Least-squares fitting of piecewise curves

The least-squares method is commonly used to estimate the parameters of a known mathematical model with a single functional form. However, in many applications, the underlying behavior is better approximated by piecewise functions composed of multiple segments. Such problems are usually addressed by empirically selecting the breakpoints and then applying a constrained least-squares method. This manual selection, nevertheless, does not always guarantee a globally optimal solution. In this work, a methodology for the simultaneous estimation of the function parameters and their breakpoints is developed. The proposed approach combines the method of indirect measurements with constraints and the Newton–Raphson method. Specifically, the breakpoints are treated as unknowns in the Newton–Raphson procedure and as parameters in the least-squares formulation. As the breakpoint estimates converge, the least-squares solution is progressively guided toward the optimal solution. Furthermore, all measurement equations retain a fixed functional structure that is piecewise-defined, enabling automatic partitioning of the measurements within the least-squares procedure. Finally, numerical examples are presented to demonstrate the proposed methodology.