What actually limits an IT2FCM-Markov chain fuzzy time series model? Type reduction, directional adjustment, and the forecast horizon

Abstract

Hybrid fuzzy time series models that combine Interval Type-2 Fuzzy C-Means (IT2FCM) partitioning with Markov chain transition weighting lag behind the series at abrupt spikes. Two explanations are available in the literature and have never been separated: the approximate type reduction used inside the clustering loop contracts the footprint of uncertainty, and the probability-weighted forecast is a convex combination of the cluster centres and so cannot leave their span. This study separates them by factorial ablation on a published architecture [10], under an expanding-window protocol, on two ambient carbon monoxide series from one station in Semarang, Indonesia: 48 readings spanning [132, 946] and 288 half-hourly readings spanning [2063, 26934], the second placing 13.7 observations in each cluster against 3.0 in the first. Neither mechanism survives scrutiny. Replacing exact Karnik–Mendel type reduction by the Nie–Tan surrogate changes RMSE by 0.07% on the short series and 1.0% on the long one, neither significant, while the exact iteration costs three to five times as much per fit. The directional term lowers RMSE by 17% on the short series and raises it on the long one, where the dynamics are more strongly mean-reverting. A third finding subsumes both. At one step ahead the full pipeline is indistinguishable from persistence on both series (p = 0.4137 and 0.4551), so the ablated configurations are collectively no better than copying the last observation forward. Extending the chain to P h shows where the model does earn its place: from two to twelve hours ahead it improves on persistence by 6.7–10.2% in RMSE (p ≤ 0.0015), and at that horizon the differences between all six configurations fall below significance. For half-hourly monitoring, the choice of type-reduction operator is not what limits these models; the choice of forecast horizon is.