Abstract Formulas for finding a weighted least-squares fit to a vector-to-vector transformation are provided in two cases: (1) when the mapping is

Transcription

1 Abstract Formulas for finding a weighted least-squares fit to a vector-to-vector transformation are provided in two cases: ( when the mapping is available as a continuous analtical function on a known domain, and (2 when the mapping function is available onl approximatel through knowledge of a finite collection of input and output sample vector pairs. The formulas in each case are ver similar to each other, and also similar to more familiar linear regression formulas described in elementar linear-algebra textbooks.

2 Continuous and discrete formulas for multi-linear(affine regression R.M. Brannon Jul 24, 20 Introduction A vector transformation (x takes a vector x as input and returns a vector as output. Figure shows a square domain,, in gra. For each x, the mapping function ma be applied to generate a transformed shape. The left side of Fig. shows, in blue, the result of appling a nonlinear transformation to the square domain. The nonlinearit is evident since the deformed (i.e., and vector b. An affine transformation will alwas transform a transformed shape is distorted into curves. An affine transformation is the generalization of the affine scalar equation mx + b; specificall, an affine transformation can alwas be written in the form M x +b, for some second- order tensor M square domain into a parallelogram. In particular, the right-hand side of Fig. depicts a least-squares best-fit of the nonlinear transformation. The goal of this document is to illustrate how to find the best affine fit to a nonlinear vectorto-vector transformation. The formulas will be ver similar to those for fitting nonlinear scalar funtions (x to a straight line. Figure : Left: continuous data. Right: continuous affine fit In numerical work, an analtic continuous mapping function (x might not be available. Instead, a discrete approximate sampling of the mapping might be 2

3 the onl information available to approximatel characterize the transformation. In other words, numerical methods often describe a mapping transformation onl b knowing how, as illustrated in Fig. 2(a, a collection of discrete points in the initial domain transform. Below, we will demonstrate that finding the best fit to this tpe of discrete data is ver similar to the continuous case. The formulas will be ver similar to classical linear regression for fitting to scalar (x, data pairs. Figure 2: Left: Discrete data. Right: continuous affine fit. When finding an approximate affine fit to data, continuous or discrete, ou might wish for the fit to be better near some points of interest. This is accomplished b specifing a weighting function w(x or, in the discrete case, b giving weights w i for each discrete input-output vector pair. The effect of weighting is illustrated in Fig. 3. Figure 3: Left: Equall weighted fit. Right: A fit that is weighted to improve the approximation near the upper-left cusp on the mapped region (with the price being, of course, a poorer fit awa from that cusp. 3

4 2 Problem Statement Suppose that ou have a nonlinear vector-to-vector function, (x, and ou seek a constant tensor M and vector b such that the affine transformation, M x +b is a best fit to the nonlinear function (x over a specified region in space, x. Below, we provide an algorithm corresponding to minimizing the weighted mean square residual, R 2 R 2 (M, b, defined b R 2 r r wdv wdv ( in which dv is the volume element for integration over the input domain x, w w(x is a scalar-valued weighting function (e.g., it could be the mass densit if ou want to have a mass-weighted residual in a mechanics problem, and r r (x ; M, b is the local residual vector field defined b r (x ; M, b (x (M x + b. (2 Note that the local residual vector r depends on the location x and on the unknown (sought tensor M and vector b. Thus, after integration over x, the resulting weighted mean square residual, R 2 depends onl on the unknowns, and ma therefore be minimized in b setting each partial derivative of R 2 with respect to the components of M and b equal to zero. This provides a set of equations solvable for these unknowns. Rather than showing this tedious analsis, this document provides onl the algorithm for the final answer. In addition to providing the algorithm for this continuous fitting exercise, we will also provide ver similar formulas for fitting discrete mapping data. Unlike the continuous fitting problem described above, which presumed that the mapping function (x was known, the discrete problem presumes that ou have onl discrete values of the mapping function s output, {,,... 2 N} corresponding to a finite discrete set of inputs {x, x,..., x }. Rather than 2 N having a continuous weighting function w(x, the discrete multi-linear regression problem presumes that ou have a table of weight values {w, w 2,..., w N } for each discrete point. For the discrete problem, the mean square residual is defined similarl to that in Eq. (, except that the integral is replaced b summation N R 2 r r w i i i N w. (3 i Here, analogous to Eq. (2, the discrete residual vectors {r, r,..., r } are 2 N defined b ( r (M i, b M x + b. (4 i i 4

5 3 Algorithm To find the desired tensor M and vector b, the calculations shown below should be performed in sequence (using the continuous or discrete formula as appropriate. Begin b evaluating the weight of the domain: W wdv w i (5 Next, evaluate the first moment vectors x and, which would be called the centers of mass in mechanics: x W W Evaluate two helper tensors: x wdv W wdv W ( g x W wdv x G ( W ( W ( x x wdv x x W x w i (6 i i (7 iw Then the final answer for the affine mapping tensor M ix w i x i (8 x x w i x ii x (9 is M g G (0 and the final answer for the intercept is b M x ( Note that the affine mapping, M x +b ma be written more intuitivel in the point-slope form, M (x x (2 4 Example Let e and e denote base vectors in a 2D space. Consider a function (x 2 that takes a vector, x x e + x e (

6 as input, and returns a vector e + 2e 2 (4 as output. In particular, consider the mapping function that was used in the graphics of Figs., 2, and 3: ( 2 2x + 5x (3x (x 2 7x (5 2 ( 2 2x 5x (x ( x (6 Carring out the above steps for a uniform weighting function (viz. w(x or w i gives the following results: W (7 {x } {0. 0, } (8 {} { , } (9 [ ] g {{ , 0. 7 }, {0. 2, 0}} (20 [ ] G {{ , 0}, {0, }} (2 [ ] M {{ 0.,. 2}, {0., 0}} (22 Here, some of the digits have been replaced with to allow this problem to be given as a homework assignment in a universit setting. Data files (x.dat,.dat, and w.dat containing the discrete x,, and w values ma be found in an online compressed archive file (2dAffineRegression.zip at the Universit of Utah CSM website. The archive also contains a data file of the non-uniform weights (wstar.dat that were used to generate the results in Fig. 3. 6