Support vector regression with modified firefly algorithm for stock price forecasting

Abstract

The support vector regression (SVR) has been employed to deal with stock price forecasting problems. However, the selection of appropriate kernel parameters is crucial to obtaining satisfactory forecasting performance. This paper proposes a novel approach for forecasting stock prices by combining the SVR with the firefly algorithm (FA). The proposed forecasting model has two stages. In the first stage, to enhance the global convergence speed, a modified version of the FA, which is termed the MFA, is developed in which the dynamic adjustment strategy and the opposition-based chaotic strategy are introduced. In the second stage, a hybrid SVR model is proposed and combined with the MFA for stock price forecasting, in which the MFA is used to optimize the SVR parameters. Finally, comparative experiments are conducted to show the applicability and superiority of the proposed methods. Experimental results show the following: (1) Compared with other algorithms, the proposed MFA algorithm possesses superior performance, and (2) The proposed MFA-SVR prediction procedure can be considered as a feasible and effective tool for forecasting stock prices.

Appendix

Proof of Theorem

Assume that, among N fireflies, their brightness are in sequential order. Namely, firefly 1 is the brightest, and firefly N is the worst.

In the following, we will prove the theorem by induction.

STEP 1: Suppose that there are two fireflies. Then we have

$$\begin{array}{@{}rcl@{}} &&\frac{r_{1, 2}(t + 1)}{r_{1, 2}(t)}\\&=&\frac{||x_{1}(t) - x_{2}(t) - \beta_{0}e^{-{\gamma} {r_{1,2}^{2}}}(x_{1}(t) - x_{2}(t)) - \alpha(t)({\mu} - \frac{1}{2})||}{||x_{1}(t)-x_{2}(t)||}\\ &\leq&1-\beta_{0}e^{-\gamma r_{1,2}^{2}}+\frac{||\alpha(t)(\mu-\frac{1}{2}||)}{||x_{1}(t)-x_{2}(t)||}. \end{array} $$

(18)

Note that

$$ {\lim_{t\to+\infty}}r_{1, 2}(t)={\lim_{t \to +\infty}}{{\prod}_{1}^{t}}\frac{r_{1,2}(t)}{r_{1,2}(t-1)}*r_{1,2}(0). $$

(19)

Then, according to (18) and (19), we have

$$\begin{array}{@{}rcl@{}} 0\leq {\lim_{t \to +\infty}}r_{1,2}(t)&\leq&{\lim_{t \to +\infty}}{{\prod}_{1}^{t}} \left( 1-\beta_{0}e^{-\gamma r_{1, 2}^{2}}\right.\\ &&+ \left. \frac{\alpha(t)(\mu-\frac{1}{2})}{||x_{1}(t)-x_{2}(t)||}\right)*r_{1, 2}(0) \end{array} $$

(20)

Because 0 < β₀ < 1, we have $0<1-\beta _{0}e^{-\gamma r_{1, 2}^{2}}<1$. Moreover, we know that $\lim \limits _{t\rightarrow +\infty }\alpha (t)= 0$. Thus, according to (20), we have

$$ 0\leq{\lim_{t\to+\infty}}r_{1,2}(t)\leq 0. $$

(21)

Furthermore, we have ${\lim \limits _{t\to +\infty }}r_{1,2}(t)= 0$. That’s to say,

$$ \lim_{t\to +\infty}x_{1}(t)=\lim_{t\to +\infty}x_{2}(t). $$

(22)

STEP 2: Let k ≥ 2 be an integer. Suppose that k fireflies are convergent. Namely, for some x_i, x_j, i,j = 1, 2,…,k, and for arbitrary 𝜖 > 0, there exists T > 0, such that t > T, implies r_ij < 𝜖.

Next, we will prove k + 1 fireflies are also convergent. Here, the reduction to absurdity is employed.

Suppose that k + 1 fireflies are not convergent, i.e., for arbitrary T > 0, there exists t > T such that r_1,k+ 1(t) > 𝜖. If the firefly k + 1 is attracted by firefly s(1 ≤ s ≤ k), then we have

$$\begin{array}{@{}rcl@{}} \frac{r_{1, k + 1}(t + 1)}{r_{1, k + 1}(t)}&=&\frac{||x_{1}(t)-x_{k + 1}(t)-\beta_{0}e^{-\gamma r_{s, k + 1}^{2}}\left( x_{s}(t)-x_{k + 1}(t)\right)-\alpha(t)(\mu-\frac{1}{2})||}{||x_{1}(t)-x_{k + 1}(t)||}\\ &=& \frac{||x_{1}(t)-x_{k + 1}(t)-\beta_{0}e^{-\gamma r_{s,k + 1}^{2}}\left( x_{1}(t)-x_{k + 1}(t)+x_{s}(t)-x_{1}(t)\right)-\alpha(t)(\mu-\frac{1}{2})||} {||x_{1}{(t)}-x_{k + 1}(t)||}\\ &=&\frac{||(1-\beta_{0}e^{-\gamma r_{s,k + 1}^{2}})\left( x_{1}(t)-x_{k + 1}(t)\right)-\beta_{0}e^{-\gamma r_{s, k + 1}^{2}}\left( x_{s}(t)-x_{1}(t)\right)-\alpha(t)(\mu-\frac{1}{2})||} {||x_{1}{(t)}-x_{k + 1}(t)||}\\ &\leq& 1-\beta_{0}e^{-\gamma r_{s,k + 1}^{2}}+\beta_{0}e^{-\gamma r_{s,k + 1}^{2}}\frac{||x_{s}{(t)}-x_{1}(t)||}{||x_{1}{(t)}-x_{k + 1}(t)||}+\frac{||\alpha(t) (\mu-\frac{1}{2})||}{||x_{1}{(t)}-x_{k + 1}(t)||}\\ &=&1-\beta_{0}e^{-\gamma r_{s,k + 1}^{2}}\left( 1-\frac{||x_{s}{(t)}-x_{1}(t)||}{||x_{1}{(t)}-x_{k + 1}(t)||}\right)+\frac{||\alpha(t) (\mu-\frac{1}{2})||}{||x_{1}{(t)}-x_{k + 1}(t)||} \end{array} $$

(23)

Note that

$$ {\lim_{t\to +\infty}}r_{1, k + 1}(t)={\lim_{t \to +\infty}}{{\prod}_{1}^{t}}\frac{r_{1,k + 1}(t)}{r_{1,k + 1}(t-1)}*r_{1,k + 1}(0) $$

(24)

Furthermore, from (23), and (24), we have

$$ 0\leq{\lim_{t\to+\infty}}r_{1,k + 1}(t)\leq{\lim_{t\to +\infty}}{{\prod}_{1}^{t}}\left\{1-\beta_{0}e^{-\gamma r_{s,k + 1}^{2}}\left( 1-\frac{||x_{s}{(t)}-x_{1}(t)||}{||x_{1}{(t)}-x_{k + 1}(t)||}\right)+\frac{||\alpha(t) (\mu-\frac{1}{2})||}{||x_{1}{(t)}-x_{k + 1}(t)||}\right\}*r_{1k + 1}(0) $$

(25)

From assumptions, we know that k fireflies are convergent, i.e., 0 < ||x₁(t) − x_s(t)|| < 𝜖, and k + 1th firefly is not convergent, namely, ||x₁(t) − x_k+ 1(t)|| > 𝜖. Namely, 0 < ||x₁(t) − x_s(t)|| < 𝜖 < ||x₁(t) − x_k+ 1(t)||. Moreover, we know that $0<1-\beta _{0}e^{-\gamma r_{s,k + 1}^{2}}<1$. Thus, we have

$$\begin{array}{@{}rcl@{}} & &{\lim_{t\to +\infty}}\left\{1-\beta_{0}e^{-\gamma r_{s,k + 1}^{2}}\left( 1-\frac{||x_{s}{(t)}-x_{1}(t)||}{||x_{1}{(t)}-x_{k + 1}(t)||}\right)\right \}\\ &&<1 \end{array} $$

(26)

Therefore, according to (26), and $\lim \limits _{t\rightarrow +\infty }\alpha (t)= 0$, we have

$$\begin{array}{@{}rcl@{}} &&{\lim_{t \to +\infty}}{{\prod}_{1}^{t}}\left\{ 1-\beta_{0}e^{-\gamma r_{s,k + 1}^{2}}\left( 1-\frac{||x_{s}{(t)}-x_{1}(t)||}{||x_{1}{(t)}-x_{k + 1}(t)||}\right)\right.\\ &&+ \left. \frac{||\alpha(t) (\mu-\frac{1}{2})||}{||x_{1}{(t)}-x_{k + 1}(t)||}\right\}*r_{1k + 1}(0)= 0 \end{array} $$

(27)

Furthermore, from (25) and (27), we have ${\lim \limits _{t \to +\infty }}r_{1, k + 1}(t)$ = 0. That’s to say, k + 1th firefly is convergent, i.e., for some 𝜖 > 0, there exists T > 0, such that t > T, implies r_1k+ 1(t) < 𝜖, which is contrary with assumption. Therefore, we have k + 1 fireflies are convergent.

Based on the STEP 1 and STEP 2, the theorem is proved. □