| Title: | Solving Linear Inverse Models |
|---|---|
| Description: | Functions that (1) find the minimum/maximum of a linear or quadratic function: min or max (f(x)), where f(x) = ||Ax-b||^2 or f(x) = sum(a_i*x_i) subject to equality constraints Ex=f and/or inequality constraints Gx>=h, (2) sample an underdetermined- or overdetermined system Ex=f subject to Gx>=h, and if applicable Ax~=b, (3) solve a linear system Ax=B for the unknown x. It includes banded and tridiagonal linear systems. |
| Authors: | Karline Soetaert [aut, cre], Karel Van den Meersche [aut], Dick van Oevelen [aut], Charles L. Lawson [ctb] (file inverse.f), Richard J. Hanson [ctb] (file inverse.f), Jack Dongarra [ctb] (files solve.f, inverse.f), Cleve Moler [ctb] (file solve.f) |
| Maintainer: | Karline Soetaert <[email protected]> |
| License: | GPL |
| Version: | 1.5.7.1 |
| Built: | 2024-11-11 04:29:22 UTC |
| Source: | https://github.com/cran/limSolve |
Functions that:
(1.) Find the minimum/maximum of a linear or quadratic function:
min or max (f(x)), where or
subject to equality constraints and/or inequality constraints
.
(2.) Sample an underdetermined- or overdetermined system
subject to , and if applicable .
(3.) Solve a linear system for the unknown x.
Includes banded and tridiagonal linear systems.
The package calls Fortran functions from LINPACK
limSolve is designed for solving linear inverse models (LIM).
These consist of linear equality and, or inequality conditions, which can be solved either by least squares or by linear programming techniques.
Amongst the possible applications are: food web quantification, flux balance analysis (e.g. metabolic networks), compositional estimation, and operations research problems.
The package contains several examples to exemplify its use
Karline Soetaert (Maintainer),
Karel Van den Meersche
Dick van Oevelen
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.
https://www.jstatsoft.org/v30/c01/
Blending, Chemtax, RigaWeb,
E_coli, Minkdiet the examples.
ldei, lsei,linp, ldp,
nnls to solve LIM
xranges, varranges to estimate ranges of
unknowns and variables
xsample, varsample to create a random sample
of unknowns and variables
Solve, Solve.banded, Solve.tridiag,
to solve non-square, banded and tridiagonal linear systems of equations.
resolution row and column resolution of a matrix
package vignette limSolve
## Not run: ## show examples (see respective help pages for details) example(Blending) example(Chemtax) example(E_coli) example(Minkdiet) ## run demos demo("limSolve") ## open the directory with original E_coli input file browseURL(paste(system.file(package="limSolve"), "/doc", sep="")) ## show package vignette with tutorial about xsample vignette("xsample") ## show main package vignette vignette("limSolve") ## End(Not run)## Not run: ## show examples (see respective help pages for details) example(Blending) example(Chemtax) example(E_coli) example(Minkdiet) ## run demos demo("limSolve") ## open the directory with original E_coli input file browseURL(paste(system.file(package="limSolve"), "/doc", sep="")) ## show package vignette with tutorial about xsample vignette("xsample") ## show main package vignette vignette("limSolve") ## End(Not run)
A manufacturer produces a feeding mix for pet animals.
The feed mix contains two nutritive ingredients and one ingredient (filler) to provide bulk.
One kg of feed mix must contain a minimum quantity of each of four nutrients as below:
| Nutrient | A | B | C | D | |
| gram | 80 | 50 | 25 | 5 | |
The ingredients have the following nutrient values and cost
| (gram/kg) | A | B | C | D | Cost/kg | |
| Ingredient 1 | 100 | 50 | 40 | 10 | 40 | |
| Ingredient 2 | 200 | 150 | 10 | - | 60 | |
| Filler | - | - | - | - | 0 | |
The problem is to find the composition of the feeding mix that minimises the production costs subject to the constraints above.
Stated otherwise: what is the optimal amount of ingredients in one kg of feeding mix?
Mathematically this can be estimated by solving a linear programming problem:
subject to
Where the Cost (to be minimised) is given by:
The equality ensures that the sum of the three fractions equals 1:
And the inequalities enforce the nutritional constraints:
and so on
The solution is Ingredient1 (x1) = 0.5909, Ingredient2 (x2)=0.1364 and Filler (x3)=0.2727.
BlendingBlending
A list with matrix G and vector H that contain the inequality
conditions and with vector Cost, defining the cost function.
Columnnames of G or names of Cost are the names of the
ingredients, rownames of G and names of H are the nutrients.
Karline Soetaert <[email protected]>.
linp to solve a linear programming problem.
# Generate the equality condition (sum of ingredients = 1) E <- rep(1, 3) F <- 1 G <- Blending$G H <- Blending$H # add positivity requirement G <- rbind(G, diag(3)) H <- c(H, rep(0, 3)) # 1. Solve the model with linear programming res <- linp(E = t(E), F = F, G = G, H = H, Cost = Blending$Cost) # show results print(c(res$X, Cost = res$solutionNorm)) dotchart(x = as.vector(res$X), labels = colnames(G), main = "Optimal blending with ranges", sub = "using linp and xranges", pch = 16, xlim = c(0, 1)) # 2. Possible ranges of the three ingredients (xr <- xranges(E, F, G, H)) segments(xr[,1], 1:ncol(G), xr[,2], 1:ncol(G)) legend ("topright", pch = c(16, NA), lty = c(NA, 1), legend = c("Minimal cost", "range")) # 3. Random sample of the three ingredients # The inequality that all x > 0 has to be added! xs <- xsample(E = E, F = F, G = G, H = H)$X pairs(xs, main = "Blending, 3000 solutions with xsample") # Cost associated to these random samples Costs <- as.vector(varsample(xs, EqA = Blending$Cost)) hist(Costs) legend("topright", c("Optimal solution", format(res$solutionNorm, digits = 3)))# Generate the equality condition (sum of ingredients = 1) E <- rep(1, 3) F <- 1 G <- Blending$G H <- Blending$H # add positivity requirement G <- rbind(G, diag(3)) H <- c(H, rep(0, 3)) # 1. Solve the model with linear programming res <- linp(E = t(E), F = F, G = G, H = H, Cost = Blending$Cost) # show results print(c(res$X, Cost = res$solutionNorm)) dotchart(x = as.vector(res$X), labels = colnames(G), main = "Optimal blending with ranges", sub = "using linp and xranges", pch = 16, xlim = c(0, 1)) # 2. Possible ranges of the three ingredients (xr <- xranges(E, F, G, H)) segments(xr[,1], 1:ncol(G), xr[,2], 1:ncol(G)) legend ("topright", pch = c(16, NA), lty = c(NA, 1), legend = c("Minimal cost", "range")) # 3. Random sample of the three ingredients # The inequality that all x > 0 has to be added! xs <- xsample(E = E, F = F, G = G, H = H)$X pairs(xs, main = "Blending, 3000 solutions with xsample") # Cost associated to these random samples Costs <- as.vector(varsample(xs, EqA = Blending$Cost)) hist(Costs) legend("topright", c("Optimal solution", format(res$solutionNorm, digits = 3)))
Input files for assessing the algal composition of a field sample, based on the pigment composition of the algal groups (measured in the laboratory) and the pigment composition of the field sample.
In the example there are 8 types of algae:
Prasinophytes
Dinoflagellates
Cryptophytes
Haptophytes type 3 (Hapto3s)
Haptophytes type 4 (Hapto4s)
Chlorophytes
Cynechococcus
Diatoms
and 12 pigments:
Perid = Peridinin,
19But = 19-butanoyloxyfucoxanthin,
Fucox = fucoxanthin,
19Hex = 19-hexanoyloxyfucoxanthin,
Neo = neoxanthin,
Pras = prasinoxanthin,
Viol = violaxanthin,
Allox = alloxanthin,
Lutein = lutein,
Zeax = zeaxanthin,
Chlb = chlorophyll b,
Chla = chlorophyll a
The input data consist of:
the pigment composition of the various algal groups, for instance determined from cultures (the Southern Ocean example -table 4- in Mackey et al., 1996)
the pigment composition of a field sample.
Based on these data, the algal composition of the field sample is estimated, under the assumption that the pigment composition of the field sample is a weighted avertage of the pigment composition of algae present in the sample, where weighting is proportional to their biomass.
As there are more measurements (12 pigments) than unknowns (8 algae), the resulting linear system is overdetermined.
It is thus solved in a least squares sense (using function lsei):
subject to
If there are 2 algae A,B, and 3 pigments 1,2,3 then the 3
approximate equalities () would be:
where and are the (unknown) proportions of algae
A and B in the field sample (S), and
is the relative amount of pigment 1 in alga A, etc...
The equality ensures that the sum of fractions equals 1:
and the inequalities ensure that fractions are positive numbers
It should be noted that in the actual Chemtax programme the problem is solved in a more complex way. In Chemtoz, the A-coefficients are also allowed to vary, while here they are taken as they are (constant). Chemtax then finds the "best fit" by fitting both the fractions, and the non-zero coefficients in A.
ChemtaxChemtax
A list with the input ratio matrix (Ratio) and a vector with the
field data (Field)
The input ratio matrix Ratio contains the pigment compositions
(columns) for each algal group (rows);
the compositions are scaled relative to Chla (last column).
The vector with the Field data contains the pigment composition
of a sample in the field, also scaled relative to Chla;
the pigments are similarly ordened as for the input ratio matrix.
The rownames of matrix Ratio are the algal group names, columnames
of Ratio (=names of Field) are the pigments
Karline Soetaert <[email protected]>.
Mackey MD, Mackey DJ, Higgins HW, Wright SW, 1996. CHEMTAX - A program for estimating class abundances from chemical markers: Application to HPLC measurements of phytoplankton. Marine Ecology-Progress Series 144 (1-3): 265-283.
Van den Meersche, K., Soetaert, K., Middelburg, J., 2008. A Bayesian compositional estimator for microbial taxonomy based on biomarkers. Limnology and Oceanography Methods, 6, 190-199.
R-package BCE
lsei, the function to solve for the algal composition of the
field sample.
# 1. Graphical representation of the chemtax example input data palette(rainbow(12, s = 0.6, v = 0.75)) mp <- apply(Chemtax$Ratio, MARGIN = 2, max) pstars <- rbind(t(t(Chemtax$Ratio)/mp) , sample = Chemtax$Field/max(Chemtax$Field)) stars(pstars, len = 0.9, key.loc = c(7.2, 1.7),scale=FALSE,ncol=4, main = "CHEMTAX pigment composition", draw.segments = TRUE, flip.labels=FALSE) # 2. Estimating the algal composition of the field sample Nx <-nrow(Chemtax$Ratio) # equations that have to be met exactly Ex=f: # sum of all fraction must be equal to 1. EE <- rep(1, Nx) FF <- 1 # inequalities, Gx>=h: # all fractions must be positive numbers GG <- diag(nrow = Nx) HH <- rep(0, Nx) # equations that must be reproduced as close as possible, Ax ~ b # = the field data; the input ratio matrix and field data are rescaled AA <- Chemtax$Ratio/rowSums(Chemtax$Ratio) BB <- Chemtax$Field/sum(Chemtax$Field) # 1. Solve with lsei method X <- lsei(t(AA), BB, EE, FF, GG, HH)$X (Sample <- data.frame(Algae = rownames(Chemtax$Ratio), fraction = X)) # plot results barplot(X, names = rownames(Chemtax$Ratio), col = heat.colors(8), cex.names = 0.8, main = "Chemtax example solved with lsei") # 2. Bayesian sampling; # The standard deviation on the field data is assumed to be 0.01 # jump length not too large or NO solutions are found! xs <- xsample(t(AA), BB, EE, FF, GG, HH, sdB = 0.01, jmp = 0.025)$X pairs(xs, main= "Chemtax, Bayesian sample")# 1. Graphical representation of the chemtax example input data palette(rainbow(12, s = 0.6, v = 0.75)) mp <- apply(Chemtax$Ratio, MARGIN = 2, max) pstars <- rbind(t(t(Chemtax$Ratio)/mp) , sample = Chemtax$Field/max(Chemtax$Field)) stars(pstars, len = 0.9, key.loc = c(7.2, 1.7),scale=FALSE,ncol=4, main = "CHEMTAX pigment composition", draw.segments = TRUE, flip.labels=FALSE) # 2. Estimating the algal composition of the field sample Nx <-nrow(Chemtax$Ratio) # equations that have to be met exactly Ex=f: # sum of all fraction must be equal to 1. EE <- rep(1, Nx) FF <- 1 # inequalities, Gx>=h: # all fractions must be positive numbers GG <- diag(nrow = Nx) HH <- rep(0, Nx) # equations that must be reproduced as close as possible, Ax ~ b # = the field data; the input ratio matrix and field data are rescaled AA <- Chemtax$Ratio/rowSums(Chemtax$Ratio) BB <- Chemtax$Field/sum(Chemtax$Field) # 1. Solve with lsei method X <- lsei(t(AA), BB, EE, FF, GG, HH)$X (Sample <- data.frame(Algae = rownames(Chemtax$Ratio), fraction = X)) # plot results barplot(X, names = rownames(Chemtax$Ratio), col = heat.colors(8), cex.names = 0.8, main = "Chemtax example solved with lsei") # 2. Bayesian sampling; # The standard deviation on the field data is assumed to be 0.01 # jump length not too large or NO solutions are found! xs <- xsample(t(AA), BB, EE, FF, GG, HH, sdB = 0.01, jmp = 0.025)$X pairs(xs, main= "Chemtax, Bayesian sample")
Input matrices and vectors for performing Flux Balance Analysis of the E.coli metabolism
(as from http://gcrg.ucsd.edu/Downloads/Flux_Balance_Analysis).
The original input file can be found in the package subdirectory
/inst/docs/E_coli.input
There are 53 substances:
GLC, G6P, F6P, FDP, T3P2, T3P1, 13PDG, 3PG, 2PG, PEP, PYR, ACCOA, CIT, ICIT, AKG, SUCCOA, SUCC, FUM, MAL, OA, ACTP, ETH, AC, LAC, FOR, D6PGL, D6PGC, RL5P, X5P, R5P, S7P, E4P, RIB, GLX, NAD, NADH, NADP, NADPH, HEXT, Q, FAD, FADH, AMP, ADP, ATP, GL3P, CO2, PI, PPI, O2, COA, GL, QH2
and 13 externals:
Biomass, GLCxt, GLxt, RIBxt, ACxt, LACxt, FORxt, ETHxt, SUCCxt, PYRxt, PIxt, O2xt, CO2xt
There are 70 unknown reactions (named by the gene encoding for it):
GLK1, PGI1, PFKA, FBP, FBA, TPIA, GAPA, PGK, GPMA, ENO, PPSA, PYKA, ACEE, ZWF, PGL, GND, RPIA, RPE, TKTA1, TKTA2, TALA, GLTA, ACNA, ICDA, SUCA, SUCC1, SDHA1, FRDA, FUMA, MDH, DLD1, ADHE2, PFLA, PTA, ACKA, ACS, PCKA, PPC, MAEB, SFCA, ACEA, ACEB, PPA, GLPK, GPSA1, RBSK, NUOA, FDOH, GLPD, CYOA, SDHA2, PNT1A, PNT2A, ATPA, GLCUP, GLCPTS, GLUP, RIBUP, ACUP, LACUP, FORUP, ETHUP, SUCCUP, PYRUP, PIUP, O2TX, CO2TX, ATPM, ADK, Growth
The lsei model contains:
54 equalities (Ax=B): the 53 mass balances (one for each substance) and one equation that sets the ATP drain flux for constant maintenance requirements to a fixed value (5.87)
70 unknowns (x), the reaction rates
62 inequalities (Gx>h). The first 28 inequalities impose bounds on some reactions. The last 34 inequalities impose that the reaction rates have to be positive (for unidirectional reactions only).
1 function that has to be maximised, the biomass production (growth).
As there are more unknowns (70) than equations (54), there exist an infinite amount of solutions (it is an underdetermined problem).
E_coliE_coli
A list with the matrices and vectors that constitute the mass balance problem:
A, B, G and H and
Maximise, with the function to maximise.
The columnames of A and G are the names of the unknown
reaction rates;
The first 53 rownames of A give the names of the components
(these rows consitute the mass balance equations).
Karline Soetaert <[email protected]>
originated from the urlhttp://gcrg.ucsd.edu/Downloads/Flux_Balance_Analysis
Edwards,J.S., Covert, M., and Palsson, B., (2002) Metabolic Modeling of Microbes: the Flux Balance Approach, Environmental Microbiology, 4(3): pp. 133-140.
# 1. parsimonious (simplest) solution pars <- lsei(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H)$X # 2. the optimal solution - solved with linear programming # some unknowns can be negative LP <- linp(E = E_coli$A, F = E_coli$B,G = E_coli$G, H = E_coli$H, Cost = -E_coli$Maximise, ispos = FALSE) (Optimal <- LP$X) # 3.ranges of all unknowns, including the central value and all solutions xr <- xranges(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H, central = TRUE, full = TRUE) # the central point is a valid solution: X <- xr[ ,"central"] max(abs(E_coli$A%*%X - E_coli$B)) min(E_coli$G%*%X - E_coli$H) # 4. Sample solution space; the central value is a good starting point # for algorithms cda and rda - but these need many iterations ## Not run: xs <- xsample(E = E_coli$A, F = E_coli$B, G = E_coli$G,H = E_coli$H, iter = 50000, out = 5000, type = "rda", x0 = X)$X pairs(xs[ ,10:20], pch = ".", cex = 2, main = "sampling, using rda") ## End(Not run) # using mirror algorithm takes less iterations, # but an iteration takes more time ; it is better to start in a corner... # (i.e. no need to use X as starting value) xs <- xsample(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H, iter = 2000, out = 500, jmp = 50, type = "mirror")$X pairs(xs[ ,10:20], pch = ".", cex = 2, main = "sampling, using mirror") # Print results: data.frame(pars = pars, Optimal = Optimal, xr[ ,1:2], Mean = colMeans(xs), sd = apply(xs, 2, sd)) # Plot results par(mfrow = c(1, 2)) nr <- length(Optimal)/2 ii <- 1:nr dotchart(Optimal[ii], xlim = range(xr), pch = 16) segments(xr[ii,1], 1:nr, xr[ii,2], 1:nr) ii <- (nr+1):length(Optimal) dotchart(Optimal[ii], xlim = range(xr), pch = 16) segments(xr[ii,1], 1:nr, xr[ii,2], 1:nr) mtext(side = 3, cex = 1.5, outer = TRUE, line = -1.5, "E coli Core Metabolism, optimal solution and ranges")# 1. parsimonious (simplest) solution pars <- lsei(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H)$X # 2. the optimal solution - solved with linear programming # some unknowns can be negative LP <- linp(E = E_coli$A, F = E_coli$B,G = E_coli$G, H = E_coli$H, Cost = -E_coli$Maximise, ispos = FALSE) (Optimal <- LP$X) # 3.ranges of all unknowns, including the central value and all solutions xr <- xranges(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H, central = TRUE, full = TRUE) # the central point is a valid solution: X <- xr[ ,"central"] max(abs(E_coli$A%*%X - E_coli$B)) min(E_coli$G%*%X - E_coli$H) # 4. Sample solution space; the central value is a good starting point # for algorithms cda and rda - but these need many iterations ## Not run: xs <- xsample(E = E_coli$A, F = E_coli$B, G = E_coli$G,H = E_coli$H, iter = 50000, out = 5000, type = "rda", x0 = X)$X pairs(xs[ ,10:20], pch = ".", cex = 2, main = "sampling, using rda") ## End(Not run) # using mirror algorithm takes less iterations, # but an iteration takes more time ; it is better to start in a corner... # (i.e. no need to use X as starting value) xs <- xsample(E = E_coli$A, F = E_coli$B, G = E_coli$G, H = E_coli$H, iter = 2000, out = 500, jmp = 50, type = "mirror")$X pairs(xs[ ,10:20], pch = ".", cex = 2, main = "sampling, using mirror") # Print results: data.frame(pars = pars, Optimal = Optimal, xr[ ,1:2], Mean = colMeans(xs), sd = apply(xs, 2, sd)) # Plot results par(mfrow = c(1, 2)) nr <- length(Optimal)/2 ii <- 1:nr dotchart(Optimal[ii], xlim = range(xr), pch = 16) segments(xr[ii,1], 1:nr, xr[ii,2], 1:nr) ii <- (nr+1):length(Optimal) dotchart(Optimal[ii], xlim = range(xr), pch = 16) segments(xr[ii,1], 1:nr, xr[ii,2], 1:nr) mtext(side = 3, cex = 1.5, outer = TRUE, line = -1.5, "E coli Core Metabolism, optimal solution and ranges")
Solves the following underdetermined inverse problem:
subject to
uses least distance programming subroutine ldp (FORTRAN) from Linpack
The model has to be UNDERdetermined, i.e. the number of independent equations < number of unknowns.
ldei(E, F, G = NULL, H = NULL, tol = sqrt(.Machine$double.eps), verbose = TRUE, lower = NULL, upper = NULL)ldei(E, F, G = NULL, H = NULL, tol = sqrt(.Machine$double.eps), verbose = TRUE, lower = NULL, upper = NULL)
E |
numeric matrix containing the coefficients of the equality
constraints |
F |
numeric vector containing the right-hand side of the equality constraints. |
G |
numeric matrix containing the coefficients of the inequality
constraints |
H |
numeric vector containing the right-hand side of the inequality constraints. |
tol |
tolerance (for singular value decomposition, equality and inequality constraints). |
verbose |
logical to print warnings and messages. |
upper, lower
|
vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H. |
a list containing:
X |
vector containing the solution of the least distance with equalities and inequalities problem. |
unconstrained.solution |
vector containing the unconstrained solution
of the least distance problem, i.e. ignoring |
residualNorm |
scalar, the sum of absolute values of residuals of equalities and violated inequalities; should be zero or very small if the problem is feasible. |
solutionNorm |
scalar, the value of the quadratic function at the
solution, i.e. the value of |
IsError |
logical, |
type |
the string "ldei", such that how the solution was obtained can be traced. |
numiter |
the number of iterations. |
One of the steps in the ldei algorithm is the creation of an orthogonal basis, constructed by Singular Value Decomposition. As this makes use of random numbers, it may happen - for problems that are difficult to solve - that ldei sometimes finds a solution or fails to find one for the same problem, depending on the random numbers used to create the orthogonal basis. If it is suspected that this is happening, trying a few times may find a solution. (example RigaWeb is such a problem).
Karline Soetaert <[email protected]>.
Lawson C.L.and Hanson R.J. 1974. Solving Least Squares Problems, Prentice-Hall
Lawson C.L.and Hanson R.J. 1995. Solving Least Squares Problems. SIAM classics in applied mathematics, Philadelphia. (reprint of book)
Minkdiet, for a description of the Mink diet example.
#------------------------------------------------------------------------------- # A simple problem #------------------------------------------------------------------------------- # minimise x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 # subject to: #-x1 + x4 + x5 = 0 # - x2 - x4 + x6 = 0 # x1 + x2 + x3 > 1 # x3 + x5 + x6 < 1 # xi > 0 E <- matrix(nrow = 2, byrow = TRUE, data = c(-1, 0, 0, 1, 1, 0, 0,-1, 0, -1, 0, 1)) F <- c(0, 0) G <- matrix(nrow = 2, byrow = TRUE, data = c(1, 1, 1, 0, 0, 0, 0, 0, -1, 0, -1, -1)) H <- c(1, -1) ldei(E, F, G, H) #------------------------------------------------------------------------------- # Imposing bounds #------------------------------------------------------------------------------- ldei(E, F, G, H, lower = 0.25) ldei(E, F, G, H, lower = c(0.25, 0.25, 0.25, NA, NA, 0.5)) #------------------------------------------------------------------------------- # parsimonious (simplest) solution of the mink diet problem #------------------------------------------------------------------------------- E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) parsimonious <- ldei(E, F, G = diag(7), H = rep(0, 7)) data.frame(food = colnames(Minkdiet$Prey), fraction = parsimonious$X) dotchart(x = as.vector(parsimonious$X), labels = colnames(Minkdiet$Prey), main = "Diet composition of Mink extimated using ldei", xlab = "fraction")#------------------------------------------------------------------------------- # A simple problem #------------------------------------------------------------------------------- # minimise x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 # subject to: #-x1 + x4 + x5 = 0 # - x2 - x4 + x6 = 0 # x1 + x2 + x3 > 1 # x3 + x5 + x6 < 1 # xi > 0 E <- matrix(nrow = 2, byrow = TRUE, data = c(-1, 0, 0, 1, 1, 0, 0,-1, 0, -1, 0, 1)) F <- c(0, 0) G <- matrix(nrow = 2, byrow = TRUE, data = c(1, 1, 1, 0, 0, 0, 0, 0, -1, 0, -1, -1)) H <- c(1, -1) ldei(E, F, G, H) #------------------------------------------------------------------------------- # Imposing bounds #------------------------------------------------------------------------------- ldei(E, F, G, H, lower = 0.25) ldei(E, F, G, H, lower = c(0.25, 0.25, 0.25, NA, NA, 0.5)) #------------------------------------------------------------------------------- # parsimonious (simplest) solution of the mink diet problem #------------------------------------------------------------------------------- E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) parsimonious <- ldei(E, F, G = diag(7), H = rep(0, 7)) data.frame(food = colnames(Minkdiet$Prey), fraction = parsimonious$X) dotchart(x = as.vector(parsimonious$X), labels = colnames(Minkdiet$Prey), main = "Diet composition of Mink extimated using ldei", xlab = "fraction")
Solves the following inverse problem:
subject to
uses least distance programming subroutine ldp (FORTRAN) from Linpack
ldp(G, H, tol = sqrt(.Machine$double.eps), verbose = TRUE, lower = NULL, upper = NULL)ldp(G, H, tol = sqrt(.Machine$double.eps), verbose = TRUE, lower = NULL, upper = NULL)
G |
numeric matrix containing the coefficients of the inequality
constraints |
H |
numeric vector containing the right-hand side of the inequality constraints. |
tol |
tolerance (for inequality constraints). |
verbose |
logical to print warnings and messages. |
upper, lower
|
vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H. |
a list containing:
X |
vector containing the solution of the least distance problem. |
residualNorm |
scalar, the sum of absolute values of residuals of violated inequalities; should be zero or very small if the problem is feasible. |
solutionNorm |
scalar, the value of the quadratic function at the
solution, i.e. the value of |
IsError |
logical, |
type |
the string "ldp", such that how the solution was obtained can be traced. |
numiter |
the number of iterations. |
Karline Soetaert <[email protected]>
Lawson C.L.and Hanson R.J. 1974. Solving Least Squares Problems, Prentice-Hall
Lawson C.L.and Hanson R.J. 1995. Solving Least Squares Problems. SIAM classics in applied mathematics, Philadelphia. (reprint of book)
ldei, which includes equalities.
# parsimonious (simplest) solution G <- matrix(nrow = 2, ncol = 2, data = c(3, 2, 2, 4)) H <- c(3, 2) ldp(G, H) # imposing bounds on the first unknown ldp(G, H, lower = c(1, NA))# parsimonious (simplest) solution G <- matrix(nrow = 2, ncol = 2, data = c(3, 2, 2, 4)) H <- c(3, 2) ldp(G, H) # imposing bounds on the first unknown ldp(G, H, lower = c(1, NA))
Solves a linear programming problem,
subject to
(optional)
This function provides a wrapper around lp (see note)
from package lpSolve, written to be consistent with the functions
lsei, and ldei.
It allows for the x's to be negative (not standard in lp).
linp(E = NULL, F = NULL, G = NULL, H = NULL, Cost, ispos = TRUE, int.vec = NULL, verbose = TRUE, lower = NULL, upper = NULL, ...)linp(E = NULL, F = NULL, G = NULL, H = NULL, Cost, ispos = TRUE, int.vec = NULL, verbose = TRUE, lower = NULL, upper = NULL, ...)
E |
numeric matrix containing the coefficients of the equality
constraints |
F |
numeric vector containing the right-hand side of the equality constraints. |
G |
numeric matrix containing the coefficients of the inequality
constraints |
H |
numeric vector containing the right-hand side of the inequality constraints. |
Cost |
numeric vector containing the coefficients of the cost function;
if |
ispos |
logical, when |
int.vec |
when not |
verbose |
logical to print warnings and messages. |
upper, lower
|
vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H. |
... |
extra arguments passed to R-function |
a list containing:
X |
vector containing the solution of the linear programming problem. |
residualNorm |
scalar, the sum of absolute values of residuals of equalities and violated inequalities. Should be very small or zero for a feasible linear programming problem. |
solutionNorm |
scalar, the value of the minimised |
IsError |
logical, |
type |
the string "linp", such that how the solution was obtained can be traced. |
If the requirement of nonnegativity are relaxed, then strictly speaking the problem is not a linear programming problem.
The function lp may fail and terminate R for very small problems that
are repeated frequently...
Also note that sometimes multiple solutions exist for the same problem.
Karline Soetaert <[email protected]>
Michel Berkelaar and others (2007). lpSolve: Interface to Lpsolve v. 5.5 to solve linear or integer programs. R package version 5.5.8.
lp the original function from package lpSolve
Blending, a linear programming problem.
#------------------------------------------------------------------------------- # Linear programming problem 1, not feasible #------------------------------------------------------------------------------- # maximise x1 + 3*x2 # subject to #-x1 -x2 < -3 #-x1 + x2 <-1 # x1 + 2*x2 < 2 # xi > 0 G <- matrix(nrow = 3, data = c(-1, -1, 1, -1, 1, 2)) H <- c(3, -1, 2) Cost <- c(-1, -3) (L <- linp(E = NULL, F = NULL, Cost = Cost, G = G, H = H)) L$residualNorm #------------------------------------------------------------------------------- # Linear programming problem 2, feasible #------------------------------------------------------------------------------- # minimise x1 + 8*x2 + 9*x3 + 2*x4 + 7*x5 + 3*x6 # subject to: #-x1 + x4 + x5 = 0 # - x2 - x4 + x6 = 0 # x1 + x2 + x3 > 1 # x3 + x5 + x6 < 1 # xi > 0 E <- matrix(nrow = 2, byrow = TRUE, data = c(-1, 0, 0, 1, 1, 0, 0,-1, 0, -1, 0, 1)) F <- c(0, 0) G <- matrix(nrow = 2, byrow = TRUE, data = c(1, 1, 1, 0, 0, 0, 0, 0, -1, 0, -1, -1)) H <- c(1, -1) Cost <- c(1, 8, 9, 2, 7, 3) (L <- linp(E = E, F = F, Cost = Cost, G = G, H = H)) L$residualNorm # Including a lower bound: linp(E = E, F = F, Cost = Cost, G = G, H = H, lower = 0.25) #------------------------------------------------------------------------------- # Linear programming problem 3, no positivity #------------------------------------------------------------------------------- # minimise x1 + 2x2 -x3 +4 x4 # subject to: # 3x1 + 2x2 + x3 + x4 = 2 # x1 + x2 + x3 + x4 = 2 # 2x1 + x2 + x3 + x4 >=-1 # -x1 + 3x2 +2x3 + x4 >= 2 # -x1 + x3 >= 1 E <- matrix(ncol = 4, byrow = TRUE, data =c(3, 2, 1, 4, 1, 1, 1, 1)) F <- c(2, 2) G <- matrix(ncol = 4, byrow = TRUE, data = c(2, 1, 1, 1, -1, 3, 2, 1, -1, 0, 1, 0)) H <- c(-1, 2, 1) Cost <- c(1, 2, -1, 4) linp(E = E, F = F, G = G, H = H, Cost, ispos = FALSE)#------------------------------------------------------------------------------- # Linear programming problem 1, not feasible #------------------------------------------------------------------------------- # maximise x1 + 3*x2 # subject to #-x1 -x2 < -3 #-x1 + x2 <-1 # x1 + 2*x2 < 2 # xi > 0 G <- matrix(nrow = 3, data = c(-1, -1, 1, -1, 1, 2)) H <- c(3, -1, 2) Cost <- c(-1, -3) (L <- linp(E = NULL, F = NULL, Cost = Cost, G = G, H = H)) L$residualNorm #------------------------------------------------------------------------------- # Linear programming problem 2, feasible #------------------------------------------------------------------------------- # minimise x1 + 8*x2 + 9*x3 + 2*x4 + 7*x5 + 3*x6 # subject to: #-x1 + x4 + x5 = 0 # - x2 - x4 + x6 = 0 # x1 + x2 + x3 > 1 # x3 + x5 + x6 < 1 # xi > 0 E <- matrix(nrow = 2, byrow = TRUE, data = c(-1, 0, 0, 1, 1, 0, 0,-1, 0, -1, 0, 1)) F <- c(0, 0) G <- matrix(nrow = 2, byrow = TRUE, data = c(1, 1, 1, 0, 0, 0, 0, 0, -1, 0, -1, -1)) H <- c(1, -1) Cost <- c(1, 8, 9, 2, 7, 3) (L <- linp(E = E, F = F, Cost = Cost, G = G, H = H)) L$residualNorm # Including a lower bound: linp(E = E, F = F, Cost = Cost, G = G, H = H, lower = 0.25) #------------------------------------------------------------------------------- # Linear programming problem 3, no positivity #------------------------------------------------------------------------------- # minimise x1 + 2x2 -x3 +4 x4 # subject to: # 3x1 + 2x2 + x3 + x4 = 2 # x1 + x2 + x3 + x4 = 2 # 2x1 + x2 + x3 + x4 >=-1 # -x1 + 3x2 +2x3 + x4 >= 2 # -x1 + x3 >= 1 E <- matrix(ncol = 4, byrow = TRUE, data =c(3, 2, 1, 4, 1, 1, 1, 1)) F <- c(2, 2) G <- matrix(ncol = 4, byrow = TRUE, data = c(2, 1, 1, 1, -1, 3, 2, 1, -1, 0, 1, 0)) H <- c(-1, 2, 1) Cost <- c(1, 2, -1, 4) linp(E = E, F = F, G = G, H = H, Cost, ispos = FALSE)
Solves an lsei inverse problem (Least Squares with Equality and Inequality Constraints)
subject to
Uses either subroutine lsei (FORTRAN) from the LINPACK package, or
solve.QP from R-package quadprog.
In case the equality constraints cannot be satisfied, a
generalized inverse solution residual vector length is obtained for .
This is the minimal length possible for .
lsei (A = NULL, B = NULL, E = NULL, F = NULL, G = NULL, H = NULL, Wx = NULL, Wa = NULL, type = 1, tol = sqrt(.Machine$double.eps), tolrank = NULL, fulloutput = FALSE, verbose = TRUE, lower = NULL, upper = NULL)lsei (A = NULL, B = NULL, E = NULL, F = NULL, G = NULL, H = NULL, Wx = NULL, Wa = NULL, type = 1, tol = sqrt(.Machine$double.eps), tolrank = NULL, fulloutput = FALSE, verbose = TRUE, lower = NULL, upper = NULL)
A |
numeric matrix containing the coefficients of the quadratic
function to be minimised, |
B |
numeric vector containing the right-hand side of the quadratic function to be minimised. |
E |
numeric matrix containing the coefficients of the equality
constraints, |
F |
numeric vector containing the right-hand side of the equality constraints. |
G |
numeric matrix containing the coefficients of the inequality
constraints, |
H |
numeric vector containing the right-hand side of the inequality constraints. |
Wx |
numeric vector with weighting coefficients of unknowns (length = number of unknowns). |
Wa |
numeric vector with weighting coefficients of the quadratic function (Ax-B) to be minimised (length = number of number of rows of A). |
type |
integer code determining algorithm to use 1= |
tol |
tolerance (for singular value decomposition, equality and inequality constraints). |
tolrank |
only used if |
fulloutput |
if |
verbose |
logical to print warnings and messages. |
upper, lower
|
vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H. |
a list containing:
X |
vector containing the solution of the least squares problem. |
residualNorm |
scalar, the sum of absolute values of residuals of equalities and violated inequalities. |
solutionNorm |
scalar, the value of the minimised quadratic function
at the solution, i.e. the value of |
IsError |
logical, |
type |
the string "lsei", such that how the solution was obtained can be traced. |
covar |
covariance matrix of the solution; only returned if
|
RankEq |
rank of the equality constraint matrix.; only returned if
|
RankApp |
rank of the reduced least squares problem (approximate
equations); only returned if |
See comments in the original code for more details; these comments are included in the ‘docs’ subroutine of the package.
Karline Soetaert <[email protected]>
K. H. Haskell and R. J. Hanson, An algorithm for linear least squares problems with equality and nonnegativity constraints, Report SAND77-0552, Sandia Laboratories, June 1978.
K. H. Haskell and R. J. Hanson, Selected algorithms for the linearly constrained least squares problem - a users guide, Report SAND78-1290, Sandia Laboratories,August 1979.
K. H. Haskell and R. J. Hanson, An algorithm for linear least squares problems with equality and nonnegativity constraints, Mathematical Programming 21 (1981), pp. 98-118.
R. J. Hanson and K. H. Haskell, Two algorithms for the linearly constrained least squares problem, ACM Transactions on Mathematical Software, September 1982.
Berwin A. Turlach R and Andreas Weingessel (2007). quadprog: Functions to solve Quadratic Programming Problems. R package version 1.4-11. S original by Berwin A. Turlach R port by Andreas Weingessel.
solve.QR the original function from package quadprog.
# ------------------------------------------------------------------------------ # example 1: polynomial fitting # ------------------------------------------------------------------------------ x <- 1:5 y <- c(9, 8, 6, 7, 5) plot(x, y, main = "Polynomial fitting, using lsei", cex = 1.5, pch = 16, ylim = c(4, 10)) # 1-st order A <- cbind(rep(1, 5), x) B <- y cf <- lsei(A, B)$X abline(coef = cf) # 2-nd order A <- cbind(A, x^2) cf <- lsei(A, B)$X curve(cf[1] + cf[2]*x + cf[3]*x^2, add = TRUE, lty = 2) # 3-rd order A <- cbind(A, x^3) cf <- lsei(A, B)$X curve(cf[1] + cf[2]*x + cf[3]*x^2 + cf[4]*x^3, add = TRUE, lty = 3) # 4-th order A <- cbind(A, x^4) cf <- lsei(A, B)$X curve(cf[1] + cf[2]*x + cf[3]*x^2 + cf[4]*x^3 + cf[5]*x^4, add = TRUE, lty = 4) legend("bottomleft", c("1st-order", "2nd-order","3rd-order","4th-order"), lty = 1:4) # ------------------------------------------------------------------------------ # example 2: equalities, approximate equalities and inequalities # ------------------------------------------------------------------------------ A <- matrix(nrow = 4, ncol = 3, data = c(3, 1, 2, 0, 2, 0, 0, 1, 1, 0, 2, 0)) B <- c(2, 1, 8, 3) E <- c(0, 1, 0) F <- 3 G <- matrix(nrow = 2, ncol = 3, byrow = TRUE, data = c(-1, 2, 0, 1, 0, -1)) H <- c(-3, 2) lsei(E = E, F = F, A = A, B = B, G = G, H = H) # ------------------------------------------------------------------------------ # example 3: bounds added # ------------------------------------------------------------------------------ lsei(E = E, F = F, A = A, B = B, G = G, H = H, lower = c(2, NA, 0)) lsei(E = E, F = F, A = A, B = B, G = G, H = H, upper = 2)# ------------------------------------------------------------------------------ # example 1: polynomial fitting # ------------------------------------------------------------------------------ x <- 1:5 y <- c(9, 8, 6, 7, 5) plot(x, y, main = "Polynomial fitting, using lsei", cex = 1.5, pch = 16, ylim = c(4, 10)) # 1-st order A <- cbind(rep(1, 5), x) B <- y cf <- lsei(A, B)$X abline(coef = cf) # 2-nd order A <- cbind(A, x^2) cf <- lsei(A, B)$X curve(cf[1] + cf[2]*x + cf[3]*x^2, add = TRUE, lty = 2) # 3-rd order A <- cbind(A, x^3) cf <- lsei(A, B)$X curve(cf[1] + cf[2]*x + cf[3]*x^2 + cf[4]*x^3, add = TRUE, lty = 3) # 4-th order A <- cbind(A, x^4) cf <- lsei(A, B)$X curve(cf[1] + cf[2]*x + cf[3]*x^2 + cf[4]*x^3 + cf[5]*x^4, add = TRUE, lty = 4) legend("bottomleft", c("1st-order", "2nd-order","3rd-order","4th-order"), lty = 1:4) # ------------------------------------------------------------------------------ # example 2: equalities, approximate equalities and inequalities # ------------------------------------------------------------------------------ A <- matrix(nrow = 4, ncol = 3, data = c(3, 1, 2, 0, 2, 0, 0, 1, 1, 0, 2, 0)) B <- c(2, 1, 8, 3) E <- c(0, 1, 0) F <- 3 G <- matrix(nrow = 2, ncol = 3, byrow = TRUE, data = c(-1, 2, 0, 1, 0, -1)) H <- c(-3, 2) lsei(E = E, F = F, A = A, B = B, G = G, H = H) # ------------------------------------------------------------------------------ # example 3: bounds added # ------------------------------------------------------------------------------ lsei(E = E, F = F, A = A, B = B, G = G, H = H, lower = c(2, NA, 0)) lsei(E = E, F = F, A = A, B = B, G = G, H = H, upper = 2)
Input data for assessing the diet composition of mink in southeast Alaska, using C and N isotope ratios (d13C and d15N).
The data consist of
the input matrix Prey, which contains the C (1st row) and N
(2nd row) isotopic values of the prey items (columns), corrected for
fractionation.
the input vector Mink, with the C and N isotopic value of
the predator, mink
There are seven prey items as food sources:
fish
mussels
crabs
shrimp
rodents
amphipods
ducks
The d13C and d15N for each of these prey items, and for mink (the predator) was assessed. The isotopic values of the preys were corrected for fractionation.
The problem is to find the diet composition of mink, e.g. the fraction of each of these food items in the diet.
Mathematically this is by solving an lsei (least squares with equalities
and inequalities) problem: subject to .
The equalities :
and inequalities :
are solved for p1,p2,...p7.
The first two equations calculate the isotopic ratio of the consumer (Mink) as a weighted average of the ratio of the food sources
Equation 3 assures that the sum of all fraction equals 1.
As there are 7 unknowns and only 3 equations, the model is UNDERdetermined, i.e. there exist an infinite amount of solutions.
This model can be solved by various techniques:
least distance programming will select the "simplest" solution.
See ldei.
the remaining uncertainty ranges of the fractions can be estimated
using linear programming. See xranges
the statistical distribution of the fractions can be estimated using
an MCMC algorithm which takes a sample of the solution space.
See xsample
MinkdietMinkdiet
a list with matrix Prey and vector Mink.
Prey contains the isotopic composition (13C and 15N) of the
7 possible food items of Mink
Mink contains the isotopic composition (13C and 15N) of Mink
columnnames of Prey are the food items, rownames of Prey
(=names of Mink) are the names of the isotopic elements.
Karline Soetaert <[email protected]>
Ben-David M, Hanley TA, Klein DR, Schell DM (1997) Seasonal changes in diets of coastal and riverine mink: the role of spawning Pacific salmon. Canadian Journal of Zoology 75:803-811.
ldei to solve for the parsimonious solution
xranges to solve for the uncertainty ranges
xsample to sample the solution space
# 1. visualisation of the data plot(t(Minkdiet$Prey), xlim = c(-25, -13), xlab = "d13C", ylab = "d15N", main = "Minkdiet", sub = "Ben-David et al. (1979)") text(t(Minkdiet$Prey)-0.1, colnames(Minkdiet$Prey)) points(t(Minkdiet$Mink), pch = 16, cex = 2) text(t(Minkdiet$Mink)-0.15, "MINK", cex = 1.2) legend("bottomright", pt.cex = c(1, 2), pch = c(1, 16), c("food", "predator")) # 2. Generate the food web model input matrices # the equalities: E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) # the inequalities (all pi>0) G <- diag(7) H <- rep(0, 7) # 3. Select the parsimonious (simplest) solution parsimonious <- ldei(E, F, G = G, H = H) # 4. show results data.frame(food = colnames(Minkdiet$Prey), fraction = parsimonious$X) dotchart(x = as.vector(parsimonious$X), labels = colnames(Minkdiet$A), main = "Estimated diet composition of Mink", sub = "using ldei and xranges", pch = 16) # 5. Ranges of diet composition iso <- xranges(E, F, ispos = TRUE) segments(iso[,1], 1:ncol(E), iso[,2], 1:ncol(E)) legend ("topright", pch = c(16, NA), lty = c(NA, 1), legend = c("parsimonious", "range")) pairs (xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 1000)$X, main = "Minkdiet 1000 solutions, using xsample")# 1. visualisation of the data plot(t(Minkdiet$Prey), xlim = c(-25, -13), xlab = "d13C", ylab = "d15N", main = "Minkdiet", sub = "Ben-David et al. (1979)") text(t(Minkdiet$Prey)-0.1, colnames(Minkdiet$Prey)) points(t(Minkdiet$Mink), pch = 16, cex = 2) text(t(Minkdiet$Mink)-0.15, "MINK", cex = 1.2) legend("bottomright", pt.cex = c(1, 2), pch = c(1, 16), c("food", "predator")) # 2. Generate the food web model input matrices # the equalities: E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) # the inequalities (all pi>0) G <- diag(7) H <- rep(0, 7) # 3. Select the parsimonious (simplest) solution parsimonious <- ldei(E, F, G = G, H = H) # 4. show results data.frame(food = colnames(Minkdiet$Prey), fraction = parsimonious$X) dotchart(x = as.vector(parsimonious$X), labels = colnames(Minkdiet$A), main = "Estimated diet composition of Mink", sub = "using ldei and xranges", pch = 16) # 5. Ranges of diet composition iso <- xranges(E, F, ispos = TRUE) segments(iso[,1], 1:ncol(E), iso[,2], 1:ncol(E)) legend ("topright", pch = c(16, NA), lty = c(NA, 1), legend = c("parsimonious", "range")) pairs (xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 1000)$X, main = "Minkdiet 1000 solutions, using xsample")
Solves the following inverse problem:
subject to
Uses subroutine nnls (FORTRAN) from Linpack
nnls(A, B, tol = sqrt(.Machine$double.eps), verbose = TRUE)nnls(A, B, tol = sqrt(.Machine$double.eps), verbose = TRUE)
A |
numeric matrix containing the coefficients of the equality
constraints |
B |
numeric vector containing the right-hand side of the equality constraints. |
tol |
tolerance (for singular value decomposition and for the "equality" constraints). |
verbose |
logical to print |
a list containing:
X |
vector containing the solution of the nonnegative least squares problem. |
residualNorm |
scalar, the sum of absolute values of residuals of violated inequalities (i.e. sumof x[<0]); should be zero or very small if the problem is feasible. |
solutionNorm |
scalar, the value of the quadratic function at the
solution, i.e. the value of |
IsError |
logical, |
type |
the string "nnls", such that how the solution was obtained can be traced. |
numiter |
the number of iterations. |
Karline Soetaert <[email protected]>
Lawson C.L.and Hanson R.J. 1974. Solving Least Squares Problems, Prentice-Hall
Lawson C.L.and Hanson R.J. 1995. Solving Least Squares Problems. SIAM classics in applied mathematics, Philadelphia. (reprint of book)
ldei, which includes equalities
A <- matrix(nrow = 2, ncol = 3, data = c(3, 2, 2, 4, 2, 1)) B <- c(-4, 3) nnls(A, B)A <- matrix(nrow = 2, ncol = 3, data = c(3, 2, 2, 4, 2, 1)) B <- c(-4, 3) nnls(A, B)
Given an input matrix or its singular value decomposition,
calculates the resolution of the equations (rows) and of the unknowns (columns) of the matrix.
resolution (s, tol = sqrt(.Machine$double.eps))resolution (s, tol = sqrt(.Machine$double.eps))
s |
either a matrix or its singular value decomposition. |
tol |
tolerance for the singular values. |
a list containing:
row |
resolution of the rows (equations). |
col |
resolution of the columns (variables). |
nsolvable |
number of solvable unknowns - the rank of the matrix. |
Karline Soetaert <[email protected]>
Dick van Oevelen<[email protected]>
Menke, W., 1989. Geophysical Data Analysis: Discrete Inverse Theory. Revised edition. International Geophysics Series. Academic Press, London.
svd, the singluar value decomposition
resolution (matrix(nrow = 3, runif(9))) #3rows,3columns resolution (matrix(nrow = 3, runif(12))) #3rows,4columns resolution (matrix(nrow = 3, runif(6))) #3rows,2columns resolution (cbind(c(1, 2, 3), c(2, 3, 4), c(3, 5, 7))) # r3=r1+r2,c3=c1+c2resolution (matrix(nrow = 3, runif(9))) #3rows,3columns resolution (matrix(nrow = 3, runif(12))) #3rows,4columns resolution (matrix(nrow = 3, runif(6))) #3rows,2columns resolution (cbind(c(1, 2, 3), c(2, 3, 4), c(3, 5, 7))) # r3=r1+r2,c3=c1+c2
Input matrices and vectors for estimating the flows in the planktonic food web of the Gulf of Riga.
(as in Donali et al. (1999)).
The original input file can be found in the package subdirectory
/inst/docs/RigaSpring.input
There are 7 functional compartments: P1,P2,B,N,Z,D,OC (two phytoplankton groups, Bacteria, Nanozooplankton, Zooplankton, Detritus and DOC).
and 2 externals: CO2 and SED (sedimentation)
These are connected with 26 flows: P1->CO2, P2->CO2, Z->CO2, N->CO2, B->CO2, CO2->P1, CO2->P2, P1->Z, P1->N, P1->DOC, P1->SED, P2->DOC, P2->Z, P2->D, P2->SED, N->DOC, N->Z, Z->DOC, Z->D, Z->SED, D->Z, D->DOC, D->SED, B->N, B->SED, DOC->B
The lsei model contains:
14 equalities (Ax=B): the 7 mass balances (one for each compartment) and 7 measurement equations
26 unknowns (x), the flow values
45 inequalities (Gx>h). The first 19 inequalities impose bounds on some combinations of flows. The last 26 inequalities impose that the flows have to be positive.
As there are more unknowns (26) than equations (14), there exist an infinite amount of solutions (it is an underdetermined problem).
RigaWebRigaWeb
A list with the matrices and vectors that constitute the mass balance problem:
A, B, G and H.
The columnames of A and G are the names of the unknown
reaction rates;
The first 14 rownames of A give the names of the components
(these rows consitute the mass balance equations).
Karline Soetaert <[email protected]>
Donali, E., Olli, K., Heiskanen, A.S., Andersen, T., 1999. Carbon flow patterns in the planktonic food web of the Gulf of Riga, the Baltic Sea: a reconstruction by the inverse method. Journal of Marine Systems 23, 251..268.
E <- RigaWeb$A F <- RigaWeb$B G <- RigaWeb$G H <- RigaWeb$H # 1. parsimonious (simplest) solution pars <- lsei(E = E, F = F, G = G, H = H)$X # 2.ranges of all unknowns, including the central value xr <- xranges(E = E, F = F, G = G, H = H, central = TRUE) # the central point is a valid solution: X <- xr[,"central"] max(abs(E%*%X - F)) min(G%*%X - H) ## Not run: # this does not work on windows i386! # 3. Sample solution space; the central value is a good starting point # for algorithms cda and rda - but these need many iterations xs <- xsample(E = E, F = F, G = G, H = H, iter = 10000, out = 1000, type = "rda", x0 = X)$X # better convergence using 50000 iterations, but this takes a while xs <- xsample(E = E, F = F, G = G, H = H, iter = 50000, out = 1000, type = "rda", x0 = X)$X pairs(xs, pch = ".", cex = 2, gap = 0, upper.panel = NULL) # using mirror algorithm takes less iterations, # but an iteration takes more time ; it is better to start in a corner... # (i.e. no need to use X as starting value) xs <- xsample(E = E, F = F, G = G, H = H, iter = 1500, output = 500, type = "mirror")$X pairs(xs, pch = ".", cex = 2, gap = 0, upper.panel = NULL, yaxt = "n", xaxt = "n") # Print results: data.frame(pars = pars, xr[ ,1:2], Mean = colMeans(xs), sd = apply(xs, 2, sd)) ## End(Not run)E <- RigaWeb$A F <- RigaWeb$B G <- RigaWeb$G H <- RigaWeb$H # 1. parsimonious (simplest) solution pars <- lsei(E = E, F = F, G = G, H = H)$X # 2.ranges of all unknowns, including the central value xr <- xranges(E = E, F = F, G = G, H = H, central = TRUE) # the central point is a valid solution: X <- xr[,"central"] max(abs(E%*%X - F)) min(G%*%X - H) ## Not run: # this does not work on windows i386! # 3. Sample solution space; the central value is a good starting point # for algorithms cda and rda - but these need many iterations xs <- xsample(E = E, F = F, G = G, H = H, iter = 10000, out = 1000, type = "rda", x0 = X)$X # better convergence using 50000 iterations, but this takes a while xs <- xsample(E = E, F = F, G = G, H = H, iter = 50000, out = 1000, type = "rda", x0 = X)$X pairs(xs, pch = ".", cex = 2, gap = 0, upper.panel = NULL) # using mirror algorithm takes less iterations, # but an iteration takes more time ; it is better to start in a corner... # (i.e. no need to use X as starting value) xs <- xsample(E = E, F = F, G = G, H = H, iter = 1500, output = 500, type = "mirror")$X pairs(xs, pch = ".", cex = 2, gap = 0, upper.panel = NULL, yaxt = "n", xaxt = "n") # Print results: data.frame(pars = pars, xr[ ,1:2], Mean = colMeans(xs), sd = apply(xs, 2, sd)) ## End(Not run)
Generalised inverse solution of
Solve (A, B = diag(nrow = nrow(A)), tol = sqrt(.Machine$double.eps))Solve (A, B = diag(nrow = nrow(A)), tol = sqrt(.Machine$double.eps))
A |
numeric matrix containing the coefficients of the equations
|
B |
numeric matrix containing the right-hand sides of the equations; the default is the unity matrix, in which case the function will return the Moore-Penrose generalized inverse of matrix A. |
tol |
tolerance for selecting singular values. |
a vector with the generalised inverse solution.
Solve uses the Moore-Penrose generalized inverse of matrix A
(function ginv from package MASS).
solve, the R default requires a square, positive
definite A. Solve does not have this restriction.
Karline Soetaert <[email protected]>
package MASS:
Venables, W. N. & Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0
ginv to estimate the Moore-Penrose generalized inverse
of a matrix, in package MASS,
solve the R default
A <- matrix(nrow = 4, ncol = 3, data = c(1:8, 6, 8, 10, 12)) # col3 = col1+col2 B <- 0:3 X <- Solve(A, B) # generalised inverse solution A %*% X - B # should be zero (except for roundoff) (gA <- Solve(A)) # generalised inverse of AA <- matrix(nrow = 4, ncol = 3, data = c(1:8, 6, 8, 10, 12)) # col3 = col1+col2 B <- 0:3 X <- Solve(A, B) # generalised inverse solution A %*% X - B # should be zero (except for roundoff) (gA <- Solve(A)) # generalised inverse of A
Solves the linear system of equations
by Gaussion elimination
where A has to be square, and banded, i.e. with the only nonzero
elements in bands near the diagonal.
The matrix A is either inputted as a full square matrix or as the non-zero
bands.
uses lapack subroutine dgbsv (FORTRAN)
Solve.banded(abd, nup, nlow, B = rep(0, times = ncol(abd)), full = (nrow(abd) == ncol(abd)))Solve.banded(abd, nup, nlow, B = rep(0, times = ncol(abd)), full = (nrow(abd) == ncol(abd)))
abd |
either a matrix containing the (nonzero) bands, rotated row-wise (anti-clockwise) only, or a full square matrix. |
nup |
number of nonzero bands above the diagonal; ignored if |
nlow |
number of nonzero bands below the diagonal; ignored if |
B |
Right-hand side of the equations, a vector with length = number
of rows of |
full |
if |
If the input matrix abd is square, it is assumed that the full,
square A is inputted, unless full is set to FALSE.
If abd is not square, then the number of columns denote the
number of unknowns, while the number of rows equals the nonzero bands,
i.e. nup+nlow+1
matrix with the solution, X, of the banded system of equations A X =B,
the number of columns of this matrix = number of columns of B.
A similar function but that requires a totally different input can now
also be found in the Matrix package
Karline Soetaert <[email protected]>
J.J. Dongarra, J.R. Bunch, C.B. Moler, G.W. Stewart, LINPACK Users' Guide, SIAM, 1979.
Solve.tridiag to solve a tridiagonal system of linear equations.
Solve the generalised inverse solution,
solve the R default
# 1. Generate a banded matrix of random numbers, full format nup <- 2 # nr nonzero bands above diagonal ndwn <- 3 # nr nonzero bands below diagonal nn <- 10 # nr rows and columns of A A <- matrix(nrow = nn, ncol = nn, data = runif(1 : (nn*nn))) A [row(A) < col(A) - nup | row(A) > col(A) + ndwn] <- 0 diag(A) <- 1 # 1 on diagonal is easily recognised # right hand side B <- runif(nrow(A)) # solve it, using the default solver and banded (inputting full matrix) Full <- solve(A, B) Band1 <- Solve.banded(A, nup, ndwn, B) # 2. create banded form of matrix A Aext <- rbind(matrix(ncol = ncol(A), nrow = nup, 0), A, matrix(ncol = ncol(A), nrow = ndwn, 0)) abd <- matrix(nrow = nup + ndwn + 1, ncol = nn, data = Aext[col(Aext) <= row(Aext) & col(Aext) >= row(Aext) - ndwn - nup]) # print both to screen A abd # solve problem with banded version Band2 <- Solve.banded(abd, nup, ndwn, B) # compare 3 methods of solution cbind(Full, Band1, Band2) # same, now with 3 different right hand sides B3 <- cbind(B, B*2, B*3) Solve.banded(abd, nup, ndwn, B3)# 1. Generate a banded matrix of random numbers, full format nup <- 2 # nr nonzero bands above diagonal ndwn <- 3 # nr nonzero bands below diagonal nn <- 10 # nr rows and columns of A A <- matrix(nrow = nn, ncol = nn, data = runif(1 : (nn*nn))) A [row(A) < col(A) - nup | row(A) > col(A) + ndwn] <- 0 diag(A) <- 1 # 1 on diagonal is easily recognised # right hand side B <- runif(nrow(A)) # solve it, using the default solver and banded (inputting full matrix) Full <- solve(A, B) Band1 <- Solve.banded(A, nup, ndwn, B) # 2. create banded form of matrix A Aext <- rbind(matrix(ncol = ncol(A), nrow = nup, 0), A, matrix(ncol = ncol(A), nrow = ndwn, 0)) abd <- matrix(nrow = nup + ndwn + 1, ncol = nn, data = Aext[col(Aext) <= row(Aext) & col(Aext) >= row(Aext) - ndwn - nup]) # print both to screen A abd # solve problem with banded version Band2 <- Solve.banded(abd, nup, ndwn, B) # compare 3 methods of solution cbind(Full, Band1, Band2) # same, now with 3 different right hand sides B3 <- cbind(B, B*2, B*3) Solve.banded(abd, nup, ndwn, B3)
Solves the linear system A*X=B where A is an almost block diagonal matrix of the form:
TopBlock
... Array(1) ... ... ...
... ... Array(2) ... ...
...
... ... ... Array(Nblocks)...
... ... ... BotBlock
The method is based on Gauss elimination with alternate row and column elimination with partial pivoting, producing a stable decomposition of the matrix A without introducing fill-in.
uses FORTRAN subroutine colrow
Solve.block(Top, AR, Bot, B, overlap)Solve.block(Top, AR, Bot, B, overlap)
Top |
the first block of the almost block diagonal matrix |
AR |
intermediary blocks; |
Bot |
the last block of the almost block diagonal matrix |
B |
Right-hand side of the equations, a vector with length = number
of rows of |
overlap |
the number of columns in which successive blocks
overlap, and where |
matrix with the solution, X, of the block diagonal system of equations Ax=B, the number of columns of this matrix = number of columns of B.
A similar function but that requires a totally different input can now
also be found in the Matrix package
Karline Soetaert <[email protected]>
J. C. Diaz , G. Fairweather , P. Keast, 1983. FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination, ACM Transactions on Mathematical Software (TOMS), v.9 n.3, p.358-375
Solve.tridiag to solve a tridiagonal system of linear equations.
Solve.banded to solve a banded system of linear equations.
Solve the generalised inverse solution,
solve the R default
# Solve the following system: Ax=B, where A is block diagonal, and # 0.0 -0.98 -0.79 -0.15 Top # -1.00 0.25 -0.87 0.35 Top # 0.78 0.31 -0.85 0.89 -0.69 -0.98 -0.76 -0.82 blk1 # 0.12 -0.01 0.75 0.32 -1.00 -0.53 -0.83 -0.98 # -0.58 0.04 0.87 0.38 -1.00 -0.21 -0.93 -0.84 # -0.21 -0.91 -0.09 -0.62 -1.99 -1.12 -1.21 0.07 # 0.78 -0.93 -0.76 0.48 -0.87 -0.14 -1.00 -0.59 blk2 # -0.99 0.21 -0.73 -0.48 -0.93 -0.91 0.10 -0.89 # -0.68 -0.09 -0.58 -0.21 0.85 -0.39 0.79 -0.71 # 0.39 -0.99 -0.12 -0.75 -0.68 -0.99 0.50 -0.88 # 0.71 -0.64 0.0 0.48 Bot # 0.08 100.0 50.00 15.00 Bot B <- c(-1.92, -1.27, -2.12, -2.16, -2.27, -6.08, -3.03, -4.62, -1.02, -3.52, 0.55, 165.08) AA <- matrix (nrow = 12, ncol = 12, 0) AA[1,1:4] <- c( 0.0, -0.98, -0.79, -0.15) AA[2,1:4] <- c(-1.00, 0.25, -0.87, 0.35) AA[3,1:8] <- c( 0.78, 0.31, -0.85, 0.89, -0.69, -0.98, -0.76, -0.82) AA[4,1:8] <- c( 0.12, -0.01, 0.75, 0.32, -1.00, -0.53, -0.83, -0.98) AA[5,1:8] <- c(-0.58, 0.04, 0.87, 0.38, -1.00, -0.21, -0.93, -0.84) AA[6,1:8] <- c(-0.21, -0.91, -0.09, -0.62, -1.99, -1.12, -1.21, 0.07) AA[7,5:12] <- c( 0.78, -0.93, -0.76, 0.48, -0.87, -0.14, -1.00, -0.59) AA[8,5:12] <- c(-0.99, 0.21, -0.73, -0.48, -0.93, -0.91, 0.10, -0.89) AA[9,5:12] <- c(-0.68, -0.09, -0.58, -0.21, 0.85, -0.39, 0.79, -0.71) AA[10,5:12]<- c( 0.39, -0.99, -0.12, -0.75, -0.68, -0.99, 0.50, -0.88) AA[11,9:12]<- c( 0.71, -0.64, 0.0, 0.48) AA[12,9:12]<- c( 0.08, 100.0, 50.00, 15.00) ## Block diagonal input. Top <- matrix(nrow = 2, ncol = 4, data = AA[1:2 , 1:4] ) Bot <- matrix(nrow = 2, ncol = 4, data = AA[11:12, 9:12]) Blk1 <- matrix(nrow = 4, ncol = 8, data = AA[3:6 , 1:8] ) Blk2 <- matrix(nrow = 4, ncol = 8, data = AA[7:10 , 5:12]) AR <- array(dim = c(4, 8, 2), data = c(Blk1, Blk2)) overlap <- 4 # answer = (1, 1,....1) Solve.block(Top, AR, Bot, B, overlap = 4) # Now with 3 different B values B3 <- cbind(B, 2*B, 3*B) Solve.block(Top, AR, Bot, B3, overlap = 4)# Solve the following system: Ax=B, where A is block diagonal, and # 0.0 -0.98 -0.79 -0.15 Top # -1.00 0.25 -0.87 0.35 Top # 0.78 0.31 -0.85 0.89 -0.69 -0.98 -0.76 -0.82 blk1 # 0.12 -0.01 0.75 0.32 -1.00 -0.53 -0.83 -0.98 # -0.58 0.04 0.87 0.38 -1.00 -0.21 -0.93 -0.84 # -0.21 -0.91 -0.09 -0.62 -1.99 -1.12 -1.21 0.07 # 0.78 -0.93 -0.76 0.48 -0.87 -0.14 -1.00 -0.59 blk2 # -0.99 0.21 -0.73 -0.48 -0.93 -0.91 0.10 -0.89 # -0.68 -0.09 -0.58 -0.21 0.85 -0.39 0.79 -0.71 # 0.39 -0.99 -0.12 -0.75 -0.68 -0.99 0.50 -0.88 # 0.71 -0.64 0.0 0.48 Bot # 0.08 100.0 50.00 15.00 Bot B <- c(-1.92, -1.27, -2.12, -2.16, -2.27, -6.08, -3.03, -4.62, -1.02, -3.52, 0.55, 165.08) AA <- matrix (nrow = 12, ncol = 12, 0) AA[1,1:4] <- c( 0.0, -0.98, -0.79, -0.15) AA[2,1:4] <- c(-1.00, 0.25, -0.87, 0.35) AA[3,1:8] <- c( 0.78, 0.31, -0.85, 0.89, -0.69, -0.98, -0.76, -0.82) AA[4,1:8] <- c( 0.12, -0.01, 0.75, 0.32, -1.00, -0.53, -0.83, -0.98) AA[5,1:8] <- c(-0.58, 0.04, 0.87, 0.38, -1.00, -0.21, -0.93, -0.84) AA[6,1:8] <- c(-0.21, -0.91, -0.09, -0.62, -1.99, -1.12, -1.21, 0.07) AA[7,5:12] <- c( 0.78, -0.93, -0.76, 0.48, -0.87, -0.14, -1.00, -0.59) AA[8,5:12] <- c(-0.99, 0.21, -0.73, -0.48, -0.93, -0.91, 0.10, -0.89) AA[9,5:12] <- c(-0.68, -0.09, -0.58, -0.21, 0.85, -0.39, 0.79, -0.71) AA[10,5:12]<- c( 0.39, -0.99, -0.12, -0.75, -0.68, -0.99, 0.50, -0.88) AA[11,9:12]<- c( 0.71, -0.64, 0.0, 0.48) AA[12,9:12]<- c( 0.08, 100.0, 50.00, 15.00) ## Block diagonal input. Top <- matrix(nrow = 2, ncol = 4, data = AA[1:2 , 1:4] ) Bot <- matrix(nrow = 2, ncol = 4, data = AA[11:12, 9:12]) Blk1 <- matrix(nrow = 4, ncol = 8, data = AA[3:6 , 1:8] ) Blk2 <- matrix(nrow = 4, ncol = 8, data = AA[7:10 , 5:12]) AR <- array(dim = c(4, 8, 2), data = c(Blk1, Blk2)) overlap <- 4 # answer = (1, 1,....1) Solve.block(Top, AR, Bot, B, overlap = 4) # Now with 3 different B values B3 <- cbind(B, 2*B, 3*B) Solve.block(Top, AR, Bot, B3, overlap = 4)
Solves the linear system of equations
where A has to be square and tridiagonal, i.e with nonzero elements only on, one band above, and one band below the diagonal.
Solve.tridiag ( diam1, dia, diap1, B=rep(0,times=length(dia)))Solve.tridiag ( diam1, dia, diap1, B=rep(0,times=length(dia)))
diam1 |
a vector with (nonzero) elements below the diagonal. |
dia |
a vector with (nonzero) elements on the diagonal. |
diap1 |
a vector with (nonzero) elements above the diagonal. |
B |
Right-hand side of the equations, a vector with length = number of rows of A, or a matrix with number of rows = number of rows of A. |
If the length of the vector dia is equal to N, then the lengths of
diam1 and diap1 should be equal to N-1
matrix with the solution, X, of the tridiagonal system of equations Ax=B.
The number of columns of this matrix equals the number of columns of B.
Karline Soetaert <[email protected]>
Solve.banded, the function to solve a banded system of
linear equations.
Solve.block, the function to solve a block diagonal system of
linear equations.
Solve the generalised inverse solution,
solve the R default
# create tridagonal system: bands on diagonal, above and below nn <- 20 # nr rows and columns of A aa <- runif(nn) bb <- runif(nn) cc <- runif(nn) # full matrix A <- matrix(nrow = nn, ncol = nn, data = 0) diag(A) <- bb A[cbind(1:(nn-1), 2:nn)] <- cc[-nn] A[cbind(2:nn, 1:(nn-1))] <- aa[-1] B <- runif(nn) # solve as full matrix solve(A, B) # same, now using tridiagonal algorithm as.vector(Solve.tridiag(aa[-1], bb, cc[-nn], B)) # same, now with 3 different right hand sides B3 <- cbind(B, B*2, B*3) Solve.tridiag(aa[-1], bb, cc[-nn], B3)# create tridagonal system: bands on diagonal, above and below nn <- 20 # nr rows and columns of A aa <- runif(nn) bb <- runif(nn) cc <- runif(nn) # full matrix A <- matrix(nrow = nn, ncol = nn, data = 0) diag(A) <- bb A[cbind(1:(nn-1), 2:nn)] <- cc[-nn] A[cbind(2:nn, 1:(nn-1))] <- aa[-1] B <- runif(nn) # solve as full matrix solve(A, B) # same, now using tridiagonal algorithm as.vector(Solve.tridiag(aa[-1], bb, cc[-nn], B)) # same, now with 3 different right hand sides B3 <- cbind(B, B*2, B*3) Solve.tridiag(aa[-1], bb, cc[-nn], B3)
Given the linear constraints
and a set of "variables" described by the linear equations
finds the minimum and maximum values of the variables by successively minimising and maximising each variable equation
varranges(E=NULL, F=NULL, G=NULL, H=NULL, EqA, EqB=NULL, ispos=FALSE, tol=1e-8, verbose=TRUE, lower=NULL, upper=NULL)varranges(E=NULL, F=NULL, G=NULL, H=NULL, EqA, EqB=NULL, ispos=FALSE, tol=1e-8, verbose=TRUE, lower=NULL, upper=NULL)
E |
numeric matrix containing the coefficients of the equalities
|
F |
numeric vector containing the right-hand side of the equalities. |
G |
numeric matrix containing the coefficients of the inequalities
|
H |
numeric vector containing the right-hand side of the inequalities. |
EqA |
numeric matrix containing the coefficients that define the variable equations. |
EqB |
numeric vector containing the right-hand side of the variable equations. |
ispos |
if |
tol |
tolerance for equality and inequality constraints. |
verbose |
logical to print warnings and messages. |
upper, lower
|
vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H. |
a 2-column matrix with the minimum and maximum value of each equation (variable)
uses linear programming function lp from package
lpSolve.
Karline Soetaert <[email protected]>
Michel Berkelaar and others (2010). lpSolve: Interface to Lp_solve v. 5.5 to solve linear/integer programs. R package version 5.6.5. http://CRAN.R-project.org/package=lpSolve
Minkdiet, for a description of the Mink diet example.
xranges, to estimate ranges of inverse unknowns.
xsample, to randomly sample the lsei problem
lp: linear programming function from package lpSolve.
# Ranges in the contribution of food 3+4+5 in the diet of Mink (try ?Minkdiet) E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) EqA <- c(0, 0, 1, 1, 1, 0, 0) # sum of food 3,4,5 (isoA <- varranges(E, F, EqA = EqA, ispos = TRUE)) # ranges of part of food 3+4+5 # The same, but explicitly imposing positivity varranges(E, F, EqA = EqA, G = diag(7), H = rep(0, 7)) # The same, but shorter - using lower bound: varranges(E, F, EqA = EqA, lower=0)# Ranges in the contribution of food 3+4+5 in the diet of Mink (try ?Minkdiet) E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) EqA <- c(0, 0, 1, 1, 1, 0, 0) # sum of food 3,4,5 (isoA <- varranges(E, F, EqA = EqA, ispos = TRUE)) # ranges of part of food 3+4+5 # The same, but explicitly imposing positivity varranges(E, F, EqA = EqA, G = diag(7), H = rep(0, 7)) # The same, but shorter - using lower bound: varranges(E, F, EqA = EqA, lower=0)
Uses random samples of an under- or overdetermined linear problem to estimate the distribution of equations
Based on a random sample of x (e.g. produced with xsample),
produces the corresponding set of "variables" consisting of linear
equations in the unknowns.
varsample (X, EqA, EqB=NULL)varsample (X, EqA, EqB=NULL)
X |
matrix whose rows contain the sampled values of the unknowns
|
EqA |
numeric matrix containing the coefficients that define the variables. |
EqB |
numeric vector containing the right-hand side of the variable equation. |
a matrix whose rows contain the sampled values of the variables.
Karline Soetaert <[email protected]>
Minkdiet, for a description of the Mink diet example.
varranges, to estimate ranges of inverse variables.
xsample, to randomly sample the lsei problem.
# The probability distribution of vertebrate and invertebrate # food in the diet of Mink # food items of Mink are (in that order): # fish mussels crabs shrimp rodents amphipods ducks # V I I I V I V # V= vertebrate, I = invertebrate # In matrix form: VarA <- matrix(ncol = 7, byrow = TRUE, data = c( 0, 1, 1, 1, 0, 1, 0, # invertebrates 1, 0, 0, 0, 1, 0, 1)) # vertebrates # first sample the Minkdiet problem E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) X <- xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 1000)$X #then determine Diet Composition in terms of vertebrate and invertebrate food DC <- varsample(X = X, EqA = VarA) hist(DC[,1], freq = FALSE, xlab = "fraction", main = "invertebrate food in Mink diet", col = "lightblue")# The probability distribution of vertebrate and invertebrate # food in the diet of Mink # food items of Mink are (in that order): # fish mussels crabs shrimp rodents amphipods ducks # V I I I V I V # V= vertebrate, I = invertebrate # In matrix form: VarA <- matrix(ncol = 7, byrow = TRUE, data = c( 0, 1, 1, 1, 0, 1, 0, # invertebrates 1, 0, 0, 0, 1, 0, 1)) # vertebrates # first sample the Minkdiet problem E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) X <- xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 1000)$X #then determine Diet Composition in terms of vertebrate and invertebrate food DC <- varsample(X = X, EqA = VarA) hist(DC[,1], freq = FALSE, xlab = "fraction", main = "invertebrate food in Mink diet", col = "lightblue")
Given the linear constraints
finds the minimum and maximum values of all elements of vector
This is done by successively minimising and maximising each x,
using linear programming.
xranges(E = NULL, F = NULL, G = NULL, H = NULL, ispos = FALSE, tol = 1e-8, central = FALSE, full=FALSE, verbose = TRUE, lower = NULL, upper = NULL)xranges(E = NULL, F = NULL, G = NULL, H = NULL, ispos = FALSE, tol = 1e-8, central = FALSE, full=FALSE, verbose = TRUE, lower = NULL, upper = NULL)
E |
numeric matrix containing the coefficients of the equalities
|
F |
numeric vector containing the right-hand side of the equalities. |
G |
numeric matrix containing the coefficients of the inequalities
|
H |
numeric vector containing the right-hand side of the inequalities. |
ispos |
if |
tol |
tolerance for equality and inequality constraints. |
central |
if TRUE, the mean value of all range solutions is also outputted. |
full |
if |
verbose |
logical to print warnings and messages. |
upper, lower
|
vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H. |
The ranges are estimated by successively minimising and maximising each
unknown, and using linear programming (based on function lp from
R-package lpSolve.
By default linear programming assumes that all unknowns are positive.
If all unknowns are indeed to be positive, then it will generally be faster
to set ispos equal to TRUE
If ispos is FALSE, then a system double the size of the
original system must be solved.
xranges outputs only the minimum and maximum value of each flow unless:
full is TRUE. In this case, all the results of the successive
minimisation and maximisation will be outputted, i.e. for each linear
programming application, not just the value of the unknown being optimised
but also the corresponding values of the other unknowns will be outputted.
If central is TRUE, then the mean of all the results of the
linear programming will be outputted.
This may be a good starting value for xsample
Note: the columns corresponding to the central value and the
full results are valid solutions of the equations
and . This is not the case for the first two columns (with
the minimal and maximal values).
a matrix with at least two columns:
column 1 and 2: the minimum and maximum value of each x
if central is TRUE: column 3 = the central value
if full is TRUE: next columns contain all valid range solutions
Karline Soetaert <[email protected]>
Michel Berkelaar and others (2010). lpSolve: Interface to Lp_solve v. 5.5 to solve linear/integer programs. R package version 5.6.5. http://CRAN.R-project.org/package=lpSolve
Minkdiet, for a description of the Mink diet example.
varranges, for range estimation of variables,
xsample, to randomly sample the lsei problem
lp: linear programming from package lpSolve
# Estimate the ranges in the Diet Composition of Mink E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) (DC <- xranges(E, F, ispos = TRUE)) # The same, but explicitly imposing positivity using G and H xranges(E, F, G = diag(7), H = rep(0, 7)) # and using lower bound xranges(E, F, lower = 0, verbose = FALSE)# Estimate the ranges in the Diet Composition of Mink E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) (DC <- xranges(E, F, ispos = TRUE)) # The same, but explicitly imposing positivity using G and H xranges(E, F, G = diag(7), H = rep(0, 7)) # and using lower bound xranges(E, F, lower = 0, verbose = FALSE)
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
where have to be met exactly, and x is distributed
according to
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$double.eps), x0 = NULL, fulloutput = FALSE, test = TRUE, verbose=TRUE, lower = NULL, upper = NULL)xsample(A = NULL, B = NULL, E = NULL, F =NULL, G = NULL, H = NULL, sdB = NULL, W = 1, iter = 3000, outputlength = iter, burninlength = NULL, type = "mirror", jmp = NULL, tol = sqrt(.Machine$double.eps), x0 = NULL, fulloutput = FALSE, test = TRUE, verbose=TRUE, lower = NULL, upper = NULL)
A |
numeric matrix containing the coefficients of the
(approximate) equality constraints, |
B |
numeric vector containing the right-hand side of the (approximate) equality constraints. |
E |
numeric matrix containing the coefficients of the (exact)
equality constraints, |
F |
numeric vector containing the right-hand side of the (exact) equality constraints. |
G |
numeric matrix containing the coefficients of the inequality
constraints, |
H |
numeric vector containing the right-hand side of the inequality constraints. |
sdB |
vector with standard deviation on B. Defaults to |
W |
weighting for |
iter |
integer determining the number of iterations. |
outputlength |
number of iterations kept in the output; at most
equal to |
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: |
tol |
tolerance for equality and inequality constraints; numbers
whose absolute value is smaller than |
x0 |
initial (particular) solution. |
fulloutput |
if |
test |
if |
verbose |
logical to print warnings and messages. |
upper, lower
|
vector containing upper and lower bounds on the unknowns. If one value, it is assumed to apply to all unknowns. If a vector, it should have a length equal to the number of unknowns; this vector can contain NA for unbounded variables. The upper and lower bounds are added to the inequality conditions G*x>=H. |
The algorithm proceeds in two steps.
the equality constraints are eliminated, and the
system , is rewritten as ,
i.e. containing only inequality constraints and where Z is a basis for
the null space of E.
the distribution of 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.
and ), 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.
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 |
p |
only returned if |
jmp |
the jump length used for the random walk. Can be used to check the automated jump length. |
Karel Van den Meersche
Karline Soetaert <[email protected]>
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.
https://www.jstatsoft.org/v30/c01/
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.
#------------------------------------------------------------------------------- # 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) xs <- xsample(E = E, F = F, lower = 0) pairs(xs) #------------------------------------------------------------------------------- # Sample the underdetermined Mink diet problem #------------------------------------------------------------------------------- E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) # Here the Requirement x > 0 is been inposed in G and H. pairs(xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 5000, output = 1000, type = "cda")$X, main = "Minkdiet 1000 solutions, - cda")#------------------------------------------------------------------------------- # A simple problem #------------------------------------------------------------------------------- # Sample the probability density function of x1,...x4 # subject to: # x1 + x2 + x4 = 3 # x2 -x3 + x4 = -1 # xi > 0 E <- matrix(nrow = 2, byrow = TRUE, data = c(1, 1, 0, 1, 0, 1, -1, 1)) F <- c(3, -1) xs <- xsample(E = E, F = F, lower = 0) pairs(xs) #------------------------------------------------------------------------------- # Sample the underdetermined Mink diet problem #------------------------------------------------------------------------------- E <- rbind(Minkdiet$Prey, rep(1, 7)) F <- c(Minkdiet$Mink, 1) # Here the Requirement x > 0 is been inposed in G and H. pairs(xsample(E = E, F = F, G = diag(7), H = rep(0, 7), iter = 5000, output = 1000, type = "cda")$X, main = "Minkdiet 1000 solutions, - cda")