1 Introduction

Various approaches are used to predict multidimensional time series (MTS), which are usually based on methods for predicting one-dimensional time series [¹, ²]. Limitations of these methods are: complexity of accounting for the indirect influence of the interdependent MTS components in the conditions of uncertainty, the non-linear nature of their interaction [³, ⁴], their lack of consistency, and the complexity of identifying and accounting for the trends of each of the MTS components [⁵, ⁶].

Methods for predicting time series using fuzzy and neural network models are being actively developed. However, their limitations are the complexity of accounting for the direct and indirect interaction of the MTS components.

Methods of fuzzy cognitive modeling allow us to take into account the direct and indirect interaction of the components of the MTS [⁷, ⁸, ⁹, ¹⁰]. However, the use of these methods to predict the MTS is limited by: the capabilities of the used system dynamics models of the MTS components; the lack of consideration of interference MTS components with their different time lags relative to each other; the lack of approaches to coordinated setting of each of the MTS components; MTS small forecasting samples; identification of the trends of the MTS components.

The study deals with the use of Neuro-Fuzzy Cognitive Temporal Models (NFCTM) for MTS predicting, taking into account the fuzzy trends of each of the MTS components. NFCTM allow predicting the MTS in conditions of non-stochastic uncertainty, non-linearity of interaction, partial inconsistency and significant interdependence of the MTS components in conditions of small samples. This takes into account the direct and indirect mutual influence of all the MTS components with different time lags relative to each other.

Proposed NFCTM include a lot of concepts (relevant to the MTS components), which are connected by subsets of arcs, weighted by fuzzy degrees of influence, arranged in chronological sequence taking into account time lags (delays) of the respective MTS components relative to each other.

To carry out the temporal changes in the individual NFCTM components MTS original neuro-fuzzy models of type RecANFIS (Recurrent Adaptive Neuro-Fuzzy Inference System/Model) are used, that: firstly, allow to save the predicted values of the MTS components in the range of “sliding-window” time series; secondly, identify fuzzy trends of the MTS components in the range of “sliding-window” time series; thirdly, take into account adaptive fuzzy trends of the MTS components based on the fuzzy mappings.

An original way of a coherent learning NFCTM is described, which lies: firstly, in training RecANFIS for each concept NFCTM (MTS component) taking into account fuzzy trends; secondly, in coherence of all RecANFISs between each other to maximize the prediction accuracy of each of the MTS components without compromising the prediction accuracy of at least one of the other MTS components.

Experimental studies have been carried out and the results of using the proposed method were obtained on the example of the problem of multidimensional analysis and forecasting of the urban environment state in Moscow.

2 The Problem of Forecasting Multidimensional Time Series

Previously in the work [¹⁰] a formalized representation of the MTS, focused on accounting for the mutual influence of the MTS components was proposed:

where S – MTS; St=(s1(t),s2(t),…,sN(t)) – time “slice” of the MTS at the t-th instant of time; sj(t) – value of the j-th MTS component at the t-th instant of time; Lji – maximum value of the time lag (retrospective) of the j-th component relative to the i-th; φi,j – operator for accounting for mutual influence of the j-th and i-th MTS components; Fi – conversion to get si(t) taking into account the fuzzy trends of the i-th MTS component, i=1,…,N,N – the number of MTS components.

The proposed formalized representation of the MTS illustrates the possibility of taking into account the direct and indirect interaction of the MTS components relative to their different time lags.

The formalized statement of the problem of multidimensional analysis and forecasting of the state of complex systems and processes for well-coordinated MTS components assumes the possibility of minimizing forecasting errors concurrently for all MTS components and is presented as follows:

where δS – error of the forecast of the MTS in general; δSi – error in predicting of the i-th MTS component; si(t) – reference value of the i-th MTS component; si (cur)t – the forecast value of the i-th MTS component; N – the number of MTS counts. Different metrics can be used to estimate errors in predicting δSi of each of the si(t) MTS components, for example, the mean-square deviation:

The multi-criteria nature of the MTS prediction makes it necessary to minimize errors concurrently for all the MTS components. However, as a rule, it is impossible to achieve in real conditions of non-stochastic uncertainty, non-linearity of interaction, partial inconsistency and significant interdependence of the MTS components.

Therefore, we apply a compromise approach to the multi-criteria estimation of the forecast error (δS) of the MTS in general based on the generalized Edgeworth-Pareto principle [¹¹], which, in relation to the problem being solved, is expressed in the fact that it is impossible to maximize the prediction accuracy of any MTS component without deterioration of prediction accuracy of at least one of the other MTS components.

3 Description of Neuro-Fuzzy Cognitive Temporal Models

The NFCTM can be presented in the following way:

where C – multiplicity of NFCTM concepts that define the values of the corresponding MTS components; F˜i – fuzzy temporal transformation implemented by the corresponding concept Ci taking into account the fuzzy trends of the corresponding MTS component; N – the number of NFCTM concepts; s˜i(t) – the predicted fuzzy value of the concept Ci at the t-th instant of time; (s′j(t−1),…,s′j(t−Lij)) – a subset of the input temporal fuzzy variables of the concept Ci, related to the corresponding output temporal fuzzy variables of the concept Cj; Ni – the number of NFCTM concepts directly related to the concept Ci; lij – time lag (delay) for the corresponding input temporal fuzzy variable s′j(t−lij) of the concept Ci, lij=0,…,Lij; W – a set of direct influence relations between all pairs of NFCTM concepts; Wij – a subset of fuzzy values that defines a chronologically ordered set of fuzzy influence degrees wij(t−lij) of the concept Cj on the concept Ci taking into account the time lag lij; φij – fuzzy operator for accounting for the degree of mutual influence of the output variable of the concept Cj on the input variable of the concept Ci, in the case of fuzzy variables, it is advisable to use the T-norm as the operator φij.

Justification of time lags lij and determination of the degree of mutual influence wij(t−lij) for the NFCTM concepts is a separate task, which can be solved using various (statistical or expert) methods of analyzing retrospective data, based on the establishment of direct interdependence between the MTS components.

For example, multiple linear regression method can be used to determine the interaction between the time lags of the MTS components [¹², ¹³].

To implement fuzzy temporal transformations F˜i, the original neuro-fuzzy models of the RecANFIS (Recurrent Adaptive Neuro-Fuzzy Inference System/Model) are further proposed.

4 Method for Designing, Training, and Using Neuro-Fuzzy Cognitive Temporal Models for Predicting Multidimensional Time Series

An example of building and using NFCTM for multidimensional forecasting of the urban environment state in Moscow in conditions of a complex epidemiological situation is under consideration.

The state of the urban environment is characterized by the state of its heterogeneous objects, systems and infrastructure (hereinafter referred to as urban environment objects): of real estate, engineering and transport infrastructure, ecosystem [¹⁴, ¹⁵]. Therefore, its assessment cannot be reduced to a single comprehensive indicator. The method includes the following stages.

Stage 1. Determination of significant MTS components.

Thus, based on the results of previous studies [¹⁶],

[¹⁷] the following most significant factors (MTS components) that characterize the state of the urban environment have been identified: C1 – ecology of the urban environment; C2 – capacity of urban environment infrastructure; C3 – income level of the population; C4 – industrial consumption of fuel and energy resources; C5 – population life quality; C6 – sanitary and epidemiological situation.

Stage 2. Determination of the mutual influence of the MTS components for different time lags.

To correctly analyze the mutual influence of the MTS components, the retrospective MTS data is normalized:

where si(t−lij) – original value; si (min⁡),si (max⁡) – the maximum and minimum values, respectively; si (norm)(t−lij) – normalized value.

Note. Hereinafter and for normalized values si (norm)(t−lij) of the MTS components, we will use a notation si(t−lij).

To analyze the mutual influence of the MTS components a fuzzy extension of the method of multiple linear regression is reasonably chosen [¹⁸] due to the peculiarities of the analyzed urban factors:

where aj(t−lij) – fuzzy regression coefficients; b˜j(t−lij) – fuzzy free terms (usually equal to 0).

Obtained values of fuzzy coefficients aj(t−lij) of regression are then normalized and reduced to a range [0, 1]. After that, time lags whose modal values of fuzzy coefficients are less than a certain threshold are excluded from consideration (for the example under consideration, less than 0.4).

And, thus, a subset of time lags is defined corresponding to these fuzzy values Wij={wij(t−lij)|lij=0,…,Lij} of the influence indicators of the source-concept Cj on the concept-receiver Ci. The formed matrix W of fuzzy relations for the NFCTM concepts of urban environment state is presented in Table 1. The formed matrix W of fuzzy relations of influence between the NFCTM concepts.

Stage 3. The formation of the NFCTM structure.

The formation of the NFCTM structure consists in setting subsets of arcs (corresponding to the time lag) between the concepts of NFCTM, weighted with fuzzy values wij(t−lij) of their influence on each other. The structure of the NFCTM for multidimensional forecasting of the urban environment state in Moscow is shown in Figure 1.

Fig. 1 The structure of the NFCTM for multidimensional forecasting of the urban environment state in Moscow 

Stage 4. The designing of RecANFISs FSi for implementing fuzzy temporal transformations F˜i.

As RecANFISs FSi, that implement fuzzy temporal transformations F˜i, original neuro-fuzzy RecANFISs are offered. These models provide the formation, storage and output of the predicted fuzzy values of the MTS components with the time delays required for NFCTM, taking into account fuzzy trends.

Input temporal fuzzy variables of the RecANFIS FSi of the concept Ci are related to the output temporal fuzzy variables of those concepts that directly affect the concept Ci. In the process, the input temporal fuzzy variables Ci are pre-“weighted” by the corresponding fuzzy degrees of influence wij(t−lij):

The output temporal fuzzy variables of the RecANFIS FSi of the Ci are intended for generating, storing, and displaying predicted values of the i-th MTS component corresponding to reasonable time lags (see Table 1).

To designing RecANFISs FSi a priori information about the MTS components received from experts can be used, as well as data obtained because of evaluation or measurements.

In the first case, it is assumed that the problem of ensuring the completeness and inconsistency of the database of fuzzy rules of the RecANFIS FSi has been solved in advance. If only experimental data is known, then the task is to identify this model. Then to solve this problem, well-known methods can be applied to extract fuzzy rules from training samples, for example, fuzzy clustering [¹⁹]; fuzzy self-organizing maps T. Kohonen [²⁰]; adaptive fuzzy associative memory B. Kosko [²⁰]; method of gradually increasing splitting of the feature space [²²] and others.

There is often a mixed case when the initial database of model rules is constructed based on heuristic assumptions, and its parametric setting (training) is carried out based on a training sample.

Let’s consider this particular case using the example of constructing the structure and then parametric configuration of the RecANFIS FS1.

The input fuzzy variables of the RecANFIS FS1 are S′1={s′3(t−1),s′3(t−3),s′4(t−3),s′5(t−3),s′1(t−3)}, and its output fuzzy variables are S1={s˜1(t),s˜1(t−1),s˜1(t−2)}.

As an example, a separate fuzzy rule is given for the RecANFIS FS1 for the concept C1 of the NFCTM:

with L˜,M˜,H˜ – fuzzy sets for the rules of the RecANFIS FS1. The Figure 2 shows an example of the structure of the proposed RecANFIS FS1.

Fig. 2 The structure of the RecANFIS 

The structure of RecANFIS FS1 consists of the following layers of elements.

Layer 1. At the output of elements of this layer, the degrees of truth are determined for the current values of input variables relative to the corresponding fuzzy statements assumptions and all model rules:

where s˜i(t−1) – fuzzy temporal variables; ∩ – the union operator of fuzzy variables; μL˜(s˜i(t−1)) – linguistic term of a fuzzy input variable; T˜it−θ,θ=1,..,τ – a fuzzy tendency detected using the element FTiθ,θ∈1,..,τ; τ – range of “sliding window” of the i-th MTS component; μ′L˜(s˜i(t−1)) – modified term based on fuzzy tendency; ψiθ – fuzzy mapping operator of a fuzzy variable taking into account the detected fuzzy tendency, θ=1..τ; L˜,M˜,H˜ – examples of linguistic terms of input variables.

The fuzzy mapping operator ψiθ modifies the fuzzy variable s˜i(t−θ) in accordance with the iddentified fuzzy tendency in the range τ of the “sliding window”. For example, the following types of fuzzy trends can be represented: growth, weak growth, stability, slight fall, fall. Fuzzy displays with fuzzy trends are illustrated in Figure 3.

Fig. 3 Illustration of fuzzy mappings depending on the fuzzy tendency 

Layer 2. Layer elements are intended for aggregation based on the T-norm operation (here, min-conjunction) of the truth degrees of rule assumptions. For the p-th rule in question (p=1,…,P).

Layer 3. Layer elements activate conclusions of the corresponding rules according to the truth degrees of their assumptions based on the operation.

For the rule in question:

Layer 4. The layer element carries out the max-disjunction operation, accumulating the activated conclusions of all the model rules:

Layer 5. Elements are designed to detect a fuzzy trend based on values Z′(s˜i(t)),…,Zt−τ(s˜i(t)) with time delay θ=1,…,τ, implemented using elements Z:

Note: Hereinafter (as before) for normalized values si,norm(t) will use the notation si(t).

In addition to the above, the value of the output fuzzy variable s˜i(t) of the componential temporal model FSi of each component Ci is defuzzified (reduced to a crisp value si(t)) using the “center of gravity” method [²³].

Thus, a set of values {si(t)|i=1,…,N} at the output of the corresponding models {FSi|i=1,…,N} comprehensively characterizes the predicted state of stability of the urban environment at instant of time t.

Notice also that in contrast to the Fuzzy Cognitive Maps of B. Kosko [²³, ²⁴], the use of the approach suggested above eliminates the need to separately account for positive and negative effects by applying the RecANFIS for each MTS component.

Stage 5. Parametric setting (training) of RecANFISs.

The training procedure for each RecANFIS is preceded by a procedure for forming a training sample to configure the corresponding models for detecting fuzzy trends FTi(t−θ),θ=1,…,τ:

After generating sets of training examples, the construction of models FSi(t−θ) for example, fuzzy rule-based models, is carried out:

The identification and use of fuzzy trends eliminates the problem of non-stationarity of the predicted time series, in addition, it is a feedback that gives system stability.

After that, a training sample for RecANFISs is built based on the prepared training sample together with the identified fuzzy trends.

Using retrospective data, training samples are built for each model FSi(t−θ) of the detection of fuzzy trends.

After detecting fuzzy trends, a training sample is built for neuro-fuzzy models of the RecANFIS. Table 2 shows a fragment of the formed training sample for the fuzzy componential temporal model FS1. Note that, for clarity, Table 2 shows only modal values (without degrees of their blurring) of the input and output temporal variables of the model FS1.

Note that for RecANFIS FSi, the configurable parameters are the parameters of the membership functions of antecedents and consequents.

Stage 6. Reconciliation of all fuzzy RecANFISs.

Reconciliation of all RecANFISs FSi,i=1,…,N of the NFCTM is carried out after their “personalized” parametric setting and consists in such a change in parameters of fuzzy degrees of influence {wij(t−lij)|lij=0,…,Lij} between NFCTM concepts to ensure the maximization of the prediction accuracy of each of the MTS components without deterioration of prediction accuracy of at least one of the other MTS components. The genetic algorithm can be used for this purpose [²⁵].

Stage 7. Prediction of MTS.

Prediction of MTS can be carried out in the following modes:

Predictive assessment in its dynamics are modeled from certain situation set by the initial values of all NFCTM concepts, with external influence on the values of concepts and/or influence relations between NFCTM concepts.

5 The Multidimensional Forecasting of the Urban Environment State in Moscow

Experiments have been carried out and the results of using the NFCTM for multi-dimensional forecasting of the urban environment state in Moscow in a complex epidemiological situation have been obtained (Figure 4).

Fig. 4 Illustration of the results of multidimensional forecasting of the urban environment state in Moscow 

Table 3 presents a comparative assessment of the results of multidimensional forecasting of the urban environment state in Moscow (based on historical data from 2000 to 2020) using an artificial neural network (ANN) and developed by the NFCTM. As a comparison, a multilayer perceptron with a single hidden layer of 16 neurons was used, which showed the best among the various variants of ANN.

The obtained results of the comparative evaluation showed that the use of NFCTM in small samples allows increasing the accuracy of the MTS forecast by compared to the multilayer perceptron with one hidden layer of 16 neurons, which showed the best among various variants of ANN.

The new type of Neuro-Fuzzy Cognitive Temporal Models is proposed. These models are focused on multidimensional prediction of MTS in conditions of non-stochastic uncertainty, non-linearity of interaction, partial inconsistency and significant interdependence of MTS components, as well as in conditions of small samples with fuzzy trends.

To implement fuzzy temporal transformations of concepts, the proposed original RecANFISs are used, which provide the formation, storage and output of the predicted fuzzy values of the corresponding MTS components with the time delays required for NFCTM, taking into account fuzzy trends.

The method of constructing, training and using NFCTM for MTS predicting taking into account fuzzy trends is considered, which includes:

6 Conclusion

Prediction of MTS is based on a structurally and parametrically configured NFCTM and can be carried out in the following modes:

Experimental studies have been carried out and the results of using the proposed NFCTM for multidimensional forecasting of the urban environment state in Moscow in conditions of a complex epidemiological situation have been obtained.

Comparative assessment showed that in conditions of small samples the use of NFCTM allows improving the forecast accuracy of MTS compared with the neural network approach, which demonstrated one of the best results in solving this task.