# Methods of Nonlinear Time Series Cycle Analysis in Big Data Environment and IoT Application

Authors: Ou Xiong; Shuxiang Li
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

There have been a lot of changes in nonlinear time series analysis techniques over the last few decades. They came from their time series background and have grown some of these techniques to try to fill in a gap in the ability to model and predict certain types of data, like chaotic and fixed systems. It is possible to find a lot of these systems all over different places in both natural and human worlds. This study explains how these techniques came to be, what they are based on, and how some of them can predict what will happen in the future. This study is trying to figure out how they work and what they could be used for. The contribution of this study is a different way to look at time series data. It looks at how the data is made and how big the time series is. Here, we talk about how the model comes together in terms of order. In this example, we look at the time series of the sectorial indices on the National Stock Exchange (India) to show that the model is getting close. Convergence also points out that, when we fit the same number of data points to the same model, we get the same thing every time. This shows that a true model correctly predicted the value of the future and that data always comes together in nature. A study found that the size of a big data time series should be taken into account when looking at the time series’ realization. This means that the DGP intervals for each study period can be the best representative intervals for that time period.


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## Body

## 1. Introduction

### 1.1. Nonlinear Time Series Analysis

We want to come up with a new way to make time series out of current data and keep going until we have enough information about how the data is made to keep going. Model convergence is a concept in this paper that refers to the size of the series. At a certain point in the series, the fitted model is the same as the model that was used. The forecasted range is used to evaluate the model. The converge model is the best way to predict what the future value will be if the current value is in the range that was forecasted for it. If not, you should look for a different model to look for instead of this one. People might think that a model does not work when it does not come together because you chose the wrong model, or you did not have enough information to reach the limiting model [1].


### 1.2. Nonlinear Time Series Analysis (NTSA)

Nonlinear time series analysis spans a set of methods that extract dynamic information from time series data. Its importance lies in the fact that for many real-world systems, information regarding their temporal behavior is available only through a scalar time series of a single variable. For dissipative dynamical systems, variables move asymptotically toward a low-dimensional attractor that tells us about how they move. We cannot usually see these attractors in the real world because they are too complicated for us to see right away. Nonlinear time series analysis tools help us to know more about the underlying attractor which controls the dynamics of the system under consideration. Phase space reconstruction is the backbone of the nonlinear time series analysis, and two pioneering papers laid the foundation for this analysis. The first one was proposed by Bellazzi et al. [2]. The Internet of Things (IoT) is aimed at making conventional objects intelligent by utilizing a wide range of cutting-edge technologies, including Internet protocols, data analytics, embedded devices, and communication technologies. The IoT’s potential economic impact is anticipated to create a number of business opportunities and hasten the growth of IoT-based services, based on a McKinsey analysis of the IoT’s influence on the world economy [3].The works of Packard et al. in 1980 [4] and another in 1981 by Takens [5] introduced the concept of phase space reconstruction into dynamical system theory. Eckmann and Ruelle also have a role in the extension of the work. Packard et al. gave the techniques regarding the reconstruction of phase space (state space) from the univariate observational data of any dissipative dynamical system as well as the method for determining the number of independent variables (dimension) needed for specifying the state of the system at any given instant. As a demonstration of the method proposed by Packard et al., here, we make use of a standard model of a dynamical system (Rössler system) that can be described by a set of three ordinary differential equations that are given by [6]:
(1)X=−y+z,Y=a+ay,Z=b+xz–cs,with values of control parameters as
(2)a=0.2,b=0.2,c=5.7.When you solve these equations, you get something called a chaotic attractor that looks like Figure1(a). According to Packard et al., the idea behind the reconstruction of the attractor is that, in order to know the state of a three-dimensional system at any given time, you should be able to measure any three separate things [7]. In other words, any set of three independent quantities that label the states of the attractor in a unique and smooth way is called diffeomorphically equivalent or the same thing. So, if you get a time series from the equation by sampling a single coordinate, you can get a variety of three different things. In one example, you could compare the current value of the coordinate with the values of the coordinate at two previous times, like xt, xt, and xt2 [8]. This is one way to look at three different things at once. This is what Packard et al. talked about in their paper. Another set of three coordinates can be found by making the delays small and taking the differences. This set of coordinates is called the “Packard coordinates” (t). Figure 1(b) shows the reconstructed Rössler attractor, which is shown in the figure. The effect of white noise on a synthetic time series is shown in Figures 1(c) and 1(d) when 2% and 4% white noise is added, respectively, to the original time series xt. It can be seen that with 4% white noise contamination, the attractor loses its topological structure in the phase space. So, the method proposed by Packard et al. is not robust to even a small percent of noise. So, it is not practical to apply this method to real-world systems as it is contaminated with different types and levels of noise. This method works fine only with the noise-free time series [9].Figure 1
Rössler chaotic attractor: (a) using the solutions of the equation; (b) reconstructed attractor using coordinatesxt, ẋt, and ẍt; (c) reconstructed attractor with 2% white noise and (d) with 4% white noise.
(a)(b)(c)(d)Taken formalized the concept of Packard et al. which is named as Taken’s theorem or delayed coordinate embedding theorem which is now conventionally used for the nonlinear time series analysis [10].


#### 1.2.1. Delay Coordinate Embedding

The regular technique for stage space recreation is delay coordinate implanting. It permits one to reproduce the total elements of a complex nonlinear framework from a univariate time series. Assuming the recreation is done accurately, it ensures that it is topologically indistinguishable from the genuine elements of the hidden framework since significant properties of dynamical frameworks are invariant under diffeomorphism. In this strategy, a progression of past upsides of a solitary variablext from a dynamical framework is utilized to shape a vector that characterizes a point in the new space. So, for building M layered space, one should make M-layered vectors R~t from M time postponed tests of the estimations xt, to such an extent that [11]
(3)R¯t=xt−T,t−2r,⋯⋯<xt−M−1.One can also use future values ofxt for making the M-dimensional vectors which are mathematically equivalent to taking past values of xt as mentioned above. But for practical applications like prediction, we normally take the past values [12].


#### 1.2.2. Estimation of Time Delay (τ)

It is very important to know what value of delayτ for a given time series gives the correct delayed coordinates for reconstructing the equivalent topological structure that holds the properties of the true underlying dynamics. The original embedding theorem only requires that τ should be nonzero and not numerous of any circle’s period. This is legitimate just on an endless clamor spare energy series information. In genuine circumstances, the time series might be loud and limited in length. In such circumstances, one requires a higher worth of τ to unfurl the elements of the hidden framework. In the event that we decide τ tiny, the M directions in every one of the vectors given in the equation are firmly associated. So, the stage space elements after remaking will lie near the principle askew of the remade stage space [13]. This is shown in Figure 2 using the solution for xt of the Rössler system given by the equation which is a univariate time series and using τ=1. Here, the reconstruction is done with coordinates xt and xt−τ. The reconstructed Rössler attractor with τ=5, 25, and 35 is also shown in Figure 2 [14]. If we choose τ improperly, the reconstruction will not unfold the subspace of the underlying system; as a result, the topology will not be conjugate to the true dynamics. So, estimating the proper value of τ for a given time series is a challenging task. In the past few decades, many methods have been proposed for estimating the inserting boundaries—delay τ and implanting aspect M. In any case, there is no general strategy for the determination of τ since the fundamental interaction is framework subordinate. Additionally, a worth of τ that functions admirably for one reason may not function admirably for another. A couple of the strategies normally utilized are referenced underneath [15].
(1)
The initial zero of the autocorrelation capacity of the time series is taken as the legitimate postponementτ, since it gives the littlest τ that amplifies the direct autonomy of the directions of the installing vector(2)
The first minimum of the autocorrelation function is taken asτ(3)
The first minima of the average mutual information [16] are taken as τ(4)
The first minima of the correlation sum [17] are taken as τ. Methods 2 and 3 mentioned above happen at τ values that expand more—general types of freedom. It is to be noticed that we really want the littlest τ in light of the fact that the recreated attractor can overlay over onto itself as the value of τ increases, which creates difficulties for the analysis. Besides the abovementioned methods, there have been some theoretical discussions that help to find the optimal τ [18]Figure 2
Reconstructed Rössler attractor with two coordinates with delayτ=1, 5, 25, and 35.


#### 1.2.3. Estimation of Embedding Dimension (M)

The second important parameter for the reconstruction is the inserting aspectM. As on account of postponement τ, bigger upsides of M are bad, since a solitary loud point in the time series will influence M number of the places in M-layered inserting. So, one ought to track down the littlest worth of M that properly unfolds the topological structure that is conjugate to the underlying dynamics. As an example, in the case of the Rössler system, the trajectory in the phase space will not cross each other. But when we do embedding in M=2, the trajectory crossing occurs as in Figure 2 [19]. Here, the embedding does not succeed in unfolding the subspace of the underlying system. So, to reconstruct properly, we have to embed in a higher dimension greater than M=2. The reconstructed Rössler attractor with delay τ=25 embedded in dimensions M=2 and M=3 is shown in Figure 3. It can be seen clearly that the crossing trajectories when embedded with M=2 are unfolded properly with M=3. So, the minimum embedding dimension for the Rössler attractor is M=3. The fractal dimension of the Rössler attractor is slightly above 2. For the values of control parameters a=0.2, b=0.2, and c=14 in the equation, the fractal dimension is estimated between 2.01 and 2.02. So, the minimum embedding dimension for unfolding the subspace of the underlying dynamics of the Rössler system is M=3 [20].Figure 3
Reconstructed Rössler attractor withM=2 (a) and M=3 (b).
(a)(b)The original embedding theorem requires thatM>2d to assure the reconstruction is proper, where d is the genuine element of the hidden elements. Yet, experiencing the same thing, d is seldom known. There are a few techniques to register the inserting aspect with basic ideas involving the essential necessity of the implanting hypothesis. A portion of these strategies is the particular worth disintegration of the example covariance grid, the immersion of some invariant amount, the technique for genuine vector fields, the cross-approval strategy, and the strategy for false nearest neighbors.We tested the cross-validation, the saturation of invariants, and the false nearest neighbor methods. The best results were obtained with the false nearest neighbor [21] (FNN) method proposed by Kennel et al. which we used in this work. It is explained in the following section.In an FNN-based method, the calculations are in light of the possibility that in the entry from aspectM to aspect M+1, one can separate between focuses on the circle xt that are “valid neighbors” and focuses on the circle xt which are “misleading neighbors.” A misleading neighbor is a point in the informational collection, that is, a neighbor exclusively in light of the fact that we notice the circle (attractor) from an installing space M lower than the genuine aspect ME of the framework (M<ME). Whenever we accomplish the appropriate inserting aspect or higher (M≥ME), all neighbors of each circle point in the multivariate stage space will be valid neighbors. This can be demonstrated with the Hénon map [22]:
(4)R2mt,n=∑K=0M−1xt+T−xxt+T2,R2Mt−n=xx2+xt+T−xxt+T+⋯⋯+xt+M9−1T−x2t+M.By increasing the dimension toM+1, the distance between the reference point and its neighbors is altered due to the new coordinates xt+Mτ and xnt+Mτ. So now, the square of the distance between them is given by [23]
(5)R2m+1xt+T−xxt+TR2mt,nt+M2−1T.The criteria for checking whether the embedding has fully unfolded the underlying dynamics are that we measure the increase in distance betweenxt and xnt when going from dimension M to M+1. The change in distance can be obtained from the equations. The criteria for deciding whether a neighbor is a true or false neighbor are that [4]
(6)R2m+1xt+T−xxt+T1/2kR2mt,n=xt+MT−xn2x2t+MRmt,n>Rtd,where Rtol is some threshold, and if 10≤Rtol≤50, the number of false neighbors identified by this criterion is constant.In Figure3, the projection of the Hénon map onto a M=1 layered stage space (the x-hub) as well as in a M=ME=2 layered is displayed to install space. Focuses B and C give off an impression of being neighbors in the projection onto the x pivot (M=1). A magnified portion of the Hénon map is shown in Figure 4 to clearly show points A, B, and C. Now, when it is embedded in dimension (M=ME=2), points A and B almost stay together, but point C moves away from point B. It shows that point C is not a true neighbor of point B as seen when embedded in M=1. So, point C is a false neighbor of point B, but since point A stay near to B in dimensions M=1, M=2, and higher, it is a true neighbor [5].Figure 4
Descriptive statistics graph.From the mathematical point of view, if we are inM dimensions, we denote the nth nearest neighbor of a reference point xt by xnt. Then, from the equation, the square of the Euclidian distance between the point xt and this neighbor xnt is given.The spearheading by Packard et al. and other people and the first paper by Takens, which was written about 20 years ago, set the stage for another way to look at complex dynamical characteristics, which is now called nonlinear time series assessment. Expansions to nonlinear stochastic models and to nonlinear quantifiable assessment are complicated and have been looked at in a lot of different ways [24].This means that if a sign is called “fixed,” it means that all change probabilities from one state of the system to the next are not affected by time while the sign is being looked at. All of the important limits should be kept the same, and the time series should show enough of the characteristics that have a place with the components so that the probabilities or other rules can be checked properly. In chaos, the most important thing is that you cannot predict what will happen next, no matter how things turn out in a deterministic way. This unpredictability comes from the fact that the game plans are naturally shaky, which is shown by what is called “sensitive dependence on early on conditions.” In two or three time steps, the small differences between the initial conditions of a large number of bearings are set off. Each small detail of the state at time zero is important for the specific method of the course in state space [16].This leads to two concepts:(i)
Loss of information related to unpredictability(ii)
Nearby trajectories separate exponentially fast over timeOn the off chance that partition over the long haul is exceptionally sluggish, this is not in itself sensational—ordinary of transcendently intermittent frameworks. Assuming that this detachment of directions after some time is quick, dramatically quick—outstanding dissimilarity, we can discuss disorder. This increment of the division after some time can be an attribute of the hidden framework [17].(1) Estimation Procedure and Sampling Rates. A period series (as some other estimation) needs to give sufficient data to decide the amount of revenue unambiguously. Calculations have a few insignificant necessities concerning how long and how exact the time series should be and how as often as possible estimations must be taken to get significant outcomes. In the event that significant outcomes are not acquired, then a more modest subpeculiarity is to be examined. Adequate testing is expected overall not simply to specific strategies for time series investigation.(2) Illustration of Adequate Examining. An illustration of adequate examining is the measurement of temperature changes on top of a house. What number of single estimations is required? Reply is it depends [5]:
(i)
For one day: check one every hour for multiweek to get a fair normal of the temperature profile(ii)
For occasional changes: fixed times, when daily for a year(iii)
For year to year: a few estimations consistently(iv)
For a total depiction of peculiarity: when 60 minutes, consistently, for a few yearsDemonstrating and prediction rely upon the estimation cycle, examining strategy, and furthermore the type of portrayal of the information.


## 1.1. Nonlinear Time Series Analysis

We want to come up with a new way to make time series out of current data and keep going until we have enough information about how the data is made to keep going. Model convergence is a concept in this paper that refers to the size of the series. At a certain point in the series, the fitted model is the same as the model that was used. The forecasted range is used to evaluate the model. The converge model is the best way to predict what the future value will be if the current value is in the range that was forecasted for it. If not, you should look for a different model to look for instead of this one. People might think that a model does not work when it does not come together because you chose the wrong model, or you did not have enough information to reach the limiting model [1].


## 1.2. Nonlinear Time Series Analysis (NTSA)

Nonlinear time series analysis spans a set of methods that extract dynamic information from time series data. Its importance lies in the fact that for many real-world systems, information regarding their temporal behavior is available only through a scalar time series of a single variable. For dissipative dynamical systems, variables move asymptotically toward a low-dimensional attractor that tells us about how they move. We cannot usually see these attractors in the real world because they are too complicated for us to see right away. Nonlinear time series analysis tools help us to know more about the underlying attractor which controls the dynamics of the system under consideration. Phase space reconstruction is the backbone of the nonlinear time series analysis, and two pioneering papers laid the foundation for this analysis. The first one was proposed by Bellazzi et al. [2]. The Internet of Things (IoT) is aimed at making conventional objects intelligent by utilizing a wide range of cutting-edge technologies, including Internet protocols, data analytics, embedded devices, and communication technologies. The IoT’s potential economic impact is anticipated to create a number of business opportunities and hasten the growth of IoT-based services, based on a McKinsey analysis of the IoT’s influence on the world economy [3].The works of Packard et al. in 1980 [4] and another in 1981 by Takens [5] introduced the concept of phase space reconstruction into dynamical system theory. Eckmann and Ruelle also have a role in the extension of the work. Packard et al. gave the techniques regarding the reconstruction of phase space (state space) from the univariate observational data of any dissipative dynamical system as well as the method for determining the number of independent variables (dimension) needed for specifying the state of the system at any given instant. As a demonstration of the method proposed by Packard et al., here, we make use of a standard model of a dynamical system (Rössler system) that can be described by a set of three ordinary differential equations that are given by [6]:
(1)X=−y+z,Y=a+ay,Z=b+xz–cs,with values of control parameters as
(2)a=0.2,b=0.2,c=5.7.When you solve these equations, you get something called a chaotic attractor that looks like Figure1(a). According to Packard et al., the idea behind the reconstruction of the attractor is that, in order to know the state of a three-dimensional system at any given time, you should be able to measure any three separate things [7]. In other words, any set of three independent quantities that label the states of the attractor in a unique and smooth way is called diffeomorphically equivalent or the same thing. So, if you get a time series from the equation by sampling a single coordinate, you can get a variety of three different things. In one example, you could compare the current value of the coordinate with the values of the coordinate at two previous times, like xt, xt, and xt2 [8]. This is one way to look at three different things at once. This is what Packard et al. talked about in their paper. Another set of three coordinates can be found by making the delays small and taking the differences. This set of coordinates is called the “Packard coordinates” (t). Figure 1(b) shows the reconstructed Rössler attractor, which is shown in the figure. The effect of white noise on a synthetic time series is shown in Figures 1(c) and 1(d) when 2% and 4% white noise is added, respectively, to the original time series xt. It can be seen that with 4% white noise contamination, the attractor loses its topological structure in the phase space. So, the method proposed by Packard et al. is not robust to even a small percent of noise. So, it is not practical to apply this method to real-world systems as it is contaminated with different types and levels of noise. This method works fine only with the noise-free time series [9].Figure 1
Rössler chaotic attractor: (a) using the solutions of the equation; (b) reconstructed attractor using coordinatesxt, ẋt, and ẍt; (c) reconstructed attractor with 2% white noise and (d) with 4% white noise.
(a)(b)(c)(d)Taken formalized the concept of Packard et al. which is named as Taken’s theorem or delayed coordinate embedding theorem which is now conventionally used for the nonlinear time series analysis [10].


### 1.2.1. Delay Coordinate Embedding

The regular technique for stage space recreation is delay coordinate implanting. It permits one to reproduce the total elements of a complex nonlinear framework from a univariate time series. Assuming the recreation is done accurately, it ensures that it is topologically indistinguishable from the genuine elements of the hidden framework since significant properties of dynamical frameworks are invariant under diffeomorphism. In this strategy, a progression of past upsides of a solitary variablext from a dynamical framework is utilized to shape a vector that characterizes a point in the new space. So, for building M layered space, one should make M-layered vectors R~t from M time postponed tests of the estimations xt, to such an extent that [11]
(3)R¯t=xt−T,t−2r,⋯⋯<xt−M−1.One can also use future values ofxt for making the M-dimensional vectors which are mathematically equivalent to taking past values of xt as mentioned above. But for practical applications like prediction, we normally take the past values [12].


### 1.2.2. Estimation of Time Delay (τ)

It is very important to know what value of delayτ for a given time series gives the correct delayed coordinates for reconstructing the equivalent topological structure that holds the properties of the true underlying dynamics. The original embedding theorem only requires that τ should be nonzero and not numerous of any circle’s period. This is legitimate just on an endless clamor spare energy series information. In genuine circumstances, the time series might be loud and limited in length. In such circumstances, one requires a higher worth of τ to unfurl the elements of the hidden framework. In the event that we decide τ tiny, the M directions in every one of the vectors given in the equation are firmly associated. So, the stage space elements after remaking will lie near the principle askew of the remade stage space [13]. This is shown in Figure 2 using the solution for xt of the Rössler system given by the equation which is a univariate time series and using τ=1. Here, the reconstruction is done with coordinates xt and xt−τ. The reconstructed Rössler attractor with τ=5, 25, and 35 is also shown in Figure 2 [14]. If we choose τ improperly, the reconstruction will not unfold the subspace of the underlying system; as a result, the topology will not be conjugate to the true dynamics. So, estimating the proper value of τ for a given time series is a challenging task. In the past few decades, many methods have been proposed for estimating the inserting boundaries—delay τ and implanting aspect M. In any case, there is no general strategy for the determination of τ since the fundamental interaction is framework subordinate. Additionally, a worth of τ that functions admirably for one reason may not function admirably for another. A couple of the strategies normally utilized are referenced underneath [15].
(1)
The initial zero of the autocorrelation capacity of the time series is taken as the legitimate postponementτ, since it gives the littlest τ that amplifies the direct autonomy of the directions of the installing vector(2)
The first minimum of the autocorrelation function is taken asτ(3)
The first minima of the average mutual information [16] are taken as τ(4)
The first minima of the correlation sum [17] are taken as τ. Methods 2 and 3 mentioned above happen at τ values that expand more—general types of freedom. It is to be noticed that we really want the littlest τ in light of the fact that the recreated attractor can overlay over onto itself as the value of τ increases, which creates difficulties for the analysis. Besides the abovementioned methods, there have been some theoretical discussions that help to find the optimal τ [18]Figure 2
Reconstructed Rössler attractor with two coordinates with delayτ=1, 5, 25, and 35.


### 1.2.3. Estimation of Embedding Dimension (M)

The second important parameter for the reconstruction is the inserting aspectM. As on account of postponement τ, bigger upsides of M are bad, since a solitary loud point in the time series will influence M number of the places in M-layered inserting. So, one ought to track down the littlest worth of M that properly unfolds the topological structure that is conjugate to the underlying dynamics. As an example, in the case of the Rössler system, the trajectory in the phase space will not cross each other. But when we do embedding in M=2, the trajectory crossing occurs as in Figure 2 [19]. Here, the embedding does not succeed in unfolding the subspace of the underlying system. So, to reconstruct properly, we have to embed in a higher dimension greater than M=2. The reconstructed Rössler attractor with delay τ=25 embedded in dimensions M=2 and M=3 is shown in Figure 3. It can be seen clearly that the crossing trajectories when embedded with M=2 are unfolded properly with M=3. So, the minimum embedding dimension for the Rössler attractor is M=3. The fractal dimension of the Rössler attractor is slightly above 2. For the values of control parameters a=0.2, b=0.2, and c=14 in the equation, the fractal dimension is estimated between 2.01 and 2.02. So, the minimum embedding dimension for unfolding the subspace of the underlying dynamics of the Rössler system is M=3 [20].Figure 3
Reconstructed Rössler attractor withM=2 (a) and M=3 (b).
(a)(b)The original embedding theorem requires thatM>2d to assure the reconstruction is proper, where d is the genuine element of the hidden elements. Yet, experiencing the same thing, d is seldom known. There are a few techniques to register the inserting aspect with basic ideas involving the essential necessity of the implanting hypothesis. A portion of these strategies is the particular worth disintegration of the example covariance grid, the immersion of some invariant amount, the technique for genuine vector fields, the cross-approval strategy, and the strategy for false nearest neighbors.We tested the cross-validation, the saturation of invariants, and the false nearest neighbor methods. The best results were obtained with the false nearest neighbor [21] (FNN) method proposed by Kennel et al. which we used in this work. It is explained in the following section.In an FNN-based method, the calculations are in light of the possibility that in the entry from aspectM to aspect M+1, one can separate between focuses on the circle xt that are “valid neighbors” and focuses on the circle xt which are “misleading neighbors.” A misleading neighbor is a point in the informational collection, that is, a neighbor exclusively in light of the fact that we notice the circle (attractor) from an installing space M lower than the genuine aspect ME of the framework (M<ME). Whenever we accomplish the appropriate inserting aspect or higher (M≥ME), all neighbors of each circle point in the multivariate stage space will be valid neighbors. This can be demonstrated with the Hénon map [22]:
(4)R2mt,n=∑K=0M−1xt+T−xxt+T2,R2Mt−n=xx2+xt+T−xxt+T+⋯⋯+xt+M9−1T−x2t+M.By increasing the dimension toM+1, the distance between the reference point and its neighbors is altered due to the new coordinates xt+Mτ and xnt+Mτ. So now, the square of the distance between them is given by [23]
(5)R2m+1xt+T−xxt+TR2mt,nt+M2−1T.The criteria for checking whether the embedding has fully unfolded the underlying dynamics are that we measure the increase in distance betweenxt and xnt when going from dimension M to M+1. The change in distance can be obtained from the equations. The criteria for deciding whether a neighbor is a true or false neighbor are that [4]
(6)R2m+1xt+T−xxt+T1/2kR2mt,n=xt+MT−xn2x2t+MRmt,n>Rtd,where Rtol is some threshold, and if 10≤Rtol≤50, the number of false neighbors identified by this criterion is constant.In Figure3, the projection of the Hénon map onto a M=1 layered stage space (the x-hub) as well as in a M=ME=2 layered is displayed to install space. Focuses B and C give off an impression of being neighbors in the projection onto the x pivot (M=1). A magnified portion of the Hénon map is shown in Figure 4 to clearly show points A, B, and C. Now, when it is embedded in dimension (M=ME=2), points A and B almost stay together, but point C moves away from point B. It shows that point C is not a true neighbor of point B as seen when embedded in M=1. So, point C is a false neighbor of point B, but since point A stay near to B in dimensions M=1, M=2, and higher, it is a true neighbor [5].Figure 4
Descriptive statistics graph.From the mathematical point of view, if we are inM dimensions, we denote the nth nearest neighbor of a reference point xt by xnt. Then, from the equation, the square of the Euclidian distance between the point xt and this neighbor xnt is given.The spearheading by Packard et al. and other people and the first paper by Takens, which was written about 20 years ago, set the stage for another way to look at complex dynamical characteristics, which is now called nonlinear time series assessment. Expansions to nonlinear stochastic models and to nonlinear quantifiable assessment are complicated and have been looked at in a lot of different ways [24].This means that if a sign is called “fixed,” it means that all change probabilities from one state of the system to the next are not affected by time while the sign is being looked at. All of the important limits should be kept the same, and the time series should show enough of the characteristics that have a place with the components so that the probabilities or other rules can be checked properly. In chaos, the most important thing is that you cannot predict what will happen next, no matter how things turn out in a deterministic way. This unpredictability comes from the fact that the game plans are naturally shaky, which is shown by what is called “sensitive dependence on early on conditions.” In two or three time steps, the small differences between the initial conditions of a large number of bearings are set off. Each small detail of the state at time zero is important for the specific method of the course in state space [16].This leads to two concepts:(i)
Loss of information related to unpredictability(ii)
Nearby trajectories separate exponentially fast over timeOn the off chance that partition over the long haul is exceptionally sluggish, this is not in itself sensational—ordinary of transcendently intermittent frameworks. Assuming that this detachment of directions after some time is quick, dramatically quick—outstanding dissimilarity, we can discuss disorder. This increment of the division after some time can be an attribute of the hidden framework [17].(1) Estimation Procedure and Sampling Rates. A period series (as some other estimation) needs to give sufficient data to decide the amount of revenue unambiguously. Calculations have a few insignificant necessities concerning how long and how exact the time series should be and how as often as possible estimations must be taken to get significant outcomes. In the event that significant outcomes are not acquired, then a more modest subpeculiarity is to be examined. Adequate testing is expected overall not simply to specific strategies for time series investigation.(2) Illustration of Adequate Examining. An illustration of adequate examining is the measurement of temperature changes on top of a house. What number of single estimations is required? Reply is it depends [5]:
(i)
For one day: check one every hour for multiweek to get a fair normal of the temperature profile(ii)
For occasional changes: fixed times, when daily for a year(iii)
For year to year: a few estimations consistently(iv)
For a total depiction of peculiarity: when 60 minutes, consistently, for a few yearsDemonstrating and prediction rely upon the estimation cycle, examining strategy, and furthermore the type of portrayal of the information.


## 1.2.1. Delay Coordinate Embedding

The regular technique for stage space recreation is delay coordinate implanting. It permits one to reproduce the total elements of a complex nonlinear framework from a univariate time series. Assuming the recreation is done accurately, it ensures that it is topologically indistinguishable from the genuine elements of the hidden framework since significant properties of dynamical frameworks are invariant under diffeomorphism. In this strategy, a progression of past upsides of a solitary variablext from a dynamical framework is utilized to shape a vector that characterizes a point in the new space. So, for building M layered space, one should make M-layered vectors R~t from M time postponed tests of the estimations xt, to such an extent that [11]
(3)R¯t=xt−T,t−2r,⋯⋯<xt−M−1.One can also use future values ofxt for making the M-dimensional vectors which are mathematically equivalent to taking past values of xt as mentioned above. But for practical applications like prediction, we normally take the past values [12].


## 1.2.2. Estimation of Time Delay (τ)

It is very important to know what value of delayτ for a given time series gives the correct delayed coordinates for reconstructing the equivalent topological structure that holds the properties of the true underlying dynamics. The original embedding theorem only requires that τ should be nonzero and not numerous of any circle’s period. This is legitimate just on an endless clamor spare energy series information. In genuine circumstances, the time series might be loud and limited in length. In such circumstances, one requires a higher worth of τ to unfurl the elements of the hidden framework. In the event that we decide τ tiny, the M directions in every one of the vectors given in the equation are firmly associated. So, the stage space elements after remaking will lie near the principle askew of the remade stage space [13]. This is shown in Figure 2 using the solution for xt of the Rössler system given by the equation which is a univariate time series and using τ=1. Here, the reconstruction is done with coordinates xt and xt−τ. The reconstructed Rössler attractor with τ=5, 25, and 35 is also shown in Figure 2 [14]. If we choose τ improperly, the reconstruction will not unfold the subspace of the underlying system; as a result, the topology will not be conjugate to the true dynamics. So, estimating the proper value of τ for a given time series is a challenging task. In the past few decades, many methods have been proposed for estimating the inserting boundaries—delay τ and implanting aspect M. In any case, there is no general strategy for the determination of τ since the fundamental interaction is framework subordinate. Additionally, a worth of τ that functions admirably for one reason may not function admirably for another. A couple of the strategies normally utilized are referenced underneath [15].
(1)
The initial zero of the autocorrelation capacity of the time series is taken as the legitimate postponementτ, since it gives the littlest τ that amplifies the direct autonomy of the directions of the installing vector(2)
The first minimum of the autocorrelation function is taken asτ(3)
The first minima of the average mutual information [16] are taken as τ(4)
The first minima of the correlation sum [17] are taken as τ. Methods 2 and 3 mentioned above happen at τ values that expand more—general types of freedom. It is to be noticed that we really want the littlest τ in light of the fact that the recreated attractor can overlay over onto itself as the value of τ increases, which creates difficulties for the analysis. Besides the abovementioned methods, there have been some theoretical discussions that help to find the optimal τ [18]Figure 2
Reconstructed Rössler attractor with two coordinates with delayτ=1, 5, 25, and 35.


## 1.2.3. Estimation of Embedding Dimension (M)

The second important parameter for the reconstruction is the inserting aspectM. As on account of postponement τ, bigger upsides of M are bad, since a solitary loud point in the time series will influence M number of the places in M-layered inserting. So, one ought to track down the littlest worth of M that properly unfolds the topological structure that is conjugate to the underlying dynamics. As an example, in the case of the Rössler system, the trajectory in the phase space will not cross each other. But when we do embedding in M=2, the trajectory crossing occurs as in Figure 2 [19]. Here, the embedding does not succeed in unfolding the subspace of the underlying system. So, to reconstruct properly, we have to embed in a higher dimension greater than M=2. The reconstructed Rössler attractor with delay τ=25 embedded in dimensions M=2 and M=3 is shown in Figure 3. It can be seen clearly that the crossing trajectories when embedded with M=2 are unfolded properly with M=3. So, the minimum embedding dimension for the Rössler attractor is M=3. The fractal dimension of the Rössler attractor is slightly above 2. For the values of control parameters a=0.2, b=0.2, and c=14 in the equation, the fractal dimension is estimated between 2.01 and 2.02. So, the minimum embedding dimension for unfolding the subspace of the underlying dynamics of the Rössler system is M=3 [20].Figure 3
Reconstructed Rössler attractor withM=2 (a) and M=3 (b).
(a)(b)The original embedding theorem requires thatM>2d to assure the reconstruction is proper, where d is the genuine element of the hidden elements. Yet, experiencing the same thing, d is seldom known. There are a few techniques to register the inserting aspect with basic ideas involving the essential necessity of the implanting hypothesis. A portion of these strategies is the particular worth disintegration of the example covariance grid, the immersion of some invariant amount, the technique for genuine vector fields, the cross-approval strategy, and the strategy for false nearest neighbors.We tested the cross-validation, the saturation of invariants, and the false nearest neighbor methods. The best results were obtained with the false nearest neighbor [21] (FNN) method proposed by Kennel et al. which we used in this work. It is explained in the following section.In an FNN-based method, the calculations are in light of the possibility that in the entry from aspectM to aspect M+1, one can separate between focuses on the circle xt that are “valid neighbors” and focuses on the circle xt which are “misleading neighbors.” A misleading neighbor is a point in the informational collection, that is, a neighbor exclusively in light of the fact that we notice the circle (attractor) from an installing space M lower than the genuine aspect ME of the framework (M<ME). Whenever we accomplish the appropriate inserting aspect or higher (M≥ME), all neighbors of each circle point in the multivariate stage space will be valid neighbors. This can be demonstrated with the Hénon map [22]:
(4)R2mt,n=∑K=0M−1xt+T−xxt+T2,R2Mt−n=xx2+xt+T−xxt+T+⋯⋯+xt+M9−1T−x2t+M.By increasing the dimension toM+1, the distance between the reference point and its neighbors is altered due to the new coordinates xt+Mτ and xnt+Mτ. So now, the square of the distance between them is given by [23]
(5)R2m+1xt+T−xxt+TR2mt,nt+M2−1T.The criteria for checking whether the embedding has fully unfolded the underlying dynamics are that we measure the increase in distance betweenxt and xnt when going from dimension M to M+1. The change in distance can be obtained from the equations. The criteria for deciding whether a neighbor is a true or false neighbor are that [4]
(6)R2m+1xt+T−xxt+T1/2kR2mt,n=xt+MT−xn2x2t+MRmt,n>Rtd,where Rtol is some threshold, and if 10≤Rtol≤50, the number of false neighbors identified by this criterion is constant.In Figure3, the projection of the Hénon map onto a M=1 layered stage space (the x-hub) as well as in a M=ME=2 layered is displayed to install space. Focuses B and C give off an impression of being neighbors in the projection onto the x pivot (M=1). A magnified portion of the Hénon map is shown in Figure 4 to clearly show points A, B, and C. Now, when it is embedded in dimension (M=ME=2), points A and B almost stay together, but point C moves away from point B. It shows that point C is not a true neighbor of point B as seen when embedded in M=1. So, point C is a false neighbor of point B, but since point A stay near to B in dimensions M=1, M=2, and higher, it is a true neighbor [5].Figure 4
Descriptive statistics graph.From the mathematical point of view, if we are inM dimensions, we denote the nth nearest neighbor of a reference point xt by xnt. Then, from the equation, the square of the Euclidian distance between the point xt and this neighbor xnt is given.The spearheading by Packard et al. and other people and the first paper by Takens, which was written about 20 years ago, set the stage for another way to look at complex dynamical characteristics, which is now called nonlinear time series assessment. Expansions to nonlinear stochastic models and to nonlinear quantifiable assessment are complicated and have been looked at in a lot of different ways [24].This means that if a sign is called “fixed,” it means that all change probabilities from one state of the system to the next are not affected by time while the sign is being looked at. All of the important limits should be kept the same, and the time series should show enough of the characteristics that have a place with the components so that the probabilities or other rules can be checked properly. In chaos, the most important thing is that you cannot predict what will happen next, no matter how things turn out in a deterministic way. This unpredictability comes from the fact that the game plans are naturally shaky, which is shown by what is called “sensitive dependence on early on conditions.” In two or three time steps, the small differences between the initial conditions of a large number of bearings are set off. Each small detail of the state at time zero is important for the specific method of the course in state space [16].This leads to two concepts:(i)
Loss of information related to unpredictability(ii)
Nearby trajectories separate exponentially fast over timeOn the off chance that partition over the long haul is exceptionally sluggish, this is not in itself sensational—ordinary of transcendently intermittent frameworks. Assuming that this detachment of directions after some time is quick, dramatically quick—outstanding dissimilarity, we can discuss disorder. This increment of the division after some time can be an attribute of the hidden framework [17].(1) Estimation Procedure and Sampling Rates. A period series (as some other estimation) needs to give sufficient data to decide the amount of revenue unambiguously. Calculations have a few insignificant necessities concerning how long and how exact the time series should be and how as often as possible estimations must be taken to get significant outcomes. In the event that significant outcomes are not acquired, then a more modest subpeculiarity is to be examined. Adequate testing is expected overall not simply to specific strategies for time series investigation.(2) Illustration of Adequate Examining. An illustration of adequate examining is the measurement of temperature changes on top of a house. What number of single estimations is required? Reply is it depends [5]:
(i)
For one day: check one every hour for multiweek to get a fair normal of the temperature profile(ii)
For occasional changes: fixed times, when daily for a year(iii)
For year to year: a few estimations consistently(iv)
For a total depiction of peculiarity: when 60 minutes, consistently, for a few yearsDemonstrating and prediction rely upon the estimation cycle, examining strategy, and furthermore the type of portrayal of the information.


## 2. Research Methodology

In the importance of big data, speed is a critical part which fabricates the size of time series. Fan et al. have commented on dimensionality which makes a practical issue and said that as the dimensionality augments then, at that point [23], relating profound computational cost moves as well as questions that it is expected to have the assessment of the most part data, and it is contributing much the same way. Since the beginning of the discussion about the huge information investigation, specialists are worried to work on scientific strategies and furthermore limit the handling time. Here, the following inquiries are significant in regard to examination:
(i)
Does a huge size time series give more accurate information to a specific model(ii)
Does the capability of the model add more insight to the model series(iii)
Is there any opportunity to get more careful information about the system/object under studyKeeping track of the economy is very important, so each country has its own way to keep track of the records of the economy. India communicates records through the BSE and NSE, the USA through NASDAQ OMX and NYSE EURONEXT, Canada through TMX GROUP, China through Shanghai SE and Shenzhen SE, the UK through the London SE pack, and many other places [13–15, 18]. According to Statsoft Inc [4], at the National Stock Exchange (NSE), trading Indian subordinates is very simple. The NSE is releasing sixteen documents about different parts of the country and the people who own them. These documents include broad market reports, sectoral reports, effective records, and framework records. Also, there are records for auto, bank, energy, FS, FMGC, IT, metal, PSU, and so forth [22].


## 3. Data Analyses

In the current review, which is an experimental examination, we have chosen eight documents from the National Stock Exchange (NSE) site to show how big data time series could look. They cover the period from January 1994 to January 2017. These words are usually spoken quickly, so this could be a factor in how quickly things grow each day. In this cycle, the information in the parent series grows less, but we should still expect it to grow in every single part of the second one as well. All things considered, speed is very high, so it is hard to show this kind of series, and old-fashioned factual instruments cannot do the job [5]. We are figuring out the series on our own by looking at a big data time series on average enlisting machines. Every time, there is a new detail added to the series, which is like the speed of big data. In rational automated data, you might see the same thing in thousands or even MBs. Here, we want to set up the framework for the evaluation. Here, we could argue that the word “enormous” is both too broad and too specific [25, 26, 27]. However, we are separating this by a direct I-7 processor and also displaying the series that comes after adding the information in a month-long range (Almulihi, 2022):
(7)Yt=logPt−logpt−1∗100.As it is more and easier to use computer software for time series models and faster computers to solve problems with them, they can be used for more things. Box Jenkins is the best way to get an ARIMA model that is as good as possible. ADF is another way we checked the model’s stationery. The test statistics values are shown in Table1 with descriptive statistics.Table 1
Descriptive statistics for sectorial index of NSE log return.
SeriesMeanSDSkewersADF testMaruti0.02351.236-0.2365-15.3363Power0.03261.258-0.2656-15.3226FMCG0.06451.326-0.6256-14.2265IT0.03261.485-42.236-15.3697Energy0.03261.369-0.3669-14.3267Coal0.01651.214-0.3657-15.2697Metal0.03251.6930.1336-12.6599PSU0.00311.369-0.1269-13.5220∗∗0.5% level of significance.At a 5% level of significance, we found that the test statistics value is important. According to other businesses, the IT sector has a lower mean and higher standard deviation than other types of businesses, which are both bad things. It is only the metal industry that is skewed in the direction of the good. This means that, except for the IT sector return series, all of the return series is about normal. The value of skewers for the IT sector is not close to zero (-42.236). Whether or not the return series is stationary is something we want to find out about. Whether the return series is flat or does not have a unit root is what we think is a set pattern. It turns out that all sectorial return series is stationary [19]. This is based on the ADF test that we used. When we are fitting a model, we add the data for one month and record the order in which it fits the best. In a chronologically recorded series, the number of records grows in real time. If the order of a series does not change, we say that the series is in model convergence, which means that it is the same. During the method, if it does not look like we can fit a series, we stop and do not do it again. This is very important. When we did our empirical analysis, we used the ARIMA model because it can take into account trends and seasonality as well as cycles, error, and the fact that it is not always stable, so we used this model. We use the forecast package in R in order to get the best ARIMA model [28]. After adding monthly observations to the original series, a cumulative series will be formed [29]. The best ARIMA model can be used to fit cumulative series and look for ARCH and GARCH effects in residual series to see how they work as shown in Figure 4.


## 4. Conclusion

If we look at a time series model, we may also want to look for convergence. If more data does not change the trend of the series or make a difference to the model or the values of the parameters, we can say that it does not matter (s). The standard error and information criterion may also change as a result of this, but this may not be a good thing to do as part of the goal of modeling, so it may not be worth it. In practise, big data analysis tools can look at very large amounts of data or very long time series without having to think about how big the model is. This paper has looked into how important size is in the analysis of time series instead of all observations or the best size. The NSE data backs up the method. You can only get forecasted values in your range of predicted values if you have convergence on your model [22]. If not, you will need to look for a model that is convergent to get the forecasted values in your range of predicted values. There may be extra costs for the analysis if the size of the series is not taken into account. This includes the cost of recording the data as well as the cost of analytics and the cost of hardware and software tools to use. Generalized models could be worked on for a long time, taking into account other models and other things [2].



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*Source: 6127601-2022-09-06.xml*