Human action recognition based on mixed gaussian hidden markov model

MATEC Web of Conferences 336, 06004 (2021) CSCNS2020 https://doi.org/10.1051/matecconf/202133606004 Human action recognition based on mixed gaussian hidden markov model Jiawei Xu1,*, and Qian Luo1 1 Beijing Information Science and Technology University, 35 North Fourth Ring Middle Road, Yayun Village Street, Chaoyang District, Beijing, China Abstract. Human action recognition is a challenging field in recent years. Many traditional signal processing and machine learning methods are gradually trying to be applied in this field. This paper uses a hidden Markov model based on mixed Gaussian to solve the problem of human action recognition. The model treats the observed human actions as samples which conform to the Gaussian mixture model, and each Gaussian mixture model is determined by a state variable. The training of the model is the process that obtain the model parameters through the expectation maximization algorithm. The simulation results show that the Hidden Markov Model based on the mixed Gaussian distribution can perform well in human action recognition. 1 Introduction Due to the development of sensors, the data used for human action recognition has become more abundant [1]. People are gradually turning their attention to the field of human action recognition. Many methods have tried to solve the problem of human action recognition, such as random forest [1], variants of random forest [2], graph convolutional neural networks [3], Deep Progressive Reinforcement Learning [4], Directed Graph Neural Networks [5] and so on. Most of these methods are based on constructing action features, and use discriminative methods to complete the recognition task. Hidden Markov model establishes the joint probability distribution of hidden variables and observation variables and introduces mixed Gaussian distribution to approximate the real distribution of human movements, which can express the distribution of the data themselves better. It can perform well in human action recognition. 2 Mixed gaussian hidden markov model Hidden Markov model is a kind of dynamic Bayesian network. It is used to process time series data. The model consists of the initial state probability vector π, the state transition probability matrix A={aij} (i=1,..,m;j=1,…,n) and the observation probability matrix B={bjk} (j=1,..,m;k=1,…,n). m is the number of state variables Z={ z1，…，zm} in the model. When * Corresponding author: © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 336, 06004 (2021) CSCNS2020 https://doi.org/10.1051/matecconf/202133606004 the hidden Markov model is based on Gaussian mixture distribution, n is the number of Gaussian distribution in Gaussian mixture model determined by state variable z（j=1,…， m）. j The process of model training is to determine the values of π, A, and B. In view of the complexity of human actions, this paper applies Gaussian mixture model to approximate the real distribution of human action data. The Gaussian mixture model is as formula (1). n p ( x ) = ∑ α (k ) N ( x | uk , Σ k ) (1) k =1 The Gaussian mixture distribution is composed of n Gaussian distributions. α (k ) is n the weight of each Gaussian distribution, where ∑α (k ) = 1 . In the mixed Gaussian k =1 hidden Markov model, each state variable zj has one Gaussian mixture model itself, and α (k ) is the element bjk of the observation probability matrix B. Meanwhile, α (k ) , uk ,and Σ k are parameter values that need to be determined during model training. The Hidden Markov Model assumes that human actions are controlled by a set of unobserved hidden variables, namely state variables Z. The state variables Z may be a set of postures in practice, but may be other unobservable factors. π and A determine the state time series Z={z1，…，zt}, and B determines the observation time series X={x1，…，xt}. The generation process of the observation sequence is to select a state variable zt at a certain time t. Next, select a specific Gaussian mixture model according to zt , then an observation sample xt is generated from the selected Gaussian mixture distribution. The generation process of observation xt+1 at time t+1 is as follows. Firstly, zt and the state transition probability matrix A determine zt+1, and then xt+1 is generated by the Gaussian mixture distribution of zt+1. The initial state is generated from the initial state probability vector π. It can be seen that the hidden Markov model uses random sampling controlled by a hidden Markov chain to explain the generation of observation sequences. 3 Expectation maximization algorithm In order to solve the model parameters, this paper uses the Expectation Maximization(EM) algorithm. The EM algorithm is essentially an iterative update algorithm. The iteration process is divided into E step and M step. E step is to find the expectation of the conditional probability of the hidden variable Z of the log-likelihood function of the complete data, as in formula (2). M step finds the parameter value θ that maximizes the expectation in E step, as in formula (3). Q(θ ,θ (i ) ) = ∑ log P( X , Z | θ ) P( Z | X ,θ (i ) ) Z θ (i +1) = arg max Q(θ ,θ (i ) ) θ (2) (3) θ (i ) represents the parameter value obtained in the i th iteration in equations (2) and (3). (i ) ( i +1) The value θ is the θ which is obtained by maximizing Q (θ ,θ ) based on the 2 MATEC Web of Conferences 336, 06004 (2021) CSCNS2020 fixed θ (i ) https://doi.org/10.1051/matecconf/202133606004 in the conditional probability P ( Z | X ,θ (i ) ) . θ (i +1) is the parameter value used in the i+1th iteration to compute Q (θ ,θ ). In the Mixed Gaussian Hidden Markov Model, the parameters needed to be solved are aij, bjk, uk and Σ k . The update process of parameters is as follows given the initial values ( i +1) π0, a0, b0, u0 and Σ 0 . T −1 ∑ ξ (i, j ) aij = t t =1 T −1 K ∑∑ γ (i, k ) t t =1 k =1 (4) T b jk = ∑ γ ( j, k ) t t =1 T K ∑∑ γ ( j, k ) t t =1 k =1 (5) T u jk = ∑ γ ( j, k ) ⋅ x t =1 t t T ∑ γ ( j, k ) t =1 t T Σ jk = ∑ γ ( j, k ) ⋅ ( x t =1 T ∑ γ ( j, k ) t =1 where ξt (i, j ) − u jk )( xt − u jk )T t t (6) t (7) represents the probability of being in state i at time t and being in state j at time t+1 given models and observations. γ t ( j, k ) is the probability that the observation is generated by the k th Gaussian component in the state j at time t. These two variables can be obtained by the forward-backward algorithm. Returning to the EM algorithm, the process of using the forward-backward algorithm to obtain the statistics ξ t (i, j ) and γ t ( j, k ) is equivalent to E step. The parameter update process of formulas (4) ~ (7) is M step. E step and M step iteratively update until convergence, and the final required model parameters are obtained. 4 The sim (...truncated)