In the article there are Neuro-Fuzzy Cognitive Temporal Models (NFCTM) described. Those provide accounting of indirect and indirect mutual impact of all the multidimensional time series (MTS) components with their temporary delays relative to each other and are oriented on forecasting of multidimensional time series. Neuro-Fuzzy Cognitive Temporal Componental Models, which provide the formation of forecasted values of the MTS components with the temporary delays demanded, are used in NFCTM concepts in order to accomplish the temporal transformation. There is the way of NFCTM coordinated training described, which consists in Neuro-Fuzzy Componental Temporary Models for each of the NFCTM component and also in coherence of these Neuro-Fuzzy Componental Temporary Models (NFCTM) between each other. There is an MTS forecasting method offered in condition of unreliability the nonlinearity of the interaction, partial inconsistency and interdependence of the MTS components, that is based on NFCTM. There are experimental studies conducted and the results of using the proposed method are presented on the example of the problem of multidimensional forecasting of the state of the urban environment in Moscow. The use of the proposed method may be in demand to provide reliable forecasting of the state of the urban environment in various regions of Russia and other countries, including into account the complex epidemiological situation.
used and the lack of consideration of the different time delays of the interdependent components of the MTS.
The article deals with Neuro-Fuzzy Cognitive Temporal Models (NFCTM) which provide direct and indirect interaction of all components of multidimensional time series (MTS) with their time delays relative to each other, and are focused on predicting multidimensional time series. The method of coordinated training of NFCTM is described, which consists in training of Neuro-Fuzzy Component Temporal Models for each NFCTM concept, as well as in matching of these Neuro-Fuzzy Component Temporal Models of NFCTM.
There are experimental studies conducted and the results of using the proposed method are presented on the example of the problem of multidimensional forecasting of the state of the urban environment in Moscow. The use of the proposed method may be in demand to provide reliable forecasting of the state of the urban environment in various regions of Russia and other countries, including into account the complex epidemiological situation. 2. Neuro-Fuzzy Cognitive Temporal Models for predicting multidimensional time series Let’s present the MTS as follows:
S = ( S1.. SN ), s1(t) = F1 ϕ1,1 ( s1(t−1)..s1(t−L11)) ) ..ϕ1,N ( sN(t−1)..sN(t−L1N ) ) , ∀t ∈{1..τ ..} St = ...
sN(t) = FN ϕ N,1 ( s1(t−1)..s1(t−L1N ) )..ϕ N,N ( sN(t−1) , ..sN(t−LNN ) ) , where S – multidimensional time series; St = ( s1(t).. s(t) ) – time «slice» of the MTS at the t-th instant of N time; s(jt) – the value of the j-th component of the MTS at the t-th instant of time; Lij – maximum time delay of the j-th component of the MTS relative to the i-th; ϕi, j – operator for accounting for the interaction between the j-th and i-th MTS components; Fi – transformation for definition si(t) , i = 1, ..., N , N – quantity of the MTS components.
Article [9] proposes a new type of NFCTM focused on MTS forecasting:
FCTM = С, W ,
(
=C , Ci : si(t) = F i s′j(t−1).. s′j(t−Lij ) | j ∈1.. Ni , i =N, 1..
W
={Wij | i, j ∈1.. N}, Wij ={wi(jt−lij ) | lij
=0..Lij }, s j =wi(jt−lij ϕij ) , s j
(t−lij ) , lij =, 0.. Lij where С – multiple NFCTM concepts corresponding to MTS components; Fi – fuzzy temporal transformation implemented by the concept Ci ; N – number of NFCTM concepts; si(t) – predicted fuzzy value of the concept Ci at the t-th instant of time; s′j(t−1).. s′j(t−Lij ) – subset of the input temporal fuzzy variables of the concept Ci , associated with the corresponding output temporal fuzzy variables of the concept C j ; Ni – number of NFCTM concepts directly related to the concept Ci ; lij – time delay for the corresponding input variable s′j(t−lij ) of the concept Ci , lij = 0.. Lij ; W – a set of fuzzy degrees of direct impact between all pairs of NFCTM concepts; Wij – a subset of fuzzy degrees of impact wi(jt−lij ) of the concept C j on the concept Ci taking into account the time delay lij ; ϕij – fuzzy operator accounting for the degree of impact of the output variable of the concept C j on the concept's input variable Ci . 3. Description of the method for predicting multidimensional time series based on Neuro-Fuzzy Cognitive Temporal Models The method of prediction of MTS based on NFCTM consists of the stages discussed below.
Stage 1. Identification of meaningful components of the MTS for determining the composition of NFCTM concepts.
The implementation of the proposed method will be considered on the example of multidimensional forecasting of the state of the urban environment in Moscow. The state of the urban environment is characterized by the state of its facilities, systems and infrastructure and cannot be reduced to any single indicator. Basing on the results of previous studies [10-12], the following meaningful factors (components of MTS) characterizing the state of the urban environment have been determined: • 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 – life quality of the population; • C6 – sanitary and epidemiologic situation.
Stage 2. Determining the fuzzy degrees of impact of the components of the MTS for different time delays and forming the structure of the NFCTM.
To determine the degree of mutual impact wi(jt−lij ) taking into account time delays lij for NFCTM concepts, various methods of data analysis can be used, based on the establishment of interdependencies between all the components of the MTS. For example, for the example under consideration (due to the different quality of the urban environment, the expert nature of their assessment, the non-linear relationship between them and the non-stochastic uncertainty), a fuzzy extension of the multiple linear regression method has been chosen [13].
In table 1 shows the formed matrix of fuzzy relations W of impact of concept sources on concept receivers of NFCTM for solved task of multidimensional forecasting of urban environment state. For clarity, only modal values of fuzzy degrees of impact are shown. 1,0 0,40 1,00 0 0 0 (t) s1
: s1(t) concepts weighted by fuzzy values wi(jt−lij ) forecasting of the state of the urban environment of Moscow is shown in figure 1.
. The formed structure of the NFCTM for multidimensional predicted fuzzy values of MTS components with required time delays [9].
The input variables of the model FSi concept Ci are related to the output variables of those concepts that have a direct impact on the concept C . At the same time input variables C are «weighted» by i i fuzzy degrees of impact wi(jt−lij ) :
s j =Тs wij j
j , li (t−lij )
j
=, 0, ..., L
i
(
The output variables of the model FSi of the concept Ci are intended to generate the predicted values of the i-th MTS component, corresponding to reasonable time delays. w(3t2−2)
w(2t4−1) w(2t5−3) w(2t3−1) w(4t2−1) w(4t2−2) w(4t2−3) To build models FSi , both expert information about the components of the MVR and experimental data can be used. Next, we will consider a mixed version, when the model's rule base is formed by an expert, and its training is carried out on the basis of a training sample. Let's consider this particular case as an example of building the structure (and later parametric configuration) of a Neuro-Fuzzy Component ′(t−1) , s′(3t−3) , s′(4t−3) , s′5(t−3) , s′1(t−3) , the Temporal Model FS1 . The input variables of the model FS1 – S1′ = s 3 output variables of this model – S1 = {s1(t) , s1(t−1) , s1(t−2)} .
Here is an example of one fuzzy production rule of the model FS1 for the concept C1 of NFCTM: If s′1(t−1) is L AND s′(3t−3) is L AND s′(4t−3) is M
AND s′(5t−3) is M AND s′1(t−3) is H ,
Then ( s1(t) is M ) AND ( s1(t−1) is M ) AND ( s1(t−2) is L ),
where L, M , H – fuzzy sets of prerequisites and conclusions of model rules FS1 .
(
µM s′1(t−1)
µH s′1(t−1) ... µL s′1(t−3)
µM s′1(t−3)
µH s′1(t−3) α1 α p α P µL ( s1(t) ) µM ( s1(t) ) µH ( s1(t) )
Z0 Z−1 Z−1 µ M ( s1(t) ) = min (α p , M ).
s1(t) = max (µ L ( s1(t) ), ..., µ M ( s1(t) ), ..., µ H ( s1(t) )).
Layer 5. Layer elements are designed to normalize and output model output variable values with required time delays: s1 (norm) =s1(t)), Z 0 ( s1 (norm) =s1(t) Z −1 ( ), s1 (norm) =s1(t−1) Z −1 ( ).
(t) (t−1) (t−2) Next, we use the notation si(t) for normalized values si(,tn)orm .
Value of the output fuzzy variable si(t) of the model FSi if necessary, is defuzzified using the «center of gravity» method [14]:
Layer 4. The layer element performs the max-disjunction operation, accumulating the activated
conclusions of all the model rules:
(
M =def ( si(t) ) =m=1 M si , M
=Supp ( si(t) ) , ∑µ (t) ( si(,tm) ) m=1 si (t) si =si(,tm) {(µ (t) ( ) / si(,tm) ) | m =}, 1, ..., M
si (t) where si – defuzzified value of the output variable si(t) of the model FSi in timepoint t; def ( si(t) ) – defuzzification operator using the «center of gravity» method; si(,t m) – m-th the discretized value of a variable si(t) , m = 1, ..., M ; µ s(t) ( si(,tm) ) – degree of the identity of the variable si(t) for the value si(,tm) ; i Supp ( si(t) ) – variable carrier si(t) .
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 the urban environment at a given time t.
Stage 3. The coordinated training of NFCTM
For coordinated training of NFCTM, a method is proposed comprising the following two procedures: firstly, training Neuro-Fuzzy Component Temporal Models for each NFCTM concept; secondly, matching of Neuro-Fuzzy Component Temperature models with each other.
Training procedure for Neuro-Fuzzy Component Temporal Models FSi is preceded by the
formation of training samples:
s′j(t−1) (k),..., s′j(t−Lij ) (k) | j ∈1,..., Ni , si(t) (k) , k =, 1, ..., K
(
For the models FSi implementing Mamdani’s inference algorithm [14], modal values and blur degrees of fuzzy sets of prerequisites and rule conclusions are configurable parameters.
Training procedure for all NFCTM models FSi includes the following steps.
Step 1. For each example of the training selection based on the values of input variables s′j(t−1) (k),..., s′j(t−Lij ) (k) | j ∈1,..., Ni the model FSi calculates the value of the output variable si(t()cur) (k) .
Step 2. For all examples of teaching sample, the error function is calculated, depending on the parameters of the model to be configured:
Ei
1 K (t) 2 =∑ ( s(t) (k) − si (cur) (k)) .
K k=1 i
(
Procedure for matching all NFCTM FSi , i = 1, ..., N consists in the following steps.
Step 1. For each example from a matching training sample based on the values of input variables s′j(t−1) (q),..., s′j(t−Lij ) (q) | j ∈1,..., Ni | i =...,N 1, all the models FSi , i = 1, ..., N calculate the values of (t) output variables si (cur) (q), i = 1, ..., N .
Step 2. For all sample examples for all models FSi , i = 1, ..., N error functions that depend on
configurable fuzzy impact parameters {wi(jt−lij ) | lij = 0, ..., Lij } between NFCTM concepts:
(
Step 3. In accordance with the learning algorithm (e.g., an error reverse propagation algorithm or a genetic algorithm), adjustments are made to the parameters to be adjusted.
Steps 1-3 are iteratively repeated, and model training is considered complete when for each of them the total error does not exceed the set threshold.
Procedure for matching all Neuro-Fuzzy Component Temporal Models FSi , i = 1, ..., N is performed after their individual training and consists in such change of modal values and degrees of blur of fuzzy degrees of impact {wi(jt−lij ) | lij = 0, ..., Lij } between concepts of NFCTM, which provides maximum increase of prediction accuracy of each component of MTS without deterioration of prediction accuracy of at least one of other components of MTS. This procedure is preceded by a teaching sample consisting of data for all TDM components: s′j(t−1) (q),..., s′j(t−Lij ) (q) | j ∈1,..., Ni , si(t) (q) | i =..., 1, N , q =..., 1, Q,
Ei = 1 Q (t) 2
∑ ( s(t) (q) − si (cur) (q)) , i = 1, ..., N.
K q=1 i
Step 3. According to the genetic algorithm used (e.g, [15]) According to the used genetic algorithm, adjustment of customizable parameters of fuzzy degrees of impact is performed {wi(jt−lij ) | lij = 0, ..., Lij } between NFCTM concepts thus, to ensure maximum increase in accuracy of forecasting each of the components of MTS without deterioration of prediction accuracy of at least one of the other MTS components.
Steps 1-3 are iteratively repeated, and the procedure for matching all NFCTM is considered successful if the total error for each of these models does not exceed a certain set threshold (For wellaligned MTS components), or for these models, the Ejworth-Pareto principle will be implemented, [14], which, in relation to consistent NFCTM training, is expressed in that it is impossible to maximize the prediction accuracy of any MTS component without deteriorating the prediction accuracy of at least one of the other MTS components.
Stage 4. MTS forecast based on trained NFCTM.
MTS forecasting is performed based on a trained NFCTM and consists in calculating the values of output model variables FSi , i = 1, ..., N by the corresponding sets of values of the input variables of these models that are set each time.
Experiments were carried out and the results of using the proposed method on the example of multidimensional and forecasting the state of the urban environment in Moscow were obtained. Figure 3 illustrates the results obtained.
Table 2 presents a comparative assessment of the results of multidimensional forecasting of the state of the urban environment in Moscow using an artificial neural network (ANN) and the developed NFCTM. As a comparison, a multilayer perceptron with a hidden layer of 24 neurons was used, which showed the best among various ANN variants.
The comparative assessment showed that the use of the proposed method based on NFCTM in small sample conditions allows to increase the accuracy of the forecast of MTS by an average of 10-15% compared to the best-performing ANN.
Experimental studies are conducted and the results of using the proposed method are presented on the example of the problem of multidimensional forecasting of the state of the urban environment in Moscow. A comparative assessment showed that using this method based on NFCTM in small sample conditions allows to improve the accuracy of the MTS forecast by an average of 10-15% compared to the best ANN results.
The use of the proposed method may also be in demand to ensure reliable forecasting of the state of the urban environment in different regions of Russia and other countries, including taking into account the difficult epidemiological situation.
The reported study was funded by RFBR, project number 19-31-90054. [14] Borisov V.V., Krugliv V.V., Fedulov A.S. Fuzzy models and networks. – M: Hotline – Telecom, 2018. 284p. (In Russian). [15] Stach W., Kurgan L., Pedrycz W., Reformat M. Genetic learning of fuzzy cognitive maps // Fuzzy
Sets and Systems. 2005. vol. 153. No. 3. pp. 371-401. [16] Noghin V.D. Edgeworth-Pareto principle // Studies in Systems, Decision and Control. 2018. Vol. 126. pp. 1-22. DOI: 10.1007/978-3-319-67873-3_1.