cmls: Solve a Constrained Multivariate Least Squares Problem
In CMLS: Constrained Multivariate Least Squares
cmls R Documentation
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,
z = 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.
z
Predictor values for the spline basis (for smoothness constraints). 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 z, 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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/BF02591962")}
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. (2019). quadprog: Functions to solve Quadratic Programming Problems. R package version 1.5-8. 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 documentation built on April 3, 2023, 5:24 p.m.
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| cmls | R Documentation |
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,
z = 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 |
Y |
Matrix of dimension |
const |
Constraint code. See |
struc |
Structural constraints (defaults to unstructured). See Note. |
z |
Predictor values for the spline basis (for smoothness constraints). 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 |
maxit |
Maximum number of iterations for back-fitting algorithm. Ignored if |
eps |
Convergence tolerance for back-fitting algorithm. Ignored if |
del |
Stability tolerance for back-fitting algorithm. Ignored if |
XtX |
Crossproduct matrix: |
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 z, 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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/BF02591962")}
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. (2019). quadprog: Functions to solve Quadratic Programming Problems. R package version 1.5-8. 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")
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