Time seriesChow testFuzzy transformEvaluative linguistic expressionsFuzzy natural logic

A new methodology for the detection of structural breaks in time series is introduced. The proposed method is based on the combination of special techniques of fuzzy modeling, namely Fuzzy Transform (F-transform) and selected methods of Fuzzy Natural Logic (FNL), and the Chow test that is the well-known statistical method for anomaly detection in time series. The concept of detection structural breaks in time series using the Chow test is based on the difference of intercept terms as well as different regression coefficients. The time series in which a structural break occurs is divided into two linear regressions models. The test of hypothesis related to the test of structural break is conducted by testing the null hypothesis. However, it is difficult to determine the changepoint in a given time series. Therefore, we apply fuzzy techniques for this purpose. Under the assumption that a time series can be additively decomposed into trend-cycle, seasonal component, and random disturbances, the fuzzy transform makes it possible to find the arbitrary shapes of the trend-cycle, and detect specific areas, or intervals of monotonous behavior. We can also estimate the slope (average value of the first derivative) of the time series in a given (imprecisely determined) area. FNL provides a mathematical model of semantics of special expressions of natural language (the so-called evaluative linguistic expressions). Using them we are able to evaluate the slope of time series or detect areas of monotonous behavior. Using this information, we are able to detect possible structural breaks. The fuzzy transform used in time series provides zero degree (absolute components) and first degree ones. The coefficients are estimations of average values of the tangents (slopes) of the function over areas characterized by the fuzzy sets.