Financial time series forecasting using support vector machines Author: Kyoung-jae Kim 2003 Elsevier B.V.

Outline Introduction to SVM Introduction to datasets Experimental settings Analysis of experimental results

Linear separability – In general, two groups are linearly separable in n- dimensional space if they can be separated by an (n − 1)-dimensional hyperplane.

Support Vector Machines Maximum-margin hyperplane maximum-margin hyperplane

Formalization Training data Hyperplane Parallel bounding hyperplanes

Objective Minimize (in w, b) ||w|| subject to (for any i=1, …, n)

A 2-D case In 2-D: – Training data: xixi cici x+2y+1=-1 -2x+2y+1=1 -2x+2y+1=0 w= b=-1 margin=sqrt(2)/2

Not linear separable No hyperplane can separate the two groups

Soft Margin Choose a hyperplane that splits the examples as cleanly as possible Still maximizing the distance to the nearest cleanly split examples Introduce an error cost C d*C

Higher dimensions Separation might be easier

Kernel Trick Build maximal margin hyperplanes in high- dimenisonal feature space depends on inner product: more cost Use a kernel function that lives in low dimensions, but behaves like an inner product in high dimensions

Kernels Polynomial – K(p, q) = (pq + c) d Radial basis function – K(p, q) = exp(-γ||p-q|| 2 ) Gaussian radial basis – K(p, q) = exp(-||p-q|| 2 /2δ 2 )

Tuning parameters Error weight – C Kernel parameters – δ 2 – d – c 0

Underfitting & Overfitting Underfitting Overfitting High generalization ability

Datasets Input variables – 12 technical indicators Target attribute – Korea composite stock price index (KOSPI) 2928 trading days – 80% for training, 20% for holdout

Settings (1/3) SVM – kernels polynomial kernel Gaussian radial basis function – δ 2 – error cost C

Settings (2/3) BP-Network – layers 3 – number of hidden nodes 6, 12, 24 – learning epochs per training example 50, 100, 200 – learning rate 0.1 – momentum 0.1 – input nodes 12

Settings (3/3) Case-Based Reasoning – k-NN k = 1, 2, 3, 4, 5 – distance evaluation Euclidean distance

Experimental results The results of SVMs with various C where δ 2 is fixed at 25 Too small C underfitting * Too large C overfitting * * F.E.H. Tay, L. Cao, Application of support vector machines in -nancial time series forecasting, Omega 29 (2001) 309–317

Experimental results The results of SVMs with various δ 2 where C is fixed at 78 Small value of δ 2 overfitting * Large value of δ 2 underfitting * * F.E.H. Tay, L. Cao, Application of support vector machines in -nancial time series forecasting, Omega 29 (2001) 309–317

Experimental results and conclusion SVM outperformes BPN and CBR SVM minimizes structural risk SVM provides a promising alternative for financial time-series forecasting Issues – parameter tuning