Systems have functions {system functions}. Linear functions, such as polynomials or series functions, sum weighted harmonic frequencies or weighted power functions. Linear functions can be scalars or vectors.
Feature values can relate in direct or inverse proportion {cross-correlation function}.
Potentials are vectors. Field or potential cross product of del and f {curl, function} describes area density. Curl is vector-field rotation rate {circulation density}, with magnitude and direction. Curl is non-linear. Using no coordinates, curl is limit, as volume goes to zero, of surface integral over closed surface, of cross product of unit outward-normal vector and function, all divided by volume. Find region-boundary function value and then divide by region volume.
Function dot product of del and f {divergence} describes vector-potential flow or flux. Divergences are scalars. Positive divergence means diverging. Negative divergence means converging. Divergence is limit, as volume goes to zero, of double integral over surface area of dot product of vector field and surface-area differential, all divided by volume. Divergence is limit, as volume goes to zero, of surface integral over closed surface of dot product of unit outward-normal vector and function, all divided by volume. Find region-boundary function value and then divide by region volume.
Function del f {gradient, function} describes field or potential changes over space. Gradients are maximum field direction and magnitude vectors. Field or potential can be scalar or vector. Gradients are non-linear. Using no coordinates, gradient is limit, as volume goes to zero, of surface integral, over closed surface, of product of unit outward-normal vector and function. Find region-boundary function value and then divide by region volume.
Multivariate functions {radial basis function} (RBF) can be weighted sums of independent linear functions.
input
Inputs can be spatial coordinates, angles, line-segment lengths, colors, segment configurations, feature binocular disparities, or texture descriptions. Training uses input data points.
dimensions
Data points have distances from coordinate means: |x - t|, where x are data-point coordinate values, and t are coordinate means. Data typically has Gaussian distribution, which can be broad or narrow, along all dimensions. Dimension number is typically less than data-point number.
training
Training assigns weights to dimensions or factors.
test sum
Test data point has sum over all weighted dimensions. Comparing sum to input data-object sums can identify test object. For narrow Gaussian distributions, RBF is like lookup table, because test objects only match if input equals mean.
Functions {hyperbolic basis function} (HBF) can allow more flexibility than radial basis functions. Networks can express weighted function f*(x) = summation over i from 1 to N of c(i) * G(transpose of (x - t(i)) * transpose of W * W * (x - t(i))) + p(x), where x are data-point values, t(i) are means or centers, c(i) are weights or coefficients, p(x) sets f(x) = 0, G is Gaussian distribution, and W is square matrix. Norm has weights: transpose of (x - t(i)) * transpose of W * W * (x - t(i)).
3-Computer Science-System Analysis
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Date Modified: 2022.0225