Build interval-valued time series forecasting model with interval cognitive map trained by principle of justifiable granularity

In the study of time series forecasting based on fuzzy cognitive maps (FCMs), the causalities between past values and future values are represented by real-valued weights in [-1,1]. However, for interval-valued time series (ITS), the causalities are affected by various uncertainties including ways of measuring and ways of intervals influencing intervals and thus involve uncertainty. Therefore, real-valued weights are no longer enough for characterizing such causalities, equipping FCMs with interval-valued weights becomes necessary and resulting in interval cognitive maps (ICMs). In this case, how to determine the interval-valued weights of an ICM becomes a crucial problem. To solve this problem, this paper first proposes the principle of justifiable granularity for interval-valued data, which is guaranteed to accumulate enough experimental evidence and effectively express the ITS, then develops a reasonable method that can optimally determine the interval-valued weights and enable the interval-valued weights having clear semantics. By means of the proposed method for determining interval-valued weights, an ICM-based ITS forecasting model is established, which can not only deal with the uncertainty of causalities between interval-valued data, but also avoid counterintuitive outputs which often appeared in existing ITS forecasting models. Experimental results show the good performance of the proposed forecasting model.