Direct Probabilistic Inference for Continuity of Piecewise Models at Weighted Mean Threshold

Music City Center 

In weighted piecewise regression (PM), an unknown threshold is usually estimated by a weighted mean and confidence interval. A key issue brought about by this is how to use probability to infer the continuity of two adjacent models at the threshold. This article will take a 2-segment linear model in a 2D space to demonstrate a method to infer the continuity of the PMs. Assuming that the fullwise model (FM) is y=a+bx, and the convex self-weighted mean (Cmean) of its absolute residuals is AR_bar_c. Then, we first take X as the segmented vattribute and the FM as the benchmark model. By keeping each pair of PMs (PM1 and PM2) homogeneously with the FM during the iteration for the threshold, we can calculate the Cmean ar_bar_(x,c,i) of the combined absolute residuals of the PMs obtained at each iteration, and then we will have a regressive weight w_x,i=(AR_bar_c - ar_bar_(x,c,i))/AR_bar_c, thus the threshold X_bar_∆=(∑x_i×w_x,i)/(∑w_x,i). The two predictions will be Y_1_hat and Y_2_hat at the threshold X_bar_∆. Thus we have Y_cv=|Y_1_hat - Y_2_hat|. Similarly for X by taking Y as the segmented one, we have X_cv=|X_1_hat - X_2_hat|. Thus, P_c=(X_cv×Y_cv)/(2×R_X×R_Y) (R is range).

Fullwise-Piecewise model

Convex Self-weighted Mean (Cmean) of Absolute residuals

Regressive weight

weighted mean threshold

Connection variation at mean threshold

Continuity probability 

IMS