Self-adaptive type-1/type-2 hybrid fuzzy reasoning techniques for two-factored stock index time-series prediction

Abstract

Considerable research outcomes on stock index time-series prediction using classical (type-1) fuzzy sets are available in the literature. However, type-1 fuzzy sets cannot fully capture the uncertainty involved in prediction because of its limited representation capability. This paper fills the void. Here, we propose four chronologically improved methods of time-series prediction using interval type-2 fuzzy sets. The first method is concerned with prediction of the (main factor) variation time-series using interval type-2 fuzzy reasoning. The second method considers secondary factor variation as an additional condition in the antecedent of the rules used for prediction. Another important aspect of the first and the second methods is non-uniform partitioning of the dynamic range of the time-series using evolutionary algorithm, so as to ensure that each partition includes at least one data point. The third method considers uniform partitioning without imposing any restriction on the number of data points in a partition. The partitions are here modeled by type-1 fuzzy sets, if there exists a single block of contiguous data, and by interval type-2 fuzzy sets, if there exists two or more blocks of contiguous data in a partition. The fourth method keeps provision for tuning of membership functions using recent data from the given time-series to influence the prediction results with the current trends. Experiments undertaken confirm that the fourth technique outperforms the first three techniques and also the existing techniques with respect to root-mean-square error metric.

Appendix

1.1 Differential evolution algorithm

The classical differential evolution (DE) algorithm (Halder et al. 2013; Das and Suganthan 2011; Das et al. 2009; Price et al. 2005) comprises four basic steps: (i) initialization of trial solutions, represented by parameter vectors, (ii) mutation, (iii) crossover or recombination and (iii) selection. The steps are briefly outlined below.

• (I) Initialization Initialize the generation number $t= 0$, and randomly initialize a population of NP trial solutions $P_{t} = \{ x_{1}(t), x_{2}(t), \ldots ,x_\mathrm{NP}\;(t)\}$ where $x_{i}\,(t) =\{x_{i,1} (t), x_{i,2} (t),\ldots ,x_{i,D}\; (t)\}$ is a d-dimensional parameter vector having the j-th component lying in a range: [$x_{i,j-\min }, x_{i,j-\max }$], for integer $i= [1, \hbox {NP}]$ and integer $j= [1, D]$.

• (II) While the stopping criterion is not reached, do Begin For $i=1$ to NP do Begin

  • (a) Mutation For each target vector $x_{i}(t)$ obtain a donor vector $V(t)=\{v_{{i,1}}(t), v_{{i,2}}(t),\ldots ,v_{{i, D}(t)}\}$ using the following transformation: $V_{i}(t)=x_{r1}(t)+F(x_{r2}(t)-x_{r3}(t))$ where $r1, r2\text { and }r3$ are randomly chosen distinct integers in [1, NP] and F is a scale factor in (0, 2).

  • (b) Crossover Obtain a trial vector $U_{i}(t)=\{u_{{i,1}}(t), u_{\mathrm{i,2}}\,(t),\ldots ,u_{{i,D}}\,(t)\}$ corresponding to the i-th target vector $x_{i}(t)$ by binomial crossover, using the following conditions:

    $$\begin{aligned} u_{i,j}(t)=v_{i,j} (t)\,\textit{if rand}\,\,(0,1)< cr, \text {a fixed crossover} \end{aligned}$$

    rate

    $$\begin{aligned} \qquad =x_{i,j}\,(t)\, { otherwise}. \end{aligned}$$

    where rand (0, 1) is a random number in (0, 1).

  • (c) Selection If $f(U_{i}(t))<f(x_{i}(t))$, Then $x_{i}(t+1)=U_{i}(t)$

    End if;

    End for;

  • (d) Increase t =t + 1.

    End while;

Two controlling parameters, crossover rate $C_{\mathrm{r}}$ and scale factor F, are introduced in the above algorithm. They are initialized once at the beginning of the algorithm. Here, $C_{\mathrm{r}}$ that lies in (0, 2) has been set as 1.2 in the program. Similarly, to satisfy stability criteria for convergence, F usually lies in (0, 2) and is fixed as 0.7 in our program.

The stopping criterion in DE is defined in many alternative ways. A few commonly used techniques are (i) setting the upper bound of the program iterations, (ii) fixing an error bound to the best-fit member/average fitness over each iterations in the population, (iii) any one of (i) and (ii) which occurs earlier (Konar and Bhattacharya 2017).