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Nonparametric Decision Theoretic Classification
Published in Sing-Tze Bow, Pattern Recognition and Image Preprocessing, 2002
A piecewise linear function is a function that is linear over subregions of the feature space. These piecewise linear discriminant functions give piecewise linear boundaries between categories as shown in Figure 3.9a. In Figure 3.9b the boundary surface between class ω1 and class ω2 is nonconvex. However, it can be broken down into piecewise linear boundaries between these two classes ω1 and ω2. The discriminant functions are given by () dk(x)=maxm=1,…,Nk[dkm(x)]k=1,…,M
Basic elements and definitions
Published in András Bárdossy, Lucien Duckstein, Fuzzy Rule-Based Modeling with Applications to Geophysical, Biological and Engineering Systems, 2000
András Bárdossy, Lucien Duckstein
Fuzzy numbers (or arbitrary fuzzy sets on the set of real numbers) can also be defined with the help of piecewise linear membership functions. For this a set of breakpoints x0 < x11 < … < xL and the corresponding membership values μ(x0), μ(x1), … < μ(xL) have to be given, and it is assumed that the membership function is linear between these points. The advantage of this formulation is that most continuous functions can be closely approximated by piecewise linear functions, and the operations on the latter are usually simple.
Scientific Computing
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
An interpolating polynomial has large degree and tends to oscillate greatly for large data sets. Piecewise polynomial approximation divides the interval into a collection of subintervals and constructs an approximating polynomial on each subinterval. Piecewise linear interpolation consists of simply joining the data points with line segments. This collection is continuous but not differentiable at the node points. Cubic spline interpolation is popular since no derivative information is needed.
Control strategies for dynamic motorway traffic subject to flow uncertainties
Published in Transportmetrica B: Transport Dynamics, 2019
Ying Li, Andy H. F. Chow, Renxin Zhong
Given the cell density, the cell transmission rule models the outflow from cell i within time step t by a piecewise linear fundamental diagram as follows: where is regarded as the ‘send’ function from cell i at time t and is the ‘receive’ function at downstream cell i+1 at the same time; is the capacity flow at cell i which corresponds to the maximum flow that can leave cell i; is the capacity flow at cell i+1 which corresponds here to the maximum flow that can enter cell i+1. The inclusion of both capacity flows at adjacent cells is due to the consideration of inhomogeneous section, where we have different capacities at different locations. Equation (4) can be regarded as a piecewise linear approximation of Equation (2). When there is no congestion, the traffic moves from one cell to the next at free-flow speed, . The notation denotes the backward shockwave speed specified by the fundamental diagram at the downstream cell i+1, and is the jam density at cell i+1. The quantity specifies the available space for incoming traffic at the downstream cell i+1 during time t. The above formulation covers both congested and uncongested regimes.