ∗ ) 

i=1 

y i − 〈w i , x i 〉 − b ≤ ε + ξ i 

〈w i , x i 〉 + b − y i ≤ ε + ξ ∗ i 

ξ i + ξ ∗ i ≥ 0

Kernels 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Radial Basis Function 

Equation 

k(x, x ′ ) = e (−γ|x−x′ | 2) , 

for γ > 0 

graphical representation: 

Polynomial Function 

Equation 

k(x, x ′ ) = (γ(x · x ′ ) + r) d , 

for γ > 0, r > 0 

graphical representation: 

go back






Introduction 


Results For 






REFERENCES 



Future Work 

APPENDIX 




References

References I 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

A. Atiya, N. Talaat, and S. Shaheen. 

An efficient stock market forecasting model using neural networks. 

Neural Networks,1997., International Conference on, 4:2112–2115 vol.4, Jun 1997. 

E. Michael Azoff. 

Neural Network Time Series forecasting of Financial Markets. 

John Wiley & Sons, 1994. 

G. E. P. Box, G. M. Jenkins, and G. C. Reinsel. 

Time Series Analysis, Forecasting, and Control. 

Prentice-Hall, Englewood Cliffs, New Jersey, third edition, 1994. 

Lijuan Cao and Francis E.H Tay. 

Financial forecasting using support vector machines. 

Neural Computing & Applications, 10(2):184–192, May 2001. 

Chon Lung Chai. 

Finding kernel function for stock market prediction with support vector regression. 

Technical report, Universiti Teknologi Malaysia, 2006. 

Wun-Hua Chen, Jen-Ying Shih, and Soushan Wu. 

Comparison of support-vector machines and back propagation neural networks in forecasting the six major asian stock 

markets. 

International Journal of Electronic Finance, 1(1):49–67, January 2006. 

Wei Cheng, Lorry Wagner, and Chien-Hua Lin. 

Forecasting the 30-year u.s. treasury bond with a system of neural networks.

References II 






Introduction 


Results For 








Future Work 

APPENDIX 




References 

Wei Cheng, Lorry Wagner, and Chien-Hua Lin. 

Forecasting the 30-year u.s. treasury bond with a system of neural networks. 

Shirley Gregor Feng Lin, Xing Huo Yu and Richard Irons. 

Time series forecasting with neural networks. 

Complexity International, 2, April 1995. 

Boyd M.S. Kermanshahi B. Kohzadi, N. and I. Kaastra. 

A comparison of artificial neural network and time series model for forecasting commodity price. 

Neurocomputing, 10:169–181, 1996. 

M. Kumar and M. Thenmozhi. 

Forecasting nifty index futures returns using neural network and arima models. 

Financial Engineering and Applications, 437, 2004. 

Jason E. Kutsurelis. 

Forecasting financial markets using neural networks: An analysis of methods and accuracy. 

Mohan Neeraj, Jha Pankaj, Laha Arnab Kumar, and Dutta Goutam. 

Artificial neural network models for forecasting stock price index in bombay stock exchange. 

IIMA Working Papers 2005-10-01, Indian Institute of Management Ahmedabad, Research and Publication Department, 

October 2005. 

available at http://ideas.repec.org/p/iim/iimawp/2005-10-01.html. 

Mohan Neeraj, Jha Pankaj, Laha Arnab Kumar, and Dutta Goutam. 

Artificial neural network models for forecasting stock price index in bombay stock exchange. 

IIMA Working Papers 2005-10-01, Indian Institute of Management Ahmedabad, Research and Publication Department, 

October 2005.

References III 






Introduction 


Results For 








Future Work 

Abhishek Seth. 

On using a multitude of time series forecasting models. 

Mtech thesis, Kanwal Rekhi School of Information Technology, IIT Bombay, 2006. 

Talaat N. Shaheen S. Atiya A. 

An efficient stock market forecasting model using neural networks. 

1997. 

Theodore B. Trafalis and Huseyin Ince. 

Support vector machine for regression and applications to financial forecasting. 

ijcnn, 06:6348, 2000. 

Chi-Cheong Chris Wong, Man-Chung Chan, and Chi-Chung Lam. 

Financial time series forecasting by neural network using conjugate gradient learning algorithm and multiple linear 

regression weight initialization. 

Computing in Economics and Finance 2000 61, Society for Computational Economics, July 2000. 

Y. Yang and H. Zou. 

Combining time series models for forecasting, 2002. 

APPENDIX 




References

FINANCIAL FORECASTING: Comparison Of ARIMA, FFNN And SVR

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?