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Concept
Piecewise linear function
Summary
In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments.
A piecewise linear function is a function defined on a (possibly unbounded) interval of real numbers, such that there is a collection of intervals on each of which the function is an affine function. (Thus "piecewise linear" is actually defined to mean "piecewise affine".) If the domain of the function is compact, there needs to be a finite collection of such intervals; if the domain is not compact, it may either be required to be finite or to be locally finite in the reals.
The function defined by
is piecewise linear with four pieces. The graph of this function is shown to the right. Since the graph of an affine() function is a line, the graph of a piecewise linear function consists of line segments and rays. The x values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots. As in many applications, this function is also continuous. The graph of a continuous piecewise linear function on a compact interval is a polygonal chain.
Other examples of piecewise linear functions include the absolute value function, the sawtooth function, and the floor function.
() A linear function satisfies by definition and therefore in particular ; functions whose graph is a straight line are affine rather than linear.
An approximation to a known curve can be found by sampling the curve and interpolating linearly between the points. An algorithm for computing the most significant points subject to a given error tolerance has been published.
Segmented regression
If partitions, and then breakpoints, are already known, linear regression can be performed independently on these partitions.
However, continuity is not preserved in that case, and also there is no unique reference model underlying the observed data. A stable algorithm with this case has been derived.
Official source
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