Abstract
We propose a new multivariate Markov chain model for adding a new categorical data sequence. The number of the parameters in the new multivariate Markov chain model is only (3s) less than the number of the parameters in the former multivariate Markov chain model. Numerical experiments demonstrate the benefits of the new multivariate Markov chain model on saving computational resources.
1. Introduction
Markov chains are of interest in a wide range of applications, for example, telecommunication systems [1, 2], remanufacturing and inventory systems [3], speech recognition [4], PageRank [5–7], microbial gene [8], and AIDS [9]. In recent years, the predictions of data sequences have become more and more useful in other applications such as sales demand prediction [10], DNA sequencing [11], credit risk [12], and stock prices [13].
Different models have been proposed for multiple categorical data sequences prediction. A multivariate Markov chain model was proposed in [10] by Ching et al.; they constructed a new matrix by means of the transition probability matrices among different sequences. An improved multivariate Markov chain model had also been studied to speed up the convergent speed for computing the stationary or steady-state solutions [14]. In the improved multivariate Markov chain model, Ching et al. incorporated positive and negative association parts. The extensions of intensity-based models for pricing credit risk and derivative securities to the simulation and valuation of portfolios were discussed in [15]. Moreover, there are many other papers contributing to the multivariate Markov chain model, for example, [16–22] and so on.
With the developments of Markov chain models and their applications, the number of the sequences may be larger. It is inevitable that a large categorical data sequence group will cause high computational cost in multivariate Markov chain model. Thus, reducing the number of parameters in the models is useful in numerical computation. For the above reasons, we present a new multivariate Markov chain model for detecting the relations between the previous data sequences and the following data sequence.
The rest of the paper is organized as follows. In Section 2, we review two lemmas and several Markov chain models. In Section 3, the new multivariate Markov chain model is proposed for adding a new categorical sequence. Moreover, the convergence of the new multivariate Markov chain model is proved. Section 4 gives parameter estimation method for the new multivariate Markov chain model. Numerical experiments on sales demand prediction and stock prices prediction are presented to test the efficiency of the new multivariate Markov chain model in Section 5. Concluding remarks are given in Section 6. The data of the stocks’ prices are provided in the Appendix.
2. A Review on Markov Chain Models
In this section, we briefly introduce two lemmas, the Markov chain model [23] and the multivariate Markov chain model [10].
Lemma 1 (see [24, Perron-Frobenius theorem]). Let be a nonnegative and irreducible matrix. Then,(1)has a positive real eigenvalue equal to its spectral radius; that is, where denotes the th eigenvalue of ;(2)to there corresponds an eigenvector of its entries being real and positive, such that ;(3) is a simple eigenvalue of .
Lemma 2 (see [22]). Let be the iterative matrix of multivariate Markov chain model and let be the state distribution at time . If is irreducible and aperiodic, then there is a unique stationary distribution satisfying and .
2.1. The Markov Chain Model
Let the state set of the categorical data sequence be . The Markov chain satisfise the following relations: where , . The conditional probability is called one-step transition probability. If we rewrite the transition probability as the Markov chain model can be presented as follows: where Here, is the initial probability distribution and is the state probability distribution at time .
2.2. The Multivariate Markov Chain Model
The multivariate Markov chain model has the following form: where is the initial probability distribution of the th sequence, is the state probability distribution of the th sequence at time , is the state probability distribution of the th sequence at time , and is the one-step transition probability matrix from the states in the th sequence at time to the states in the th sequence at time . In the matrix form, (5) has Entries can be obtained directly from the categorical data sequences, and can be got by the linear programming [10].
3. A New Multivariate Markov Chain Model
In order to reduce the number of the parameters in multivariate Markov chain model, a new multivariate Markov chain model is proposed. Moreover, the convergent property of the new model is also analyzed.
Suppose that there are categorical data sequences and each of the sequences has possible states in . The multivariate Markov chain model for categorical data sequences has the form where where ,, and are defined the same as those in Section 2.2. In the matrix form, (8) is Transition probability matrix can be obtained directly by the categorical data sequences. The parameters can be solved from the corresponding linear programming.
Assuming that the multivariate Markov chain model for previous sequences is obtained, we add a new sequence at the back of the previous sequences. For detecting the relations between the previous categorical data sequences and the new categorical data sequence, a new multivariate Markov chain model is proposed and has the following form:where In the matrix form, (11) has Let Equation (13) is abbreviated as
Theorem 3. Let . If , then the iterative matrix has an eigenvalue equal to one and the modulus of all its eigenvalues are less than or equal to one.
Proof. Suppose that
From (9) and (12), each column sum of this matrix is equal to one and the matrix is nonnegative and irreducible. According to Lemma 1, there exists a positive vector
satisfying
Let . Since is a transition probability matrix, it has
It is clear that
with an eigenvalue of .
Now, our aim is to prove that the modulus of all the eigenvalues ofare less than or equal to one. Suppose that , satisfying . is similar to . From (20), it has . Then
Theorem 4. Assume that is irreducible,, and . Then there is a vector satisfying and
Proof. The proof is similar to Proposition 2 in [1] and therefore it is omitted.
To keep the irreducibility of , we fill the column of with when the column sum of is zero.
Theorem 5. Let be the stationary probability of the new multivariate Markov chain model. Then and .
Proof. From Lemma 2, our goal is to prove that is irreducible and aperiodic. Since is connected, is irreducible. Then we only need to prove that is aperiodic. Let ,. There exists such that . From the form of , we obtain . There exists , , satisfying , . Therefore, . Because is a positive matrix. Then is aperiodic. According to the above results, the conclusions of this theorem are obtained.
4. Parameter Estimation Method of the New Multivariate Markov Chain Model
Let be the states set and let be frequency from the state in the th sequence at time to the state in the th sequence at time . The transition frequency matrix is The transition probability matrix can be obtained by normalizing the transition frequency matrix as follows: where
Subsequently, the way of estimating the parameter will be introduced. Consider to be a joint stationary probability distribution. can be presented as satisfying One would expect that Certainly, (28) can be interpreted as where is small enough.
One way of estimating is to transform (29) into a minimization problem as the following form: The minimization problem (30) is identical to the following form: where is the th entry of the vector. Let the norm be . The above problem can be rewritten as a linear programming problem where
5. Numerical Experiments
In this section, numerical experiments with different multivariate Markov chain models on sales demand prediction and stock prices prediction are given. We report on numerical results obtained with a Matlab 7.0.1 implementation on Windows XP with 2.93 GHz 64-bit processor and 1 GB memory.
5.1. Sales Demand Prediction
In this section, the sales demand sequences are presented to show the benefits of the new multivariate Markov chain model. Since the requirement of the market fluctuates heavily, the production planning and the inventory control directly affect the estate cost. Thus, studying the interplay between the storage space requirement and the overall growing sales demand is a pressing issue for the company. Here, our goal is to predict the sales demand of the market for minimizing the estate cost. Assume that products are classified into six possible states ; for example, no sale volume, very low sale volume, low sale volume, standard sale volume, high sale volume, and very high sale volume. The customers’ sales demand data of five important products can be found in [10].
The multivariate Markov chain model of four categorical data sequences, , , , and , will be given. By computing the proportions of the occurrences of each state in each sequence, we formulate the initial probability distributions of four categorical data sequences The transition probability matrix can be obtained after normalizing the transition frequency matrix. By solving the corresponding linear programming problem, one can obtain . The multivariate Markov chain model is presented as follows: In order to uncover the relations of , , , , and , we add a new categorical data sequence at the back of the original categorical data sequences. With the data sequence of, the initial probability distribution of is obtained as follows:
In the multivariate Markov chain model, can be calculated by a corresponding linear programming problem. The multivariate Markov chain model is
After calculating by a corresponding linear programming problem, the new multivariate Markov chain model can be presented as follows: From the results of the new multivariate Markov model, and are closely related. Moreover, the sales demand of depends strongly on . The reason is that the chemical nature of and is the same, only used for different packaging of marketing purposes. , , and are closely related. The fact is that and have the same product flavor, only different in packaging.
In the following, we use the new multivariate Markov chain model and the multivariate Markov chain model to predict the state of the th sequence at time . The maximum probability, is taken as the state at time . For evaluating the effectiveness of the new multivariate Markov chain model, prediction results are measured by the prediction accuracy defined as where is the length of the data sequence and
Note that “” is CPU time, “” is the object function value of the corresponding linear programming problem, “” is the number of the parameters in the models, and the prediction accuracies of , , , , and are “,” “,” “,” “,” and “,” respectively. Suppose that the results of the multivariate Markov chain model of , , , and are obtained. The new multivariate Markov chain model for , , , and adding is denoted as “,,, add .” The multivariate Markov chain model is denoted as “Mmodel.” Stop criterion can be found in Matlab function . The results are presented in Table 1.
Observing the results from Table 1, we find that the object function values of the new model and the multivariate Markov chain model are the same. The prediction accuracies of our new models are comparable to the prediction accuracy of the multivariate Markov chain model. The new multivariate Markov chain model performs better than the multivariate Markov chain model in time consumption and controlled parameters.
5.2. Stock Prices Prediction
The data of 12 American stocks’ price from December 17, 2013, to January 16, 2014, are given in the Appendix. They are divided equally into 6 regions as 6 states between maximum price and minimum price of the stocks. The state set of 12 stocks is . The data of 12 stocks in the Appendix are transformed into categorical data sequences.
“,” “,” ,” and “” are denoted the same as those in Section 5.2. Note that ” is the multivariate Markov chain model of all stocks except AMAP. Suppose that the results of ” are obtained. The new multivariate Markov chain model which is denoted as ” can detect the relations of BIDU, CTRP, GA, EDU, SINA, SOHU, YOKU, XRS, QIHU, HTHT, HMIN, and AMAP. Stop criterion can be found in Matlab order . The results are presented in Table 2.
From Table 2, the object function values of the new multivariate Markov chain model and the multivariate Markov chain model are nearly the same. The CPU time of the new multivariate Markov chain model is the CPU time of the multivariate Markov chain model’s. The number of the parameters in the new multivariate Markov chain model is one-third of those in the multivariate Markov chain model. The new multivariate Markov chain model is better than the multivariate Markov chain model in time consumption and controlled parameters.
6. Conclusions
In this paper, a new multivariate Markov chain model is proposed. The convergence of the new model is proved. With the results of the multivariate Markov chain model for categorical data sequences, the relations of the categorical data sequences and the new sequence can be detected by our new model. The new multivariate Markov chain model only needs parameters less than which is the number of the parameters in multivariate Markov chain model. Numerical experiments illustrate the benefits of our new model in saving computational resources. The performances of the new multivariate Markov chain model are nearly the same as the multivariate Markov chain model in prediction. Certainly, our model can also be applied in credit risk and other research areas.
Appendix
Consider
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research is supported by the Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020).