A TOPSIS-ELM framework for stock index price movement prediction

Abstract

In recent years Extreme Learning Machine (ELM) has gained the interest of various researchers due to its superior generalization and approximation capability. The network architecture and type of activation functions are the two important factors that influence the performance of an ELM. Hence in this study, a Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) oriented multi-criteria decision making (MCDM) framework is suggested for analyzing various ELM models developed with distinct activation functions with respect to sixteen evaluation criteria. Evaluating the performance of the ELM with respect to multiple criteria instead of single criterion can help in designing a more robust network. The proposed framework is used as a binary classification system for pursuing the problem of stock index price movement prediction. The model is empirically evaluated by using historical data of three stock indices such as BSE SENSEX, S&P 500 and NIFTY 50. The empirical study has disclosed promising results by evaluating ELM with different activation functions as well as multiple criteria.

Keywords

ELM MCDM TOPSIS time series forecasting

1. Introduction

In recent years, financial markets are becoming populated with companies and investors in an unprecedented speed. With increase in technology and availability of data, investors are becoming more cognizant about the stock market and its working principle. This may be a main reason of adequate involvement of them in different investments related to stock market. Equity, stocks, bonds, currency price, gold price etc. are some of the products or commodities that the investors buy and sell on a daily basis in stock market. Although people are aware of high future profit of these investments, the risk of investment cannot be eliminated. Hence predicting the upcoming prices and its movement are extremely important to reduce this risk. Again the stock market is very volatile and non-linear in nature, due to its dependency upon external factors like inflation, deflation, gold price, economic policies of a country, current trends etc. [1]. So the predictability of a stock’s upcoming price is a very demanding yet problematic affair.

In literature a nimble elevation in application of computational techniques are appearing for different aspects of stock market analysis such as stock price prediction, its movement prediction, its volatility prediction, stock index return prediction, stock trading and soon. Out of these techniques, Artificial Neural Network (ANN) is one popular stream, which has widely applied in different stock market analysis. Though different types of ANN have performed well from time to time, most of them suffer from lower generalization, high complexity, network architecture, rigorous parameter tuning, human intervention and higher training time. Deciding the structure of the network, activation functions to be used and learning algorithm have always been challenging for researchers. In existing literature, the simple Feed Forward Neural Network (FFNN) with a single hidden layer has applied extensively for various prediction and classification task. Back propagation (BP) algorithm is one commonly used training algorithm for it. The BP is a popular one but it suffers from local minima and has slow convergence rate. So researchers have introduced a variety of nature inspired learning algorithms for ANN. Though it solves the local minima problem but it increases the time of learning. In both approaches, the unknown weights of the network are adjusted iteratively through the whole learning process, which has a major role in increasing the training time of the network. To solve these issues, Huang et al. [15] have proposed a new training algorithm noted as Extreme Learning Machine (ELM) for training the FFNN with a single hidden layer. This simple FFNN uses two sets of weights. One set is related to the connections between the input and hidden layer, which are assigned randomly initially during network creation. These are not tuned further in the learning phase. The other set of weights is associated with the connections between hidden and output layer, which is induced by a pseudo inverse least square computation. Unlike the other traditional training algorithms, ELM completes the network learning within a single iteration. Due to this, the overall model execution time is reduced greatly making it a preferable alternative over the other training models. Being a competent and immensely fast learning algorithm, ELM has successfully applied for solving various regression and classification problems in financial domain. However, the choice of the number of hidden layer neurons and the activation functions used in the hidden layer, involved in modelling an ELM highly influences the overall outcome of ELM. These components need to be examined thoroughly before modelling an ELM for any application. As the performance of ELM varies for different activation functions with respect to different performance measures, it is impractical to model an ELM with a specific activation function for all purposes. Hence, developing an efficient ELM by investigating the impact of different activation functions that will able to produce better result with respect to multiple criteria can be considered as a Multi Criteria Decision Making (MCDM) problem.

In this study, TOPSIS, one of the in demands MCDM technique is suggested to examine the performance of an ELM, modelled with distinct activation functions with respect to miscellaneous evaluation criteria. Estimating the capability of the ELM with respect to multiple criteria instead of a single criterion can help in designing a more robust network. The proposed framework is used as a binary classification system for pursuing the problem of stock index price movement prediction. The TOPSIS has contributed in ranking ELMs modelled separately by using fifteen activation functions such as Sigmoid, Gaussian, Sinusoid, TanH, ArcTan, ArcSinH, Rectified Linear Unit (ReLU), Leaky ReLU (LreLU), Parametric ReLU (PReLU), SinC, SoftPlus, Exponential Linear Unit (ELU), Inverse Square Root Unit(ISRU), ElliotSig, and Square Nonlinearity (SQNL). The outcome of all the fifteen ELM classifiers are observed for sixteen evaluation criteria such as True Positive (TP), True Negative (TN), Accuracy (ACC), Precision (PREC), Recall (REC), F-Measure (FM), G-Mean (GM), Specificity (SPEC), Negative Prediction Value (NPV), False Positive Rate (FPR), False Negative Rate (FNR), Youden’s Index (YI), Markedness (MK), Balanced Classification Rate (BCR), Half Total Error Rate (HTER) and Jaccard (JACC). Afterward it is analyzed as a 15 $\times$ 16 MCDM problem having 15 models and 16 criteria. The models are ranked separately in both training and testing phases. Further analysis is carried out to find the activation functions performing well both during training and testing phases. The empirical authentication of the model is carried out by applying historical stock index prices of three mainstream stock indices such as BSE SENSEX, S&P 500 and NIFTY 50 accumulated from 10 March, 2014 to 8 March, 2019. This is the first paper to perform such a detailed analysis of ELM with a dense set of activation functions and evaluation criteria in stock index price movement prediction problem. The main contribution of the paper is outlined as follows:

•
Fifteen distinct ELM models are developed using fifteen different activation functions in the hidden layer of a SLFN for stock index price movement prediction.
•
The ELM models are ranked separately in training and testing phase to identify the activation functions that performs well in designing an ELM based classifier.
•
A TOPSIS based approach is applied for ranking the ELMs by considering sixteen evaluation criteria.

The rest of the paper is assembled as follows: Section 2 provides a concise overview of the literature associated with ELM prediction models as well as TOPSIS. Section 3 enumerates the particulars of suggested model for stock index price movement prediction by implementing various activation functions in ELM models and ranking these models using TOPSIS and obtaining the best ELM model for this experiment. Section 4 specifies the empirical result obtained from classification as well as TOPSIS. Section 5 incorporates the concluding observations and future prospects.
2. Literature survey

The advancement in technology, computing power and complex yet good algorithms have made the work of prediction more effective than ever. There are several approaches for predicting the stock index movement as well as stock price. Researchers have tried to tackle this problem by using statistical methods and using intelligent computing techniques. As the computers now a days are more powerful than their ancestors, so more complex yet efficient prediction models such as Artificial Neural Networks (ANN) [2, 3, 4, 5, 8], Support Vector Machine (SVM) [6, 7], Decision Tree (DT) [10, 11], Naïve Bayes (NB) [12], K-Nearest Neighbour (KNN) [13, 14] and soon are being used by researchers for such analysis. In [3] Guresen et al. have proposed a MLP neural network based time series forecasting model for NASDAQ stock index. They compared the results of their proposed model with two benchmark models such as generalized autoregressive conditional heteroscedasticity (GARCH) and dynamic artificial neural network (DAN2) and found MLP to be the better performing one. In [4] Mostafa et al. have hypothesized that neural network models are better suited for time series forecasting than traditional statistical models. They advocated a forecasting model which employs MLP and generalized regression neural network (GRNN) for predicting the price movements of Kuwait stock exchange. Huang et al. presented a SVM model in [6] for predicting the directions of NIKKEI 225 index and exchange rate of USD. They used random walk model (RM), linear discriminant analysis (LDA), elman back propagation neural network (EBNN) and quadratic discriminant analysis (QDA) models as benchmark models and found SVM to be the prevailing model. The inclusion of kernel functions and the capacity control of the decision function have represented SVM as a promising trend prediction model in [7]. A Radial Basis Function Network (RBFN) trained using ridge ELM (RELM) is proposed in [8] for future stock index movement prediction of two popular stock indices such as BSE SENSEX and S&P500. The authors have investigated the performance of RBFN with 7 different basis functions by considering three different learning algorithms. The results reveal the better performance of the model trained using RELM with Inverse Multi Quadratic and Gaussian radial basis functions for both the dataset. Another Computational efficient functional link artificial neural network (CEFLANN) model trained using ELM is come forth as a promising model for classifying future stock index price movements of BSE SENSEX and S&P500 indices in [9]. The model has shown better results in comparison to Chebyshev FLANN and RBFN model. In [10] another SVM based trend prediction system is presented, in which DT is used for feature selection. The parameters of DT and SVM are optimized by applying genetic algorithm. Tiwari et al. implemented a hybrid model comprising of decision tree rough set and Hierarchical Hidden Markov Model (HHMM) in [11] to predict trends in BSE SENSEX. They also compared the result of the proposed model with hybrid decision tree rough set based model and standalone rough set based model and identified the superiority of the proposed model over benchmark models. An automation model based on NB is proposed in [12], which helps in alerting users for taking valuable decisions regarding buy or sell company’s stock. The model is validated over real-time as well as dummy datasets. Though the above models and methods give very promising result in terms of their applied area, they fail to impress the researchers due to its high computational time used for model training.

In 2004, Huang et al. proposed a fast learning algorithm for SLFNs termed as ELM [15]. According to their experiment, ELM overcomes the drawbacks such as parameter turning, local minima problem and slower convergence of traditional gradient based learning algorithms. In their experimentation they showed the superiority of ELM over FFNN trained with BP, RBF, SVM, bagging and boosting methods. The authors used real world medical diagnostics dataset and US forest service dataset for binary class classification. ELM outperformed other algorithms in terms of accuracy, parameter setup and training speed by a very large margin. Further, the authors have examined the implications of ELM on various fields of research in [16, 17]. Due to better generalization and extremely faster training speed, ELM has shown promising outcomes in real world problems such as regression, medical diagnostics, large and complex applications, signal processing [18], image processing [19], financial time series prediction [20, 39], automatic control [21] and medical diagnostics [22]. ELM also overcomes the trivial issues like local minima, over-fitting and irregular learning rate encountered by traditional gradient based learning algorithms. In [23] the authors have used the ELM to predict reservoir permeability, through the contrast analysis with SVM. In [24] ELM is applied for remote sensing image fusion. Here it is applied to get the training sample regression relationship between the remote sensing image data to improve the real quality of a thermal infrared image data. Another feasible and effective application of ELM is found in lithologic identification, and the establishment of the lithologic classification model in [25]. Huang et al. also advocated in [26], that unlike classic gradient based learning algorithms which only work with differentiable activation functions, ELM can work with a variety of activation functions including non-differentiable activation functions. According to their proposal, ELM is able to approximate any continuous function and can implement any classification problem. SLFNs do not require parameter tuning; instead random parameters can yield better performance [27]. Several other variations of ELMs such as Fully complex ELM [28], Online Sequential ELM (OS-ELM) [29], Optimally pruned ELM (OP-ELM) [30], Error Minimized ELM (EM-ELM) [31], Multiple Kernel Based ELM (MK-ELM) [32], Incremental ELM (I-ELM) [33] have been proposed by several researchers for the purpose of classification and regression and these ELMs have better generalization and faster convergence than conventional learning algorithms. ELM’s literature suggests that they are very much dependent on the activation functions used in their hidden layer neurons. There are a several activation functions available for implementation which are better as well as worse than one another for several test cases. It is quite uncertain to decide which activation function is best suited for ELM. So in this research, few of the commonly used activation functions have been selected from the literature, to analyse each one’s performance when it is applied on the hidden layer neurons of ELM.

As suggested in [34], evaluating the performance of a classification model with respect to different evaluation criteria is more reliable and robust. In their proposal they analyzed that different models having same accuracy can have different performance characteristics. So, evaluation of classification by employing multiple performance criteria can be favourable. In literature, it can be observed that the majority of the researches have considered one or two performance measures to show the validity and reliability of their proposed models. However in real world when there are multiple alternatives present and it is necessary to evaluate those alternatives with respect to multiple criteria, which is currently gaining priority in several domain as a MCDM problem. It can also be noted that evaluating the performance of a classifier with respect to multiple criteria can give us vital information regarding various decision, behaviour and functioning of the classifier and in the end can produce a more valid and reliable classification. The following literatures have described various MCDM approaches to evaluate multiple alternatives and criteria and produce a mathematically sound ranking system so that the user can choose the best alternative from all the alternatives. Dash et al. presented a TOPSIS based classifier ranking approach for stock index movement prediction in [35]. Their experiment shows that by evaluating classification performance of different classifiers with respect to multiple evaluation criteria and ranking them using a MCDM model can give robust and reliable prediction. In [36] the authors have examined various MCDM models such as TOPSIS, VIKOR, PROMETHEE II, Elimination and Choice Expressing REality (ELECTRE) III and Grey Relational Analysis (GRA) with the help of Spearman’s rank correlation coefficient to find the better MCDM model among them. A PROMETHEE based assessment of 11 common classifiers with few performance measures is presented in [37]. Further a ranking of classifiers using PROMETHEE and TOPSIS is presented in [38] by considering 10 classifier models and 6 performance evaluation criteria. An extension of that work is appeared in [40], where the TOPSIS has applied for selection of well performed base classifiers for a crow search based classifier ensemble. However the primary aim of the MCDM is to intensify the subjective evaluation of models considering multiple conflicting criteria.

Exploring the literature in domain of stock index movement prediction, application areas of ELM and MCDM, it is observed that prior to this proposed work, any other study has not checked the performance of ELM with diverse set of activation functions. As the performance of ELM varies for different activation functions with respect to different performance measures, it is impractical to model an ELM with a specific activation function for all purposes. Hence, it is essential to analyze the performance of ELMs with different activation functions. Again instead of selecting and ranking the well performed one on basis of a single criterion, taking the decision with respect to multiple criteria can help in designing a more robust predictor. Ranking of ELMs using TOPSIS approach, for predicting future stock index movements is also a novel contribution by the authors.

Figure 1.

An ELM network architecture.

3. Related work

This section provides a brief theoretical description about ELM, different activation functions, evaluation criteria and TOPSIS used in the proposed work.

3.1 Extreme learning machine

ELM is learning algorithm, which is originally proposed for a SLFN by Huang et al. in [15] and then, is extended to generalized SLFNs, where no tuning of hidden layers are required. Analytical calculation of output weights in a single iteration, following the random assignment of weights between inner layer and hidden layer nodes is one of the important features of ELM. This arbitrary weight assignment can produce better generalization than traditional learning algorithms. Again the negligible parameter tuning and lesser user intervention make the learning speed much faster than the other learning algorithms. ELM, as seen in the literature is very useful for function approximation, regression and real world classification problems and they are better suited for applications in which faster prediction and response is needed. Due to its superior performance, several other variations of ELMs have been applied in several real world problem solving. Using ELM for classification and regression is advantageous due to the fact that it trains faster, needs lesser intervention for parameter tuning and it can reinforce various activation functions. The network architecture used for ELM learning is represented in Fig. 1. The network is created using $N$ number of input neurons, $p$ number of hidden layer neurons and one output neuron. Considering $S(x_{i},y_{i})$ a given training dataset having $N$ elements, $i=1,2,3,\ldots,N$ where each $x_{i}$ consists of $d$ -dimensional input elements and $y_{i}$ is the actual output, $P$ number of hidden layer neurons and a linear activation function in the output neuron, the output function for ELM is represented as follows:

$\displaystyle o_{i}=\sum_{j=0}^{P}{W_{j}}h_{j}(x_{i})$ (1)

where, $W$ is the output weight matrix associated with hidden layer neurons and output layer neurons, $h_{j}(x)$ is the activation function associated with hidden layer.

Equation (1) can be written as follows:

$\displaystyle O=WH$ (2)

where $O$ is the target output matrix and $H$ is the hidden layer output matrix. We can determine the output weight $W$ from Eq. (2) as,

$\displaystyle W=H^{\psi}O$ (3)

where $H^{\psi}$ is the Moore-Penrose Generalization inverse of matrix $H$ .

In our proposed model, the output weights linking hidden layer neurons and output neurons are acquired mathematically using identical concepts of ELM.

Here $(x_{1},x_{2},\ldots,x_{N})$ are the inputs, $w$ and $b$ are individual weights and biases respectively which are generated randomly which are given input to the hidden layer neurons, $(h_{1},h_{2},\ldots,h_{P})$ are the initial output of hidden layer calculated using the following formula:

$\displaystyle h_{i}=W_{i}x_{i}+b_{i}$ (4)

$(H_{1},H_{2},\ldots,H_{P})$ are the final output of hidden layer neurons where $(h_{1},h_{2},\ldots,h_{P})$ are transformed using desired activation functions. Both $h_{1}$ and $H_{1}$ are calculated in the same step, the figure shown above gives a descriptive view of these two steps. $(W_{1},W_{2},\ldots,W_{P})$ are the output weights connecting hidden neurons and output neuron which are summed with hidden layer outputs generating the final output of ELM analytically.

3.2 Activation functions

Activation function acts as a mathematical gateway, which are used to regulate the end product of an artificial neural network. These functions are associated with each individual neuron in the neural network and decide whether that neuron should fire or not. Another key responsibility of an activation function is to normalize the output values of each neuron within a certain range. Activation functions get activated or deactivated based on certain rules or threshold value. These functions can simply be a step function that sets the output of a neuron as on or off. With increase in the complexity of the network, the strain on the activation function also increases. In general two types of activation functions are used such as linear activation functions and non-linear activation functions. Linear activation functions do not give that much flexibility in terms of training the model. It has very restricted ability to grasp complex datasets. Now a days,neural networks are implementing non-linear activation functions which have the ability to handle complex datasets, calculate and comprehend any function. These non-linear functions allow the network to construct composite mappings in between inputs and outputs in the network.

While constructing a neural network model in real-world, determining a suitable activation function is a crucial task. Evaluating several activation functions for a certain problem will help in achieving superior outcome. Hence in this study, fifteen different activation functions are taken in to account for analyzing the performance of ELM with respect to each individual one. Figure 2 presents the graphical representation of these activation functions and the details of these are listed as follows:

Figure 2.

Activation functions.

3.2.1 Sigmoid

It is one of the most common nonlinear activation functions used by researchers in several domains. Its output values lie between the range on 0 and 1. Sigmoid function has a smooth gradient and is also known as logistic regression. The sigmoid transformation of any value $y$ can be calculated by using the formula given below:

$\displaystyle f(y)=\frac{1}{1+e^{-y}}$ (5)

3.2.2 Inverse square root unit (ISRU)

ISRU has smooth and continuous derivatives and it is faster than TanH. The mathematical expression of ISRU is given below:

$\displaystyle f(y)=\frac{y}{\sqrt{1+\beta y^{2}}}$ (6)

3.2.3 Gaussian

Gaussian function is another one of the most popular activation function whose output ranges between 0 and 1. It is a continuous function and has a bell shaped curve. The Gaussian transformation can be carried out by using the following formula:

$\displaystyle f(y)=e^{-y^{2}}$ (7)

3.2.4 Sinusoid

Sinusoid or sine wave is a mathematical curve and is continuous in nature. Due to its periodic nature, sine wave can give rise to a ‘rippling’ cost function making it difficult to train. The output values of sinusoid activation function ranges between $-$ 1 and 1. Its mathematical formula is represented in Eq. (8) as follows:

$\displaystyle f(y)=\sin(y)$ (8)

3.2.5 TanH

TanH activation function is a non-linear function that transforms real valued numbers into a range between 1 and 1. Due to its stronger gradient, it is more preferred than sigmoid activation function. After a certain value, TanH suffers from ‘vanishing gradient problem’. Its mathematical equation is as follows:

$\displaystyle f(y)=\frac{e^{y}-e^{-y}}{e^{y}+e^{-y}}$ (9)

3.2.6 ArcTan

ArcTan or Tan ${}^{-1}$ transforms real numbers into a range between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ . Unlike TanH, ArcTan has a slightly flatter graph which enables it in differentiating similar inputs suitably. The transformation function is represented in Eq. (10).

$\displaystyle f(y)=\tan^{-1}(y)$ (10)

3.2.7 ArcSinH

It is also known as Hyperbolic sine and it transforms real values in to the range between $-\infty$ to $\infty$ . The mathematical equation of ArcSinH is as follows:

$\displaystyle f(y)=\ln(y+\sqrt{y^{2}+1})$ (11)

3.2.8 Rectified linear unit (ReLU)

ReLU is one of the most popular activation functions being used. It is non-linear in nature and its output value ranges between 0 and $\infty$ . Due to its simpler mathematical expression, it is less computationally expensive as compared to sigmoid and TanH etc. However, it might suffer from dying ReLU problem. The mathematical expression for ReLU activation function is as follows.

$\displaystyle f(y)=\begin{cases}0&\text{ for }y<0\\ y&\text{ for }y\geqslant 0\end{cases}$ (12)

3.2.9 Leaky ReLU (LReLU)

As compared to ReLU, instead of making the value of output 0 at $y<$ 0, Leaky ReLU allows a small non-zero constant gradient $\alpha$ . In general, $\alpha$ value is taken as 0.01. Its transformed output value ranges between $-\infty$ and $\infty$ . Due to its $\alpha$ factor, it might overcome the dying ReLU problem which ReLU suffers. However, it is linear in nature which makes it less suitable for complex classification problems. It also lags behind sigmoid and TanH. Its mathematical equation is represented below.

$\displaystyle f(y)=\begin{cases}0.01y&\text{ for }y<0\\ y&\text{ for }y\geqslant 0\end{cases}$ (13)

3.2.10 Parametric ReLU (PReLU)

Unlike LReLU where a predetermined slope 0.01 has been used, PReLU introduces a $\beta$ parameter which should be explored by the neural network. So, it might be more suitable to use LReLU instead of PReLU in some cases.

$\displaystyle f(\beta,y)=\begin{cases}\beta y&\text{ for }y<0\\ y&\text{ for }y\geqslant 0\end{cases}$ (14)

3.2.11 SinC

In SinC activation function, the output decreases in ratio of the distance from the origin. When the input value $y=0$ , it has an exception to make the output of transformation function $f(y)=1$ . SinC transforms real values into a range between $\approx-$ 0.217234 and 1. The mathematical expression of SinC is represented below:

$\displaystyle f(y)=\begin{cases}1&\text{ for }y=0\\ {\displaystyle\frac{\sin(y)}{y}}&\text{ for }y\neq 0\end{cases}$ (15)

3.2.12 Softplus

Softplus transforms real valued inputs into the range between 0 and $\infty$ . The curve of softplus is smooth and differentiable while the input value is near 0. The derivative of Softplus function results in the Sigmoid function. Equation (15) represents the mathematical formula for softplus activation function.

$\displaystyle f(y)=\ln(1+e^{y})$ (16)

3.2.13 Exponential linear unit (ELU)

ELU is one of the most commonly used activation functions. It has a $\beta$ constant factor having positive value. ELU’s output lies in the range between $-\beta$ and $\infty$ . The following equation represents ELU activation function:

$\displaystyle f(\beta,y)=\begin{cases}\beta(e^{y}-1)&\text{ for }y<0\\ y&\text{ for }y\geqslant 0\end{cases}$ (17)

3.2.14 ElliotSig

Ellioitsig or Softsign doesn’t require any trigonometric or exponential functions for calculation which makes it faster and simple. Its input and output values range between $[{-\infty,\infty}]$ and $[-1,1]$ respectively. Due to its simple mathematical calculation, it might require more training iterations or more hidden neurons for achieving superior performance. Its main disadvantage is its output flattens out for large inputs. Elliotsig transformation can be achieved by using the equation given below.

$\displaystyle f(y)=\frac{y}{1+|y|}$ (18)

3.2.15 Square nonlinearity (SQNL)

SQNL is a non-linear activation function which transforms real values into the range between $-$ 1 and 1. It has a better generalization and its convergence is faster than Elliotsig. The following equation represents SQNL activation function:

$\displaystyle f(\beta,y)=\begin{cases}1&\text{for }y>2.0\\ y-{\displaystyle\frac{y^{2}}{4}}&\text{for }0\leqslant y\leqslant 2.0\\ y+{\displaystyle\frac{y^{2}}{4}}&\text{for }-2.0\leqslant y<0\\ -1&\text{for }y<-2.0\end{cases}$ (19)

3.3 Evaluation criteria

The performance assessment is a vital process while evaluating the performance of a classifier. These performance criteria help in the process of training the classifier and provide useful information regarding the classifier’s ability to handle unknown data during the testing phase too. A single performance measure may not always provide all the key information for post classification decision making. Some cases it may lead to ambiguous outcome. For an example, two different classifiers can have the exact accuracy value but they might have taken different positive and negative decisions during classification. Although their accuracy is identical, other performance measures may or may not be identical. So in many cases, single criterion classification is not the suitable path to follow. Hence in this study, sixteen performance measures such as TP, TN, ACC, PREC, REC, FM, GM, SPEC, NPV, FPR, FNR, YI, MK, BCR, HTER and JACC are calculated and used for evaluating the classification performance of each individual ELM. The respective definitions, significance and formulas are given as follows.

3.3.1 True positive

If the sample is having the actual class value as 1 and it is also classified as 1, then it is a correctly classified positive sample and is denoted as true positive. A higher value of TP signifies a better classification model.

$\displaystyle TP=\text{Correctly classified positive class}$ (20)

3.3.2 True negative

If the sample is having 0 as actual class label and is classified as 0 too, it is a correctly classified negative sample and denoted as true negative. A higher value of TN signifies a better classification model.

$\displaystyle\textit{TN}=\text{Correctly classified negative class}$ (21)

3.3.3 Sensitivity

The sensitivity value represents the correctly classified true positive samples to the total number of positive samples. Hence it can be considered as another form of accuracy representing the positive samples only. A higher value of sensitivity signifies a better classification model.

$\displaystyle\text{Sensitivity}=\textit{TPR}=\text{Recall}=\frac{\textit{TP}}{% \textit{TP}+\textit{FN}}$ (22)

3.3.4 Specificity

Similar to sensitivity, specificity represents the correctly classified true negative samples to the total number of negative samples and can be considered as the representation accuracy in terms of negative samples. A higher value of specificity signifies a better classification model.

$\displaystyle\text{Specificity}=\textit{TNR}=\frac{\textit{TN}}{\textit{FP}+% \textit{TN}}$ (23)

3.3.5 Accuracy

It is the ratio between the correctly classified negative and positive samples to the total number of samples. Accuracy can face some problem when two classifiers yield same accuracy but may provide different positive and negative decisions. A higher value of accuracy signifies a better classification model.

$\displaystyle\text{Accuracy}=\frac{\textit{TP}+\textit{TN}}{\textit{TP}+% \textit{TN}+\textit{FP}+\textit{FN}}$ (24)

3.3.6 Precision

Positive prediction value or precision represents the proportion of true positive samples to the total number of positive predicted samples. A higher value of precision signifies a better classification model.

$\displaystyle\text{Precision}=\textit{PPV}=\frac{\textit{TP}}{\textit{FP}+% \textit{TP}}$ (25)

3.3.7 Negative prediction value

Negative prediction value represents the proportion of true negative samples to the total number of negative predicted samples. A higher value of NPV signifies a better classification model.

$\displaystyle\text{NPV}=\frac{\textit{TN}}{\textit{FN}+\textit{TN}}$ (26)

3.3.8 F-measure

F1-score or F-measure represents the harmonic mean of precision and recall. It ranges from 0 to 1. A higher value of F-measure signifies a better classification model.

$\displaystyle F\text{-measures}=\frac{2\times\textit{PPV}\times\textit{TPR}}{% \textit{PPV}+\textit{TPR}}$ (27)

3.3.9 Jaccard

Jaccard or Tanimoto similarity coefficient ignores the correct classification of negative samples as shown in the Eq. (28). A higher value of JACC signifies a better classification model.

$\displaystyle\text{Jaccard}=\frac{\textit{TP}}{\textit{TP}+\textit{FP}+\textit% {FN}}$ (28)

3.3.10 Markedness

Markedness depends on the PPV and NPV values hence any changes in those two values results in variation of markedness. A higher value of markedness signifies a better classification model.

$\displaystyle\text{Markedness}=\textit{PPV}+\textit{NPV}-1$ (29)

3.3.11 Youden’s index

Youden’s index combines specificity and sensitivity and ranges from 0 to 1. The YI values closer to 0 represent poor classification and values closer to 1 represent perfect classification. A higher value of YI signifies a better classification model.

$\displaystyle\text{Youden's Index}=\textit{TPR}+\textit{TNR}-1$ (30)

3.3.12 Balanced classification rate

Balanced classification rate combines the values of specificity and sensitivity as represented in Eq. (31). A higher value of BCR signifies a better classification model.

$\displaystyle\text{BCR}=\frac{1}{2}(\textit{TPR}+\textit{TNR})$ (31)

3.3.13 Half total error rate

The opposite of BCR is called as Half Total Error Rate and is represented as follows:

$\displaystyle\text{HTER}=1-\textit{BCR}$ (32)

A higher value of HTER signifies a better classification model.

3.3.14 False positive rate

False positive rate or false alarm rate or fallout is the ratio between the incorrectly classified negative samples to the total number of negative samples. In general it is the proportion of the negative samples that are incorrectly classified and is the complement of specificity. A lower value of FPR signifies a better classification model.

$\displaystyle\text{FPR}=1-\textit{TNR}=\frac{\textit{FP}}{\textit{FP}+\textit{% TN}}$ (33)

3.3.15 False negative rate

False negative rate or miss rate is the proportion of positive samples that are incorrectly classified. Hence it is the complement of sensitivity. A lower value of FNR signifies a better classification model.

$\displaystyle\text{FNR}=1-\textit{TPR}=\frac{\textit{FN}}{\textit{FN}+\textit{% TP}}$ (34)

3.3.16 Geometric-mean

As the aim of classification is to improve the sensitivity without sacrificing specificity, Geometric mean combines these two and is calculated using the formula as follows:

$\displaystyle G\text{-mean}=\sqrt{\textit{TPR}\times\textit{TNR}}$ (35)

A higher value of GM signifies a better classification model.

3.3.17 Technique for order of preference by similarity to ideal solution

Evaluating the performance of ELMs on the basis of sixteen different criteria, is clearly appearing as a multi-criteria decision making (MCDM) problem. Hence in this study, TOPSIS, one of the in demands MCDM technique is suggested to examine the performance of an ELM, modelled with distinct activation functions with respect to miscellaneous evaluation criteria. TOPSIS is developed on the foundation of figuring out the best substitute by increasing its distance from a negative-ideal solution, whereas decreasing its distance from an ideal solution. The alliance of outstanding performance values exhibited by any alternative for each criterion represents the ideal solution and the combination of least values produces the negative-ideal solution. The steps involved in TOPSIS are summarized as follows:

•
A decision matrix ( $M$ ) is obtained by arranging the outcomes corresponding to 16 evaluation criteria for each classifier as a row vector and finally merging the entire 15 row vectors one for each ELM model in a matrix of size 15 $\times$ 16.
•
A standardized decision matrix (SM) is obtained from $M$ using Eq. (3.3.17) as follows:

$\displaystyle\textit{SM}_{ij}=\frac{M_{ij}}{\sqrt{\sum_{j=1}^{K}{M_{ij}^{2}}}},$ $\displaystyle j=1,\ldots,K;i=1,\ldots,N$ (36)

$M_{ij}=$ Decision matrix, $\textit{SM}_{ij}=$ Standardized decision matrix ${}_{K}=$ Number of alternatives, $N=$ Number of criterion
•
A weighted standardized decision matrix (WM) is obtained from SM as follows:

$\displaystyle WM_{ij}=w_{i}\times SM_{ij},$ $\displaystyle j=1,\ldots,K;i=1,\ldots,N$ (37)

where, is the weight of the $i^{\text{th}}$ criterion and $\sum_{i=1}^{n}{w_{i}}=1$ .
•
The ideal solution ( $A^{+}$ ) is created from the alliance of outstanding performance values observed by any alternative for each criterion and the negative ideal solution ( $A^{-}$ ) is created from the combination of the least values using the following formula:

$\displaystyle A^{+}=\{WM_{1}^{+},\ldots,WM_{N}^{+}\}=\{(\max_{j}WM_{ij}|i\in b% ),(\min_{j}WM_{ij}|i\in c)\}$ $\displaystyle A^{-}=\{WM_{1}^{-},\ldots,WM_{N}^{-}\}=\{(\max_{j}WN_{ij}|i\in b% ),(\min_{j}WN_{ij}|i\in c)\}$ (38)

where, $b$ is the set of benefit criteria having the property of more is better and $c$ is the set of negative criteria having the property of less is better.
•
The separation measures $S^{+}$ and $S^{-}$ are calculated from $A^{+}$ and $A^{-}$ using the following formula:

$\displaystyle S_{j}^{+}=\sqrt{\sum_{i=1}^{N}{(WM_{ij}-WM_{i}^{+})^{2}}},$ $\displaystyle j=1,\ldots,K$ $\displaystyle S_{j}^{-}=\sqrt{\sum_{i=1}^{N}{(WM_{ij}-WM_{i}^{-})^{2}}},$ $\displaystyle j=1,\ldots,K$ (39)
•
The relative closeness ( $R^{+}_{j}$ ) to the ideal solution is calculated using following formula:

$\displaystyle R_{j}^{+}=\frac{S_{j}^{-}}{(S_{j}^{+}+S_{j}^{-})},∼{}j=1,\ldots,K$ (40)
•
Finally, the alternatives are ranked according to the decreasing order of their ( $R^{+}_{j}$ ) value.

Figure 3.
Proposed TOPSIS-ELM stock index price movement prediction model.

4. Proposed work

This section describes the detailed steps of stock index price movement prediction using ELM classifier. To understand the response of the dataset towards various activation functions being used in the hidden layer neurons in the ELM, 15 activation functions are taken into consideration. These 15 activation functions based ELMs are further analysed with the help of 16 performance evaluation criteria. Here multiple classifier models are evaluated using multiple criteria making it a suitable condition to implement a MCDM approach known as TOPSIS for generating ranks for each distinct activation function based ELM models. The proposed model is illustrated in Fig. 3. The model works in five phases.

Initially the historical prices of three benchmark stock indices BSE SENSEX, S&P 500 and NIFTY 50 are collected from 10 ${}^{\text{th}}$ March 2014 to 8 ${}^{\text{th}}$ March 2019. Each set includes a total of 1234 days of stock index opening; closing, highest and lowest prices. These raw data are put forward for data pre-processing in the second phase, where the data is converted to appropriate input and output vectors for presenting before classifier. In the input preparation step, few technical indicators are calculated and then normalized. The technical indicators of a stock index are comprised of the volumes and prices of stock data which are designed to represent the short-term movements of stock index price. Out of thousands of technical indicators, traders have to identify and utilize those technical indicators which are more suitable for them. As implemented by the authors in [8, 35], in this study six commonly referred technical indicators such as Simple Moving Average (SMA), Moving Average Convergence and Divergence(MACD), Stochastic K, D, R% and Relative Strength Index (RSI) are taken into consideration. To bring all the values of six technical indicators into a certain range i.e. between 0 and 1, min-max normalization technique has been applied by using the following equation:

$\displaystyle N_{\text{value}}=\frac{O_{\text{value}}-V_{\min}}{V_{\max}-V_{% \min}}$ (41)

where, $N_{\text{value}}=$ normalized value, $O_{\text{value}}=$ original value, $V_{\min}=$ minimum value and $V_{\max}=$ maximum value in given data elements.

The output of the prediction model is the upward or downward movement of the stock index prices i.e., if the stock index’s closing price of current day is higher than that of the previous day, it is moving in upward direction and if its closing price of current day is lower than previous day, it is moving in downward direction. To translate this movement into a suitable mathematical expression, the upward movement is labelled as 1 and the downward movement as 0 hence making this a binary classification problem where the two classes being 0 and 1. The movement of the stock index has been derived by using the formula,

$\displaystyle\textit{IND}_{\text{movement}}(i)=$ $\displaystyle\begin{cases}1∼{}(\text{increasing})&\text{for }\textit{CP}(i)>% \textit{CP}(i-1)\\ 0∼{}(\text{decreasing})&\text{for }\textit{CP}(i)<\textit{CP}(i-1)\end{cases}$ (42)

where, $\textit{CP}(i)=$ closing price of $i^{\text{th}}$ row and $\textit{IND}_{\text{movement}}$ $(i)=$ index movement of $i^{\text{th}}$ row

After pre-processing, 1209 data fields for each stock index have been extracted for classification out of which 847 data elements are utilized for training and the remaining 362 for testing.

Table 1

Dataset description

Sl. No.	Stock index	Duration	Total samples	Total increase	Total decrease
1	BSE SENSEX	10 ${}^{\text{th}}$ Mar. 2015–8 ${}^{\text{th}}$ Mar. 2019	1234	658	576
2	S&P 500			696	538
3	NIFTY 50			662	572

Table 2

Classifiers and activation functions

Sl. No.	Activation function name	Classifier name	Sl. No.	Activation function name	Classifier name
1	Sigmoid	ELM1	9	Leaky ReLU (LReLU)	ELM9
2	Inverse square root unit (ISRU)	ELM2	10	Square nonlinearity (SQNL)	ELM10
3	TanH	ELM3	11	ElliotSig	ELM11
4	Rectified linear unit (ReLU)	ELM4	12	ArcSinH	ELM12
5	Softplus	ELM5	13	Parametric ReLU (PReLU)	ELM13
6	Sinusoid	ELM6	14	Exponential linear unit (ELU)	ELM14
7	SinC	ELM7	15	ArcTan	ELM15
8	Gaussian	ELM8

In the third phase of the model, 15 separate SLFNs are created with six input layer neurons, one output layer neuron for predicting day ahead stock index price movement. Each network is trained using ELM. Keeping the architecture and the training approach of the networks fixed, the models are varied by using different activation functions in the hidden layer neurons. The activation functions used in the hidden layers in an ELM plays a vital role and influences the performance of the model hugely. Hence in this study, fifteen commonly used activation functions as explained in Section 3.2 have been implemented on the hidden layer neurons resulting fifteen distinct ELMs. Furthermore, the stock index price movement of each day is empirically predicted using these 15 ELMs. The number of hidden layer neurons used in each ELM model during training phase is kept fixed to 4, which is found from prior simulations. The weights and biases linking the input layer and hidden layer neurons are measured randomly where the values lie between $-$ 1 to 1 and the weights linking hidden layer neurons to the output layer are calculated using Moore-Penrose pseudo inverse matrix applying Eq. (3).

In the fourth phase the performance of fifteen ELMs are observed over sixteen evaluation measures described in Section 3.3. The measures used for evaluating the classification performance are categorized as type 1 and type 2. Type 1 set includes those measures, in which the more the value of the performance measure the higher the classification performance. Whereas type 2 set includes those, in which the less value implies higher classification performance. In this analysis a combination of these two types of performance measures has been taken into consideration. Here, type 1 performance measures are accuracy, precision, recall, FM, TP, TN, GM, specificity, NPV, YI, BCR, HTER and JACC. Similarly type 2 performance measures are FPR, FNR and MK. Further based on their steps of post calculation involved from the confusion matrix obtained from the network output, all the 16 measures are categorized into three groups. Group 1 consists of TP, TN, TPR and TNR measures that can be directly obtained from confusion matrix. So these are considered as initial classification performance measures. By using these four basic values, various other performance measures can be calculated. Group 2 consists of YI, BCR, HTER, GM, FPR and FNR that can be obtained from group 1 measures. Group 3 consists of ACC, JACC, NPV, PPV, FM and MK measures which can be calculated in two steps from the original output. The grouping of these criteria is done to provide different weight during ranking of the ELMs.

Finally in the last phase the ranking of ELMs are done using TOPSIS approach. The weights used in Eq. (3.3.17) to find weighted standardized decision matrix are normally assigned as per the user’s convenience. But in this study, the weights are assigned according to their precedence. According to their group precedence, each one has been assigned with a certain weight value. 40% of total weight value is given to group1 and 30% is given to each group 2 and group 3. The performance measures present in each group are given equal importance according to that group’s total weight. Again the sum of all weights is kept equal to 1. Applying the Eqs (3.3.17) to (40) over the derived WM with this new weighted scheme, finally the relative closeness is obtained for each ELM which has been used for ranking of them. The models are ranked separately over training and testing set to result in an appropriate conclusion.

5. Empirical analysis

This section presents the day ahead stock index price movement for three benchmark stock indices such as BSE SENSEX, NIFTY 50 and S&P 500 observed using fifteen ELM models, their corresponding outcomes during training and testing phase followed by a detailed analysis. The main purpose of this analysis is to evaluate various activation functions used in ELM models’ hidden layer neurons for the prediction of forthcoming stock index price movements. To examine these models, the historical price values of the three benchmark stock indices are collected across 10 ${}^{\text{th}}$ March 2015 to 8 ${}^{\text{th}}$ March 2019. These raw datasets are forwarded for data pre-processing. Initially the raw dataset consists of 1234 days of stock index opening, closing, highest and lowest values for each individual day which is then reduced to 1209 data samples after data pre-processing. These 1209 data samples hold the values of 6 technical indicators which are in a normalized form. The class labels of each 1209 data sample represents either a 0 or 1 value making it a binary classification problem. Class label 0 indicates that the stock index value is decreasing and class label 1 indicates that the stock index value is increasing for the considered trading day. The description of the dataset prior to and post data pre-processing is presented in Table 1. A 7:3 ratio of the dataset is used for training and testing purpose. Accordingly the training sample holds 847 rows of data and testing sample holds 362 rows of data in which each row contains the normalized values of 6 technical indicators. The analysis over the experimental outcomes observed for training and testing set is presented as follows.

5.1 Result analysis over training samples

As this is a supervised learning method, all 15 ELM models are first trained using the training sample set in which 6 normalized technical indicator values are given as input and their respective class labels are given as output to the classifiers. Therefore a total of 15 ELM classifier models having distinct activation functions in their hidden layer neurons are trained for predicting the future stock index movements. The training performance of these classifier models are averaged over 20 runs and is evaluated using 16 performance evaluation criteria. The ELM classifier models and their respective activation functions are represented in Table 2. Tables 3 to 5 depict the classification results over training samples for BSE SENSEX, NIFTY 50 and S&P500 respectively.

Table 3
Training classification result of BSE SENSEX

Classifier	ACC	PREC	REC	FM	TP	TN	GM	SPEC	NPV	FPR	FNR	YI	MK	BCR	HTER	JACC
ELM1	0.7306	0.7099	0.7157	0.7124	283.4000	335.4000	0.7291	0.6386	0.7042	0.3614	0.3056	0.3330	0.3320	0.6665	0.3335	0.4919
ELM2	0.7406	0.7209	0.7283	0.7241	288.4000	338.9000	0.7393	0.6940	0.6834	0.3060	0.3662	0.3279	0.3287	0.6639	0.3361	0.4700
ELM3	0.7239	0.7077	0.6997	0.7032	277.1000	336.0500	0.7216	0.7339	0.6968	0.2661	0.3636	0.3703	0.3743	0.6851	0.3149	0.4884
ELM4	0.6992	0.6931	0.6477	0.6670	256.5000	335.7000	0.6924	0.7073	0.6787	0.2927	0.3813	0.3260	0.3286	0.6630	0.3370	0.4640
ELM5	0.7225	0.7040	0.7058	0.7045	279.5000	332.4500	0.7210	0.5965	0.6170	0.4035	0.4217	0.1747	0.1742	0.5874	0.4126	0.3962
ELM6	0.7312	0.7190	0.6992	0.7085	276.9000	342.4500	0.7282	0.7273	0.7054	0.2727	0.3460	0.3813	0.3834	0.6907	0.3093	0.4990
ELM7	0.7048	0.6853	0.6837	0.6834	270.7500	326.2000	0.7021	0.6519	0.6504	0.3481	0.3990	0.2529	0.2530	0.6264	0.3736	0.4304
ELM8	0.7145	0.6848	0.7317	0.7048	289.7500	315.4500	0.7126	0.8448	0.8355	0.1552	0.1894	0.6554	0.6565	0.8277	0.1723	0.6888
ELM9	0.7272	0.7125	0.6987	0.7043	276.7000	339.2500	0.7238	0.8625	0.8330	0.1375	0.1970	0.6656	0.6698	0.8328	0.1672	0.6943
ELM10	0.7160	0.6961	0.6996	0.6967	277.0500	329.4000	0.7136	0.7273	0.7257	0.2727	0.3131	0.4141	0.4143	0.7071	0.2929	0.5241
ELM11	0.7458	0.7364	0.7129	0.7237	282.3000	349.4000	0.7426	0.8780	0.8302	0.1220	0.2045	0.6735	0.6815	0.8368	0.1632	0.6984
ELM12	0.7356	0.7188	0.7146	0.7165	283.0000	340.0500	0.7339	0.8071	0.8035	0.1929	0.2247	0.5823	0.5827	0.7912	0.2088	0.6356
ELM13	0.7175	0.6968	0.7005	0.6982	277.4000	330.3000	0.7158	0.7805	0.7788	0.2195	0.2525	0.5280	0.5281	0.7640	0.2360	0.5980
ELM14	0.7379	0.7216	0.7154	0.7181	283.3000	341.7000	0.7358	0.7871	0.7837	0.2129	0.2475	0.5397	0.5400	0.7698	0.2302	0.6057
ELM15	0.7300	0.7097	0.7191	0.7138	284.7500	333.5500	0.7287	0.7361	0.7233	0.2639	0.3207	0.4154	0.4166	0.7077	0.2923	0.5223

Table 4

Training classification result of NIFTY50

Classifier	ACC	PREC	REC	FM	TP	TN	GM	SPEC	NPV	FPR	FNR	YI	MK	BCR	HTER	JACC
ELM1	0.7259	0.7083	0.7113	0.7095	283.8000	331.0500	0.7246	0.6295	0.7050	0.3705	0.2957	0.3337	0.3336	0.6669	0.3331	0.4973
ELM2	0.7396	0.7218	0.7316	0.7261	291.9000	334.5500	0.7386	0.6763	0.6704	0.3237	0.3734	0.3029	0.3033	0.6515	0.3485	0.4596
ELM3	0.7189	0.7033	0.6992	0.7009	279.0000	329.9500	0.7173	0.7299	0.7002	0.2701	0.3509	0.3790	0.3818	0.6895	0.3105	0.4981
ELM4	0.6927	0.6860	0.6477	0.6643	258.4500	328.3000	0.6874	0.6964	0.6753	0.3036	0.3759	0.3205	0.3221	0.6602	0.3398	0.4654
ELM5	0.7166	0.7001	0.7028	0.7011	280.4000	326.6000	0.7154	0.5871	0.6116	0.4129	0.4185	0.1685	0.1680	0.5843	0.4157	0.3973
ELM6	0.7277	0.7164	0.7019	0.7086	280.0500	336.3500	0.7255	0.7031	0.7016	0.2969	0.3358	0.3673	0.3674	0.6836	0.3164	0.4981
ELM7	0.7006	0.6842	0.6825	0.6820	272.3000	321.1500	0.6983	0.6540	0.6629	0.3460	0.3734	0.2806	0.2802	0.6403	0.3597	0.4513
ELM8	0.7080	0.6821	0.7249	0.7001	289.2500	310.4000	0.7060	0.8415	0.8286	0.1585	0.1955	0.6460	0.6474	0.8230	0.1770	0.6830
ELM9	0.7207	0.7059	0.6981	0.7012	278.5500	331.8500	0.7183	0.8638	0.8395	0.1362	0.1855	0.6784	0.6814	0.8392	0.1608	0.7065
ELM10	0.7090	0.6887	0.7010	0.6936	279.7000	320.8000	0.7072	0.7188	0.7269	0.2813	0.3033	0.4155	0.4150	0.7077	0.2923	0.5295
ELM11	0.7420	0.7343	0.7105	0.7216	283.5000	344.9500	0.7391	0.8817	0.8316	0.1183	0.2005	0.6812	0.6891	0.8406	0.1594	0.7058
ELM12	0.7311	0.7150	0.7145	0.7146	285.1000	334.1000	0.7298	0.8013	0.8013	0.1987	0.2231	0.5783	0.5783	0.7891	0.2109	0.6352
ELM13	0.7125	0.6936	0.6996	0.6961	279.1500	324.3500	0.7113	0.7813	0.7625	0.2188	0.2732	0.5081	0.5099	0.7540	0.2460	0.5835
ELM14	0.7354	0.7208	0.7179	0.7189	286.4500	336.4500	0.7338	0.7790	0.7790	0.2210	0.2481	0.5309	0.5309	0.7654	0.2346	0.6024
ELM15	0.7238	0.7049	0.7172	0.7103	286.1500	326.9000	0.7227	0.7210	0.7162	0.2790	0.3208	0.4002	0.4005	0.7001	0.2999	0.5172

Table 5

Training classification result of S&P500

Classifier	ACC	PREC	REC	FM	TP	TN	GM	SPEC	NPV	FPR	FNR	YI	MK	BCR	HTER	JACC
ELM1	0.7113	0.6760	0.6238	0.6484	225.8000	376.6500	0.6955	0.8041	0.6915	0.1959	0.4807	0.3235	0.3558	0.6617	0.3383	0.4114
ELM2	0.7481	0.7134	0.6867	0.6992	248.6000	385.0500	0.7378	0.8722	0.8360	0.1278	0.2293	0.6429	0.6542	0.8214	0.1786	0.6580
ELM3	0.7196	0.6871	0.6291	0.6556	227.7500	381.7500	0.7024	0.7835	0.7143	0.2165	0.4199	0.3636	0.3810	0.6818	0.3182	0.4497
ELM4	0.7018	0.6649	0.6095	0.6334	220.6500	373.8000	0.6829	0.6907	0.7412	0.3093	0.3232	0.3675	0.3614	0.6838	0.3162	0.4785
ELM5	0.7054	0.6671	0.6192	0.6418	224.1500	373.3500	0.6898	0.7505	0.7041	0.2495	0.4227	0.3279	0.3374	0.6639	0.3361	0.4327
ELM6	0.7021	0.6634	0.6116	0.6347	221.4000	373.2500	0.6841	0.7546	0.7546	0.2454	0.3287	0.4259	0.4259	0.7130	0.2870	0.5052
ELM7	0.7063	0.6679	0.6195	0.6408	224.2500	373.9500	0.6890	0.7093	0.7414	0.2907	0.3315	0.3778	0.3732	0.6889	0.3111	0.4811
ELM8	0.7021	0.6572	0.6417	0.6453	232.3000	362.4000	0.6889	0.7918	0.7385	0.2082	0.3757	0.4161	0.4296	0.7080	0.2920	0.4881
ELM9	0.7088	0.6766	0.6102	0.6397	220.9000	379.4500	0.6890	0.7856	0.7713	0.2144	0.3122	0.4734	0.4766	0.7367	0.2633	0.5343
ELM10	0.7249	0.6930	0.6380	0.6636	230.9500	383.0500	0.7089	0.8598	0.8160	0.1402	0.2597	0.6001	0.6137	0.8001	0.1999	0.6233
ELM11	0.7230	0.6875	0.6463	0.6651	233.9500	378.4500	0.7090	0.7876	0.7208	0.2124	0.4088	0.3788	0.3958	0.6894	0.3106	0.4602
ELM12	0.7516	0.7266	0.6695	0.6962	242.3500	394.2500	0.7369	0.8515	0.7958	0.1485	0.2928	0.5587	0.5762	0.7794	0.2206	0.5899
ELM13	0.7160	0.6792	0.6376	0.6575	230.8000	375.6500	0.7024	0.7216	0.7231	0.2784	0.3702	0.3515	0.3512	0.6757	0.3243	0.4588
ELM14	0.7393	0.7039	0.6746	0.6883	244.2000	382.0000	0.7284	0.8289	0.8187	0.1711	0.2459	0.5830	0.5856	0.7915	0.2085	0.6135
ELM15	0.7523	0.7230	0.6827	0.7016	247.1500	390.0500	0.7403	0.7196	0.7286	0.2804	0.3591	0.3605	0.3590	0.6802	0.3198	0.4659

From Table 3 it is observed that, for BSE SENSEX dataset though ELM11 is producing better ACC, PREC, TN, GM, SPEC, FPR, Y1, MK, BCR, JACC values but ELM2 is producing better FM value, ELM5 is producing better HTER value and ELM8 is producing better REC, TP, NPV and FNR. Similarly analyzing Table 4 it is observed that for NIFTY 50 dataset ELM11 is producing better ACC, PREC, TN, GM, SPEC, FPR, Y1, MK, BCR values but ELM2 is producing better REC, FM, TP values, ELM5 is producing better HTER value and ELM9 is producing better NPV, FNR, JACC values. From outcomes of S&P 500 dataset represented in Table 5 it is observed that, ELM15 is producing better ACC, FM, GM values, but ELM2 is producing better REC, TP, SPEC, NPV, FPR, FNR, Y1, MK, BCR, JACC values, ELM1 is producing better HTER value and ELM12 is producing better PREC and TN values. After analysing all the training performance values of each ELM model across 16 performance evaluation criteria, it is found that no ELM model gives the best result for all performance measures. Although few ELMs perform superior in few criteria, they fail to impress for the remaining criteria. Thus, it points to a sensible situation of forming a more reliable decision considering multiple criteria. As several alternate ELM models are evaluated with respect to several criteria, TOPSIS a MCDM approach is further implemented to analyse and rank all these 15 alternate with respect to 16 criteria. Accordingly, the Tables 3 to 5 are considered as a decision matrix of size 15 $\times$ 16 for BSE SENSEX, NIFTY 50 and S&P 500 dataset respectively. Then the SM and WM are calculated using Eqs (3.3.17) and (3.3.17) respectively. As described in Section 4, there are two types of performance measures being considered in this study i.e. type 1and type 2. For finding the ideal solutions ( $A^{+}$ ) the highest value among all type 1 performance measures of 15 ELMs is considered and the lowest value among all the type 2 performance measures is considered. Similarly for finding the negative ideal solution ( $A^{-}$ ), the lowest value among all the type 1 performance measures and the highest value among all the type 2 performance measures is considered for all the 15 ELMs. The separation measures from ideal solutions ( $S^{+}$ ) and negative ideal solutions ( $S^{-}$ ) are calculated using Eq. (3.3.17). Furthermore, the relative closeness ( $R^{+}$ ) is calculated using Eq. (40) and the ELMs are ranked with respect to their relative closeness values in a decreasing order. The TOPSIS ranking of 15 ELMs over training samples is shown in Table 6 out of which the top 5 best ELMs and their respective activation functions are represented in Table 7 for all three benchmark datasets.

Table 6

TOPSIS ranking of ELM models with respect to their training classification result

Classifiers	TOPSIS ranking of ELM models: Training
	BSE SENSEX	NIFTY 50	S&P 500
ELM1	12	11	11
ELM2	11	12	1
ELM3	10	9	10
ELM4	13	13	13
ELM5	15	15	15
ELM6	9	10	7
ELM7	14	14	12
ELM8	3	3	6
ELM9	2	2	5
ELM10	8	7	2
ELM11	1	1	8
ELM12	4	4	4
ELM13	6	6	14
ELM14	5	5	3
ELM15	7	8	9

Table 7

Top five ELM model with respect to their training classification result

Classifiersrank	Training ranking of ELM models
	BSE SENSEX	NIFTY 50	S&P 500
1	ELM11- Elliotsig	ELM11- Elliotsig	ELM2-ISRU
2	ELM9-LReLU	ELM9-LReLU	ELM12-SQNL
3	ELM8-Gaussian	ELM8-Gaussian	ELM14-ELU
4	ELM12-ArcSinH	ELM12-ArcSinH	ELM10-ArcSinH
5	ELM14-ELU	ELM14-ELU	ELM9-LReLU

From Table 7, it is found that Elliotsig is having rank 1 for BSE SENSEX, NIFTY 50 and ISRU is having rank 1 for S&P 500. However, from further analysis, it is observed that, ELU, ArcSinH and LReLU are the three common performers in the top 5 ranked list for all the three dataset. From these observations, it is still uncertain that which activation functions are comparatively more suitable for our benchmark datasets. To observe the capability of ELMs in terms of generalization and handling unknown data, further the validity and reliability of these models are tested by introducing some testing samples.

5.2 Result analysis over testing samples

In real world classification, the classifiers have to deal with unknown data samples. The testing performance of a classifier notifies and ensures about the generalization capability of the classifier. Hence in the next step of analysis, the considered ELMs are introduced to 362 number of testing data samples each consisting of 6 columns holding the normalized values of 6 technical indicators. The testing classification performance of these 15 ELM models with respect to 16 performance

Table 8
Testing classification result of BSE SENSEX

Classifier	ACC	PREC	REC	FM	TP	TN	GM	SPEC	NPV	FPR	FNR	YI	MK	BCR	HTER	JACC
ELM1	0.8315	0.9580	0.6706	0.7889	114.00	187.00	0.8082	0.9740	0.7695	0.0260	0.3294	0.6445	0.7275	0.8223	0.1777	0.6514
ELM2	0.6823	0.5993	0.9765	0.7427	166.00	81.00	0.6418	0.4219	0.9529	0.5781	0.0235	0.3983	0.5522	0.6992	0.3008	0.5907
ELM3	0.7459	0.8679	0.5412	0.6667	92.00	178.00	0.7083	0.9271	0.6953	0.0729	0.4588	0.4683	0.5632	0.7341	0.2659	0.5000
ELM4	0.5608	0.5168	0.9941	0.6801	169.00	34.00	0.4196	0.1771	0.9714	0.8229	0.0059	0.1712	0.4882	0.5856	0.4144	0.5152
ELM5	0.7790	0.6907	0.9588	0.8030	163.00	119.00	0.7709	0.6198	0.9444	0.3802	0.0412	0.5786	0.6351	0.7893	0.2107	0.6708
ELM6	0.8536	0.8095	0.9000	0.8524	153.00	156.00	0.8551	0.8125	0.9017	0.1875	0.1000	0.7125	0.7113	0.8563	0.1438	0.7427
ELM7	0.8011	0.9083	0.6412	0.7517	109.00	181.00	0.7775	0.9427	0.7479	0.0573	0.3588	0.5839	0.6563	0.7919	0.2081	0.6022
ELM8	0.8066	0.8425	0.7235	0.7785	123.00	169.00	0.7980	0.8802	0.7824	0.1198	0.2765	0.6037	0.6249	0.8019	0.1981	0.6373
ELM9	0.8785	0.8539	0.8941	0.8736	152.00	166.00	0.8792	0.8646	0.9022	0.1354	0.1059	0.7587	0.7561	0.8794	0.1206	0.7755
ELM10	0.7155	0.8851	0.4529	0.5992	77.00	182.00	0.6552	0.9479	0.6618	0.0521	0.5471	0.4009	0.5469	0.7004	0.2996	0.4278
ELM11	0.8177	0.7524	0.9118	0.8245	155.00	141.00	0.8183	0.7344	0.9038	0.2656	0.0882	0.6461	0.6563	0.8231	0.1769	0.7014
ELM12	0.8646	0.8380	0.8824	0.8596	150.00	163.00	0.8655	0.8490	0.8907	0.1510	0.1176	0.7313	0.7287	0.8657	0.1343	0.7538
ELM13	0.8177	0.8467	0.7471	0.7938	127.00	169.00	0.8109	0.8802	0.7972	0.1198	0.2529	0.6273	0.6438	0.8136	0.1864	0.6580
ELM14	0.8757	0.9085	0.8176	0.8607	139.00	178.00	0.8706	0.9271	0.8517	0.0729	0.1824	0.7447	0.7602	0.8724	0.1276	0.7554
ELM15	0.8122	0.7931	0.8118	0.8023	138.00	156.00	0.8121	0.8125	0.8298	0.1875	0.1882	0.6243	0.6229	0.8121	0.1879	0.6699

Table 9

Testing classification result of NIFTY 50

Classifier	ACC	PREC	REC	FM	TP	TN	GM	SPEC	NPV	FPR	FNR	YI	MK	BCR	HTER	JACC
ELM1	0.8287	0.9402	0.6667	0.7801	110.00	190.00	0.8019	0.9645	0.7755	0.0355	0.3333	0.6311	0.7157	0.8156	0.1844	0.6395
ELM2	0.7928	0.7083	0.9273	0.8031	153.00	134.00	0.7942	0.6802	0.9178	0.3198	0.0727	0.6075	0.6261	0.8037	0.1963	0.6711
ELM3	0.5028	0.4781	0.9939	0.6457	164.00	18.00	0.3014	0.0914	0.9474	0.9086	0.0061	0.0853	0.4255	0.5427	0.4573	0.4767
ELM4	0.5470	0.5015	0.9879	0.6653	163.00	35.00	0.4189	0.1777	0.9459	0.8223	0.0121	0.1655	0.4475	0.5828	0.4172	0.4985
ELM5	0.7569	0.6559	0.9818	0.7864	162.00	112.00	0.7471	0.5685	0.9739	0.4315	0.0182	0.5503	0.6298	0.7752	0.2248	0.6480
ELM6	0.8066	0.7230	0.9333	0.8148	154.00	138.00	0.8086	0.7005	0.9262	0.2995	0.0667	0.6338	0.6492	0.8169	0.1831	0.6875
ELM7	0.8425	0.8857	0.7515	0.8131	124.00	181.00	0.8310	0.9188	0.8153	0.0812	0.2485	0.6703	0.7010	0.8351	0.1649	0.6851
ELM8	0.8204	0.8289	0.7636	0.7950	126.00	171.00	0.8142	0.8680	0.8143	0.1320	0.2364	0.6317	0.6432	0.8158	0.1842	0.6597
ELM9	0.8453	0.7795	0.9212	0.8444	152.00	154.00	0.8486	0.7817	0.9222	0.2183	0.0788	0.7029	0.7016	0.8515	0.1485	0.7308
ELM10	0.7983	0.9035	0.6242	0.7384	103.00	186.00	0.7677	0.9442	0.7500	0.0558	0.3758	0.5684	0.6535	0.7842	0.2158	0.5852
ELM11	0.8122	0.7299	0.9333	0.8191	154.00	140.00	0.8144	0.7107	0.9272	0.2893	0.0667	0.6440	0.6570	0.8220	0.1780	0.6937
ELM12	0.8619	0.8108	0.9091	0.8571	150.00	162.00	0.8646	0.8223	0.9153	0.1777	0.0909	0.7314	0.7261	0.8657	0.1343	0.7500
ELM13	0.8177	0.7647	0.8667	0.8125	143.00	153.00	0.8204	0.7766	0.8743	0.2234	0.1333	0.6433	0.6390	0.8217	0.1783	0.6842
ELM14	0.8757	0.8409	0.8970	0.8680	148.00	169.00	0.8772	0.8579	0.9086	0.1421	0.1030	0.7548	0.7495	0.8774	0.1226	0.7668
ELM15	0.8177	0.7676	0.8606	0.8114	142.00	154.00	0.8202	0.7817	0.8701	0.2183	0.1394	0.6423	0.6376	0.8212	0.1788	0.6827

Table 10

Testing classification result of S&P 500.

Classifier	ACC	PREC	REC	FM	TP	TN	GM	SPEC	NPV	FPR	FNR	YI	MK	BCR	HTER	JACC
ELM1	0.8453	0.8581	0.7964	0.8261	133.00	173.00	0.8406	0.8872	0.8357	0.1128	0.2036	0.6836	0.6938	0.8418	0.1582	0.7037
ELM2	0.6796	0.9811	0.3114	0.4727	52.00	194.00	0.5566	0.9949	0.6278	0.0051	0.6886	0.3062	0.6090	0.6531	0.3469	0.3095
ELM3	0.7845	0.6878	0.9760	0.8069	163.00	121.00	0.7782	0.6205	0.9680	0.3795	0.0240	0.5966	0.6558	0.7983	0.2017	0.6763
ELM4	0.8619	0.7970	0.9401	0.8626	157.00	155.00	0.8645	0.7949	0.9394	0.2051	0.0599	0.7350	0.7363	0.8675	0.1325	0.7585
ELM5	0.7017	0.6081	0.9940	0.7545	166.00	88.00	0.6698	0.4513	0.9888	0.5487	0.0060	0.4453	0.5968	0.7226	0.2774	0.6058
ELM6	0.8149	0.7577	0.8802	0.8144	147.00	148.00	0.8174	0.7590	0.8810	0.2410	0.1198	0.6392	0.6387	0.8196	0.1804	0.6869
ELM7	0.8204	0.7898	0.8323	0.8105	139.00	158.00	0.8212	0.8103	0.8495	0.1897	0.1677	0.6426	0.6392	0.8213	0.1787	0.6814
ELM8	0.7486	0.6532	0.9701	0.7807	162.00	109.00	0.7364	0.5590	0.9561	0.4410	0.0299	0.5290	0.6094	0.7645	0.2355	0.6403
ELM9	0.6878	0.9821	0.3293	0.4933	55.00	194.00	0.5724	0.9949	0.6340	0.0051	0.6707	0.3242	0.6161	0.6621	0.3379	0.3274
ELM10	0.8315	0.9141	0.7006	0.7932	117.00	184.00	0.8131	0.9436	0.7863	0.0564	0.2994	0.6442	0.7004	0.8221	0.1779	0.6573
ELM11	0.7845	0.6996	0.9341	0.8000	156.00	128.00	0.7831	0.6564	0.9209	0.3436	0.0659	0.5905	0.6204	0.7953	0.2047	0.6667
ELM12	0.8729	0.8954	0.8204	0.8563	137.00	179.00	0.8678	0.9179	0.8565	0.0821	0.1796	0.7383	0.7519	0.8692	0.1308	0.7486
ELM13	0.8481	0.7692	0.9581	0.8533	160.00	147.00	0.8499	0.7538	0.9545	0.2462	0.0419	0.7119	0.7238	0.8560	0.1440	0.7442
ELM14	0.7403	0.6431	0.9820	0.7773	164.00	104.00	0.7237	0.5333	0.9720	0.4667	0.0180	0.5154	0.6151	0.7577	0.2423	0.6357
ELM15	0.8923	0.9051	0.8563	0.8800	143.00	180.00	0.8891	0.9231	0.8824	0.0769	0.1437	0.7794	0.7874	0.8897	0.1103	0.7857

evaluation criteria have been shown in Tables 8 to 10 for BSE SENSEX, NIFTY 50 and S&P500 respectively.

From Table 8, it is observed that, for BSE SENSEX dataset ELM1 is producing better PREC, TN, SPEC, FPR, values, ELM4 is producing better REC,TP, NPV, FNR, HTER values, ELM9 is producing better ACC, FM,GM, Y1, BCR, JACC values and ELM14 is producing better MK value. Similarly analyzing Table 9 it is observed that for NIFTY 50 dataset ELM1 is producing better PREC, TN, SPEC, FPR values, ELM3 is producing better REC,TP, FNR, HTER values, ELM5 is producing better NPV value and ELM14 is producing better ACC, FM, GM, Y1, MK, BCR, JACC values. From outcomes of S&P 500 dataset represented in Table 10 it is observed that, ELM2 is producing better TN, SPEC, FPR, HTER values, ELM5 is producing better REC, TP, NPV, FNR values, ELM9 is producing better PREC,TN, SPEC, FPR values and ELM15 is producing better ACC, FM, GM, Y1, BCR and JACC values. Over testing samples, it is also observed that no ELM model gives the best result for all performance measures. Hence considering it as a MCDM problem further the TOPSIS ranking of all ELMs over test samples is obtained and depicted in Table 11.

After the TOPSIS ranking, it is found that ELMs 9, 12, 14, 6 and 11 are the top 5 performers for BSE SENSEX during testing. Similarly ELMs 14, 12, 9, 11 and 6 for NIFTY 50 and ELMs 15, 4, 12, 13, and 1 for S&P500 are found to be the best 5 performing ELM models during testing. Table 12 represents the top 5 ELM performers for all the three dataset over test samples. From Table 12 it is observed that, ELM 12 which uses ArcSinH as the activation function in the hidden layer neurons is the common performer among all three datasets. Apart from ARCSinH, ELM 9 (LReLU), ELM 11(ElliotSig) and ELM 6 (Sinusoid) are common for BSE SENSEX and NIFTY 50 dataset. By analysing the training and the testing rankings represented in Tables 7 and 12 respectively, it is observed that ELM 12 using ArcSinH activation function is coming among the top performers in both training and testing samples. Beside it ELM 9 using LReLU activation function is also producing better result commonly in training and testing sample for BSE SENSEX and NIFTY 50 dataset.

Table 11

TOPSIS ranking of ELM models with respect to their testing classification result

Classifiers	TOPSIS ranking of ELM models: Testing
	BSE SENSEX	NIFTY 50	S&P 500
ELM1	8	12	5
ELM2	13	8	15
ELM3	12	15	10
ELM4	15	14	2
ELM5	10	10	13
ELM6	4	5	7
ELM7	11	9	6
ELM8	9	11	11
ELM9	1	3	14
ELM10	14	13	8
ELM11	5	4	9
ELM12	2	2	3
ELM13	7	6	4
ELM14	3	1	12
ELM15	6	7	1

Table 12

Top five ELM models with respect to their testing classification result

Classifiers rank	Testing ranking of ELM models
	BSE SENSEX	NIFTY 50	S&P 500
1	ELM9-LReLU	ELM14-ELU	ELM15-ArcTan
2	ELM12-ArcSinH	ELM12-ArcSinH	ELM4-ReLU
3	ELM14-ELU	ELM9-LReLU	ELM12-ArcSinH
4	ELM6-Sinusoid	ELM11- Elliotsig	ELM13-PReLU
5	ELM11- Elliotsig	ELM6-Sinusoid	ELM1-Sigmoid

6. Result discussion

After observing the training classification results and testing classification results, it is come to the fact that, no single ELM performs superior over all the criteria. Over training samples, ELU, ArcSinH and LReLU are the only three activation functions that are commonly appeared among the top 5 best performing models according to TOPSIS ranking across all three datasets. However, in testing ArcSinH is the only one that is common in all three data sets’ top 5 performing models. Although LReLU is common for BSE SENSEX and NIFTY 50 datasets while testing, it is not present for S&P 500. So ArcSinH is the only activation function that is showing promising result and is common for both training and testing classification. From the observations it is inferred that, ArcSinH has better implications in terms of stock index movement prediction for BSE SENSEX, NIFTY 50 and S&P 500 datasets as compared to other activation functions for the considered time period. From the above empirical analysis, it can be observed that the selection of activation functions for any ELM model is a crucial task. ELMs having different activation functions yield varying classification result. Also, the evaluation of the classification results in terms of multiple criteria can give a better understanding about the performance of the classifier.

7. Conclusion

Understanding and predicting the stock index price movements accurately can help stock traders in their financial decision making. As stock market data is very diverse, volatile and dynamic in nature, it is a challenging task to predict the future outcomes of the stock index movement. ELM being a fast learning algorithm shows very promising result in stock index price movement prediction. However, the classification result of an ELM model is heavily affected by the number of neurons in the hidden layer and the activation function being used in the hidden layer. As determining the best activation functions for an ELM model is very significant to achieve better classification results, in our experiment three benchmark stock indices i.e. BSE SENSEX, S&P500 and NIFTY 50 are empirically analysed. Here fifteen activation functions have been used for training distinct ELM models for predicting the upcoming movements of stock indices. The training performance of each model has been assessed with respect to sixteen performance evaluation criteria. The main reason behind evaluating the classification performance with respect to more than one criterion is to achieve a better performing classifier and a robust classification. These classification models are further ranked using an MCDM framework i.e. TOPSIS to determine the best performing models on training dataset and the testing dataset. The above empirical analysis shows that certain activation functions such as ArcSinH and LReLU are performing better while training the classifiers for all three benchmark datasets whereas only ArcSinH is elected as common better performer over testing sample of all three datasets. As classifiers are dataset dependent and ELM classifier is dependent on the activation function being used in the hidden layer, it can be concluded that classification using ELM is activation function dependent. So choosing the suitable activation function for any ELM classifier is a crucial task and this experiment describes one of the many ways to complete this task in a robust and accurate way. This experiment shows that ELMs are dataset dependent and their generalization performance depends on the dataset, activation functions being used.

The proposed work can be summarized into the following key points:

An ELM model has been considered for stock index price movement prediction.

To understand the response of the network, fifteen activation functions are applied on the hidden layer neurons for distinct models.

Input weights and biases are assigned randomly and output weights are calculated analytically.

Empirical analysis of the model has been done over three distinct stock index datasets such as BSE SENSEX, S&P 500 and NIFTY 50 collected from 10 ${}^{\text{th}}$ March 2015 to 8 ${}^{\text{th}}$ March 2019.

A MCDM framework denoted as TOPSIS has been implemented to rank the training and testing performance of the fifteen classification models with respect to sixteen criteria.

The activation functions that are performing better for each individual stock index datasets are represented separately for training and testing samples.

Although the proposed TOPSIS-ELM framework is helpful in identifying the suitable activation functions in designing ELM based classifier, still it includes a number of challenges to be addressed in more details. Further work can be done on ELMs with respect to stock index price movement prediction by modifying the number of hidden layer neurons, exploring other learning approach used in ELMs, optimizing the structure of the ELMs using effective optimization algorithms and so on. The generalization ability of ELM can also be validated over more authentic datasets.

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Dash

Rautray

. An integrated TOPSIS crow search based classifier ensemble: In application to stock index price movement prediction. Applied Soft Computing; 2019. doi: 10.1016/j.asoc.2019.105784.

A TOPSIS-ELM framework for stock index price movement prediction

Abstract

Keywords

1. Introduction

3.1 Extreme learning machine

3.3.1 True positive

5.1 Result analysis over training samples

Table 3 Training classification result of BSE SENSEX

Table 8 Testing classification result of BSE SENSEX

7. Conclusion

References

Table 3
Training classification result of BSE SENSEX

Table 8
Testing classification result of BSE SENSEX