An Optimized Deep LSTM Model for Human Action Recognition

Detection of actions from the human considering the video data poses several applications that include user interfaces, entertainment and video annotation. Provided different actions, the issue can be modelled by categorizing actions. In general, the group of actions poses certain meaning in a domain. Automated comprehending of the behaviour of human and its communiqué with the platform poses active research domain in previous years, because of its impending application in several areas. To attain these kind of complex process, various kinds of research domains are being focused that includes human behaviour modelling under different facets, such as relational attitudes, emotions, and actions and so on. Thus, the detection of person’s behaviour is imperative whenever analyzing complicated actions.

Hence, more interest is provided for recognizing the human actions particularly in real-world platforms [1]. The recognition of human actions considering the set of videos is of huge significance that includes indexing of videos, visual surveillance and various other computer-vision areas. In spite of widespread research, the advancements are done in areas, like recognition of objects and filling the gap amidst the present abilities and the need of applications became large. Moreover, the recognition of actions is complex, because of considerable alterations in the video that are occurred, because of altering aspects that includes scale and viewpoints [2]. The recognition of actions and prediction techniques authorize several real-world applications. The existing techniques minimize the human interference in evaluating huge-scale video data and offers better understanding on future and current states of video data [3].

The recognition of human actions poses several applications, like human computer interfaces, video surveillance and retrieval of videos. In spite outstanding research efforts and several cheering advancements in the past decade, the precise detection of human action is termed to be complex process. There exist two main problems in recognizing the human actions. First is sensory input and the next one is detecting human actions, which are vibrant, unclear and communicative amidst objects. The movement of human is modelled in nature. This complexity hugely limited the efficiency of video-assisted human actions [4]. In previous years, the majority of techniques are devised to recognize the human actions considering the monocular Red Green and Blue (RGB) video series. Unluckily, the monocular RGB data is extremely responsive to several aspects, such as changes in illumination, changes in stands-points and occlusions. In addition, the monocular video sensors are not able to fully observe the motions of human are in 3D space. Thus, in spite the imperative research domains over the few domains, the recognition of actions is a complex issue [5]. There exist several key issues in recognizing the human actions that tend to be stay inexplicable [6].

Recently, several types of action modelling techniques are devised that includes global and local features on the basis of spatial and temporal changes [7], trajectory features on the basis of tracking key point [8], motion alterations on the basis of information of depths [9], and features related to actions on the basis of human pose alterations [10]. The newly devised depth sensors open several capabilities for addressing this issue by offering 3D depth data. This data not only provides a strong human motion capturing model, but also made it liable to effectively place human-object interactions and variations of intra-class [11]. Considering the triumphant tools of deep learning to classification of image and detection of objects, the majority of researchers has adapted deep model on recognizing the human actions. This facilitates action characteristics to be automatically learnt from video samples [12]. Even though, extensively utilized in several tools, precise and effective recognition of human action tends to be a complex research domain of computer vision. The majority of recent surveys have concentrated on narrow issues that include recognition of human motions, 3D-skeleton data and still image data. However, there exists no particular survey to recognize the human actions. There exist several temporal techniques for recognizing the human actions. One adapts generative models, like Conditional Random Field (CRF) and Hidden Markov model.

The recognition of human action is the most imperative domain in the area of artificial intelligence. However, the huge alterations of human actions make the process more complex process. Moreover, accurate recognition of actions is a complex process because of changes in the variations of viewpoint and cluttered backgrounds. These challenges are considered as a motivation for developing a new model for human action recognition (HAR). The aim this research is to design a model for HAR with optimized Deep LSTM. The major contributions include:

IIWBPR-based Deep LSTM for HAR: The IIWBPR-based Deep LSTM is employed for recognizing the human actions. Deep LSTM is trained with IIWBPR in order to produce optimum weights for recognizing the actions of humans.

The proposed IIWBPR algorithm is devised by combining improved invasive weed optimization (IIWO) and Poor rich optimization (PRO) algorithm.

Paper is orchestrated as: Section 2 exemplifies techniques employed for HAR. Section 3 illustrates proposed model to recognize the human actions using IIWBPR-based Deep LSTM. Section 4 illustrates outcomes analysis and section 5 provides conclusion with IIWBPR-based Deep LSTM.

Aim is to provide a model for human action recognition with optimized Deep LSTM. The objective relies in devising an optimization driven technique for determining the actions of human considering a set of videos. At first, videos are collected, and subjected to video frame mining in which frames are mined using inputted video. Thereafter, the feature mining is executed to extract significant features from input video. Feature extraction helps to reduce dimensionality. Here, the features, like SLIF, SLBT, local Texton XOR pattern, LGBP [21], Shape Index histogram, LGXP [21] are extracted. Finally, the human action recognition is done using Deep LSTM [22] and its training is done using IIWBPR algorithm. The IIWBPR algorithm is devised by combining improved IWO [23] and PRO algorithm [24]. The configuration of human action recognition model with IIWBPR-based Deep LSTM is exemplified in Figure 1.

1.png

Figure 1. Human action recognition model with IIWBPR-based Deep LSTM

3.1 Acquire the input images

The first step in recognizing the actions of human from the videos is determining the actions from each frame of video. The actions in the videos represents that the person in videos are performing some kind of activity. Assume a video database M that contains r videos wherein persons are doing various activities and it can be modelled as:

$M=\left\{ {{F}_{1}},\,{{F}_{2}},\ldots ,{{F}_{t}},\ldots ,{{F}_{r}} \right\}$     (1)

where, F_t is t^ih video, and r refers total video count.

3.2 Obtain the video frames

Video comprises several frames and hence it is crucial to mine frames considering the videos for enhanced processing. Frames are extracted as one can discover the activity that occurs in the video just by observing few frames. Inputted video F_t is fed to video frame mining for extracting frames and each frame is employed by C and can be expressed by:

$C=\left\{ {{C}_{1}},\,{{C}_{2}},....{{C}_{p}},...{{C}_{q}} \right\}\,;\,{{C}_{p}}\in {{F}_{t}}$     (2)

where, C_p is mined frame from video F_t and q express total video frames count such that p= {1, 2, ..., q}.

3.3 Acquire vital features

Vital features are obtained from the video frames C wherein the apt features, namely SLIF, SLBT, Local Texton XOR patterns, LGBP, shape index histogram, LGXP and statistical features are obtained, and each feature are briefly exemplified below.

(i) SLIF

SLIF [25] represents feature descriptor, which considers an elite depiction sampling template using spider's orb-web model. A group of vectors $D=\left\{D^1, D^2, \ldots, D^\lambda\right\}$ linked to a group of targeted points $\left\{E 1, E_2, . ., E_\lambda\right\}$ from image. Each point $E_f$ is modelled through feature scale $\alpha_f$ and coordinate point $\left(l_f, m_f\right)$. At first, the orientation $\phi_f$ is allocated to each point $E_f \in E$. Then, an orb web model $Q_f$ is described for each point $E_a$. Web weights $K_f^{n, o}$ considers all web nodes $(n, o) f$. SLIF feature $D$ is obtained by combining 8-bit binary string series linked to its key point $E_a$ and can be expressed as:

${{D}_{f}}=\left[ \begin{align}  & G_{1,1}^{f},G_{2,1}^{f},...,G_{n,1}^{f},G_{1,2}^{f},G_{2,2}^{f}, \\ & ...,G_{n,2}^{f},G_{1,2}^{f},G_{1,2}^{f},...,G_{n,o}^{f} \\\end{align} \right]$     (3)

where, $G_{n, o}^f$ expresses 8-bit binary array. The SLIF feature is designated as Q₁.

(ii) SLBT

SLBT features [26] unifies both texture and shape information. Assume I=[I₁, I₂, .., I_u] be set of image provided u indicating total sets of training, and R=[R₁, R₂, .., R_u] represents shape landmark points. Thus, landmark points are assigned and alterations of shape are acquired wherein the shape vector R in training set is modelled by:

$R\approx \bar{R}+{{S}_{x}}{{T}_{x}}$     (4)

${{T}_{x}}=S_{x}^{N}(R-\bar{R})$     (5)

where, $\bar{R}$ represents mean shape, $T_x$ symbolize shape or weight attributes, $x$ depicts shape in $T_x, S_x$ represents eigen vector of highest eigen values. In addition, SLBT adapts LBP to acquire illumination and noise characteristics. The mining of LBP is quicker and simpler compared to Gabor wavelet. By considering the center pixel by $\left(\ell_j, \varpi_j\right)$ and intensity by $L_j$, function $A(b)$ is expressed as:

$A(b)=\left\{ \begin{align}  & 1,\,\,\,\,\,\,\,b>0 \\ & 0,\,\,\,\,\,\,b\le 0 \\\end{align} \right.$     (6)

In center pixel, LBP is acquired as:

$LBP({{\ell }_{j}},{{\varpi }_{j}})=\sum\limits_{x=0}^{7}{H({{\ell }_{x}}}-{{\ell }_{j}}{{)}^{{{2}^{\ell }}}}$     (7)

where, $\ell_x$ represents grey values of eight adjoining pixels having $\ell_x(x=0,1,2,3,4,5,6,7)$. Assume $B=\left[B_l, B_2, \ldots, B_u\right]$ is employed as LBP feature histogram, which is trained. Modeling of Texture hence provided as:

${{T}_{y}}=S_{y}^{N}(B-\bar{B})$     (8)

where, T_y represents texture model, y signifies texture in T_y, eigen vector express S_y, and $\bar{B}$ defines mean vector. In addition, the shape and texture are produced by employing Principal Component Analysis (PCA) on unified vector attribute, which is provided by:

${{T}_{xy}}=\left( \begin{align}  & {{H}_{x}}{{T}_{x}} \\ & \,\,\,{{T}_{y}} \\\end{align} \right)$     (9)

$D=S_{xy}^{N}({{T}_{xy}}-{{\bar{T}}_{xy}})$     (10)

where, H_x refers diagonal matrix, D stands for shape texture attribute, S_xy describes eigen vectors, and $\bar{T}_{x y}$ elucidates mean vector. SLBT feature is expressed as Q₂.

(iii) Local XOR Texton pattern

The aim of Local XOR Texton pattern [27] is to categorize image textures and it is feasible to apply on various areas, like image retrieval and so on. It relies on correlation amidst the corresponding pixel and intermediate pixel in an image. The Local XOR Texton pattern is manipulated as:

${{\ell }_{X,Z}}=\sum\limits_{x=1}^{X}{{{2}^{(x-1)}}\times {{k}_{3}}\left( T({{q}_{j}})\otimes T\left( {{q}_{l}} \right) \right)}$     (11)

${{k}_{3}}\left( \nu \otimes \varepsilon  \right)=\left\{ \begin{align}  & 1\,\,\,\nu \ne \varepsilon  \\ & 0\,\,else \\\end{align} \right.$     (12)

where, T(q_j) exemplifies Texton shape for a neighbouring pixel q_j and T(q_l) stands for Texton shape for a neighbouring pixel q_l and $\otimes$ symbolizes XOR operation amidst variables. The Local XOR Texton pattern feature is designated as Q₃.

(iv) LGBP

LGBP [21] is acquired through encoding values of magnitude in LBP operator. Consider value of center by $\hbar_c$ and adjoining pixel by $\hbar_d$, then LGBP evaluation is manifested by:

$\Im \left( {{\hbar }_{d}}-{{\hbar }_{c}} \right)=\left\{ \begin{matrix}   \begin{matrix}   1, & {{\hbar }_{d}}\ge {{\hbar }_{c}}  \\\end{matrix}  \\   \begin{matrix}   0, & {{\hbar }_{d}}<{{\hbar }_{c}}  \\\end{matrix}  \\\end{matrix} \right.$     (13)

In addition, the pattern of LBP with value of pixel is evaluated by allocating binomial factor d of each $\Im\left(\hbar_d-\hbar_c\right)$, and provided by:

$LBP=\sum\limits_{d=0}^{7}{\Im \left( {{\hbar }_{d}}-{{\hbar }_{c}} \right)\,{{2}^{d}}}$      (14)

Expression (14) depicts spatial pattern of local image texture. In addition, LGBP feature is expressed by Q₄.

v) Shape index histogram

It represents an image geometry that captured second-order phase in continues time and permits one to analyze the curvature histogram distribution. The shape index histogram [28] is produced by selecting the group of n_b bin centers b₁b_n consistently dispersed along the shape index interval π/2π/2. Considering a bin B centered at b, one can compute its total contributions considering the weighted sum of both curvedness metric c and spatial donut weighting D and is provided as:

$B\left( \sigma ,x,\gamma ,r,b,\beta  \right)=\frac{\sum\nolimits_{x}{Dc\exp \left( -\frac{{{(b,s)}^{2}}}{2{{\beta }^{2}}} \right)}}{\sum\nolimits_{x}{D}}$      (15)

where, β symbolizes tonal scale parameter that helps to adjust smoothing, σ stands for scale, $s$ represents shape, and x,r,γ exemplifies spatial pooling. The shape index histogram feature is designated as Q₅.

(vi) LGXP

LGXP [29] stages are firstly quantized to huge variety and then LXP operator is used for quantizing stages of innermost pixel and each of its neighbours. At last, the result binary labels are combined equally and local structures of central pixel. In addition, the patterns of LGXP in decimal and binary structure are expressed as:

$\begin{align}  & {{\varepsilon }_{2}}={{\tau }_{\upsilon ,\kappa }}\left( {{\aleph }_{\nu }} \right) \\ & ={{\left[ \tau _{\upsilon ,\kappa }^{\beta },\tau _{\upsilon ,\kappa }^{\beta -1},...,\tau _{\upsilon ,\kappa }^{1} \right]}_{\eta }}={{\left[ \sum\limits_{\alpha =1}^{\beta }{{{2}^{\alpha -1}}}.\tau _{\upsilon ,\kappa }^{\alpha } \right]}_{\mu }} \\\end{align}$      (16)

where, $\eta$ expresses binary and $\mu$ is decimal, innermost pixel location in Gabor stage map with scale $v$ and orientation $\kappa$ is expressed by $\aleph_v$ while dimension of neighbourhood is expressed by $\beta$ and $\tau_{v, \kappa}^\alpha(\alpha=1,2, \ldots, \beta)$ indicates a structure evaluated amidst $\aleph_v$ and neighbour $\aleph_\alpha$, and represented as:

$\tau _{\upsilon ,\kappa }^{\alpha }=\vartheta \left( {{\theta }_{\upsilon ,\kappa }}\left( {{\aleph }_{\nu }} \right) \right)\otimes \vartheta \left( {{\theta }_{\upsilon ,\kappa }}\left( {{\aleph }_{\alpha }} \right) \right),\,\alpha =1,2,...,\beta $     (17)

where, θ_υ,κ(•) expresses phase, LXP operator is denoted as ⊗which is devised using XOR operator. ϑ(•) refers quantization operator which computes quantized code, which is modelled by:

$\nu \otimes \gamma =\left\{ \begin{align}  & 0,\,\,\,\,\,\,\,if\,\nu =\gamma  \\ & 1,\,\,\,\,\,\,\,\,\,\,else \\\end{align} \right.$     (18)

$\begin{align}  & \vartheta \left( {{\theta }_{\upsilon ,\kappa }}\left( \bullet  \right) \right)=\alpha ;\, \\ & \,if\,\frac{360*\alpha }{\partial }\le {{\theta }_{\upsilon ,\kappa }}\left( \bullet  \right)<\frac{360*\left( \alpha +1 \right)}{\partial },\,\,\,\,\,\, \\ & \alpha =0,1,....,\partial -1 \\\end{align}$     (19)

where, $\partial$ represents number of phase ranges. The LGXP feature is designated as Q₆.

(vii) Statistical features

Some of the statistical features adapted as imperative feature are listed:

(i) Mean

It exemplifies the complete values of images with respect to complete number of pixel values and it articulated as:

${{Q}_{7}}=\frac{1}{o}\sum\limits_{\psi =1}^{o}{{{\rho }_{\psi }}}$     (20)

where, o exposes total images count and number of pixel values is obtained as ρ_ψ.

(ii) Variance

It indicates squared standard deviations wherein values of inputted image depicts variance and uttered as:

${{Q}_{8}}=ƛ_\chi^2=\frac{1}{o}\sum\limits_{\psi =1}^{o}{\left( {{\rho }_{\psi }}-{{Q}_{7}} \right)}$     (21)

where, Q₇ specifies mean.

(iii) Standard deviation

It demonstrates variance square root, that can be modelled as:

${{Q}_{9}}=\sqrt{\frac{1}{o}\sum\limits_{\psi =1}^{o}{\left( {{\rho }_{\psi }}-{{Q}_{7}} \right)}}$     (22)

where, standard deviation is denoted by Q₉.

(iv) Kurtosis

Statistical evaluated utilized for defining experimented distribution of image through a mean is termed as kurtosis and represented as:

${{Q}_{10}}=\frac{\sum{{{\left( {{Q}_{7}}-\overline{{{Q}_{7}}} \right)}^{4}}}}{o{{Q}_{9}}}$     (23)

where, Q₇ is mean and Q₉ expresses standard deviation.

(v) Skewness

It describes the quantification of distorted images from definite images and is modelled as:

${{Q}_{11}}=\frac{\sum\nolimits_{\psi }^{o}{\,\left( {{\xi }_{o}}-\overline{{{Q}_{7}}} \right)}}{\left( o-1 \right)*{{Q}_{10}}}$     (24)

where, o defines number of images, ξ_o reveals arbitrarily selected image. Hence, the obtained feature vector is manipulated as:

$Q=\{{{Q}_{1}},{{Q}_{2}},\ldots ,{{Q}_{11}}\}$     (25)

3.4 Recognition of human activity using IIWBPR-based Deep LSTM

The human activity is recognized considering the IIWBPR-Deep LSTM which is trained with IIWBPR with feature vector Q. The deep LSTM has the capability for learning the features automatically from the raw sensor data. Here, the IIWBPR is obtained by unifying the merits of both IWO and PRO algorithm. Deep LSTM and IIWBPR steps are defined herewith.

a) Deep LSTM structure

Deep LSTM [30] is employed for managing diminution and assortment of data by utilizing using feature vector Q and input parameter set U. The inputted node V_e adapts input U using deep input and through hidden states a_e-1. Thus, the data forecast becomes non-linear, and hence technique employed to evaluate output is termed to be non-linear, that offers output with highest accuracy. The outcome of cluster-assisted topology routing U and a_e-1 is fed to tanh function is represented as:

${{V}_{e}}=\tanh \left( \delta .{{g}_{V\delta }}+{{O}_{e-1}}{{g}_{V\delta }}+{{P}_{sa}} \right)$     (26)

where, g_Vδ depicts weight matrix, O_e-1 displays input of hidden state, and P_sa signifies bias to inputted node.

Input gate process is represented as:

$\mathrm{T}_e=\lambda\left(\delta \cdot g_{V \delta}+a_{e-1} g_{V a}+P_s\right)$      (27)

where, T_e refers inputted gate at instance e, λ signifies sigmoidal activation function, and P_s denotes bias to inputted gate and is provided by:

$\mathrm{T}_e=\lambda\left(\delta \cdot g_{V \delta}+a_{e-1} g_{V a}+P_s\right)$      (28)

where, V_e is interior state at instance e, and k_e-1 is internal state at instance e-1. The forget gate W is utilized to reinitiate memory cell internal state, and definedby:

${{W}_{e}}=\lambda \left( \delta .{{g}_{W\delta }}+{{a}_{e-1}}{{g}_{Ws}}+{{P}_{f}} \right)$     (29)

where, W refers forget state at instance $e$, Θ signifies linear operator, g_Wδ stands for weight among forget and input layer, g_Ws symbolizes weight among forget and hidden state, and P_f depicts forget gate bias.

Output gate O_e is presented by:

${{O}_{e}}=\lambda \left( e.{{g}_{a\delta }}+{{a}_{e-1}}{{g}_{Oa}}+{{P}_{o}} \right)$      (30)

where, g_aδ exemplifies weight among output and input layers, g_Oa denotes weight amidst output and hidden states, and P_o denotes output gate bias.

Furthermore, the finalized output obtained through memory cell is manifested by:

${{a}_{e}}=\tanh \left( {{k}_{e}} \right)\Theta {{O}_{e}}$      (31)

b) Training with IIWBPR

Deep LSTM training is implemented with IIWBPR wherein is generated by combining IWO and PRO. IWO [23] represents population-assisted evolutionary technique motivated from weed colony features. It relies on chaos theory, which aids to attain a quick rate of convergence and elevated accuracy. It maximizes population diversity and is utilized to update location to weed. It uses sigma that helps to create quick variations over the iterations. On the other hand, the PRO [24] is obtained by two sets of poor and rich that tries to attain wealth and enhance its economic conditions. It had effective performance in solving the real-time problems and finds best parameter values. It discovers optimum parameters by controlling the conditions of each problem. Hence, the incorporation of PRO in IWO is done to improve whole performance. Hence, developed IIWBPR steps are described herewith.

Step 1: Initialization

Fundamental step includes initialization of solution and expressed as Y with total Ψ solution such that 1≤Φ≤Ψ.

$Y=\{{{Y}_{1}},{{Y}_{2}},\ldots ,{{Y}_{\Phi }},\ldots ,{{Y}_{\Psi }}\}$      (32)

where, Ψ depicts total solution, and Y_Φ referred Φ^th solution.

Step 2: Locate error

The error is generated based on mean square error and the optimal results are obatined based on the minimum error value.

$E r r=\frac{1}{\mathrm{E}} \sum_{o=1}^{\mathrm{E}}\left[\varpi_o-O\right]^2$    (33)

where, Err is error function, E expresses total instances, ϖ_ο defines predicted output, and O is Deep LSTM output.

Step 3: Specify logistic chaotic map

According to IWO [23], the admired chaotic maps indicates logistic chaotic map and is employed as a second order polynomial and this logistic chaotic map is represented by:

${{Y}_{h+1}}=v{{Y}_{h}}(1-{{Y}_{h}})$     (34)

where, v is capricious number and Y_h is arbitrary number amidst [0, 1]. Hence 0<v≤4, and h=0, 1, 2, … and Y_h∈[0,1].

According to IWO, the optimum weed is utilized to finest solution and is represented by:

$Y_{w}^{z+1}=\chi (z)\times Y_{w}^{z}+({{Y}_{best}}-Y_{w}^{z})$     (35)

where, $Y_w^{z+1}$ depicts new weed location, $Y_w^z$ reveals current weed location, Y_best denotes best weed discovered, and χ(z) expose presents standard deviation.

The standard deviation is provided by:

$\begin{align}  & \chi (z)={{\left( \frac{Z-z}{Z} \right)}^{v}}\left( {{\chi }_{initial}}-{{\chi }_{final}} \right) \\ & +{{\chi }_{final}}\times \gamma (z) \\\end{align}$      (36)

where, γ(z) refers chaotic mapping in z^th iteration, Z indicates highest iteration.

$Y_{w}^{z+1}=Y_{w}^{z}(\chi (z)-1)+{{Y}_{best}}$      (37)

For obtaining better efficiency, PRO is used. According to PRO [24], the update expression is manifested by:

$\vec{Y}_{rich,z}^{new}=\vec{Y}_{rich,z}^{old}+i\left( \vec{Y}_{rich,z}^{old}-\vec{Y}_{poor,best}^{old} \right)$     (38)

where, $i$ stands for class gap parameter, $\vec{Y}_{\text {poor,best }}^{\text {old }}$ is present position of best member of the poor population, $\vec{Y}_{\text {rich }, z}^{o l d}$ stands for present position of rich member and $\vec{Y}_{r i c h, z}^{n e w}$ is new position of rich member.

Assume $\vec{Y}_{\text {rich }, z}^{\text {new }}=Y_z^{w+1}$ and $\vec{Y}_{\text {rich }, z}^{\text {old }}=Y_z^w$ and $\vec{Y}_{\text {poor }, \text { best }}^{\text {old }}=$ $Y_{\text {pbest }}^w$

Now equation becomes:

$Y_{z}^{w+1}=Y_{z}^{w}+i\left( Y_{z}^{w}-Y_{pbest}^{w} \right)$     (39)

$Y_{z}^{w+1}=Y_{z}^{w}+iY_{z}^{w}-iY_{pbest}^{w}$     (40)

$Y_{z}^{w+1}=Y_{z}^{w}\left( 1+i \right)-iY_{pbest}^{w}$     (41)

$Y_{z}^{w}=\frac{Y_{z}^{w+1}+iY_{pbest}^{w}}{1+i}$     (42)

Substitute expression (42) in expression (37):

$Y_{w}^{z+1}=\left[ \frac{Y_{z}^{w+1}+iY_{pbest}^{w}}{1+i} \right](\chi (z)-1)+{{Y}_{best}}$     (43)

$\begin{align}  & Y_{w}^{z+1}=\frac{Y_{z}^{w+1}}{1+i}(\chi (z)-1) \\ & +\left[ \frac{iY_{pbest}^{w}}{1+i} \right](\chi (z)-1)+{{Y}_{best}} \\\end{align}$     (44)

$\begin{align}  & Y_{w}^{z+1}-\frac{Y_{z}^{w+1}}{1+i}(\chi (z)-1) \\ & =\left[ \frac{iY_{pbest}^{w}}{1+i} \right](\chi (z)-1)+{{Y}_{best}} \\\end{align}$     (45)

$\begin{align}  & Y_{w}^{z+1}\left[ 1-\frac{(\chi (z)-1)}{1+i} \right] \\ & =\left[ \frac{iY_{pbest}^{w}}{1+i} \right](\chi (z)-1)+{{Y}_{best}} \\\end{align}$      (46)

$\begin{align}  & Y_{w}^{z+1}\left[ \frac{1+i-\chi (z)+1}{1+i} \right] \\ & =\left[ \frac{iY_{pbest}^{w}}{1+i} \right](\chi (z)-1)+{{Y}_{best}} \\\end{align}$     (47)

$\begin{align}  & Y_{w}^{z+1}\left[ \frac{2+i-\chi (z)}{1+i} \right] \\ & =\left[ \frac{iY_{pbest}^{w}}{1+i} \right](\chi (z)-1)+{{Y}_{best}} \\\end{align}$     (48)

$\begin{align}  & Y_{w}^{z+1}\left[ 2+i-\chi (z) \right] \\ & =\left[ iY_{pbest}^{w} \right](\chi (z)-1)+(1+i){{Y}_{best}} \\\end{align}$     (49)

Thus, the update expression of IIWBPR is given as:

$Y_{w}^{z+1}=\frac{iY_{pbest}^{w}(\chi (z)-1)+{{Y}_{best}}(1+i)}{2+i-\chi (z)}$     (50)

Step 4: Re-estimate error to generate optimum solution

The most favourable solution is enumerated with error, and solution having least error is known as finest solution.

Step 5: Termination

Aforesaid steps are repeated until highest iteration is attained to expose finest solution. The pseudo-code of IIWBPR is elucidated in Table 1.

Table 1. Pseudo code of IIWBPR