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cmls

Solve a Constrained Multivariate Least Squares Problem

Description

Finds the p x m matrix B that minimizes the multivariate least squares problem

sum(( Y - X %*% B )^2)

subject to the specified constraints on the rows of B.

Usage

cmls(X, Y, const = "uncons", struc = NULL, 
     df = 10, degree = 3, intercept = TRUE,
     backfit = FALSE, maxit = 1e3, eps = 1e-10, 
     del = 1e-6, XtX = NULL, mode.range = NULL)

Arguments

X

Matrix of dimension n x p.
Y

Matrix of dimension n x m.
const

Constraint code. See const for the 24 available options.
struc

Structural constraints (defaults to unstructured). See Note.
df

Degrees of freedom for the spline basis (for smoothness constraints). See Note.
degree

Polynomial degree for the spline basis (for smoothness constraints). See Note.
intercept

Logical indicating whether the spline basis should contain an intercept (for smoothness constraints). See Note.
backfit

Estimate B via back-fitting (TRUE) or vectorization (FALSE). See Details.
maxit

Maximum number of iterations for back-fitting algorithm. Ignored if backfit = FALSE.
eps

Convergence tolerance for back-fitting algorithm. Ignored if backfit = FALSE.
del

Stability tolerance for back-fitting algorithm. Ignored if backfit = FALSE.
XtX

Crossproduct matrix: XtX = crossprod(X).
mode.range

Mode search ranges (for unimodal constraints). See Note.

Details

If backfit = FALSE (default), a closed-form solution is used to estimate B whenever possible. Otherwise a back-fitting algorithm is used, where the rows of B are updated sequentially until convergence. The backfitting algorithm is determined to have converged when

mean((B.new - B.old)^2) < eps * (mean(B.old^2) + del),

where B.old and B.new denote the parameter estimates at iterations t and t+1 of the backfitting algorithm.

Value

Returns the estimated matrix B with attribute "df" (degrees of freedom), which gives the df for each row of B.

Note

Structure constraints (struc) should be specified with a p x m matrix of logicals (TRUE/FALSE), such that FALSE elements indicate a weight should be constrained to be zero. Default uses unstructured weights, i.e., a p x m matrix of all TRUE values.

Inputs df, degree, and intercept are only applicable when using one of the 12 constraints that involves a spline basis, i.e., "smooth", "smonon", "smoper", "smpeno", "ortsmo", "orsmpe", "monsmo", "mosmno", "unismo", "unsmno", "unsmpe", "unsmpn".

Input mode.range is only applicable when using one of the 8 constraints that enforces unimodality: "unimod", "uninon", "uniper", "unpeno", "unismo", "unsmno", "unsmpe", "unsmpn". Mode search ranges (mode.range) should be specified with a 2 x p matrix of integers such that

1 <= mode.range[1,j] <= mode.range[2,j] <= m for all j = 1:p.

Default is mode.range = matrix(c(1, m), 2, p).

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

Goldfarb, D., & Idnani, A. (1983). A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming, 27, 1-33.

Helwig, N. E. (in prep). Constrained multivariate least squares in R.

Ten Berge, J. M. F. (1993). Least Squares Optimization in Multivariate Analysis. Volume 25 of M & T Series. DSWO Press, Leiden University. ISBN: 9789066950832

Turlach, B. A., & Weingessel, A. (2013). quadprog: Functions to solve Quadratic Programming Problems. R package version 1.5-5. https://CRAN.R-project.org/package=quadprog

See Also

const prints/returns the contraint options.

cv.cmls performs k-fold cross-validation to tune the constraint options.

mlsei and mlsun are used to implement several of the constraints.

Examples

######***######   GENERATE DATA   ######***######

# make X
set.seed(2)
n <- 50
m <- 20
p <- 2
Xmat <- matrix(rnorm(n*p), nrow = n, ncol = p)

# make B (which satisfies all constraints except monotonicity)
x <- seq(0, 1, length.out = m)
Bmat <- rbind(sin(2*pi*x), sin(2*pi*x+pi)) / sqrt(4.75)
struc <- rbind(rep(c(TRUE, FALSE), each = m / 2),
               rep(c(FALSE, TRUE), each = m / 2))
Bmat <- Bmat * struc

# make noisy data
set.seed(1)
Ymat <- Xmat %*% Bmat + rnorm(n*m, sd = 0.25)


######***######   UNCONSTRAINED   ######***######

# unconstrained
Bhat <- cmls(X = Xmat, Y = Ymat, const = "uncons")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unconstrained and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "uncons", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")


######***######   NON-NEGATIVITY   ######***######

# non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "nonneg")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "nonneg", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")


######***######   PERIODICITY   ######***######

# periodic
Bhat <- cmls(X = Xmat, Y = Ymat, const = "period")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# periodic and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "period", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# periodic and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "pernon")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# periodic and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "pernon", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")


######***######   SMOOTHNESS   ######***######

# smooth
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smooth")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# smooth and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smooth", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# smooth and periodic
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smoper")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# smooth and periodic and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smoper", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# smooth and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smonon")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# smooth and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smonon", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# smooth and periodic and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smpeno")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# smooth and periodic and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "smpeno", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")


######***######   ORTHOGONALITY   ######***######

# orthogonal
Bhat <- cmls(X = Xmat, Y = Ymat, const = "orthog")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# orthogonal and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "orthog", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# orthgonal and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "ortnon")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# orthgonal and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "ortnon", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# orthogonal and smooth
Bhat <- cmls(X = Xmat, Y = Ymat, const = "ortsmo")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# orthogonal and smooth and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "ortsmo", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# orthogonal and smooth and periodic
Bhat <- cmls(X = Xmat, Y = Ymat, const = "orsmpe")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# orthogonal and smooth and periodic and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "orsmpe", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")


######***######   UNIMODALITY   ######***######

# unimodal
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unimod")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unimod", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "uninon")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "uninon", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and periodic
Bhat <- cmls(X = Xmat, Y = Ymat, const = "uniper")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and periodic and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "uniper", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and periodic and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unpeno")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and periodic and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unpeno", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")


######***######   UNIMODALITY AND SMOOTHNESS   ######***######

# unimodal and smooth
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unismo")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and smooth and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unismo", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and smooth and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unsmno")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and smooth and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unsmno", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and smooth and periodic
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unsmpe")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and smooth and periodic and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unsmpe", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and smooth and periodic and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unsmpn")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# unimodal and smooth and periodic and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "unsmpn", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")


######***######   MONOTONICITY   ######***######

# make B
x <- 1:m
Bmat <- rbind(1 / (1 + exp(-(x - quantile(x, 0.5)))), 
              1 / (1 + exp(-(x - quantile(x, 0.8)))))
struc <- rbind(rep(c(FALSE, TRUE), c(1 * m, 3 * m) / 4),
               rep(c(FALSE, TRUE), c(m, m) / 2))
Bmat <- Bmat * struc               

# make noisy data
set.seed(1)
Ymat <- Xmat %*% Bmat + rnorm(m*n, sd = 0.25)

# monotonic increasing
Bhat <- cmls(X = Xmat, Y = Ymat, const = "moninc")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# monotonic increasing and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "moninc", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# monotonic increasing and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "monnon")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# monotonic increasing and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "monnon", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# monotonic increasing and smooth
Bhat <- cmls(X = Xmat, Y = Ymat, const = "monsmo")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# monotonic increasing and smooth and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "monsmo", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# monotonic increasing and smooth and non-negative
Bhat <- cmls(X = Xmat, Y = Ymat, const = "mosmno")
mean((Bhat - Bmat)^2)
attr(Bhat, "df")

# monotonic increasing and smooth and non-negative and structured
Bhat <- cmls(X = Xmat, Y = Ymat, const = "mosmno", struc = struc)
mean((Bhat - Bmat)^2)
attr(Bhat, "df")
CMLS

Constrained Multivariate Least Squares

v1.0-0
GPL (>= 2)
Authors
Nathaniel E. Helwig <helwig@umn.edu>
Initial release
2018-06-06

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