CN107704944B - Construction method of stock market fluctuation interval prediction model based on information theory learning - Google Patents

Construction method of stock market fluctuation interval prediction model based on information theory learning Download PDF

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CN107704944B
CN107704944B CN201710696129.4A CN201710696129A CN107704944B CN 107704944 B CN107704944 B CN 107704944B CN 201710696129 A CN201710696129 A CN 201710696129A CN 107704944 B CN107704944 B CN 107704944B
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李春光
翟一帆
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Abstract

The invention discloses a construction method of a stock market fluctuation interval prediction model based on information theory learning. The corresponding method comprises the following steps: acquiring and compartmentalizing stock index data; establishing an interval neural network prediction model; designing a cost function based on information theory learning; and training the interval neural network prediction model through a maximized cost function, and using the trained model for forecasting the stock market fluctuation interval. The invention realizes modeling and prediction of the stock index fluctuation interval, overcomes the limitation that the common minimum mean square error criterion depends on the Gaussian premise hypothesis, has better accuracy and robustness, and has wider application range.

Description

Construction method of stock market fluctuation interval prediction model based on information theory learning
Technical Field
The invention relates to the field of machine learning, data prediction and finance, in particular to a construction method of a stock market fluctuation interval prediction model based on information theory learning.
Background
Modeling and forecasting of returns and volatility to stock markets is a key issue in applications such as portfolio management, derivative security pricing, and risk management. The current major economic prediction models are based on point data, such as choosing the opening price or closing price of a stock market in a day for analysis. However, the stock index in a day is changing, and with the rapid development of electronic trading and data storage technology, more accurate daily trading data can be obtained, which is generally summarized into the highest price and the lowest price in the day, thereby forming a stock index interval. A large number of studies prove that such intervals are an effective way to measure the volatility of the stock market within a certain time interval, and the volatility within a day can be ignored when only opening prices or closing prices are adopted. The prediction of the interval value data has the advantages that the fluctuation or inaccuracy of the data can be reflected, so that more information is provided than the conventional point data prediction, and better results and performance are obtained. Many existing conventional prediction models can be applied to the prediction of the stock market fluctuation interval, such as an exponential smoothing method, an ARIMA model, a neural network model, and the like. Among them, the neural network model has a prominent advantage for solving the nonlinear problem, and is called an interval neural network in the context of interval value data.
In the training of a stock market fluctuation interval prediction model based on a neural network, the currently common method is based on second-order statistics. For example, using the minimum mean square error criterion, the form of the cost function is based on the following definition of the distance between two interval value data:
Figure GDA0002961106500000011
however, these methods have a strong dependence on the premise assumption of gaussian, i.e., optimal results are obtained only under the condition of gaussian noise. However, in the stock market fluctuation interval prediction, the noise of the stock index data does not necessarily meet the Gaussian distribution and is often non-Gaussian, and the accuracy and robustness of prediction by applying the method under the condition are often not ideal.
Disclosure of Invention
The invention aims to provide a stock market fluctuation interval prediction method based on information theory learning, aiming at the problem that the prediction performance is not ideal under the non-Gaussian noise condition when the existing neural network model is applied to stock market fluctuation interval prediction.
In order to solve the technical problems, the invention is realized by the following technical scheme:
(1) acquiring and compartmentalizing strand finger data;
(2) establishing an interval neural network prediction model;
(3) designing a cost function based on information theory learning;
(4) and (4) training the interval neural network prediction model by maximizing the cost function in the step (3), and using the trained model for predicting the fluctuation interval of the stock market.
The acquisition and interval of the strand finger data in the step (1) comprise the following specific processes:
obtaining historical data of the highest price and the lowest price of the stock index in a certain time period from various financial databases, and arranging the historical data into a stock market fluctuation interval form, namely X ═ XL,xH]Or X ═<xC,xR>Wherein x isHIs the highest price, xLIs the lowest price of the stock index, xCIs the midpoint of the fluctuation interval, xRIs the radius of the fluctuation interval.
Establishing an interval neural network prediction model in the step (2) as follows:
the method for predicting the fluctuation interval by establishing the interval neural network model comprises the following steps: the device comprises an input layer of n input units, one or more hidden layers containing h nonlinear activation function units and an output layer of m output units. The input and output of the model are the stock index data in the form of interval values, and the model has two layers of weight values alphaj、wjiRequires training, where αjIs the weight, w, from the hidden layer to the output layerjiIs the weight of the input layer to the hidden layer.
The cost function based on the information theory learning in the step (3) is designed as follows:
and (4) introducing relevant entropy in information theory learning to define a new cost function for training the model. Two random variables, X and Y, are given, with the entropy of correlation between them defined as:
V(X,Y)＝E[κ(X,Y)]＝∫∫κ(x,y)fXY(x,y)dxdy (1)
where κ (x, y) is the kernel function, fXY(X, Y) is a joint probability distribution function of X and Y. In practical cases, the joint distribution of the random variables X and Y is usually unknown, and only some limited values of them are known, in which case we can estimate the correlation entropy by using a sample averaging method:
Figure GDA0002961106500000021
where N is the number of samples, σ is the width of the kernel, X (i) and Y (i) are the i-th sample observations of X and Y, respectively; when the correlation entropy is maximized, the probability that the error between the estimated value and the expected value is 0 is the largest, and thus the maximum correlation entropy criterion is defined as:
Figure GDA0002961106500000022
wherein, e (i) x (i) -y (i) is set as the estimation output interval
Figure GDA0002961106500000023
The expected output interval is Y', the parameter training of the model is carried out under the maximum correlation entropy criterion (MCC), and the cost function E is adoptedMCCIs a weighted sum of the center and radius dependent entropies. Where t represents the sequence number of the training samples, and the calculation result is the average error of p training samples.
Figure GDA0002961106500000031
Wherein
Figure GDA0002961106500000032
JC、JRThe associated entropy of the center and radius, respectively, beta is the weight,
Figure GDA0002961106500000033
is a regular term. The goal is to maximize EMCCEquivalent to minimizing the cost function E ═ EMCC。
Training the interval neural network prediction model by maximizing the cost function in the step (3), and using the trained model for predicting the fluctuation interval of the stock market, wherein the specific process is as follows:
initializing the weight alpha of each layerj、wjiIs a random number between (-0.5,0.5), in each iteration process, randomly selecting p samples from the training set, for each sample, calculating the predicted fluctuation interval according to the model structure and formula, and the training input in each time is n interval value data { X }i1, wherein
Figure GDA0002961106500000034
Is the ith strand-finger interval,
Figure GDA0002961106500000035
is the center of the interval and,
Figure GDA0002961106500000036
is the radius of the interval. The inputs to the hidden layer unit are:
Figure GDA0002961106500000037
the hidden layer unit contains a non-linear activation function g (S)j) Then its output is:
Aj＝g(Sj) (6)
final output fluctuation interval
Figure GDA0002961106500000038
Is a linear combination of hidden layer cell output and bias:
Figure GDA0002961106500000039
according to the output interval
Figure GDA00029611065000000310
Calculating the cost function designed in the step (3) for two layers of weight values alphaj、wjiThen calculating the sum of the gradients of p samples
Figure GDA00029611065000000311
The parameters are adjusted according to the learning rate as follows:
Figure GDA00029611065000000312
Figure GDA00029611065000000313
in which eta is gradient descent learning rateAnd (4) rate. And adjusting the parameters until an iteration stop condition is reached, namely calculating the total error on the training set after each adjustment, stopping iteration when the total error is smaller than the training precision or the iteration times is larger than the maximum iteration times, and otherwise, continuing the next iteration. After the iteration is stopped, the optimized parameter alpha is obtained at the momentj、wjiThe stock index data can be tested and applied to predict the fluctuation interval of the stock market. Index interval { X) n days before inputiN, and obtaining a predicted fluctuation interval X according to calculation formulas (5), (6) and (7) of the interval neural network prediction outputn+1。
The stock market fluctuation interval prediction model based on the maximum correlation entropy criterion has the advantages that the stock market fluctuation interval prediction model based on the maximum correlation entropy criterion can realize modeling and prediction of a stock index interval, and more information can be provided compared with a traditional single-value stock market prediction method. The model uses the criterion based on the information theory during training, and overcomes the limitation that the common minimum mean square error criterion depends on the Gaussian premise hypothesis. From the test effect of the model in practical application, the method has better accuracy and robustness under the condition of non-Gaussian noise, and the application range is wider.
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FIG. 1 is a block diagram of a neural network model according to the present invention;
FIG. 2 is a flow chart of the present invention for forecasting the fluctuation interval of stock market;
FIG. 3 is a graph showing the predicted results of example 1 of the present invention;
Detailed Description
In order to make the advantages and features of the present invention more clear, embodiments of the present invention are further described below with reference to the accompanying drawings. The invention comprises the following steps:
1. acquisition and processing of stock index data
Obtaining daily highest price and lowest price historical data of stock indexes in a certain time period from various financial databases, and recording the daily highest price as xHThe lowest price is denoted as xLThen, the data can be arranged into a fluctuation interval X ═ X of the stock marketL,xH]It is also possible to write the center and radius form of the interval, i.e. X ═<xC,xR>Wherein x isCIs the midpoint of the fluctuation interval, xRIs the radius of the fluctuation interval;
2. establishment and initialization of interval neural network prediction model
The structure of the model in the invention is shown in fig. 1, and the Interval Neural network model described in the literature [ a.m.roque, c.mate, j.aroyo, et al, "iMLP: Applying Multi-Layer vertices to Interval-Valued Data," Neural Processing Letters,2007 ] "is adopted, and the structure of the model is a three-Layer Neural network, and comprises: the device comprises an input layer of n input units, one or more hidden layers containing h nonlinear activation function units and an output layer of m output units. Without loss of generality, we consider a hidden layer with h active units and an output layer with an output unit, so the model has two layers of weights αj、wjiNeeds to be removed, where αjIs the weight, w, from the hidden layer to the output layerjiIs the weight of the input layer to the hidden layer. These weights are all single-valued, while the n inputs and one output are index data in the form of interval values.
After building the basic structure of the model, the model is trained to optimize the parameters. Firstly, basic parameters of a model are required to be set, and the basic parameters include an input layer unit number n, a hidden layer unit number h, an output layer unit number m, a sample batch size p, a cost function weight beta of a center and a radius, a learning rate eta, a model training precision tol, a maximum iteration number max _ iters and a width sigma of a Gaussian kernel function in a cost function.
3. Designing cost function based on information theory learning
In order to adjust the weight of the model, a reasonable cost function needs to be defined to measure the error between the estimated output and the expected output of the model. The invention introduces the concept of the related entropy in the information theory learning, provides two random variables X and Y, and the related entropy between the two random variables is defined as:
V(X,Y)＝E[κ(X,Y)]＝∫∫κ(x,y)fXY(x,y)dxdy (1)
where κ (x, y) is the kernel function, fXY(X, Y) is a joint probability distribution function of X and Y. In practical cases, the joint distribution of the random variables X and Y is usually unknown, and only some limited values of them are known, in which case we can estimate the correlation entropy by using a sample averaging method:
Figure GDA0002961106500000051
where N is the number of samples, X (i) and Y (i) are the i-th sample observations of X and Y, respectively; when the correlation entropy is maximized, the probability that the error between the estimated value and the expected value is 0 is the largest, and thus the maximum correlation entropy criterion is defined as:
Figure GDA0002961106500000052
wherein
Figure GDA0002961106500000053
Is a gaussian kernel, σ is the width of the gaussian kernel, where e (i) ═ x (i) -y (i).
The parameter training of the model is carried out under the maximum correlation entropy criterion (MCC), the cost function adopted by the model is the weighted sum of the center correlation entropy and the radius correlation entropy, wherein t represents the sequence number of the training sample, and the calculation result is the average error of p training samples.
Figure GDA0002961106500000054
Wherein
Figure GDA0002961106500000055
JC、JRThe associated entropy of the center and radius, respectively, beta is the weight,
Figure GDA0002961106500000056
is a regularization term, our goal is to maximize EMCCIs equivalent to the maximumReducing cost function E ═ EMCC. The regular term in the cost function has the functions of constraining the parameters and introducing the prior knowledge of the parameters to be estimated, so that the over-fitting phenomenon is prevented, and the generalization capability of the model is improved.
4. Updating model parameters by adopting a random gradient descent method and predicting
We iterate the parameters using an error back-propagation algorithm similar to the standard multi-layer perceptron, where the gradient descent process uses a small batch of random gradient descent. During each iteration, a small number of samples, called Batch Size (Batch Size), are randomly selected from all training examples and input into the model. Then, the cost functions on the output and the samples are calculated, and then the derivative of the parameter is subjected to gradient descent according to the cost functions.
The specific parameter update process can be described as follows: training data is input, and random numbers with weights of (-0.5,0.5) in each layer are initialized. In each iteration process, p samples are randomly selected from a training set, and each sample consists of n input intervals and one output interval. For each sample, its forward output is based on the network weight and formula
Figure GDA0002961106500000061
The calculation is as follows:
each training input is n interval value data { X i1, wherein
Figure GDA0002961106500000062
Is the ith strand-finger interval,
Figure GDA0002961106500000063
is the center of the interval and,
Figure GDA0002961106500000064
is the radius of the interval. Then the output of the jth hidden layer unit is a weighted linear combination of the n inputs and the offset, and the output result is:
Figure GDA0002961106500000065
according to the interval arithmetic addition A + B ═ aL+bL,aU+bU]＝<aC+bC,aR+bR>And interval arithmetic multiplication, since wjiMay be negative, so special attention should be paid to the multiplication of an interval with a real number.
Figure GDA0002961106500000066
The output result is thus written in the form of a center and radius:
Figure GDA0002961106500000067
the output also needs to go through the activation function g (S)j) Processing to obtain an activation output Aj＝g(Sj)。
We use the tanh function as the activation function for the hidden layer unit, and since this function is monotonic, the output result of the activation process can be calculated as:
Figure GDA0002961106500000068
finally, the whole neural network outputs
Figure GDA0002961106500000071
From the linear combination of the activation outputs of all hidden units and the bias:
Figure GDA0002961106500000072
according to network output
Figure GDA0002961106500000073
Heshi (Chinese character of' HeshiCalculating cost function E about two-layer weight alphaj、wjiGradient values of (omitting the regularization term):
Figure GDA0002961106500000074
Figure GDA0002961106500000075
wherein the values of the respective derivatives are:
Figure GDA0002961106500000076
Figure GDA0002961106500000077
Figure GDA0002961106500000078
Figure GDA0002961106500000079
Figure GDA00029611065000000710
determining the sum of the gradients of p samples
Figure GDA0002961106500000081
The parameters are then adjusted according to the learning rate as follows:
Figure GDA0002961106500000082
Figure GDA0002961106500000083
and calculating the total error on the training set after the parameters are adjusted, stopping iteration when the total error is smaller than the training precision or the iteration times is larger than the maximum iteration times, and otherwise, continuing the next iteration. And obtaining optimized parameters after iteration is finished, and finishing the training of the model. And then, stock index test data can be input, and the model is used for predicting the stock index fluctuation interval.
It can be seen from the above steps that, when the stock market is predicted, the volatility of the stock index can be reflected by the interval data. And the maximum correlation entropy criterion in the information theory learning is adopted, the type of data noise is not limited, the application range is wider, and the robustness is stronger.
Example 1
In order to verify the performance of the stock market fluctuation interval prediction method based on information theory learning and compare the effect with the effect of the interval neural network prediction method based on MSE, the method is applied to the prediction of national upper syndrome indexes. We selected the highest and lowest prices of the upper syndrome index (code 000001) of 539 days from 1/2015 to 3/2017/3/21 as data sets, 428 days from 1/2015 to 9/2016 and 30 as training sets, and 111 days from 10/2016 to 3/2017/21 as test sets. The number of input layer units n is 8, the number of hidden layer units h is 15, the number of output layer units m is 1, the sample batch size p is 5, the weight β of the cost function of the center and the radius is 0.5, and the width σ of the correlation entropy kernel function is 0.3. The prediction results and the actual interval values on the test set are shown in figure 3, and it can be seen that the method of the invention can better predict the fluctuation interval of the stock market. The following table 1 compares the method of the present invention with the MSE-based interval neural network prediction method, and the results show that the errors of the upper bound, the lower bound, the center and the radius of the prediction interval of the present invention are all smaller than the MSE-based interval neural network prediction method, so that a better prediction effect is obtained.
TABLE 1
Figure GDA0002961106500000084
The scope of the invention is not limited to the description of the embodiments.

Claims (2)

1. A construction method of a stock market fluctuation interval prediction model based on information theory learning is characterized in that the method is used for establishing an interval neural network prediction model, and the process is as follows:
the method for predicting the fluctuation interval by establishing the interval neural network model comprises the following steps: the input layer of n input units, one or more hidden layers containing h nonlinear activation function units, the output layer of m output units, the input and output of the model are interval value-form stock index data, and there are two layers of weight values alphaj、wjiRequires training, where αjIs the weight, w, from the hidden layer to the output layerjiIs the weight from the input layer to the hidden layer;
the correlation entropy in the information theory learning is introduced to define a new cost function for training the model, two random variables X and Y are given, and the correlation entropy between the two random variables X and Y is defined as:
V(X,Y)＝E[κ(X,Y)]＝∫∫κ(x,y)fXY(x,y)dxdy (1)
where κ (x, y) is the kernel function, fXY(X, Y) is a joint probability distribution function of X and Y, and the correlation entropy is estimated by adopting a sample averaging method:
Figure FDA0002961106490000011
where N is the number of samples, σ is the width of the kernel, X (i) and Y (i) are the i-th sample observations of X and Y, respectively; when the correlation entropy is maximized, the probability that the error between the estimated value and the expected value is 0 is the largest, and thus the maximum correlation entropy criterion is defined as:
Figure FDA0002961106490000012
setting the estimated output interval as
Figure FDA0002961106490000013
The expected output interval is Y', the parameter training of the model is carried out under the maximum correlation entropy criterion (MCC), and the cost function E is adoptedMCCThe weighted sum of the center and radius associated entropies, where t represents the number of training samples, is calculated as the average error of p training samples:
Figure FDA0002961106490000014
wherein
Figure FDA0002961106490000015
JC、JRThe associated entropy of the center and radius, respectively, beta is the weight,
Figure FDA0002961106490000016
is a regularization term with the goal of maximizing EMCCEquivalent to minimizing the cost function E ═ EMCC。
2. The method for constructing the stock market fluctuation interval prediction model based on the information theory learning as claimed in claim 1, wherein the interval neural network prediction model is trained by a maximized cost function, and the trained model is used for stock market fluctuation interval prediction, and the specific process is as follows:
initializing the weight alpha of each layerj、wjiIs a random number between (-0.5,0.5), in each iteration process, randomly selecting p samples from the training set, for each sample, calculating the predicted fluctuation interval according to the model structure and formula, and the training input in each time is n interval value data { X }i1, wherein
Figure FDA0002961106490000021
Is the ith strand-finger interval,
Figure FDA0002961106490000022
is the center of the interval and,
Figure FDA0002961106490000023
is the radius of the interval; the inputs to the hidden layer unit are:
Figure FDA0002961106490000024
the hidden layer unit contains a non-linear activation function g (S)j) Then its output is:
Aj＝g(Sj) (6)
final output fluctuation interval
Figure FDA0002961106490000025
Is a linear combination of hidden layer cell output and bias:
Figure FDA0002961106490000026
according to the output interval
Figure FDA0002961106490000027
Calculating the cost function designed in the step (3) for two layers of weight values alphaj、wjiThen calculating the sum of the gradients of p samples
Figure FDA0002961106490000028
The parameters are adjusted according to the learning rate as follows until an iteration stop condition is reached:
Figure FDA0002961106490000029
Figure FDA00029611064900000210
wherein eta is the learning rate of the gradient descent method, the total error on the training set after the parameters are adjusted is calculated, the iteration is stopped when the total error is smaller than the training precision or the iteration times is larger than the maximum iteration times, otherwise, the next iteration is continued; after the iteration is stopped, the optimized parameter alpha is obtained at the momentj、wjiThe stock index data can be tested and applied to predict the fluctuation interval of the stock market: index interval { X) n days before inputiN, and obtaining a predicted fluctuation interval X according to calculation formulas (5), (6) and (7) of the interval neural network prediction outputn+1。
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