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Many of the ideas presented on this page can be generalized to directed graphs. [1] The edge weight function associates to every edge (,) a real value (,) >. For both mathematical and application specific reasons, the weight function on the edges is often required to be strictly positive and on this page it will be assumed to be so unless ...
The knee of a curve can be defined as a vertex of the graph. This corresponds with the graphical intuition (it is where the curvature has a maximum), but depends on the choice of scale. The term "knee" as applied to curves dates at least to the 1910s, [1] and is found more commonly by the 1940s, [2] being common enough to draw criticism.
Extrapolation refers to the use of a fitted curve beyond the range of the observed data, [16] and is subject to a degree of uncertainty [17] since it may reflect the method used to construct the curve as much as it reflects the observed data. For linear-algebraic analysis of data, "fitting" usually means trying to find the curve that minimizes ...
In any graph, the degree of a vertex is defined as the number of edges that have as an endpoint. For graphs that are allowed to contain loops connecting a vertex to itself, a loop should be counted as contributing two units to the degree of its endpoint for the purposes of the handshaking lemma. [2]
A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point [1] [2] and exactly one inflection point. Properties
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
It is the incidence matrix of any bidirected graph that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph. The column of a negative edge has either a 1 or a −1 in both rows.
The method consists of plotting the explained variation as a function of the number of clusters and picking the elbow of the curve as the number of clusters to use. The same method can be used to choose the number of parameters in other data-driven models, such as the number of principal components to describe a data set.