A Fuzzy Approach to Parameter Estimation in Autoregressive Models

In this paper, we conducted an in-depth study on the estimation of fuzzy parameters of the autoregressive (AR) process using the expected value method. Fuzzy set theory, introduced by Zadeh in 1965, opened the way to new solutions for dealing with this type of problem. By definition, a process is said to be stationary when its stochastic parameters, namely the expected value, the variance, and the autocorrelation function, are invariant over time. These three parameters define the conditions for stationarity of an autoregressive process. Our main objective was to examine whether an AR model with fuzzy coefficients respects these stationarity conditions. After a rigorous analysis, we found that the introduction of fuzzy coefficients into the model does not alter the stationarity conditions of the process. However, the major challenge lies in determining the membership degree of the fuzzy parameters. To solve this problem, we resort to Zadeh's extension theory, which allows us to define the membership degree of fuzzy variables. In order to validate our results, we carried out several illustrations using AR models with fuzzy coefficients. The simulations confirm that the estimated stochastic parameters remain consistent with the stationarity conditions. Furthermore, we studied the application of the Yule-Walker equations in a fuzzy context. These equations, which allow the estimation of the model coefficients from the autocorrelation (or autocovariance) function, retain their validity even in a fuzzy environment. However, as before, the difficulty lies in the evaluation of the degree of membership, which requires the use of fuzzy arithmetic. Through our examples, we used Zadeh's fuzzy theory to perform these calculations. This study shows that the introduction of fuzziness into the coefficients of the AR model does not call into question its stationarity, but requires the use of specific tools for estimating the degrees of membership, such as fuzzy set theory. After making a thorough analysis of the stationarity of the fuzzy autoregressive process. We concluded that the model parameter does not influence the stationarity conditions and the white noise, but rather it modifies the autocorrelation function by a multitude of partial and total models.