Probabilistic criteria for time-series predictability estimation

Assessing the time series predictability is necessary for forecasting models validating, for classifying series to optimize the choice of the model and its parameters, and for analyzing the results. The difficulties in assessing predictability occur due to large heteroscedasticity of errors obtained when predicting several series of different nature and characteristics. In this work, the internal predictability of predictive modeling objects is investigated. Using the example of time series forecasting, we explore the possibility of quantifying internal predictability in terms of the probability (frequency) of obtaining a forecast with an error greater than some certain level. We also try to determine the relationship of such a measure with the characteristics of the time series themselves. The idea of the proposed method is to estimate the internal predictability by the probability of an error exceeding a predetermined threshold value. The studies were carried out on data from open sources containing more than seven thousand time series of stock market prices. We compare the probability of errors which exceed the allowable value (miss probabilities) for the same series on different forecasting models. We show that these probabilities differ insignificantly for different forecasting models with the same series, and hence, the probability can be a measure of predictability. We also show the relationship of the miss probability values with entropy, the Hurst exponent, and other characteristics of the series according to which the predictability can be estimated. It has been established that the resulting measure makes it possible to compare the predictability of time series with pronounced heteroscedasticity of forecast errors and when using different models. The measure is related to the characteristics of the time series and is interpretable. The results can be generalized to any objects of predictive modeling and forecasting quality scores. It can be useful to developers of predictive modeling algorithms, machine learning specialists in solving practical problems of forecasting.