Time-series data prediction problem analysis through multilayered intuitionistic fuzzy sets.

For several years, time-series prediction seems to have been a popular research topic. Sales plans, ECG forecasts, meteorological circumstances, and even COVID-19 spreading projections are among its uses. These implementations have inspired several scientists to develop an optimum forecasting method; however, the modeling method varies as the implementation domain evolves. Telemetry data prediction is an important component of networking and information center control software. As a generalization of such a fuzzy system, the concept of an intuitionistic fuzzified set was created, which has proven to become a highly valuable tool in dealing with indeterminacy (hesitation) as in-network. Indeterminacy is frequently overlooked in applying fuzzified time-series prediction for no obvious cause. We introduce the concept of intuitionistic fuzzified time series within a current study to deal with non-determinism with time-series prediction. Also, it seems to be an intuitionistic fuzzified time-series prediction framework. Using time-series information, the suggested intuitionistic fuzzified time-series predicting approach employs intuitionistic fuzzified logical relationships. The suggested method's effectiveness is tested using two-time sequence data sets. By contrasting the predicted result with some other intuitionistic timing series predicting techniques utilizing root-mean-square inaccuracy and averaged predicting errors, the usefulness of the suggested intuitionistic fuzzified time-series predicting approach is demonstrated.

Introduction

Modern advancements in huge analytics and IoT have resulted in a rise in the data flow that can be evaluated to help any firm make better decisions. However, this time-series information is one sort of information that is abundant. Among the most difficult difficulties facing today’s data researchers and academics is evaluating and predicting timing series information. Timing series statistic collection will collect statistical points that indicate metrics that change over time. It is described as a collection of coordinates X(t), where t seems to be 0, 1, 2,…n, whereas if several variables altering with regard to period seems to be one, then time series as shown to be unitary. Multifunctional timing series information will be referred to as a data collection with greater than one factor.

Time series seems to be useful data in various industries, including economics, biology, and experimental studies. Furthermore, the time-series study has piqued the interest of numerous scholars (Ji et al. 2019a). The main principle behind timing series forecasting will be to summarize primary data from timing series for forecasting arriving pieces of information (Wang et al. 2019), while many methods for evaluating and forecasting sequential information have now been suggested (Luo et al. 2019). Time series are typically analyzed using probabilistic and machine learning concepts, like the ARIMA framework (Ji et al. 2019b) and the SVM framework (Wang et al. 2017). As a result, numerous time-series forecasting algorithms have been successfully implemented to industrial productivity and economics, among other fields.

Zadeh (1965) presented fuzzy sets that can be used in various applications (Hao et al. 2016). As such an essential knowledge-based framework, the fuzzified cognitive mapping is properly used in estimating series data. In contrast to time-series analysis techniques depending on analytical modeling, fuzzy cognitive mappings can adapt from professional expertise or be educated utilizing relevant data. However, FCMs have the comprehensibility and capacity to deal with ambiguous and imprecise input compared to time-series predicting based on artificial learning techniques like neural networks and supporting vector machines. Given the variety of time series, this is important to construct cognitive mappings explicitly from unprocessed data, to separate informational granules as vertices in previous time series and acquire the basic dependence interactions among nodes.

Data granulation serves a crucial part in the construction of fuzzy conceptual mappings. Granular coding (Leonid and Jayaparvathy 2022) seems to be a concept-level method that requires the selection of arithmetical depictions of granules (like fuzzy systems), the discovery of a method to granulate relevant data (including a data tensile method), as well as the processing of knowledge granules. The syntax and meanings of the informational granules can also be deduced from the applicable logic once they’ve been constructed. The informational granule is separated from the data set and employed like a node (idea) to construct a mapping framework in time-series forecasting. Linguistic labels and fuzzy systems are used in the methods mentioned above to make the system more understandable and descriptive. For instance, the informational granularity for the time series seems to aggregate time series onto stable clusters employing fuzzified C-means grouping in such a two-dimensional vector having magnitude and variation of magnitude. The effectiveness assessment is performed on a detailed level, employing the developed framework for logic.

The expansion of fuzzy system offers intuitionistic fuzzy setting (IFS) that incorporates membership (expressing the level of affiliation to that setting), non-membership (expressing level to which one does not match in such setting), as well as hesitation. Fuzzified settings with intuitionistic properties get a variety of applications, particularly in decision-making (Arqub et al. 2016). The key benefit of IFS seems to be that it will account for the level of hesitation of components in the subset, which will be usually determined by expert knowledge. Intuitionistic fuzzy intellectual mappings have been suggested by incorporating intuitionistic fuzzy subsets with fuzzy cognitive mappings, which again increases decision-making implementation. The IFCM has been developed for healthcare decision-making by acquiring uncertainty and weights matrices depending upon that expert’s knowledge. Doctors can determine whether such a patient has pneumonia based on the seriousness of such patient’s indications, and the hesitancy in assessing the signs can be based on professional views.

In this article, we define intuitionistic fuzzified time series as well as suggest a primarily intuitionistic fuzzified time-series predicting framework having intuitionistic fuzzification as well as intuitionistic defuzzification, as opposed to the partly intuitionistic fuzzified time-series predicting designs suggested by Kumar and Joshi as well as Kumar and Gangwar (2015), wherein the extent of hesitation has been taken into account during fuzzification of time-series information to manage non-determinism. Regarding intuitionistic defuzzification, we employ the weighted mean technique in this work. The suggested intuitionistic fuzzified time-series predicting method was tested on two databases: (1) University at Alabama student’s attendance information, and (2) SBI share market pricing information upon that BSE in India. The suggested method's efficiency is contrasted to that of other approaches.

Literature survey

The fuzzified time-series (FTS) approach in 1993 was developed by Chakraverty et al. (2016) and Bargiela and Pedrycz (2016), which replaces time-series elements with fuzzy subsets to create fuzzy associations between observations. Because establishing fuzzy relationships necessitates a sophisticated matrix procedure, it requires a longer period to compute. To make the model easier to implement, Markkandan et al. (2021) updated it by using an easier arithmetic procedure instead of a sophisticated max–min approach.

The following three stages make up fuzzified time-series designs: (1) divide the world of discursive U into specified periods as well as fuzzification for such traditional timing series, (2) create fuzzified logical interactions among fuzzified systems (Ai), and (3) defuzzification and predicting. Unfortunately, the FTS systems listed above contain no data on how the duration of periods is calculated therefore it is done at random. As a result, numerous FTS designs have been developed to evaluate the period duration and improve the predictive ability of the prototype. For example, two algorithmic FTS methods rely on the average of distinctions and dispersion of distinctions (Cheng 2018; Rashid et al. 2018) offered a technique that relies on proportions rather than identical interval durations. Genetic methods were used by the researchers of Prasanna et al. (2018) and Suresh and Aanandha (2021) used particle swarm planning. Song and Chissom (1993) and Arulaalan and Nithyanandan (2016) utilized such an optimal strategy with a unique variable limitation. Like reality, for establishing the proper length of periods, (Huarng and Yu 2006) used the sliding window method in conjunction using the type-2 fuzzified time-series method. Chen and Chung (2006) also presented an intermediate type-2 fuzzified logical system (IT2FLS) relying upon a fuzzified timing series. Integrating the concept with methods like network traffic anomalous recognition system (Lee et al. 2007), complicated network analytics (Arulaalan and Nithyanandan 2016), as well as C-means segmentation method (Mao and Xiao 2019), a fuzzified time-series framework has been built.

In many types of research on fuzzified timing series, a domain of such speech will be partitioned into comparable periods. Furthermore, whenever the dispersion of the world for such speech is not regular, fuzzified timing series methods will not give pretty predicting outcomes. To divide the world of discourse, numerous researches employed clustering approaches to establish the dispersion of world discourse from themselves. They found a distinct fuzzy partitioning at various spacing duration (Sang et al. 2018). However, clustering methods make finding an optimum division number challenging in many circumstances. As per the research described above, establishing an optimum and suitable partition strategy for the world of speech raises some fascinating difficulties.

Several time-series techniques in the research use the topologies of different artificial neural systems to identify relationships. Dincer and Akkuş (2018) created this feed-forward arbitrary neural system relying on a multilayered perceptron to resolve the time series. In addition, to analyze this time series, Rumelhart (1986) suggested such high-level order neural system relying upon such convolutional pi-sigma neural networks. Several more time-series techniques (Shin and Ghosh 1991) depend on such convolutional neural networks. They are additionally several FTS approaches in the Research that use the topologies of different artificial neural systems to identify fuzzy relationships. To anticipate all linear and nonlinear timing series, an FTS system framework was suggested. FTS system relies (Zhou et al. 2016) on such a convolutional neural network using a single exponential neuron concept. It is a pi-sigma neuro system-based high-level order fuzzified time-series technique (Aladag 2013). Additional artificial neural network-based FTS approaches are included (Yu and Huarng 2010).

Intuitionistic fuzzified time series (IFTS) seems to be a new topic for time-series research that only started in 2013. While FTS approaches use membership functions, the primary strategy of IFTS will use combined membership and non-membership variables in generating fuzzy relationships. As a result, it is reasonable to conclude that IFTS systems utilize more data than the FTS technique. Using IFTS modeling improves predicting capacity to solve real-world time series. As a result, many studies on IFTS approaches have been presented. The core principles for IFTS and the initial IFTS prediction system (Kocak 2015). As in the fuzzification phase of such IFTS research (Zheng et al. 2013), Chari’s intuitionistic fuzzified C-means (IFCM) method is recommended to determine the quantities of membership and non-membership. Many academics who have developed new IFTS prediction designs in the journals still employ the IFCM method. Three different long-term IFTS predicting methods have been suggested (Kumar and Gangwar 2015), an FCM-based different-stage IFTS high-level order system (Fan et al. 2017). Division of world of such discourse is utilized (Abhishekh and Singh 2018) to derive membership and non-membership qualities.

System model

LSTM (long short-term memory) has not been used in IFTS research. As a result, the subject of our research seems to be an LSTM relying on IFTS prediction models. This paper proposes DIFTS-LSTM, a unique high-order deeper intuitionistic fuzzified time-series approach relying on LSTM. The suggested technique is a predicting algorithm employed in “Internet of Things” situations like the machine learning approach. The suggested technique, for instance, can be utilized as a weather prediction model and integrated into an online application or network for weather prediction. The described methods can be used in various additional “Internet of Things” scenarios. For DIFTS-LSTM, an IFCM method is applied within the fuzzification phase to acquire membership quantities then non-membership. The LSTM convolutional neural system is employed to identify fuzzy relationships.

The terms intuitionistic fuzzified timing series, probabilistic fuzzified C-means clustered technique, LSTM convolutional neural networks, and suggested approach are all defined in this chapter. Several IFTS concepts were thoroughly clarified. Principles 1 and 2 provide two aspects. The essential premise of IFTS is explained in Description 1, and the high-order singular factor IFTS prediction algorithm is expressed in Description 2. Equation (2), which provided Description 2, is employed in the study's suggested technique.

The fuzzy subset notion was established, which is described as next:

Assume that X is a consistent set. Fuzzy subset A in X has been described like in which seems to be the intuitionistic fuzzy subset A's membership value  and seems to be an extent of x membership vector in A. An alternate implementation of fuzzy system with is as follows:

Atanassov further generalized the notion of the fuzzy system and established IFS as regards: an IFS I from X is described as an entity of the given type:with the constraint for every x from X as well as a membership value as well as then non-membership functional .

If seems to be the extent of indeterminacy upon indeterminacy for every IFS I within X,

Intuitionistic fuzzified time series (IFTS)

Enabling pretend that It seems to be the time series. On the global set, are probabilistic fuzzified subsets. , membership functions of , as well as non-membership elements for makeup , a multidimensional time series. The following is a description of the term:

High-order single variable IFTS predicting model

Let be such the fuzzified timing series with intuitionistic properties. On the global set, represent probabilistic fuzzy sets. The membership, as well as non-membership indices for such tth witness to such jth probabilistic fuzzified subset, is represented by . The following seems to be the high-order only factor intuitionistic fuzzified time-series prototype:

As Eq. (5), G seems to be a linear but rather nonlinear functional, seems to be a zero-mean erroneous component, represent delayed factors of , as well as , are delayed membership as well as non-membership variables determined from , correspondingly. In this research, we used intuitionistic fuzzified C-means (IFCM) to calculate membership and non-membership variables. The LSTM convolutional neural network can be used to generate G.

Intuitionistic fuzzified C-means (IFCM) algorithm

In this research, IFCM applied such fuzzification step for the suggested approach. As a result, the following is what IFCM has suggested:

Step 1 Eq. (6) is used to calculate the membership functions . as well as seem to be membership elements and arbitrary values produced from homogenous dispersion with coefficients (0, 1) for such kth occurrence of such , accordingly, in Eq. (6), serial from an ith clustering.

Step 2 Eqs. (7) and (8) are used to calculate hesitation levels but also intuitionistic fuzzified estimates , accordingly. The values of intuitionistic membership are stored in a matrix.. In Eqs. (7) and (8), and represent the hesitating levels and probabilistic fuzzified membership estimates of such kth observations as on ith clustering, accordingly, in Eqs. (7) as well as (8). Furthermore, the membership levels collected in Step 1 have been represented by .

Step 3 Utilizing acquired in Step 2 through Eq. (9), the centers of clustering are computed.

represents time-series data, seem to be intuitionistic membership levels produced in Step 2, while f seems to be the fuzziness factor in Eq. (9).

Step 4 Eq. (10) modifies the membership functions ; it was acquired in Step 1.

In such Eqs. (10) as well as (11), seems to be fuzzified membership functions of such kth witness for ith clustering produced at Stage 1, seems to be the Euclidean proximity metric of such kth witness in ith cluster center, f seems to be the fuzziness factor, while seems to be the ith clustering center point found from Step 3.

Step 5 Equations (7) and (8) are used to change the hesitation grades and the intuitionistic fuzzified membership variables . In such a matrix named , the updated intuitionistic membership variables are kept.

Step 6 Such non-membership estimates are calculated employing updated values from Step 5 and Eq. (12). The unique intuitionistic non-membership attributes are therefore stored in such a V matrix.

Step 7 Equation (13) is used to verify stopping requirements (13). The procedure is terminated if the requirement is met; else, set and proceed to Step 3. seems to be the smallest positive integer info Eq. (13) as well as seems to be the standard.

An LSTM convolutional neural network will be used to evaluate fuzzy interactions in this research. Figure 1 depicts the chosen LSTM architectural design.

Fig. 1

Architecture of the LSTM

This paper proposes DIFTS-LSTM, an innovative high depth intuitionistic fuzzified time-series methodology depending on LSTM. DIFTS-LSTM strategy is usually a probabilistic fuzzified timing-series approach that will utilize Eq. (5) at Description 2. As in such DIFTS-LSTM technique, the IFCM will be described on Method 2 will be employed in the fuzzification phase to acquire the parameters of membership then non-membership, as well as the LSTM convolutional neural networks, which is a deep technique of training, is employed in step of identifying fuzzy relationships. Moreover, because such DIFTS-LSTM approach results are true predictions, the suggested technique does not require the defuzzification step. Methodology 2 shows the DIFTS-LSTM method's protocol series of steps.

Algorithm 2 Proposed method: DIFTS-LSTM method

The LSTM’s sources will be divided into three categories. The primary set comprises every intuitionistic fuzzy group's combined membership functions. The other group comprises every intuitionistic fuzzy group’s combined non-membership elements. Furthermore, these time series' delayed crisp factors make up the last group. To further comprehend the LSTM's sources, an instance of such a time sequence is shown. Set p = 2, = 3 and the membership but also non-membership variables be calculated using the IFCM technique in Table 1 from Step 2.

Table 1

Time-series fuzzification

t	X(t)/10	Member values × 100			Non-member values × 100
		A1	A2	A3	A1	A2	A3
1	2	95	7	2	5	90	96
2	3	94	6	100	5	89	97
3	4	45	72	9	44	18	87
4	3	94	6	100	5	89	97
5	2	95	7	2	5	5	96
6	5	0	100	0	100	100	100
7	6	10	82	30	85	85	61
8	8	0	100	99	99	98	1

To begin, membership variables can be used to get the first set of input data. The training samples can be determined from t = 3 till t = 8 since the model’s ordering is 2. The following formula can be used to generate combined membership functions for t = 3:

As a result, the following are the combined membership functions for t = 3; .

Utilizing the training samples generated in Stage 3, the LSTM gets learned, and the ideal values are determined. The LSTM convolutional neural network is used in this research, which is done with the help of the MATLAB software's neural networking toolkit. To educate LSTM, the “Adam” method is utilized, and a unique output architectural layout is desired. The highest amount of repetitions in the Adam method is set to 250. The beginning learning value is again calculated as 0.005 and every 125 repetitions. This value is reduced by multiplied by 0.2. As in the deployment of such Adam method, operations are performed upon that CPU without using a graphics processor. The erroneous measures are derived using results and actual results of the timing series and the results of the training LSTM, which are generated for such testing set.

Result and discussion

The formulas of the erroneous measures employed in our work’s implementations are presented in Eqs. (14) and (15) and, accordingly, evaluate how well the suggested intuitionistic fuzzified 319-time sequence prediction model performs.

RMSE, as well as MAE, seems to be the root-mean-square errors as well as mean actual errors, correspondingly, in Eqs. (14) and also (15). The major technique used in this research is to choose the optimal modeling based on the technique with the least RMSE values. Additionally, a lower MAE number for a methodology shows that it performs well in forecasting.

Different time-series prediction techniques related to the suggested method are listed under in the implementations:

Mechanism of MLP-ANN: Such an approach (MLP-ANN) seems to be a feed-forward multilayered perceptron convolutional neural network used to anticipate or forecast information in the research. For usage in the implementations of this research, the MATLAB m-file of such MLP-ANN approach has been developed.

Mechanism of PS-ANN: It is a greater pi-sigma convolutional neural network used to anticipate or forecast information in the research. The PS-ANN technique MATLAB m-file is produced for usage in that study’s implementations.

C-FTS: One such technique will be the initial fuzzy timing series predicting methodology described in this research, and also it will be dependent upon fuzzy logic interactions. It is the C-FTS system. For utilization in the implementations of this work, the MATLAB m-file of such C-FTS technique has been developed.

For all techniques, a basic trial-and-error approach is used for hyperparameter choices and summarized in the accompanying terms. The amount of delayed factors (p) is started trying successive integer values varying about 1–5, and the amount of intuitionistic fuzzified sets is started trying successive numerical values varying about 3–10, as well as the amount of concealed layers of such convolutional neural networks  has been attempted successive numerical values varying about 1–5. Every time series gets separated into two sections: the training phase and the testing sets. The suggested approach and other predicting abilities are evaluated for such training sets in all scenarios.

To begin, the suggested technique and different techniques are performed to everyday lowest temperature temporal series collected by Giresun Meteorological Provincial Directorate at Giresun, Turkey, during January 1, 2017, and April 30, 2017. Figure 2 depicts the chart of such a time series.

Fig. 2

Minimum temperature for the time series on a daily basis

Table 2 shows the best predictions from all techniques used in our investigation and the RMSE and MAE estimates for the latest 7 measurements of Giresun's everyday lowest warmth time series.

Table 2

Forecasting results of the minimum temperature for the time series daily

Test data	MLP-ANN	PS-ANN	C-FTS	FTS-N	DIFTS-LSTM
15.45	10.15	10.65	10.45	11.40	10.70
14.55	11.50	13.55	14.95	14.75	10.85
09.70	11.25	09.80	06.00	12.45	11.75
13.05	09.05	07.85	10.15	08.75	10.35
10.25	11.15	12.65	10.45	13.15	10.40
08.05	09.75	09.15	10.45	09.60	08.80
08.51	08.15	07.95	07.35	08.65	08.45
RMSE	2.45	2.35	2.35	2.25	1.95
MAE	1.95	1.80	1.75	1.75	1.35

Whenever the range of delayed parameters (p) is 10, the amount of intuitionistic fuzzy subsets seems to be 6, as well as the number of concealed layers of artificial neural networks  will be 5, the suggested technique seems to have the least RMSE rating of 1.95 as well as the least MAE rating of 1.37 for such testing dataset ( = 14). Table 2 shows that the suggested approach has lesser RMSE and MAE readings than all different techniques investigated in this research. As a result, the suggested technique seems to have the greatest predictive ability for everyday minimum thermal time series at Giresun, Turkey. In Fig. 3, the suggested methodology predicts, and the test set’s facts are combined in such graphs. Excluding the actual beliefs in the first, second, third, and fourth days, predictions are near to the true readings in Fig. 3. As a result, the suggested technique performs well in predicting everyday minimal thermal time series for Giresun, Turkey.

Fig. 3

Comparison of the proposed methods with the test set

Conclusion

The suggested technique, dubbed DIFTS-LSTM, seems to be a high-deep intuitionistic fuzzified period predicting system that employs the IFCM technique in fuzzification and the LSTM technique for identifying probabilistic fuzzy relationships. As in fuzzified time-series research, IFTS systems will be a more recent topic; they are comparable alternatives to FTS systems. In addition to employing real facts, membership data, and non-membership data with non-membership in assessing a time series, IFTS prediction models are thought to contain more data. In contrast, FTS approaches examine membership functions in resolving that time series. Within that article, implementations for two real-life time sequences are presented, and it is demonstrated that the suggested technique outperforms alternative ways such as basic fuzzified time-series designs, timing series designs, fuzzy serial data, and probabilistic fuzzified timing series depending upon conventional neural networks in terms of predicting effectiveness. As a result, academics may develop new IFTS algorithms depending upon deeper learning approaches for solving diverse real-world time series as in upcoming. The suggested approach can be employed in the Internet of Things projects as a beneficial tool. The method can be integrated into a platform to produce accurate predictions.