# xsample: Randomly samples an underdetermined problem with linear... In limSolve: Solving Linear Inverse Models

## Description

Random sampling of inverse linear problems with linear equality and inequality constraints. Uses either a "hit and run" algorithm (random or coordinate directions) or a mirroring technique for sampling.

The Markov Chain Monte Carlo method produces a sample solution for

Ex=f

Ax~=B

Gx>=h

where Ex=F have to be met exactly, and x is distributed according to p(x)~e^(-.5||W(Ax-B)||^2)

## Usage

 ```1 2 3 4 5``` ```xsample(A = NULL, B = NULL, E = NULL, F =NULL, G = NULL, H = NULL, sdB = NULL, W = 1, iter = 3000, outputlength = iter, burninlength = NULL, type = "mirror", jmp = NULL, tol = sqrt(.Machine\$double.eps), x0 = NULL, fulloutput = FALSE, test = TRUE) ```

## Arguments

 `A ` numeric matrix containing the coefficients of the (approximate) equality constraints, Ax~=B. `B ` numeric vector containing the right-hand side of the (approximate) equality constraints. `E ` numeric matrix containing the coefficients of the (exact) equality constraints, Ex=F. `F ` numeric vector containing the right-hand side of the (exact) equality constraints. `G ` numeric matrix containing the coefficients of the inequality constraints, Gx>=H. `H ` numeric vector containing the right-hand side of the inequality constraints. `sdB ` vector with standard deviation on B. Defaults to `NULL`. `W ` weighting for Ax~=B. Only used if `sdB=NULL` and the problem is overdetermined. In that case, the error of B around the model Ax is estimated based on the residuals of Ax~=B. This error is made proportional to 1/W. If sdB is not NULL, W=diag(sdB^-1). `iter ` integer determining the number of iterations. `outputlength` number of iterations kept in the output; at most equal to `iter`. `burninlength` a number of extra iterations, performed at first, to "warm up" the algorithm. `type ` type of algorithm: one of: "mirror", (mirroring algorithm), "rda" (random directions algorithm) or "cda" (coordinates directions algorithm). `jmp ` jump length of the transformed variables q: x=x0+Zq (only if `type`=="mirror"); if jmp is `NULL`, a reasonable value is determined by xsample, depending on the size of the NULL space. `tol ` tolerance for equality and inequality constraints; numbers whose absolute value is smaller than `tol` are set to zero. `x0 ` initial (particular) solution. `fulloutput ` if `TRUE`, also outputs the transformed variables q. `test ` if `TRUE`, xsample will test for hidden equalities (see details). This may be necessary for large problems, but slows down execution a bit.

## Details

The algorithm proceeds in two steps.

1. the equality constraints Ex=F are eliminated, and the system Ex=f, Gx>=h is rewritten as G(p+Zq)>= h, i.e. containing only inequality constraints and where Z is a basis for the null space of E.

2. the distribution of q is sampled numerically using a random walk (based on the Metropolis algorithm).

There are three algorithms for selecting new samples: `rda`, `cda` (two hit-and-run algorithms) and a novel `mirror` algorithm.

• In the `rda` algorithm first a random direction is selected, and the new sample obtained by uniformly sampling the line connecting the old sample and the intersection with the planes defined by the inequality constraints.

• the `cda` algorithm is similar, except that the direction is chosen along one of the coordinate axes.

• the `mirror` algorithm is yet unpublished; it uses the inequality constraints as "reflecting planes" along which jumps are reflected. In contrast to `cda` and `rda`, this algorithm also works with unbounded problems (i.e. for which some of the unknowns can attain Inf).

For more information, see the package vignette `vignette(xsample)` or the file xsample.pdf in the packages ‘docs’ subdirectory.

Raftery and Lewis (1996) suggest a minimum of 3000 iterations to reach the extremes.

If provided, then `x0` should be a valid particular solution (i.e. E*x0=b and G*x0>=h), else the algorithm will fail.

For larger problems, a central solution may be necessary as a starting point for the `rda` and `cda` algorithms. A good starting value is provided by the "central" value when running the function `xranges` with option `central` equal to `TRUE`.

If the particular solution (`x0`) is not provided, then the parsimonious solution is sought, see `ldei`.

This may however not be the most efficient way to start the algorithm. The parsimonious solution is usually located near the edges, and the `rda` and `cda` algorithms may not get out of this corner. The `mirror` algorithm is insensitive to that. Here it may be even better to start in a corner (as this position will always never be reached by random sampling).

The algorithm will fail if there are hidden equalities. For instance, two inequalities may together impose an equality on an unknown, or, inequalities may impose equalities on a linear combination of two or more unknowns.

In this case, the basis of the null space Z will be deficient. Therefore, `xsample` starts by checking if such hidden equalities exist. If it is suspected that this is NOT the case, set `test` to `FALSE`. This will speed up execution slightly.

It is our experience that for small problems either the `rda` and `cda` algorithms are often more efficient. For really large problems, the `mirror` algorithm is usually much more efficient; select a jump length (`jmp`) that ensures good random coverage, while still keeping the number of reflections reasonable. If unsure about the size of jmp, the default will do.

See `E_coli` for an example where a relatively large problem is sampled.

## Value

a list containing:

 `X ` matrix whose rows contain the sampled values of x. `acceptedratio ` ratio of acceptance (i.e. the ratio of the accepted runs / total iterations). `Q ` only returned if `fulloutput` is `TRUE`: the transformed samples Q. `p ` only returned if `fulloutput` is `TRUE`: probability vector for all samples (e.g. one value for each row of `X`). `jmp ` the jump length used for the random walk. Can be used to check the automated jump length.

## Author(s)

Karel Van den Meersche

Karline Soetaert <[email protected]>

## References

Van den Meersche K, Soetaert K, Van Oevelen D (2009). xsample(): An R Function for Sampling Linear Inverse Problems. Journal of Statistical Software, Code Snippets, 30(1), 1-15.

`Minkdiet`, for a description of the Mink diet example.

`ldei`, to find the least distance solution

`lsei`, to find the least squares solution

`varsample`, to randomly sample variables of an lsei problem.

`varranges`, to estimate ranges of inverse variables.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```#------------------------------------------------------------------------------- # A simple problem #------------------------------------------------------------------------------- # Sample the probability density function of x1,...x4 # subject to: # x1 + x2 + x4 = 3 # x2 -x3 + x4 = -1 # xi > 0 E <- matrix(nrow = 2, byrow = TRUE, data = c(1, 1, 0, 1, 0, 1, -1, 1)) F <- c(3, -1) G <- diag (nrow = 4) H <- rep(0, times = 4) xs <- xsample(E = E, F = F, G = G, H = H) pairs(xs) #------------------------------------------------------------------------------- # Sample the underdetermined Mink diet problem #------------------------------------------------------------------------------- E <- rbind(Minkdiet\$Prey, rep(1, 7)) F <- c(Minkdiet\$Mink, 1) pairs(xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 5000, output = 1000, type = "cda")\$X, main = "Minkdiet 1000 solutions, - cda") ```

limSolve documentation built on Aug. 14, 2017, 3:01 p.m.