Optimized Clustering Techniques for Gait Profiling in Children with Cerebral Palsy for Rehabilitation

Optimized Clustering Techniques for Gait Profiling in Children with Cerebral Palsy for... Abstract Cerebral palsy (CP) is a neuro-development disease in children. It is quite an intricate task to categorize gait pattern into normal and CP based pathology. In this study, nature-inspired meta-heuristic algorithms are explored on a publicly available gait dataset of 156 subjects for automatic gait profiling of children with cerebral palsy. Five cases are considered to explore the feature selection criteria before applying clustering technique. Finding the optimal number of clusters is a challenging task in the unsupervised learning area. In this study, an optimal number of gait profiles in the datasets is identified based on voting from mean square error, silhouette coefficient and Dunn index. The study demonstrates that optimized based gait profile clusters could assist quantitatively in clinical rehabilitation evaluation for the children affected by CP. 1. INTRODUCTION Cerebral palsy (CP) is a neuro-development disease, common in children. It is associated with the floppy or rigid limbs, exaggerated reflexes and involuntary motion, poor speech and learning ability which, is considered as a non-progressive disease [1, 2]. The major reason for this motor disability is abnormal brain development, which often occurs before the birth of a child. On behalf of motor impairment of the limb, CP is classified into three major categories; spastic, ataxic and athetoid [3]. Different studies, around the globe highlight the severity of this disease and projects that worldwide CP cases vary from 1.5 to 4 per 1000 children. The condition is more severe in the developing countries [1, 4]. Thus a huge amount of money is involved in the rehabilitation and intervention policies related to the diagnosis of gait pathology related to cerebral palsy [5, 6]. The diversity in the gait pattern from children suffering from CP makes it a challenging task for researchers in the physical therapy and surgical community. This will enable health-care professionals to differentiate gait pattern into clinically significant categorize that assist in diagnosis, assessment, evaluation of the treatment outcomes and clinical decision-making [7–11]. All pathological pattern cannot be captured through just visual observation. There is a need to take the aid of statistical (frequency domain characteristics, spectral components, harmonic content, the coefficient of variation, etc.) and machine learning techniques [12]. Clinical gait analysis is a method to identify the hidden pattern in gait such as stride length, step length, cadence, stance and swing phase, etc. Clinicians can utilize gait classification concepts in their routine clinical practice to evaluate a patient’s status, treatment and rehabilitation for CP disorders using the spatio-temporal and kinematics parameters [7, 13, 14]. There are several attempts made by researchers to characterize the gait pattern and categorize them into normal and gait pathology using computational techniques [15–19]. Supervised learning is a popular choice for the research community to classify the gait pattern of CP pathology for automated analysis. Neural network is most widely used for normal gait analysis, robotic rehabilitation, sports monitoring, and tactics, geriatric care, surveillance, activity recognition [7, 20–23]. Supervised learning technique is a dominant methodology when labeled training data points are available. Zheng et al. [24] implemented the trained neural network classifier to identify the abnormality in older person gait. Gait pathology can be identified using ground reaction force with learning vector quantization. Multilayer perceptron, Linear discriminant analysis classifiers, kernel Fisher discriminant and Bayesian classifier have been used to classify gait pattern in [25, 26]. Support vector machine (SVM) is used for the study of normal and age-related differences, normal and abnormal gait pathology. The authors in [27, 28] showed that it exhibits good results in CP pathology. SVM is a powerful classifier suitable from small to the medium dataset. The limitation of these supervised approaches is that their accuracy depends on the training sample. The larger the training sample, the more accurate the system is. Another limitation is that these techniques cannot be applied to the data where no prior information of the data is given. To overcome these limitations, researchers have explored unsupervised, or clustering approaches [29, 13, 30]. Cluster models are not only popular in signal and image processing, but they also have wide applications in web mining and pattern recognition, sensor network, robotics, seismology, medical science in disease clustering, etc [31, 32]. In gait analysis, clustering techniques are most widely used for categorizing gait data into groups of disorders based on common hidden features in the subject dataset. K-means, fuzzy c-means, hierarchical clustering, self-organizing map (SOM) are some of the examples of clustering techniques that have been explored in diagnosing CP related gait abnormalities [7]. Finding the optimal number of clusters in a dataset is a challenging task in the unsupervised learning area [31]. Selection of number of clusters is considered as obscure as the interpretations not only depend on the shape and distribution of the data points but also on the desired clustering resolution of the user. In this study, authors have proposed a hypothesis that nature-based clustering approaches can help in finding the optimal number of gait profiles in the datasets. The resulting analysis suggests that the proposed approach is better and closer to ground truth than the clustering techniques (k-means, fuzzy clustering mean (FCM), genetic algorithm (GA) and particle swarm optimization (PSO)) used in this study. Results indicate that it helps in diagnosis, assessment and evaluation of treatment outcomes. It acts as an assistive tool to the doctors working in this area. This paper is organized as follows. In Section 2, problem formulation is presented. Section 3 discusses the experimental methodology. Sections 4 and 5 present the results and conclusion with a future scope. 2. PROBLEM FORMULATION 2.1. Gait dataset description The gait dataset used in this research is taken from the publicly accessible data from O’Malley et al. [13]. These gait data consist of two groups, first consisting of 88 children with the spastic diplegia, CP and another is the collection of 68 neurologically intact children and have no history of any motor disease. Data are collected using six cameras of Vicon System and processed with Vicon Clinical Manager at Motion Analysis Laboratory, University of Virginia [13]. The spatio-temporal gait parameters considered in the study are stride length and cadence. Age and limb length are considered as anthropometric parameters and are used to normalize the stride length and cadence. O’Malley et al. used fuzzy based clustering techniques for the classification of control and CP children based on stance and cadence parameters. For study and comparison, in this study, the authors have used the same gait parameters as in [13]. Authors considered these parameters after averaging of data measured through at least three trials of both legs. The mean age of children with CP is 9.89 years while it is 7.09 years for the control group. Table 1 presents the overview of the CP gait dataset considered in this study. Table 1. Anthropometric and the spatio-temporal gait parameters of the participating subjects [13]. 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 View Large Table 1. Anthropometric and the spatio-temporal gait parameters of the participating subjects [13]. 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 View Large 2.2. Traditional clustering approaches Partitional, overlapping and hierarchical are the main three types of clustering approaches. Automated clustering techniques play an important role in image classification, intrusion detection, document clustering and medical imaging [33]. The objective of clustering in this gait profiling problem is to group the considered CP gait dataset X=(X1,X2,…,XN) of N objects into K groups such that K≤N ⁠. Consider F features for each object, then dataset X can be represented as a matrix of size N×F ⁠, where N is the number of rows and F is number of columns. The purpose of clustering is to find K clusters each having a centroid point Ck in such a way that the similarity between data objects within the cluster to the centroid of the same cluster is minimum. 2.2.1. K-means The K-means algorithm is the simplest and most popular partition clustering approach. It is also known as hard clustering approach. In K-means, the overall distance between the cluster members and the cluster centroids (intra-cluster distance) needs to be optimized [34]. Mean square error (MSE) is considered as the objective function illustrated in Equation (1). 1N∑k=1K∑xiϵck∣∣xi−Ck∣∣2 (1) where K is the total number of pre-defined clusters, k represents the cluster index and Ck is the centroid of the cluster k. This expression is needed to be minimized. The algorithm of K-means is presented in Algorithm 1. Data points assigned to a cluster are based on their degree of closeness, measured by the Euclidean distance from the points to the cluster’s center. K-means approach is easy to implement but the time complexity increases with larger datasets and very sensitive to the initially provided centroids. Algorithm 1 K-means Algorithm 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for View Large Algorithm 1 K-means Algorithm 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for View Large Algorithm 2 Fuzzy c-means algorithm 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for View Large Algorithm 2 Fuzzy c-means algorithm 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for View Large Algorithm 3 Pseudo-code for the hybrid-GA-based clustering 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for View Large Algorithm 3 Pseudo-code for the hybrid-GA-based clustering 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for View Large Algorithm 4 Hybrid-PSO 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids View Large Algorithm 4 Hybrid-PSO 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids View Large 2.2.2. Fuzzy c-means Fuzzy logic is introduced by Zadeh during 1960s for handling the uncertain and imprecise knowledge in real-world applications [35]. The fuzzy c-means algorithm uses the reciprocal of distances between data in instances to decide the cluster centers. It is considered as the extension of the K-means and also known as soft clustering technique. The centroid of a cluster in a fuzzy c-means method is calculated as the mean of all points value, weighted by their degree of belongingness to the cluster. When the nature of cluster is overlapping then fuzzy clustering is preferred [35]. In FCM, to determine the best value of partition matrix U the following objective function Jk(U,Cj) is minimized Jk(U,Cj)=∑i=1k∑j=1cjμijm(∥Xi−Cj∥)21<m<∞ (2) where m is any real number greater than one, μij is the degree of membership of Xi in the cluster kth. Degree of membership μij and cluster center Cj is given by Equations (3) and (4), respectively: μij=1∑p=1k∥Xi−Cj∥∥Xi−Cp∥2m−1 (3) Cj=∑i=1NμijmXi∑i=1Nμijm. (4) In 1997, O’Malley et al. [13] examined FCM on cerebral palsy children based on 168 subjects, spatio-temporal parameters (stride length and cadence). Carriero et al. [36] demonstrate the possibility of principal component analysis in quantitative classification of CP gait pattern. FCM analysis is used to cluster the data of 40 subjects (20 healthy and 20 soastic diplegic CP patients) with 27 parameters each. 2.2.3. Other approaches In 1983, Wong et al. [29] explore k-nearest neighbor to classify the gait pathologies using walking speed, hip, and ankle movement of 62 CP patients, considering k as 5. Self-organizing feature map (SOM), Hierarchical clustering and K-means are used to differentiate normal and pathological gait pattern based on stride length and cadence [30]. Hierarchical clustering is used to cluster healthy group from pathological patients [37]. In 2007, Taro et al. apply hierarchical clustering on the sagittal kinematic gait data of 67 subjects (56 CP, 11 healthy). Thirteen gait clusters are formulated using sagittal plane hip, knee, and ankle kinematics [38]. The issue with the clustering techniques is that number of the cluster is required to be known in advance. Researchers have also explored the nature-inspired optimization algorithm for the clustering purpose [31]. 3. EXPERIMENTAL METHODOLOGY In this section, the methodology used and nature-inspired optimized based clustering approaches are discussed followed by the five different cases, clustering evaluation indices and parameter setting. Clustering techniques are employed on the CP dataset and based on the best possible number of cluster identified in this study, the condition after the post-surgery is examined to validate the clustering result. Figure 1 illustrates the work flow of the proposed methodology. Figure 1. View largeDownload slide Clustering-based methodology used for gait profiling of CP children. Figure 1. View largeDownload slide Clustering-based methodology used for gait profiling of CP children. In K-means clustering approach, the result depends on the choice of the randomly selected cluster centroids. If they are close, takes the time to converge, while if the distance is on two extreme sides, then better cluster formation takes place. Thus the selection of the first cluster head points can be optimally selected using meta-heuristic approaches. In this study, nature-inspired optimization algorithms are used for the clustering purpose [31]. GA and PSO are the most common in evolutionary algorithms. In this study GA, PSO and a hybrid version of the both are proposed as clustering techniques with the following objective function [39]. The objective function used in this study is the combination of intra and inter-cluster distance. Intra-cluster distance (⁠ Dintercn ⁠) is the overall distance between the cluster members and its centroid. While inter-cluster distance Dintracn is the overall distance between the centroids. Mathematically, it can be modeled as a multi-objective optimization problem to find a cluster centroid Ck that has maximum similarity. The fitness function f() can be represented as Equation (5): f(XN×F,Ck)=∑n=1K{w1∗Dinter(cn)−w2∗Dintra(cn)}whereDintercn=max∣∣Ci−Cj∣∣,Herei,jϵ[1,2,3…k],i≠j,∀k=1,2…KDintracn=∑i=1NMin{∥Xi×F−Ck×F∥2}MaxCkOptmize (5) where i is the data point of the k cluster, K is the total number of pre-defined clusters, k represents the cluster index, Ck is the centroid of the cluster k. w1 and w2 are weighted parameters. In this study, we have considered both w1 and w2 as 0.5 for the study purpose [39]. This expression needed to be maximized. 3.1. GA and proposed variant Evolutionary techniques are a good approach to the optimization problem. Based on Darwin’s principle of natural selection, GA is first meta-heuristic optimization algorithm proposed by Goldberg and Holland in 1988 [40]. In 1994, Bezdek et al. [41] explore basic GA to be used as a clustering approach. Similar to basic GA, individual features are characterized in the form of strings called chromosomes. Based on the number of clusters (⁠ K ⁠) and the population size (⁠ N ⁠), the chromosomes are considered as the cluster centroids candidate. The algorithm initiates by creating the random set of chromosomes in the search space followed by an iterative process of selection (maximum number of iterations maxiter ⁠), crossover probability (⁠ Cprob ⁠) and mutation probability (⁠ Mprob ⁠) to find the optimal K solution (cluster center (centroids)) in the search space. 3.1.1. Hybrid GA variant In 1999, Krishna and Murty [42] proposed a novel hybrid Genetic K-means algorithm. K-means help in the search operation during crossover. In this study, GA is hybridized with K-means algorithm to get the optimal cluster centers and effective fitness values as shown in Algorithm 3 [43]. Hybrid GA optimizes the location of starting centroid (cluster centers) from the population with the minimum fitness function. Use these cluster centers as the initial centroids in the k-mean and iterate, the steps until no significant changes in consecutive cluster centers. 3.2. PSO and proposed variant PSO technique mimics swarm (birds, fish, etc.) social behavior for food searching and is developed by Eberhart and Kennedy [44]. Because it has a fast convergence rate and easy to implement, it is a popular choice among the researchers. Omran et al. [45], proposed cluster analysis using PSO for image processing. One year later, Merwe explored PSO for data clustering on different datasets [46]. In PSO-based clustering algorithm, swarm are defined in search space according to the constraints. The pseudo-code of PSO is presented. Population, initial weights, iteration is defined initially. Each particle position is defined as xi=[μi1,μi1,…,μiK] (6) where K cluster centroid vectors. The cluster Cik has μik as the cluster center point (centroid). Fitness function is considered as the Equation (5). Based on this, previous position is considered as particle best position Pb as Pb=[Pb1,Pb2,…,PbK]. (7) In the beginning, Pb is consider as Xi as X=(X1,X2,…,XN) ⁠. Global solution Gb is the best position of swarm in the next iteration (t) and represented as follows. Gb=[Gb1,Gb2,…,Gbt]. (8) Cluster centroids positions are updated with updation of velocity (⁠ vik ⁠) and positions (⁠ xi ⁠) of the particles as shown below. vik(t+1)=w×vik(t)+c1×r1×Pbk(t)−xik(t))+c2×r2×(Gb(t)−xik(t)) (9) xi(t+1)=xi(t)+vi(t+1) (10) where t is the number of iteration, r1 and r2 are the random numbers between [0,1], w is the inertia weight, c1 and c2 are the acceleration constant. In this study, inertia weight is taken as a function of time which varies from w is 0.9–0.5. This iteration continues till it reaches pre-defined number of iteration or there is no further change in the centroid of the clusters. 3.2.1. Hybrid PSO variant for clustering approach Van der Merwe and Engelhrecht [46] developed a hybrid algorithm based on K-means and PSO in 2003. In this study, combination of PSO and k-means is used as expressed in Algorithm 4. 3.3. Case studies considered In this study, five different cases are considered as shown in Table 2, to explore the best possible features selection. Table 2. Cases considered in this study. CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ View Large Table 2. Cases considered in this study. CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ View Large Case 1: Considering all the four gait parameters (stride length, cadence, leg length and age) without normalization and scaling. Case 2: The first two gait parameters (stride length and cadence) in their original form are considered. In [13, 15, 36], authors suggest that stride length and cadences are clinically more significant for CP analysis than another kinematics parameter when focused on single joints, in both classification and clustering study. Stride length and cadences could be affected by age and leg length. Thus before applying any machine learning techniques, normalization model is required to remove trends that/if exist, in the CP and normal dataset with respect to age and leg length. Case 3: Considering stride length and cadence, after Polynomial normalization with leg length and age respectively. O’Malley et al. [13] suggest that detendring normalization technique is better than offset and decorrelation based normalization methods. They suggested first and second order polynomial model to normalize stride length with respect to leg length and cadence concerning age, respectively for each subject. In our study, we consider stride length and cadence after normalization and scaling procedure suggested by [13]. They are statistically independent and significant in CP analysis. Table 3 presents an overview of the case considered in the study. Table 3. Testing gait data of four subjects A, B, C and D. Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery View Large Table 3. Testing gait data of four subjects A, B, C and D. Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery View Large Case 4: Finding the optimal number of clusters in a dataset is an open research area. In the case of rehabilitation, the number of gait profiling of the patients can play a vital role. As case 4, an optimal number of gait profiles in the dataset are identified. Case 5: Evaluation of the test subject is necessary to demonstrate the significance of the surgery. To validate the clustered gait profile, four subjects are taken as case 5 as presented in Table 3 from [13]. A and B are normal subjects while C and D are examined before and after surgery. 3.4. Clustering evaluation indices The clustering performance is evaluated using six clustering performances indices, including both internal (based on intrinsic characteristics, thus considered as unsupervised) measures and external (based on previous knowledge about data, thus can be considered as supervised) cluster validity indices. One external validity index (cluster purity index (CPI)) and five internal validity measures (distance Measures (intra-cluster distance and inter-cluster distance), MSE, silhouette coefficient (SC) and Dunn index (DI)) [31] are considered in this study. The external cluster validity index is considered to be close to user’s semantic [32]. Affiliated probability index (API) is defined as the probability of belongingness of a given test sample Tsi over the given set of cluster C and represented as AP(Tsi,Cj)=1D(Tsi,Cj)∑m=1K1D(Tsi,Cj) (11) where D(Tsi,Cj) is the affiliation distance between a testing subject and a given cluster. 3.5. Optimization parameter setting The parameters affecting the performance of optimized clustering are taken as follows. In this study, population size is taken as 100 and 40 for GA and PSO, respectively. Crossover rate and mutation are chosen as 0.3 and 0.2. In this study, a time-varying inertia weight w varies from 0.9 to 0.5 is considered. w1 and w2 are 0.5 in objective function. All these parameters are tuned by using sensitivity analysis [47]. 4. RESULT ANALYSIS In this section, the analysis of the cases considered is discussed. On the basis of analysis of Table 4, it is found that the result of proposed variant (H-GA) and H-PSO is the same as the first cluster is optimally selected and same for both cases. Table 4. Clustering result for case 1 and 2; when four and first two gait parameters (stride length, cadence, leg length and age) are considered respectively without normalization and scaling for k=2 ⁠. Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 View Large Table 4. Clustering result for case 1 and 2; when four and first two gait parameters (stride length, cadence, leg length and age) are considered respectively without normalization and scaling for k=2 ⁠. Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 View Large Case 1: Table 4 presents clustering result for case 1. FCM yields the best external evaluation index; CPI when compared with the mean. Considering other internal evaluation measures, GA reports the best result in this case. The best one among all is highlighted in the Table 4. The value reported here is mean of 25 runs. Case 2: H-GA-based optimized clustering algorithms performed best on four internal clustering performances indices. The best CPI is given by FCM as reported in Table 4 case 2. The overlapping nature of CP gait data with the normal children may be reasoned for this. Considering other evaluation indices, H-GA-based clustering outperforms other traditional partitioning clustering techniques. Case 2 reported Minimum MSE than case 1. Case 3: Fig. 2, illustrates the significance of the polynomial normalization for leg and age on stride length and cadence respectively. For this study, stride length can be considered a significant factor in discriminating the children with CP from the control group (neurologically intact children) when visualized by plotting a Figure. The red dot represents the children with CP, and blue is neurologically intact children. A clear separation in the dataset is observed after normalization, considered case 3 for k=2 ⁠. Figure 2. View largeDownload slide 2D plot of gait data for (a) CP and normal children natural data; (b) polynomial normalization and scale gait data with CP and normal children and (c) gait profiling after clustering approach. Figure 2. View largeDownload slide 2D plot of gait data for (a) CP and normal children natural data; (b) polynomial normalization and scale gait data with CP and normal children and (c) gait profiling after clustering approach. Table 4 presents the result of K-means, GA, PSO, H-GA and PSO optimized clustering on gait data for k=2 ⁠. Although classical performance metrics such as mean square error, silhouette coefficient and Dunn index are suitable methods of comparing the algorithms, they are not sufficient to find a difference in performance of the algorithms. To aggregate the performance comparison and statistical significance of computational intelligence algorithms, the popularity of parametric and nonparametric tests has increased in last few years. T-test (parametric) carried out for comparing different algorithms. The t-test assesses whether the mean of two groups of results is statistically different from each other or not. For testing, the two-tailed t-test is adopted with 5% significance level. The negative t-value with PSO as base algorithm along with low P-value and h-value of 1 w.r.t. all the other algorithms prove PSO to be significantly better than other algorithms including GA. The further comparison is made in Table 5 in the article. Table 5. Result of t-test considering MSE on case 3. Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 View Large Table 5. Result of t-test considering MSE on case 3. Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 View Large Case 4: The result of different cluster sizing on three internal cluster validity indices (MSE, SC and DI) in case of 2≤k≤7 is presented in Fig. 3. Figure 3a–c is the plot of K-means for MSE, SC and DI, respectively. Figure 3d–f is the plot of FCM for MSE, SC and DI, respectively. Figure 3g–i plot is for GA-based clustering with MSE, SC and DI, respectively. Figure 3j–l is the plot of Hybrid GA clustering for MSE, SC and DI, respectively. Figure 3m–o is the plot of particle swam optimization based clustering for MSE, SC and DI, respectively. For each approach, cluster size is decided by majority voting. Lower the value of the MSE, better is the quality of clustering, and larger is the SC and DI value, the better is the cluster quality. Optimal cluster number (gait profiles) is chosen by majority voting among these validity indices [31]. Figure 3. View largeDownload slide Composite error and indices plot for case 4 in terms of MSE, SC and DI for K-means, GA, H-GA and PSO clustering in case of 2≤k≤7 ⁠. (a) K-means and MSE, (b) K-means and SC, (c) K-means and DI, (d) FCM and MSE, (e) FCM and SC, (f) FCM and DI, (g) GA and MSE, (h) GA and SC, (i) GA and DI, (j) H-GA and MSE, (k) H-GA and SC, (l) H-GA and DI, (m) PSO and MSE, (n) PSO and SC and (o) PSO and DI. Figure 3. View largeDownload slide Composite error and indices plot for case 4 in terms of MSE, SC and DI for K-means, GA, H-GA and PSO clustering in case of 2≤k≤7 ⁠. (a) K-means and MSE, (b) K-means and SC, (c) K-means and DI, (d) FCM and MSE, (e) FCM and SC, (f) FCM and DI, (g) GA and MSE, (h) GA and SC, (i) GA and DI, (j) H-GA and MSE, (k) H-GA and SC, (l) H-GA and DI, (m) PSO and MSE, (n) PSO and SC and (o) PSO and DI. K-means, FCM and H-GA-based clustering votes for five clusters as optimal choice, while GA and PSO-based clustering account four as the optimal clustering size. Figure 3 indicates that cluster size 5 exhibits a better quality than other given k setting. Thus conclusively, to validate test subject profile, assessment is carried out by considering k as 5 in the following analysis. Case 5: Evaluation of the test sample is necessary to demonstrate the significance of the surgery as considered in case 5. This section discusses the analysis of generated CP gait profile, considering the optimal cluster sizing as k=5 from Fig. 3, on four test subjects (brief description is in Table 1(b)) using affiliated probability index (API). Figure 4 presents the affiliated probability distribution for two neurological intact subjects A, B and two patients C and D from pre and post-surgery in case of five clusters. Extensive analysis shows that cluster 5 in the Fig. 4 is for the subjects with the no gait pathology, the control group, followed by Cluster 3, Cluster 1, Cluster 4 and Cluster 2. Cluster 4 and 2 can be classified as gait pattern with CP cases. Figure 4. View largeDownload slide Gait profile plots for gait profile considering 5 clusters using affiliation probability. Figure 4. View largeDownload slide Gait profile plots for gait profile considering 5 clusters using affiliation probability. The result confirms the gait pattern of subject A as normal. The study shows that subject B exhibits deviated gait pattern. After intensive analysis, it is found that the mean age of control group is 7.09 (2–13 age range), while the test subject considered is of 19-year-old. Thus this gait pattern is misclassified as cluster 4 instead of cluster 5; the control cluster. Three observations are taken for both patients C and D. First observation of the third subject (C1) is performed before surgery, affiliated probability based gait profile also confirms the CP based pattern. In the second (C2) and third (C3) observations of the subject, C shifted from cluster 4 to cluster 3 and then to cluster 5 (the second and third observation are after post-surgery case). Gait profile based on affiliated probability index confirms that surgery helps the subject C to shift toward normal gait profile. Similar kind of observation is observed in patient D, where the first observation is before surgery, and D2 and D3 are post-surgery observations. The D1 and D2 in the plot exhibit that the problem is increased even after surgery. But after 2 years of surgery, observation D3 presents that the gait of the subject could be classified as normal gait profile. Thus health-care professionals can take help of optimized clustering approaches to evaluate the recovery progress of patients after surgery. Doctors can monitor the recovery progress and if required change the strategy for treatment. But here it is important to note that the gait profile from automated optimization based clustering is an indicator of the gait state, thus aid doctors in making decisions regarding rehabilitation. 5. CONCLUSION AND FUTURE SCOPE The current study shows that optimization based clustering approaches can be used for gait profiling for rehabilitation in children with Cerebral Palsy. Different cases regarding the data pre-processing are considered in this study. The dataset used is publicly available as CP gait dataset of 156 subjects having the age range from 2 to 20 for CP cases and 2 to 13 for the control group. K-means, FCM, GA, PSO and hybrid of both GA and PSO-based clustering approaches are used to find the gait profiles for the considered subjects. Best possible clustering sizing is selected based on voting based out of mean square error, silhouette coefficient, and Dunn index. The performance of the proposed clustering methods is evaluated using internal and external cluster performance index. This is validated with the unseen test samples, which undergoes surgery. The result indicates that optimized clustering technique outperforms the traditional and hybrid clustering approaches and can help in diagnosis, assessment and evaluation of treatment outcomes. For study and comparison, the authors have used the same gait parameters as in [13] publication and there may be some missing factors (such as the number of volunteers to validate results may not be enough) and should be given prime consideration as future work. In diagnosis-related cases, there is no clear boundary between healthy and CP cases. The subject can lie in a different cluster at a time. As a future scope, one can analyze nature-inspired based optimized fuzzy clustering approach for treatment intervention for individuals with a disability. 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For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Computer Journal Oxford University Press

Optimized Clustering Techniques for Gait Profiling in Children with Cerebral Palsy for Rehabilitation

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Abstract

Abstract Cerebral palsy (CP) is a neuro-development disease in children. It is quite an intricate task to categorize gait pattern into normal and CP based pathology. In this study, nature-inspired meta-heuristic algorithms are explored on a publicly available gait dataset of 156 subjects for automatic gait profiling of children with cerebral palsy. Five cases are considered to explore the feature selection criteria before applying clustering technique. Finding the optimal number of clusters is a challenging task in the unsupervised learning area. In this study, an optimal number of gait profiles in the datasets is identified based on voting from mean square error, silhouette coefficient and Dunn index. The study demonstrates that optimized based gait profile clusters could assist quantitatively in clinical rehabilitation evaluation for the children affected by CP. 1. INTRODUCTION Cerebral palsy (CP) is a neuro-development disease, common in children. It is associated with the floppy or rigid limbs, exaggerated reflexes and involuntary motion, poor speech and learning ability which, is considered as a non-progressive disease [1, 2]. The major reason for this motor disability is abnormal brain development, which often occurs before the birth of a child. On behalf of motor impairment of the limb, CP is classified into three major categories; spastic, ataxic and athetoid [3]. Different studies, around the globe highlight the severity of this disease and projects that worldwide CP cases vary from 1.5 to 4 per 1000 children. The condition is more severe in the developing countries [1, 4]. Thus a huge amount of money is involved in the rehabilitation and intervention policies related to the diagnosis of gait pathology related to cerebral palsy [5, 6]. The diversity in the gait pattern from children suffering from CP makes it a challenging task for researchers in the physical therapy and surgical community. This will enable health-care professionals to differentiate gait pattern into clinically significant categorize that assist in diagnosis, assessment, evaluation of the treatment outcomes and clinical decision-making [7–11]. All pathological pattern cannot be captured through just visual observation. There is a need to take the aid of statistical (frequency domain characteristics, spectral components, harmonic content, the coefficient of variation, etc.) and machine learning techniques [12]. Clinical gait analysis is a method to identify the hidden pattern in gait such as stride length, step length, cadence, stance and swing phase, etc. Clinicians can utilize gait classification concepts in their routine clinical practice to evaluate a patient’s status, treatment and rehabilitation for CP disorders using the spatio-temporal and kinematics parameters [7, 13, 14]. There are several attempts made by researchers to characterize the gait pattern and categorize them into normal and gait pathology using computational techniques [15–19]. Supervised learning is a popular choice for the research community to classify the gait pattern of CP pathology for automated analysis. Neural network is most widely used for normal gait analysis, robotic rehabilitation, sports monitoring, and tactics, geriatric care, surveillance, activity recognition [7, 20–23]. Supervised learning technique is a dominant methodology when labeled training data points are available. Zheng et al. [24] implemented the trained neural network classifier to identify the abnormality in older person gait. Gait pathology can be identified using ground reaction force with learning vector quantization. Multilayer perceptron, Linear discriminant analysis classifiers, kernel Fisher discriminant and Bayesian classifier have been used to classify gait pattern in [25, 26]. Support vector machine (SVM) is used for the study of normal and age-related differences, normal and abnormal gait pathology. The authors in [27, 28] showed that it exhibits good results in CP pathology. SVM is a powerful classifier suitable from small to the medium dataset. The limitation of these supervised approaches is that their accuracy depends on the training sample. The larger the training sample, the more accurate the system is. Another limitation is that these techniques cannot be applied to the data where no prior information of the data is given. To overcome these limitations, researchers have explored unsupervised, or clustering approaches [29, 13, 30]. Cluster models are not only popular in signal and image processing, but they also have wide applications in web mining and pattern recognition, sensor network, robotics, seismology, medical science in disease clustering, etc [31, 32]. In gait analysis, clustering techniques are most widely used for categorizing gait data into groups of disorders based on common hidden features in the subject dataset. K-means, fuzzy c-means, hierarchical clustering, self-organizing map (SOM) are some of the examples of clustering techniques that have been explored in diagnosing CP related gait abnormalities [7]. Finding the optimal number of clusters in a dataset is a challenging task in the unsupervised learning area [31]. Selection of number of clusters is considered as obscure as the interpretations not only depend on the shape and distribution of the data points but also on the desired clustering resolution of the user. In this study, authors have proposed a hypothesis that nature-based clustering approaches can help in finding the optimal number of gait profiles in the datasets. The resulting analysis suggests that the proposed approach is better and closer to ground truth than the clustering techniques (k-means, fuzzy clustering mean (FCM), genetic algorithm (GA) and particle swarm optimization (PSO)) used in this study. Results indicate that it helps in diagnosis, assessment and evaluation of treatment outcomes. It acts as an assistive tool to the doctors working in this area. This paper is organized as follows. In Section 2, problem formulation is presented. Section 3 discusses the experimental methodology. Sections 4 and 5 present the results and conclusion with a future scope. 2. PROBLEM FORMULATION 2.1. Gait dataset description The gait dataset used in this research is taken from the publicly accessible data from O’Malley et al. [13]. These gait data consist of two groups, first consisting of 88 children with the spastic diplegia, CP and another is the collection of 68 neurologically intact children and have no history of any motor disease. Data are collected using six cameras of Vicon System and processed with Vicon Clinical Manager at Motion Analysis Laboratory, University of Virginia [13]. The spatio-temporal gait parameters considered in the study are stride length and cadence. Age and limb length are considered as anthropometric parameters and are used to normalize the stride length and cadence. O’Malley et al. used fuzzy based clustering techniques for the classification of control and CP children based on stance and cadence parameters. For study and comparison, in this study, the authors have used the same gait parameters as in [13]. Authors considered these parameters after averaging of data measured through at least three trials of both legs. The mean age of children with CP is 9.89 years while it is 7.09 years for the control group. Table 1 presents the overview of the CP gait dataset considered in this study. Table 1. Anthropometric and the spatio-temporal gait parameters of the participating subjects [13]. 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 View Large Table 1. Anthropometric and the spatio-temporal gait parameters of the participating subjects [13]. 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 88 Children with spastic-diplegia, CP 68 neurologically intact children Min Max Mean Std Min Max Mean Std Age (years) 2 20 9.89 4.34 2 13 7.09 2.89 Leg length (m) 0.41 0.94 0.67 0.14 0.34 0.86 0.57 0.12 Stride Length (m) 0.31 1.25 0.74 0.21 0.67 1.44 1.03 0.19 Cadence (step/min) 10.46 210.24 120.00 33.56 104.88 174.24 136.84 15.81 View Large 2.2. Traditional clustering approaches Partitional, overlapping and hierarchical are the main three types of clustering approaches. Automated clustering techniques play an important role in image classification, intrusion detection, document clustering and medical imaging [33]. The objective of clustering in this gait profiling problem is to group the considered CP gait dataset X=(X1,X2,…,XN) of N objects into K groups such that K≤N ⁠. Consider F features for each object, then dataset X can be represented as a matrix of size N×F ⁠, where N is the number of rows and F is number of columns. The purpose of clustering is to find K clusters each having a centroid point Ck in such a way that the similarity between data objects within the cluster to the centroid of the same cluster is minimum. 2.2.1. K-means The K-means algorithm is the simplest and most popular partition clustering approach. It is also known as hard clustering approach. In K-means, the overall distance between the cluster members and the cluster centroids (intra-cluster distance) needs to be optimized [34]. Mean square error (MSE) is considered as the objective function illustrated in Equation (1). 1N∑k=1K∑xiϵck∣∣xi−Ck∣∣2 (1) where K is the total number of pre-defined clusters, k represents the cluster index and Ck is the centroid of the cluster k. This expression is needed to be minimized. The algorithm of K-means is presented in Algorithm 1. Data points assigned to a cluster are based on their degree of closeness, measured by the Euclidean distance from the points to the cluster’s center. K-means approach is easy to implement but the time complexity increases with larger datasets and very sensitive to the initially provided centroids. Algorithm 1 K-means Algorithm 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for View Large Algorithm 1 K-means Algorithm 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for 1: Initialize number of cluster as (K) and max_iter 2: Select initial centroids randomly 3: for iter=0:max_iter or other termination criteria do 4:  Allocate each data-point to a cluster based on the similarity measure 5:  Update the centroids by taking means of all points in the particular cluster 6: end for View Large Algorithm 2 Fuzzy c-means algorithm 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for View Large Algorithm 2 Fuzzy c-means algorithm 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for 1: Initialize number of cluster as k 2: Initialize μij for all i,j from (4) 3: for Until termination criteria is met do 4:  Calculate cluster centers Vj 5:  Update μij for all i,j 6: end for View Large Algorithm 3 Pseudo-code for the hybrid-GA-based clustering 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for View Large Algorithm 3 Pseudo-code for the hybrid-GA-based clustering 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for 1: Initialize K ⁠, N ⁠, maxiter ⁠, Cprob and Mprob 2: Generate initial population randomly where each chromosomes act as set of cluster centers 3: for iter=0:maxiterdo 4:  Apply selection in the population 5:  Apply crossover in the population 6:  Apply mutation in the population 7: end for 8: Output is the K optimal cluster center with minimum MSE 9: Apply K-means by initializing max_iter_k 10: Select initial centroids as output in step 8 11: for iter=0:max_iter_k or other termination criteria do 12:  Allocate each data-point to a cluster based on the similarity measure 13:  Update the centroids using eq. (5) 14: end for View Large Algorithm 4 Hybrid-PSO 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids View Large Algorithm 4 Hybrid-PSO 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids 1: Initialize number of clusters as k 2: Initialize population size (⁠ N ⁠), maximum number of iterations maxiter and inertia weight w 3: Generate initial population randomly with centroids as its parameters 4: for iter=0:maxiterdo 5:  Find fitness function value for all particles 6:  Update velocity of particles by eq. (9) 7:  Update position of particles by eq. (10) 8:  Update gb and pbi 9: end for 10: Apply K-means Algorithm to return the centroids View Large 2.2.2. Fuzzy c-means Fuzzy logic is introduced by Zadeh during 1960s for handling the uncertain and imprecise knowledge in real-world applications [35]. The fuzzy c-means algorithm uses the reciprocal of distances between data in instances to decide the cluster centers. It is considered as the extension of the K-means and also known as soft clustering technique. The centroid of a cluster in a fuzzy c-means method is calculated as the mean of all points value, weighted by their degree of belongingness to the cluster. When the nature of cluster is overlapping then fuzzy clustering is preferred [35]. In FCM, to determine the best value of partition matrix U the following objective function Jk(U,Cj) is minimized Jk(U,Cj)=∑i=1k∑j=1cjμijm(∥Xi−Cj∥)21<m<∞ (2) where m is any real number greater than one, μij is the degree of membership of Xi in the cluster kth. Degree of membership μij and cluster center Cj is given by Equations (3) and (4), respectively: μij=1∑p=1k∥Xi−Cj∥∥Xi−Cp∥2m−1 (3) Cj=∑i=1NμijmXi∑i=1Nμijm. (4) In 1997, O’Malley et al. [13] examined FCM on cerebral palsy children based on 168 subjects, spatio-temporal parameters (stride length and cadence). Carriero et al. [36] demonstrate the possibility of principal component analysis in quantitative classification of CP gait pattern. FCM analysis is used to cluster the data of 40 subjects (20 healthy and 20 soastic diplegic CP patients) with 27 parameters each. 2.2.3. Other approaches In 1983, Wong et al. [29] explore k-nearest neighbor to classify the gait pathologies using walking speed, hip, and ankle movement of 62 CP patients, considering k as 5. Self-organizing feature map (SOM), Hierarchical clustering and K-means are used to differentiate normal and pathological gait pattern based on stride length and cadence [30]. Hierarchical clustering is used to cluster healthy group from pathological patients [37]. In 2007, Taro et al. apply hierarchical clustering on the sagittal kinematic gait data of 67 subjects (56 CP, 11 healthy). Thirteen gait clusters are formulated using sagittal plane hip, knee, and ankle kinematics [38]. The issue with the clustering techniques is that number of the cluster is required to be known in advance. Researchers have also explored the nature-inspired optimization algorithm for the clustering purpose [31]. 3. EXPERIMENTAL METHODOLOGY In this section, the methodology used and nature-inspired optimized based clustering approaches are discussed followed by the five different cases, clustering evaluation indices and parameter setting. Clustering techniques are employed on the CP dataset and based on the best possible number of cluster identified in this study, the condition after the post-surgery is examined to validate the clustering result. Figure 1 illustrates the work flow of the proposed methodology. Figure 1. View largeDownload slide Clustering-based methodology used for gait profiling of CP children. Figure 1. View largeDownload slide Clustering-based methodology used for gait profiling of CP children. In K-means clustering approach, the result depends on the choice of the randomly selected cluster centroids. If they are close, takes the time to converge, while if the distance is on two extreme sides, then better cluster formation takes place. Thus the selection of the first cluster head points can be optimally selected using meta-heuristic approaches. In this study, nature-inspired optimization algorithms are used for the clustering purpose [31]. GA and PSO are the most common in evolutionary algorithms. In this study GA, PSO and a hybrid version of the both are proposed as clustering techniques with the following objective function [39]. The objective function used in this study is the combination of intra and inter-cluster distance. Intra-cluster distance (⁠ Dintercn ⁠) is the overall distance between the cluster members and its centroid. While inter-cluster distance Dintracn is the overall distance between the centroids. Mathematically, it can be modeled as a multi-objective optimization problem to find a cluster centroid Ck that has maximum similarity. The fitness function f() can be represented as Equation (5): f(XN×F,Ck)=∑n=1K{w1∗Dinter(cn)−w2∗Dintra(cn)}whereDintercn=max∣∣Ci−Cj∣∣,Herei,jϵ[1,2,3…k],i≠j,∀k=1,2…KDintracn=∑i=1NMin{∥Xi×F−Ck×F∥2}MaxCkOptmize (5) where i is the data point of the k cluster, K is the total number of pre-defined clusters, k represents the cluster index, Ck is the centroid of the cluster k. w1 and w2 are weighted parameters. In this study, we have considered both w1 and w2 as 0.5 for the study purpose [39]. This expression needed to be maximized. 3.1. GA and proposed variant Evolutionary techniques are a good approach to the optimization problem. Based on Darwin’s principle of natural selection, GA is first meta-heuristic optimization algorithm proposed by Goldberg and Holland in 1988 [40]. In 1994, Bezdek et al. [41] explore basic GA to be used as a clustering approach. Similar to basic GA, individual features are characterized in the form of strings called chromosomes. Based on the number of clusters (⁠ K ⁠) and the population size (⁠ N ⁠), the chromosomes are considered as the cluster centroids candidate. The algorithm initiates by creating the random set of chromosomes in the search space followed by an iterative process of selection (maximum number of iterations maxiter ⁠), crossover probability (⁠ Cprob ⁠) and mutation probability (⁠ Mprob ⁠) to find the optimal K solution (cluster center (centroids)) in the search space. 3.1.1. Hybrid GA variant In 1999, Krishna and Murty [42] proposed a novel hybrid Genetic K-means algorithm. K-means help in the search operation during crossover. In this study, GA is hybridized with K-means algorithm to get the optimal cluster centers and effective fitness values as shown in Algorithm 3 [43]. Hybrid GA optimizes the location of starting centroid (cluster centers) from the population with the minimum fitness function. Use these cluster centers as the initial centroids in the k-mean and iterate, the steps until no significant changes in consecutive cluster centers. 3.2. PSO and proposed variant PSO technique mimics swarm (birds, fish, etc.) social behavior for food searching and is developed by Eberhart and Kennedy [44]. Because it has a fast convergence rate and easy to implement, it is a popular choice among the researchers. Omran et al. [45], proposed cluster analysis using PSO for image processing. One year later, Merwe explored PSO for data clustering on different datasets [46]. In PSO-based clustering algorithm, swarm are defined in search space according to the constraints. The pseudo-code of PSO is presented. Population, initial weights, iteration is defined initially. Each particle position is defined as xi=[μi1,μi1,…,μiK] (6) where K cluster centroid vectors. The cluster Cik has μik as the cluster center point (centroid). Fitness function is considered as the Equation (5). Based on this, previous position is considered as particle best position Pb as Pb=[Pb1,Pb2,…,PbK]. (7) In the beginning, Pb is consider as Xi as X=(X1,X2,…,XN) ⁠. Global solution Gb is the best position of swarm in the next iteration (t) and represented as follows. Gb=[Gb1,Gb2,…,Gbt]. (8) Cluster centroids positions are updated with updation of velocity (⁠ vik ⁠) and positions (⁠ xi ⁠) of the particles as shown below. vik(t+1)=w×vik(t)+c1×r1×Pbk(t)−xik(t))+c2×r2×(Gb(t)−xik(t)) (9) xi(t+1)=xi(t)+vi(t+1) (10) where t is the number of iteration, r1 and r2 are the random numbers between [0,1], w is the inertia weight, c1 and c2 are the acceleration constant. In this study, inertia weight is taken as a function of time which varies from w is 0.9–0.5. This iteration continues till it reaches pre-defined number of iteration or there is no further change in the centroid of the clusters. 3.2.1. Hybrid PSO variant for clustering approach Van der Merwe and Engelhrecht [46] developed a hybrid algorithm based on K-means and PSO in 2003. In this study, combination of PSO and k-means is used as expressed in Algorithm 4. 3.3. Case studies considered In this study, five different cases are considered as shown in Table 2, to explore the best possible features selection. Table 2. Cases considered in this study. CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ View Large Table 2. Cases considered in this study. CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ CASE Gait parameter/s considered Data size Normalization Scaling Case 1 Stride length, cadence, leg length and age 156×4 × × Case 2 Stride length and cadence 156×2 × × Case 3 Stride length and cadence 156×2 ✓ Polynomial ✓ Case 4 Stride length and cadence and different cluster size 2≤K≤7 156×2 ✓ Polynomial ✓ Case 5 Validate test case using stride length and cadence and cluster size from case 4. 156×2 ✓ Polynomial ✓ View Large Case 1: Considering all the four gait parameters (stride length, cadence, leg length and age) without normalization and scaling. Case 2: The first two gait parameters (stride length and cadence) in their original form are considered. In [13, 15, 36], authors suggest that stride length and cadences are clinically more significant for CP analysis than another kinematics parameter when focused on single joints, in both classification and clustering study. Stride length and cadences could be affected by age and leg length. Thus before applying any machine learning techniques, normalization model is required to remove trends that/if exist, in the CP and normal dataset with respect to age and leg length. Case 3: Considering stride length and cadence, after Polynomial normalization with leg length and age respectively. O’Malley et al. [13] suggest that detendring normalization technique is better than offset and decorrelation based normalization methods. They suggested first and second order polynomial model to normalize stride length with respect to leg length and cadence concerning age, respectively for each subject. In our study, we consider stride length and cadence after normalization and scaling procedure suggested by [13]. They are statistically independent and significant in CP analysis. Table 3 presents an overview of the case considered in the study. Table 3. Testing gait data of four subjects A, B, C and D. Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery View Large Table 3. Testing gait data of four subjects A, B, C and D. Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery Subject State/pathology Stride length (m) Cadence (m) Leg length (m) Age (year) Treatment condition A Neurologically Intact 1.29 112.8 0.78 13 Normal B Neurologically Intact 1.29 122.6 0.79 19 Normal C1 Spastic-diplegia, CP 0.59 134.0 0.66 8 Prior to surgery C2 Spastic-diplegia, CP 0.89 110.0 0.67 9 1 year post-surgery C3 Spastic-diplegia, CP 1.04 119.0 0.71 11 3 year post-surgery D1 Spastic-diplegia, CP 0.20 49.5 0.45 3 Prior to surgery D2 Spastic-diplegia, CP 0.51 74.0 0.47 4 1 year post-surgery D3 Spastic-diplegia, CP 0.76 131.0 0.52 5 2 year post-surgery View Large Case 4: Finding the optimal number of clusters in a dataset is an open research area. In the case of rehabilitation, the number of gait profiling of the patients can play a vital role. As case 4, an optimal number of gait profiles in the dataset are identified. Case 5: Evaluation of the test subject is necessary to demonstrate the significance of the surgery. To validate the clustered gait profile, four subjects are taken as case 5 as presented in Table 3 from [13]. A and B are normal subjects while C and D are examined before and after surgery. 3.4. Clustering evaluation indices The clustering performance is evaluated using six clustering performances indices, including both internal (based on intrinsic characteristics, thus considered as unsupervised) measures and external (based on previous knowledge about data, thus can be considered as supervised) cluster validity indices. One external validity index (cluster purity index (CPI)) and five internal validity measures (distance Measures (intra-cluster distance and inter-cluster distance), MSE, silhouette coefficient (SC) and Dunn index (DI)) [31] are considered in this study. The external cluster validity index is considered to be close to user’s semantic [32]. Affiliated probability index (API) is defined as the probability of belongingness of a given test sample Tsi over the given set of cluster C and represented as AP(Tsi,Cj)=1D(Tsi,Cj)∑m=1K1D(Tsi,Cj) (11) where D(Tsi,Cj) is the affiliation distance between a testing subject and a given cluster. 3.5. Optimization parameter setting The parameters affecting the performance of optimized clustering are taken as follows. In this study, population size is taken as 100 and 40 for GA and PSO, respectively. Crossover rate and mutation are chosen as 0.3 and 0.2. In this study, a time-varying inertia weight w varies from 0.9 to 0.5 is considered. w1 and w2 are 0.5 in objective function. All these parameters are tuned by using sensitivity analysis [47]. 4. RESULT ANALYSIS In this section, the analysis of the cases considered is discussed. On the basis of analysis of Table 4, it is found that the result of proposed variant (H-GA) and H-PSO is the same as the first cluster is optimally selected and same for both cases. Table 4. Clustering result for case 1 and 2; when four and first two gait parameters (stride length, cadence, leg length and age) are considered respectively without normalization and scaling for k=2 ⁠. Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 View Large Table 4. Clustering result for case 1 and 2; when four and first two gait parameters (stride length, cadence, leg length and age) are considered respectively without normalization and scaling for k=2 ⁠. Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 Case Algorithm Cluster purity index Intra-cluster Inter-cluster MSE Silhouette Coefficient Dunn Index Case 1 K-means Mean 0.647435897 727.414639437 2249.398883024 1454.829278874 0.534981035 0.021440854 FCM Mean 0.660256410 747.354298589 2218.037446776 1464.736588728 0.516406269 0.004728326 GA Mean 0.636538462 720.143215298 2397.168829263 1440.286430596 0.543759910 0.046664080 H-GA Mean 0.639743590 722.851700278 2453.155978148 144 5.703400555 0.553784374 0.045306570 PSO Mean 0.626923077 723.916326896 2254.787022798 1447.832653792 0.523427378 0.036291725 H-PSO Mean 0.639743590 722.851700278 2453.155978148 1445.703400555 0.553784374 0.045306570 Case 2 K-means Mean 0.637179487 702.160532015 2190.654305937 1404.321064030 0.546380125 0.011359487 FCM Mean 0.666666667 722.581050726 2221.344900625 1451.894712461 0.532371425 0.005553906 GA Mean 0.647435897 689.652285216 2636.076130618 1379.304570433 0.589329956 0.026969577 H-GA Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 PSO Mean 0.647435897 689.766697516 2642.220564311 1379.533395032 0.589329956 0.026969577 H-PSO Mean 0.647435897 689.520056309 2640.796659385 1379.040112617 0.589329956 0.026969577 Case 3 K-means Mean 0.857051282 708.519282126 1969.690460032 354.259641063 0.600939744 0.018785763 Stan Dev. 0.003096448 5.862232143 24.266868512 2.931116071 0.003002781 0.000000000 FCM Mean 0.857905983 717.704684628 870.750398306 358.852342314 0.592637055 0.010628756 Stan Dev. 0.002616976 9.006015910 1036.169813098 4.503007955 0.008819135 0.007630181 GA Mean 0.880128205 649.313594100 1202.618384945 324.656797050 0.430536190 0.008211998 Stan Dev. 0.022623352 2.793767682 113.016913587 1.396883841 0.000047791 0.004224293 H-GA Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 PSO Mean 0.895512821 648.108629885 1192.770592444 324.054314943 0.430516403 0.006462975 Stan Dev. 0.015716389 1.185202640 66.566954312 0.592601320 0.000031287 0.002765449 H-PSO Mean 0.858974359 720.253433800 1952.655520133 360.126716900 0.596759234 0.018785763 Stan Dev. 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000 View Large Case 1: Table 4 presents clustering result for case 1. FCM yields the best external evaluation index; CPI when compared with the mean. Considering other internal evaluation measures, GA reports the best result in this case. The best one among all is highlighted in the Table 4. The value reported here is mean of 25 runs. Case 2: H-GA-based optimized clustering algorithms performed best on four internal clustering performances indices. The best CPI is given by FCM as reported in Table 4 case 2. The overlapping nature of CP gait data with the normal children may be reasoned for this. Considering other evaluation indices, H-GA-based clustering outperforms other traditional partitioning clustering techniques. Case 2 reported Minimum MSE than case 1. Case 3: Fig. 2, illustrates the significance of the polynomial normalization for leg and age on stride length and cadence respectively. For this study, stride length can be considered a significant factor in discriminating the children with CP from the control group (neurologically intact children) when visualized by plotting a Figure. The red dot represents the children with CP, and blue is neurologically intact children. A clear separation in the dataset is observed after normalization, considered case 3 for k=2 ⁠. Figure 2. View largeDownload slide 2D plot of gait data for (a) CP and normal children natural data; (b) polynomial normalization and scale gait data with CP and normal children and (c) gait profiling after clustering approach. Figure 2. View largeDownload slide 2D plot of gait data for (a) CP and normal children natural data; (b) polynomial normalization and scale gait data with CP and normal children and (c) gait profiling after clustering approach. Table 4 presents the result of K-means, GA, PSO, H-GA and PSO optimized clustering on gait data for k=2 ⁠. Although classical performance metrics such as mean square error, silhouette coefficient and Dunn index are suitable methods of comparing the algorithms, they are not sufficient to find a difference in performance of the algorithms. To aggregate the performance comparison and statistical significance of computational intelligence algorithms, the popularity of parametric and nonparametric tests has increased in last few years. T-test (parametric) carried out for comparing different algorithms. The t-test assesses whether the mean of two groups of results is statistically different from each other or not. For testing, the two-tailed t-test is adopted with 5% significance level. The negative t-value with PSO as base algorithm along with low P-value and h-value of 1 w.r.t. all the other algorithms prove PSO to be significantly better than other algorithms including GA. The further comparison is made in Table 5 in the article. Table 5. Result of t-test considering MSE on case 3. Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 View Large Table 5. Result of t-test considering MSE on case 3. Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 Algorithms K-means FCM GA H-GA PSO H-PSO K-means t 0.0000 −2.2820 28.8308 −6.3298 31.9412 −6.3298 h 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 P 1.0000 0.0349 0.0000 0.0000 0.0000 0.0000 FCM t 2.2820 0.0000 24.6364 −1.7860 26.2677 −1.7860 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0349 1.0000 0.0000 0.0910 0.0000 0.0910 GA t −28.8308 −24.6364 0.0000 −80.2971 1.2556 −80.2971 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 1.0000 0.0000 0.2253 0.0000 H-GA t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 PSO t −31.9412 −26.2677 −1.2556 −192.4919 0.0000 −192.4919 h 1.0000 1.0000 0.0000 1.0000 0.0000 1.0000 P 0.0000 0.0000 0.2253 0.0000 1.0000 0.0000 H-PSO t 6.3298 1.7860 80.2971 0.0000 192.4919 0.0000 h 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 P 0.0000 0.0910 0.0000 1.0000 0.0000 1.0000 View Large Case 4: The result of different cluster sizing on three internal cluster validity indices (MSE, SC and DI) in case of 2≤k≤7 is presented in Fig. 3. Figure 3a–c is the plot of K-means for MSE, SC and DI, respectively. Figure 3d–f is the plot of FCM for MSE, SC and DI, respectively. Figure 3g–i plot is for GA-based clustering with MSE, SC and DI, respectively. Figure 3j–l is the plot of Hybrid GA clustering for MSE, SC and DI, respectively. Figure 3m–o is the plot of particle swam optimization based clustering for MSE, SC and DI, respectively. For each approach, cluster size is decided by majority voting. Lower the value of the MSE, better is the quality of clustering, and larger is the SC and DI value, the better is the cluster quality. Optimal cluster number (gait profiles) is chosen by majority voting among these validity indices [31]. Figure 3. View largeDownload slide Composite error and indices plot for case 4 in terms of MSE, SC and DI for K-means, GA, H-GA and PSO clustering in case of 2≤k≤7 ⁠. (a) K-means and MSE, (b) K-means and SC, (c) K-means and DI, (d) FCM and MSE, (e) FCM and SC, (f) FCM and DI, (g) GA and MSE, (h) GA and SC, (i) GA and DI, (j) H-GA and MSE, (k) H-GA and SC, (l) H-GA and DI, (m) PSO and MSE, (n) PSO and SC and (o) PSO and DI. Figure 3. View largeDownload slide Composite error and indices plot for case 4 in terms of MSE, SC and DI for K-means, GA, H-GA and PSO clustering in case of 2≤k≤7 ⁠. (a) K-means and MSE, (b) K-means and SC, (c) K-means and DI, (d) FCM and MSE, (e) FCM and SC, (f) FCM and DI, (g) GA and MSE, (h) GA and SC, (i) GA and DI, (j) H-GA and MSE, (k) H-GA and SC, (l) H-GA and DI, (m) PSO and MSE, (n) PSO and SC and (o) PSO and DI. K-means, FCM and H-GA-based clustering votes for five clusters as optimal choice, while GA and PSO-based clustering account four as the optimal clustering size. Figure 3 indicates that cluster size 5 exhibits a better quality than other given k setting. Thus conclusively, to validate test subject profile, assessment is carried out by considering k as 5 in the following analysis. Case 5: Evaluation of the test sample is necessary to demonstrate the significance of the surgery as considered in case 5. This section discusses the analysis of generated CP gait profile, considering the optimal cluster sizing as k=5 from Fig. 3, on four test subjects (brief description is in Table 1(b)) using affiliated probability index (API). Figure 4 presents the affiliated probability distribution for two neurological intact subjects A, B and two patients C and D from pre and post-surgery in case of five clusters. Extensive analysis shows that cluster 5 in the Fig. 4 is for the subjects with the no gait pathology, the control group, followed by Cluster 3, Cluster 1, Cluster 4 and Cluster 2. Cluster 4 and 2 can be classified as gait pattern with CP cases. Figure 4. View largeDownload slide Gait profile plots for gait profile considering 5 clusters using affiliation probability. Figure 4. View largeDownload slide Gait profile plots for gait profile considering 5 clusters using affiliation probability. The result confirms the gait pattern of subject A as normal. The study shows that subject B exhibits deviated gait pattern. After intensive analysis, it is found that the mean age of control group is 7.09 (2–13 age range), while the test subject considered is of 19-year-old. Thus this gait pattern is misclassified as cluster 4 instead of cluster 5; the control cluster. Three observations are taken for both patients C and D. First observation of the third subject (C1) is performed before surgery, affiliated probability based gait profile also confirms the CP based pattern. In the second (C2) and third (C3) observations of the subject, C shifted from cluster 4 to cluster 3 and then to cluster 5 (the second and third observation are after post-surgery case). Gait profile based on affiliated probability index confirms that surgery helps the subject C to shift toward normal gait profile. Similar kind of observation is observed in patient D, where the first observation is before surgery, and D2 and D3 are post-surgery observations. The D1 and D2 in the plot exhibit that the problem is increased even after surgery. But after 2 years of surgery, observation D3 presents that the gait of the subject could be classified as normal gait profile. Thus health-care professionals can take help of optimized clustering approaches to evaluate the recovery progress of patients after surgery. Doctors can monitor the recovery progress and if required change the strategy for treatment. But here it is important to note that the gait profile from automated optimization based clustering is an indicator of the gait state, thus aid doctors in making decisions regarding rehabilitation. 5. CONCLUSION AND FUTURE SCOPE The current study shows that optimization based clustering approaches can be used for gait profiling for rehabilitation in children with Cerebral Palsy. Different cases regarding the data pre-processing are considered in this study. The dataset used is publicly available as CP gait dataset of 156 subjects having the age range from 2 to 20 for CP cases and 2 to 13 for the control group. K-means, FCM, GA, PSO and hybrid of both GA and PSO-based clustering approaches are used to find the gait profiles for the considered subjects. Best possible clustering sizing is selected based on voting based out of mean square error, silhouette coefficient, and Dunn index. The performance of the proposed clustering methods is evaluated using internal and external cluster performance index. This is validated with the unseen test samples, which undergoes surgery. The result indicates that optimized clustering technique outperforms the traditional and hybrid clustering approaches and can help in diagnosis, assessment and evaluation of treatment outcomes. For study and comparison, the authors have used the same gait parameters as in [13] publication and there may be some missing factors (such as the number of volunteers to validate results may not be enough) and should be given prime consideration as future work. In diagnosis-related cases, there is no clear boundary between healthy and CP cases. The subject can lie in a different cluster at a time. As a future scope, one can analyze nature-inspired based optimized fuzzy clustering approach for treatment intervention for individuals with a disability. 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For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Journal

The Computer JournalOxford University Press

Published: Nov 1, 2018

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