Stock Market Trend Analysis Using Hidden Markov Model and Long Short Term Memory

Gaussian Mixture-Hidden Markov Model (GMM-HMM) has been widely used in automatic speech recognition (ASR) and has made much success.

Motivated by its application, our task is: given a observation sequence, find most likely hidden state sequence, and then find relation between hidden state and stock market trend by using Long-Short Term Memory.

Denote observation sequence $O=O_{1}O_{2}...O_{T}$, where $O_{t}$ is the observation of time t. Hidden state sequence $S=S_{1}S_{2}...S_{T}$, where $S_{t}$ is the hidden state of time t. The most likely hidden state is:

In this section, we will introduce core algorithms of GMM-HMM in predicting stock price and its practical operation

Here are two basic assumptions when applying the model. (1) The state at any time $t$ depends on its state at time $t-1$, and is independent of any other moment.

(2) The observation of any time $t$ depends on its state at time $t$, and is independent of any others.

GMM is a statistical model which can model $N$ sub population normally distributed. HMM is a statistical Markov model with hidden states. When the data is continuous, each hidden state is modeled as Gaussian distribution. The observation sequence is assumed to be generated by each hidden state according to a Gaussian mixture distribution [1].

The hidden Markov model is determined by the initial probability distribution vector $\pi$, the transition probability distribution matrix A, and the observation probability distribution called emission matrix B. $\pi$ and A determine the sequence of states, and B determines the sequence of observations.

In GMM-HMM, the parameter B is a density function of observation probability, which can be approximated as a combination of products of mixed Gaussian distributions. By evaluating each Gaussian probability density function weight $w_{jk}$, mean and covariance matrix, the continuous probability density function can be expressed as:

in which $w_{jk}$ represents the weight of each part. Fig. 4 describes the basic structure of GMM-HMM model.

Here we introduce the process of GMM-HMM algorithm.

1. 1)

   Determine the number of hidden states N

2. 2)

   Estimate model parameters $\lambda=(A,B,\pi)$ by using Baum-Welh algorithm at a given observation sequence O. Record model parameters. Record model parameters $\lambda=(A,B,\pi)$ as gmm-hmm.

3. 3)

   Estimate probability $P\{S_{t}=i\},i=\{1,2,..,N\},t=\{0,1,..,T-1\}$ by using Viterbi algorithm in gmm-hmm.

In order to test the effect of our model, we select a number of stocks for model training and determine the various parameters of the model.

For better understanding, we choose one of the stocks Fengyuan Pharmaceutical (000153.XSHE) which is from 2007-01-04 to 2013-12-17 to do the visualization.

We set N=3, and use market data and predict the state at each moment. We use market information as a feature and set new model as gmm-mm1.

Then put the predicted state for time $t$ in gmm-mm1 and its close price together to make visualization. The results of the training set are shown in Fig 6 and 6.

As can be seen from the figure, red represents the rising state, green represents the falling state, blue represents the oscillation, and intuitively gmm-mm1 works well on the training set.

Use the data of the stock Red Sun (000525.XSHE) from 2014-12-01 to 2018-05-21 as a test set, use the predicted state for time $t$ in gmm-hmm1 and its close price together to make visualization. From the Fig. 6 we can find that gmm-mm1 works well on the test set.

Then, we trained seven multi-factor features that were filtered to get the model $gmm-hmm_{i}$, $i=2,3,...8$, to prepare for the latter LSTM model.

In the above GMM-HMM model, we assume that the emission probability B is a Gaussian mixture model. Our team believes that in addition to the GMM distribution, we can use the scalable end-to-end tree propulsion system XGBoost to estimate the emission probability B.

In this section, we introduce a XGB-HMM model which exploits XGBoost instead of Gaussian mixture model (GMM) in estimating the emission probabiltiies $P\{O_{t}|S_{t}\}$. We will explain the training algorithms of XGBoost.

In order to test the effect of our model, we select one of the stocks for model training and determine the various parameters of the model.

We choose Fengyuan Pharmaceutical (000153.XSHE), the training set is from 2007-01-04 to 2013-12-17. By setting N=3, and use market data as a feature, we predict the state at each moment. The results of the XGB-HMM training set are shown in Fig. 10. The results of the XGB-HMM test set are shown in Fig. 10. The iteration diagram of the XGB-HMM algorithm is shown in Fig. 10.

As can be seen from the Fig. 10, orange represents the rising state, green represents the falling state, blue represents the oscillation, and intuitively XGB-HMM works well on the training set, and the model is $xgb-hmm_{0}$.

Use the data of the stock Red Sun (000525.XSHE) from 2014-12-01 to 2018-05-21 as a test set, use model as $xgb-hmm_{0}$ see in Fig. 10.

As can be seen from the Fig. 10, as the number of iterations increases, log-likelihood increases as well. When iteration reach about 250, log-likelihood tends to be stable. At the same time, as the number of iterations increases, score function of features increases.

Fig. 12 and 12 is the comparison of GMM-HMM and XGB-HMM results on test set and training set.

It can be seen that the result of using the XGB-HMM model is better, which makes the discrimination of the three hidden states higher, which means that XGB may have a better effect in fitting the observed data.

Moreover the results of XGB-HMM tends to be better than GMM-HMM.

The algorithm was first published by Sepp Hochreiter and Jurgen Schmidhuber on Neural Computation. The Long Short Term Memory model consists of a dedicated memory storage unit that controls the state of each memory storage unit through carefully designed forgetting gates, input gates and output gates. The control of each gate ensures the previous information can continue to propagate backwards without disappearing while the hidden layer is continuously superimposed with input sequence in the new time state [5].

LSTM can be regarded as a mapping from X to Y, in which X is a n $\times$ k matrix, Y is a vector with n columns, different X can map to one Y.

The symbols used in the figure have the following meanings:

The long and short memory network structure is mainly composed of three parts:

1. 1)

   Forget gate layer. The first step in LSTM is to decide what information should be thrown away from the cell state. The decision is made by a sigmoid layer called ‘forget gate layer’. After linearly combining the output information $h_{t-1}$ of the previous layer and the current information $X_{t}$, the function value is compressed by the activation function to obtain a threshold value between 0 and 1. The closer the function value is to 1, the more information the memory retains. The closer to 0, the more information the memory loses [3].

   

2. 2)

   Input gate layer. The input gate determines how much new information is added to the unit state. There are two steps to accomplish this: First, the sigmoid neural layer of the input gate determines which information needs to be updated; a tanh layer generates a vector that is used as an alternative to update the content $C_{t}$. Then, we combine the two parts and update the unit state.

   

   

3. 3)

   Output gate layer. Finally, the output gate determines what value to output. This output is mainly dependent on the unit state $C_{t}$ and also requires a filtered version. First, the sigmoid neural layer determines which part of the information in $C_{t}$ will be output. Next, $C_{t}$ passes a tanh layer to assign values between -1 and 1, and then multiplies the output of the tanh layer by the weight calculated by the sigmoid layer as the result of the final output [3].

   

On the previous work, we have done $gmm-hmm_{i},i=1,2,...,8$. For each $gmm-hmm_{i},i=1,2,...,8$, we get one $state\_proba,i=1,2,...,8$, and then put them together to get X.

1. 1)

   Run GMM-HMM,and get $P\{S_{t}=i\},i=\{1,2,..,N\},t=\{0,1,..,T-1\}$

2. 2)

   Use probability $P\{S_{t}=i\},i=\{1,2,..,N\},t=\{0,1,..,T-1\}$ to construct the training set X for LSTM

   $$X=\begin{pmatrix}P\{S_{1}=1\}......P\{S_{T-1}=1\}\\ P\{S_{1}=2\}......P\{S_{T-1}=2\}\\ P\{S_{1}=3\}......P\{S_{T-1}=3\}\end{pmatrix}$$

3. 3)

   Input X and Y into LSTM training

The stock selected by our group is Fengyuan Pharmaceutical, and the data from 2007-01-04 to 2013-12-17 is selected as the training set. The stock Red Sun data from 2014-12-01 to 2018-05-21 is selected as a test set.

The GMM-HMM model is trained on 8 training factors on the training set, and 8 state_proba matrices are generated to obtain X.

Start the training by entering this X into the LSTM model. The model completed in this training is $gmm-hmm+lstm_{0}$.

On the test set, run the $gmm-hmm+lstm_{0}$ model and output the result of the lstm model: the accuracy rate is 76.1612738%.

Above, our team has constructed the model $xgb-hmm_{i},i=1,2,...,8$. For each $xgb-hmm_{i},i=1,2,...,8$, you can get a $state\_proba,i=1,2,...,8$, and then merge them can get the matrix X.

1. 1)

   Run XGB-HMM, and get $P\{S_{t}=i\},i=\{1,2,..,N\},t=\{0,1,..,T-1\}$.

2. 2)

   Use probability $P\{S_{t}=i\},i=\{1,2,..,N\},t=\{0,1,..,T-1\}$ to construct the training set X for LSTM.

   $$X=\begin{pmatrix}P\{S_{1}=1\}......P\{S_{T-1}=1\}\\ P\{S_{1}=2\}......P\{S_{T-1}=2\}\\ P\{S_{1}=3\}......P\{S_{T-1}=3\}\end{pmatrix}$$

3. 3)

   Input X and Y into LSTM training.

The stock selected by our group is Fengyuan Pharmaceutical, and the data from 2007-01-04 to 2013-12-17 is selected as the training set. The stock Red Sun data from 2014-12-01 to 2018-05-21 is selected as a test set.

The XGB-HMM model is trained on 8 training factors on the training set, and 8 state_proba matrices are generated to obtain X.

Start the training by entering this X into the LSTM model. The model completed in this training is $xgb-hmm+lstm_{0}$.

On the test set, run the $xgb-hmm+lstm_{0}$ model and output the result of the lstm model: the accuracy rate is 80.6991611%.