Piecewise linear function

In mathematics, a piecewise linear or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments.

f(λx)=λf(x) and therefore in particular

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.

If partitions are not known, the residual sum of squares can be used to choose optimal separation points. However efficient computation and joint estimation of all model parameters (including the breakpoints) may be obtained by an iterative procedure currently implemented in the package segmented for the R language.

A variant of decision tree learning called model trees learns piecewise linear functions.

A piecewise linear function of two arguments (top) and the convex polytopes on which it is linear (bottom)

The notion of a piecewise linear function makes sense in several different contexts. Piecewise linear functions may be defined on n-dimensional Euclidean space, or more generally any vector space or affine space, as well as on piecewise linear manifolds and simplicial complexes (see simplicial map). In each case, the function may be real-valued, or it may take values from a vector space, an affine space, a piecewise linear manifold, or a simplicial complex. (In these contexts, the term “linear” does not refer solely to linear transformations, but to more general affine linear functions.)

In dimensions higher than one, it is common to require the domain of each piece to be a polygon or polytope. This guarantees that the graph of the function will be composed of polygonal or polytopal pieces.

Splines generalize piecewise linear functions to higher-order polynomials, which are in turn generalized by piecewise-smooth and piecewise-differentiable functions.

Important sub-classes of piecewise linear functions include the continuous piecewise linear functions and the convex piecewise linear functions. In general, for every n-dimensional continuous piecewise linear function f(x→)=minΣ∈Πmax(a→,b)∈Σa→⋅x→+b.

If f(x→)=max(a→,b)∈Σa→⋅x→+b.

Crop response to depth of the watertable Example of crop response to soil salinity

In agriculture piecewise regression analysis of measured data is used to detect the range over which growth factors affect the yield and the range over which the crop is not sensitive to changes in these factors.

The image on the left shows that at shallow watertables the yield declines, whereas at deeper (> 7 dm) watertables the yield is unaffected. The graph is made using the method of least squares to find the two segments with the best fit.

The graph on the right reveals that crop yields tolerate a soil salinity up to ECe = 8 dS/m (ECe is the electric conductivity of an extract of a saturated soil sample), while beyond that value the crop production reduces. The graph is made with the method of partial regression to find the longest range of "no effect", i.e. where the line is horizontal. The two segments need not join at the same point. Only for the second segment method of least squares is used.

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