Recognition of amyotrophic lateral sclerosis disease using factorial hidden Markov model

Gait data description

The gait data provided by Hausdorff et al. were used in this study [14, 15]. These data can be accessed via the physionet web page (http://www.physionet.org) [25]. The dataset includes the gait data from 16 healthy and 13 ALS subjects. The mean (standard deviation) age of ALS patients and healthy subjects who participated in the aforementioned study were 55.6 (12.8) and 39.3 (18.5), respectively. In order to extract the gait information, force sensors were located in the foot’s shoe of each subject. Then, the stride interval time representing the time from the contact of a foot to the ground to the following contact of the same foot was extracted from these data.

In order to remove the unwanted artifacts in the beginning of the movements, the first 20s of the stride time signals were removed. Based on the experimental setup described in Ref. [14], subjects were requested to walk through a hallway during the recording of the gait signals. Because of the limited length of the hallway for walking, the subjects had to turn around at the end of the hallway, and this produced large peaks at the extracted gait signals. To remove the outliers, the samples with 3 SDs larger or less than the median value were substituted with the mean value of the related gait signals [12].

Pattern recognition

The general structure of the proposed model for classification of stride interval times is depicted in Figure 1. After preprocessing of gait data and removing outliers from them, the related features that represent the most important information about data should be extracted from the raw preprocessed gait data. In the next step, these features are used as the inputs of classifiers to distinguish the gait data of ALS subjects from the healthy ones.

Figure 1:

The general structure of the algorithm for distinguishing ALS patients from healthy subjects.

FHMM method

The general structure of standard HMM model is depicted in Figure 2A. As can be seen, a standard HMM model is composed of two separate layers including the hidden state and observation layers. In a standard HMM, each state in time depends on only its previous state. Furthermore, the observation in each time depends only on its current state. In each time, the HMM system has one state, and the transition between states is defined based on the related probability. Furthermore, the connections between each state and observations are defined based on the associated probability. For further information about the HMM, refer to Ref. [27].

Figure 2:

The structure of two HMM model.

(A) A typical structure of standard HMM. (B) A typical structure of a FHMM with three layers.

There are other frameworks of HMM such as the FHMM, which has a more complex structure than the standard HMM and can be used to obtain better performance. This method was introduced by Ghahramani et al. for the first time for the modeling of stochastic random processes [11]. The FHMM has a complex structure that is composed of different layers. As can be seen in Figure 2B, there is no connection between the hidden layers, and so each layer is independent from the other layers. Moreover, the observation layer depends on the current state of all the hidden layers. So, in the FHMM, each state variable is composed of a combination of states, and by considering the M hidden layers, we have now a new state structure named “meta-state” in the following form:

Here, S_t, S_t^(M) represent the “meta-state”, the state of the m_th layer at a time, respectively. In the simple form of FMMM, for each layer, the same number of states can be considered. For example, for a FHMM with M layers, a structure with M k*k is required. By this definition, the FHMM structure can be considered as a standard HMM with a K^M*K^M transition matrix. Moreover, by considering the previous assumption that each “meta-state” is independent from other state variables, the conditional probability of states results to the following form:

For the calculation of this probability, the Ghahramani et al. method has been used in the current study [11]. According to their work, this probability can be modeled using a Gaussian PDF with a common mean and covariance. Equation 3 shows this PDF:

where μm|St represents the mean of layer M, and C represents its covariance. In order to estimate the parameters of the FHMM model, the expectation maximization (EM) algorithm can be used. These parameters are the mean and covariance of all states in each layer, the transition and the prior probability matrices. The details about this method are given in Ref. [11].

Gait data analysis

In this section, the results of the analysis of the gait data to distinguish between healthy and ALS subjects are introduced. In Figure 3, two examples of the raw stride interval time signal corresponding to an ALS patient and a healthy subject are shown. As can be seen in Figure 3A, two apparent features that show the difference between ALS and healthy subjects can be identified. Furthermore, as can be seen in Figure 3B, this difference is not completely significant for the raw gait data of all the subjects. However, the case shown in Figure 3B is rather rare in the gait dataset.

Figure 3:

Two example of stride interval time series of healthy and ALS subjects.

(A) An example of stride interval time series obtained from a patient with ALS (blue) and a healthy subject (red). (B) Another example of stride interval time series obtained from a healthy and ALS subject.

First, the average of the stride time during walking in the ALS subject is much higher than the average of the stride time in the healthy subject. Second, the variance of the stride times in the ALS patient is higher than the variance of the stride times in the healthy subject. According to Table 1, the Wilcoxon rank sum test shows that the average and variance of the stride interval times in the ALS subjects have been significantly different from those in healthy subjects (p<0.001). The p-value obtained from the Wilcoxon rank sum test is an indicator for measuring the significant difference of the statistical parameters. The comparison between p-the values for the mean and variance parameters shows that the mean parameter may represent the difference between the ALS and healthy gait signals better than the variance parameter.

As was told previously, the HMM-based classifiers are used to classify gait patterns. So, in this work, two features extracted from the raw stride interval time signals in terms of local features have been used as the inputs of classifiers. First, the local average of the raw gait data is selected as the input feature. For extracting this feature, the raw gait data of both ALS and healthy subjects have been segmented to 50 segments, and the corresponding average of each segment was selected for the classification task. Second, the local variance of the raw gait data is selected as an optimal feature for classification. For extracting this feature, the gait data of both the ALS and healthy subjects have been segmented to 50 segments, and the corresponding variance of each segment was selected for classification. So, the selected feature as the input of each classifier is the sequences of 50 points. In Figure 4, the local variances of the gait data corresponding to all subjects (13 ALS and 16 healthy subjects) are shown. As can be seen in this figure, the difference of the obtained local variances between the ALS and healthy gait data is significant in some cases.

Figure 4:

The comparison between the local variance of all 13 ALS and 16 healthy stride interval time series obtained from segmentation of raw gait data to 50 segments and computation of variance in each segment.

Classification results

In this section, the results of the classification of the gait data in terms of stride interval times of both the ALS and healthy subjects are presented. The classification step can be used to show the effectiveness of the proposed gait features. As was told in the previous section, two different features were selected for the training of classifiers. In this study, we applied these features as the input of the standard HMM and FHMM classifiers separately, and then, the performance of each classifier based on these features have been compared in terms of accuracy of recognition. The classification based on the standard HMM was performed by the Murphy Toolbox in the Matlab environment (The Mathworks, Inc, Natick, MA, USA). Also, the FHMM method was implemented using the Ghahramani Toolbox written in Matlab.

To optimize the hyper-parameters of the HMM model, in each phase, the gait data of 28 subjects were used for training, and the parameters that resulted in the highest classification performance were selected. So, with 29 subjects, we have optimized the hyper-parameters 29 times, each time using data from 28 subjects. The ranges of these parameters such as number of Gaussian mixtures and number of states were selected 3–8, 3–8, respectively. In Table 2, the structure of the HMM model is presented. The same procedure was used to optimize the hyper-parameters of the FHMM model, too. In Table 3, the structure of the FHMM model is shown.

The classification accuracy results of the standard HMM and FHMM on the first features (local mean of stride interval times) are shown in Table 4 and the results on the second features (local variance of stride interval times) in Table 5. In order to investigate the performance of each classifier, the leave-one-out (LOO) cross-validation method was used in the current study [9]. In this paper, we have the data from 29 subjects, and so in each step, the data from 28 subjects were used for training, and the data from the remaining one subject was used for testing. This procedure was repeated 29 times, and as mentioned before each time, the hyper-parameters were selected separately, and then, the percent of the true classification rate was calculated.

Furthermore, for evaluation of the performance of each classifier on the detection of ALS and healthy subjects separately, the sensitivity and specificity parameters were used. In the current study, the sensitivity parameter measures the percentage of ALS subjects who are correctly recognized as ALS patients. Furthermore, the specificity parameter defines the percentage of healthy subjects who are correctly recognized as healthy. According to Table 4, using the local mean of stride interval time signal as the input features, one ALS and one healthy subject were classified incorrectly by the FHMM (sensitivity: 92.31%; specificity: 93.75%). For the standard HMM method on the same features, two ALS and one healthy subject were classified incorrectly (sensitivity: 84.62%; specificity: 93.75%). These results show that the performance of the FHMM is better than the standard HMM in recognition of the true class of each subject when local mean features have been used as the input of each classifier.

Furthermore, in Table 5, the results of the classification of the ALS and healthy subjects based on the local variance features are shown. As can be seen, using these features, the FHMM shows better performance than the standard HMM in distinguishing ALS patients from the healthy subjects. For both the FHMM and standard HMM, only one healthy subject is recognized incorrectly, but the accuracy of the FHMM method in recognition of the ALS subjects is better than the HMM method. The Fisher’s exact test was used to investigate the statistical significance of the obtained accuracies that resulted from the HMM and FHMM classifiers evaluated by LOO cross validation. The results show that the performance of the FHMM method is significantly better than the HMM method in both cases (p<0.05).

In order to compare our results with other studies worked on these gait dataset, the best classification results obtained from the FHMM method on the local mean features are compared with the LS-SVM algorithm represented in Ref. [30]. In summary, based on the algorithm proposed in Ref. [30], at the first step, the PDFs of the stride interval times series corresponding to 13 ALS and 16 healthy subjects were estimated by the Parzen-window method, and then, the mean of the left foot stride interval times and the Kullback-Leibler divergence (MKLD) parameter were selected as the inputs of the LS-SVM classifier with Gaussian kernels. The parameters of the classifiers, such as the type of kernels, variance parameter, and regularization parameter, were selected in the first phase when the gait data corresponding to all the subjects were used for both training and test of the classifier. After choosing the optimum parameters, the LOO cross-validation method was used for evaluation of the classification.

According to the investigated results shown in Table 6, the performance of the FHMM method in the detection of both the ALS and healthy subject is better than that of the LS-SVM algorithm. Furthermore, the comparison of the overall accuracy shows that the performance of the proposed method based on the local mean of the gait data as the input features is superior than that of the LS-SVM algorithm used in Ref. [30].