In wangq326/SC19074: Reduced rank regression via stability approach
Overview
SC19074 is R package developed to solve the tuning parameter $\lambda$ selection problem in reduced rank regression model. Seven functions are considered, namely, get_coef_subspace_sim (obtain the variance under each lambda by the cofficient matrices) and rrr.sim6 (generate random samples) and stability_ann_sim(RRR via stability approach) and so on.
Background
Given $n$ observations of the response $y_i\in\mathcal{R}^q$ and predictor $x_i\in\mathcal{R}^p$, we consider themultivariate linear regression model $$Y =XC_0 + E,$$ where $Y =(y_1,...,y_n)^T, X =(x_1,...,x_n)^T$, $C_0$ is a $p\times q$ coefficient matrix, and $E =(e_1,...,e_n)^T$ is an $n\times q$ matrix of independently and identically distributed random errors with mean zero and variance $\sigma^2$. Throughout, we write $p∧q=min(p,q), n∧q=min(n,q)$, $r=r(C_0)$ and $r_x=r(X)$, where $r(·)$ denotes the rank of a matrix.
The adaptive nuclear norm of a matrix $C\in\mathcal{R}^{p×q}$ is defined as a weighted sum of its singular values: $$\Vert C\Vert_{∗w} =\sum_{i=1}^{p∧q}w_id_i(C).$$
Kun Chen propose to estimate $C_0$ by minimizing $$\frac12\Vert Y-XC\Vert^2_F+\lambda\Vert XC\Vert_{*w}.$$ He provided the solution function rrr in the R package rrpack. Based on this function in my package, I modified the return value and wrote it into the rrr2 function for later research.
In the reduced rank regression model via adaptive nuclear norm penalization proposed by Kun Chen, there is no tuning parameter selection method. We apply the stability method.
For a given $\lambda$, we can get its instability as follows:
1. Perform m non-replacement subsampling from the sample, the size of the subsample is about $\sqrt n$;
2. Apply the RRR model proposed by Kun Chen for each sub-sampling to get the coefficient matrix $C_{1},...,C_ {m}.$
3. Find the instability of $ C_{1}, ..., C_{m}$.
We define the variance of these matrices as instability. The variance is defined as follows:
$$var(\hat C_{1},\hat C_{2},...\hat C_{n})=(C_{2}^{n})^{-1}\sum_{i<j}d(\hat C_{i},\hat C_{j}).$$
The distance between the two matrices is defined as $$d(\mathcal{A},\mathcal{B})=max{\sigma_{1}(A_{\bot}^{H}B),\sigma_{1}(B_{\bot}^{H}A)},$$ Where $A_{\bot}^{H}$ is the basis of the orthogonal complement of $\mathcal{A}$, and $A$ is a set of orthogonal basis of $\mathcal{A}$.
The main results
library(tidyverse)
library(rrpack)
library(knitr)
library(mvtnorm)
library(gridExtra)
library(MASS)
library(xtable)
set.seed(6419)
n = 38 # samples
q = 44 # number of response variables
rho = 0.5 # correlation between Xs
rX = 8 # rank of X
r = 5 # rank of C
p = 23 # number of explanatory variables
b set to 0.05
library(SC19074)
sim21 <- rrr.sim6(n, p, q, r, b = 0.05, rho, rX)
sim_ann21 <- rrr2(sim21$Y, sim21$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_res21 <- stability_ann_sim(sim21$Y, sim21$X, sim_ann21$lambda, subset = 0.7, ntimes = 100)
b set to 0.1
sim22 <- rrr.sim6(n, p, q, r, b = 0.1, rho, rX)
sim_ann22 <- rrr2(sim22$Y, sim22$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_res22 <- stability_ann_sim(sim22$Y, sim22$X, sim_ann22$lambda, subset = 0.7, ntimes = 100)
b set to 0.3
sim23 <- rrr.sim6(n, p, q, r, b = 0.3, rho, rX)
sim_ann23 <- rrr2(sim23$Y, sim23$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_res23 <- stability_ann_sim(sim23$Y, sim23$X, sim_ann23$lambda, subset = 0.7, ntimes = 100)
b set to 0
# Rank 0
# Generate X
p = 46
sim4 <- rrr.sim6(n, p, q, r, b = 0, rho, rX)
sim_ann4 <- rrr2(sim4$Y, sim4$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_res4 <- stability_ann_sim(sim4$Y, sim4$X, sim_ann4$lambda, subset = 0.7, ntimes = 100)
detach("package:rrpack", unload = TRUE)
detach("package:MASS", unload=TRUE)
# ---- stability-s2n-table ----
# Table of signal to noise ratios for each simulation
s2n_table <- data.frame(Simulation = 1:4, n = n, p = p, q = q, b = c(0, 0.05, 0.1, 0.3),
"Signal to noise ratio" = c(sim4$s2n, sim21$s2n, sim22$s2n, sim23$s2n),
check.names = FALSE)
print(xtable(s2n_table, label = "tab:stability-s2n-table",
caption = "Parameters and signal to noise ratios for the four simulated datasets.",
digits = c(0,0,0,0,0, 2, 3)),
include.rownames = FALSE)
# ---- stability-sims-plots ----
stability_combine_sim <- function(stability_ann_results){
st_plot <- data.frame(rank = c(stability_ann_results$st_res),
lambda = rep(stability_ann_results$lambda, each = nrow(stability_ann_results$st_res)))
st_group <- group_by(st_plot, by = lambda)
st_sum <- summarize(st_group, var = var(rank), avg = mean(rank)) %>% rename(lambda = by)
st_sum <- full_join(st_sum, stability_ann_results$st_coef, by = "lambda")
}
stability_plots_sim <- function(sim_results, rank, title = "sim_results", subtitle = ""){
st_combined <- stability_combine_sim(sim_results)
# Plot mean vs variance
varmeanplot <- ggplot(st_combined, aes(x = avg, y = sqrt(var))) +
geom_point() +
geom_vline(aes(xintercept = rank)) +
xlab("Mean estimated rank") +
ylab("Standard deviation of subspace distance") +
ggtitle(title, subtitle = subtitle) +
theme_bw()
# Plot mean vs variance of coefficients
coefplot <- ggplot(st_combined, aes(x = avg, y = sqrt(var.subspace.coef))) +
geom_point() +
geom_vline(aes(xintercept = rank)) +
xlab("Mean estimated rank") +
ylab(expression(sqrt("Mean of coefficient ranks"))) +
ggtitle(title, subtitle = subtitle) +
theme_bw()
return(list(rplot = varmeanplot, coefplot = coefplot))
}
p21 <- stability_plots_sim(sim_res21, rank = r, title = "Simulation 2", subtitle = "b = 0.05")
p22 <- stability_plots_sim(sim_res22, rank = r, title = "Simulation 3", subtitle = "b = 0.1")
p23 <- stability_plots_sim(sim_res23, rank = r, title = "Simulation 4", subtitle = "b = 0.3")
p4 <- stability_plots_sim(sim_res4, rank = 0, title = "Simulation 1", subtitle = "b = 0")
# ---- stability-sims-rplots ----
grid.arrange(p4$rplot, p21$rplot, p22$rplot, p23$rplot, nrow = 2)
# ---- stability-sims-coefplots ----
grid.arrange(p4$coefplot, p21$coefplot, p22$coefplot, p23$coefplot, nrow = 2)
# ---- stability-sims-r0-plots ----
grid.arrange(p4$rplot, p4$coefplot, nrow = 1)
# ---- rank1-sims ----
# Simulations and plots for rank 1 matrices
set.seed(9988999)
n = 38 # samples
q = 44 # number of response variables
rho = 0.5 # correlation between Xs
rX = 8 # rank of X
p = 23 # number of explanatory variables
simr1 <- rrr.sim6(n, p, q, r=1, b = 0.3, rho, rX)
simr1$s2n
sim_annr1 <- rrr2(simr1$Y, simr1$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_resr1 <- stability_ann_sim(simr1$Y, simr1$X, sim_annr1$lambda, subset = 0.7, ntimes = 100)
simr12 <- rrr.sim6(n, p, q, r=1, b = 0.05, rho, rX)
simr12$s2n
sim_annr12 <- rrr2(simr12$Y, simr12$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_resr12 <- stability_ann_sim(simr12$Y, simr12$X, sim_annr12$lambda, subset = 0.7, ntimes = 100)
simr13 <- rrr.sim6(n, p, q, r=1, b = 0.1, rho, rX)
simr13$s2n
sim_annr13 <- rrr2(simr13$Y, simr13$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_resr13 <- stability_ann_sim(simr13$Y, simr13$X, sim_annr13$lambda, subset = 0.7, ntimes = 100)
simr14 <- rrr.sim6(n, p, q, r=1, b = 0, rho, rX)
simr14$s2n
sim_annr14 <- rrr2(simr14$Y, simr14$X, penaltySVD = "ann",
modstr = list(gamma = 2))
sim_resr14 <- stability_ann_sim(simr14$Y, simr14$X, sim_annr14$lambda, subset = 0.7, ntimes = 100)
pr1 <- stability_plots_sim(sim_resr1, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr1$s2n, 2)))
pr12 <- stability_plots_sim(sim_resr12, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr12$s2n, 2)))
pr13 <- stability_plots_sim(sim_resr13, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr13$s2n, 2)))
pr14 <- stability_plots_sim(sim_resr14, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr14$s2n, 2)))
grid.arrange(pr1$coefplot, pr12$coefplot, pr13$coefplot, pr14$coefplot, nrow = 2)
wangq326/SC19074 documentation built on Jan. 2, 2020, 8:48 p.m.
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Overview
SC19074 is R package developed to solve the tuning parameter $\lambda$ selection problem in reduced rank regression model. Seven functions are considered, namely, get_coef_subspace_sim (obtain the variance under each lambda by the cofficient matrices) and rrr.sim6 (generate random samples) and stability_ann_sim(RRR via stability approach) and so on.
Background
Given $n$ observations of the response $y_i\in\mathcal{R}^q$ and predictor $x_i\in\mathcal{R}^p$, we consider themultivariate linear regression model $$Y =XC_0 + E,$$ where $Y =(y_1,...,y_n)^T, X =(x_1,...,x_n)^T$, $C_0$ is a $p\times q$ coefficient matrix, and $E =(e_1,...,e_n)^T$ is an $n\times q$ matrix of independently and identically distributed random errors with mean zero and variance $\sigma^2$. Throughout, we write $p∧q=min(p,q), n∧q=min(n,q)$, $r=r(C_0)$ and $r_x=r(X)$, where $r(·)$ denotes the rank of a matrix.
The adaptive nuclear norm of a matrix $C\in\mathcal{R}^{p×q}$ is defined as a weighted sum of its singular values: $$\Vert C\Vert_{∗w} =\sum_{i=1}^{p∧q}w_id_i(C).$$
Kun Chen propose to estimate $C_0$ by minimizing $$\frac12\Vert Y-XC\Vert^2_F+\lambda\Vert XC\Vert_{*w}.$$ He provided the solution function rrr in the R package rrpack. Based on this function in my package, I modified the return value and wrote it into the rrr2 function for later research.
In the reduced rank regression model via adaptive nuclear norm penalization proposed by Kun Chen, there is no tuning parameter selection method. We apply the stability method.
For a given $\lambda$, we can get its instability as follows: 1. Perform m non-replacement subsampling from the sample, the size of the subsample is about $\sqrt n$; 2. Apply the RRR model proposed by Kun Chen for each sub-sampling to get the coefficient matrix $C_{1},...,C_ {m}.$ 3. Find the instability of $ C_{1}, ..., C_{m}$.
We define the variance of these matrices as instability. The variance is defined as follows: $$var(\hat C_{1},\hat C_{2},...\hat C_{n})=(C_{2}^{n})^{-1}\sum_{i<j}d(\hat C_{i},\hat C_{j}).$$ The distance between the two matrices is defined as $$d(\mathcal{A},\mathcal{B})=max{\sigma_{1}(A_{\bot}^{H}B),\sigma_{1}(B_{\bot}^{H}A)},$$ Where $A_{\bot}^{H}$ is the basis of the orthogonal complement of $\mathcal{A}$, and $A$ is a set of orthogonal basis of $\mathcal{A}$.
The main results
library(tidyverse) library(rrpack) library(knitr) library(mvtnorm) library(gridExtra) library(MASS) library(xtable) set.seed(6419)
n = 38 # samples q = 44 # number of response variables rho = 0.5 # correlation between Xs rX = 8 # rank of X r = 5 # rank of C p = 23 # number of explanatory variables
b set to 0.05
library(SC19074) sim21 <- rrr.sim6(n, p, q, r, b = 0.05, rho, rX) sim_ann21 <- rrr2(sim21$Y, sim21$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_res21 <- stability_ann_sim(sim21$Y, sim21$X, sim_ann21$lambda, subset = 0.7, ntimes = 100)
b set to 0.1
sim22 <- rrr.sim6(n, p, q, r, b = 0.1, rho, rX) sim_ann22 <- rrr2(sim22$Y, sim22$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_res22 <- stability_ann_sim(sim22$Y, sim22$X, sim_ann22$lambda, subset = 0.7, ntimes = 100)
b set to 0.3
sim23 <- rrr.sim6(n, p, q, r, b = 0.3, rho, rX) sim_ann23 <- rrr2(sim23$Y, sim23$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_res23 <- stability_ann_sim(sim23$Y, sim23$X, sim_ann23$lambda, subset = 0.7, ntimes = 100)
b set to 0
# Rank 0 # Generate X p = 46 sim4 <- rrr.sim6(n, p, q, r, b = 0, rho, rX) sim_ann4 <- rrr2(sim4$Y, sim4$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_res4 <- stability_ann_sim(sim4$Y, sim4$X, sim_ann4$lambda, subset = 0.7, ntimes = 100)
detach("package:rrpack", unload = TRUE) detach("package:MASS", unload=TRUE) # ---- stability-s2n-table ---- # Table of signal to noise ratios for each simulation s2n_table <- data.frame(Simulation = 1:4, n = n, p = p, q = q, b = c(0, 0.05, 0.1, 0.3), "Signal to noise ratio" = c(sim4$s2n, sim21$s2n, sim22$s2n, sim23$s2n), check.names = FALSE) print(xtable(s2n_table, label = "tab:stability-s2n-table", caption = "Parameters and signal to noise ratios for the four simulated datasets.", digits = c(0,0,0,0,0, 2, 3)), include.rownames = FALSE)
# ---- stability-sims-plots ---- stability_combine_sim <- function(stability_ann_results){ st_plot <- data.frame(rank = c(stability_ann_results$st_res), lambda = rep(stability_ann_results$lambda, each = nrow(stability_ann_results$st_res))) st_group <- group_by(st_plot, by = lambda) st_sum <- summarize(st_group, var = var(rank), avg = mean(rank)) %>% rename(lambda = by) st_sum <- full_join(st_sum, stability_ann_results$st_coef, by = "lambda") } stability_plots_sim <- function(sim_results, rank, title = "sim_results", subtitle = ""){ st_combined <- stability_combine_sim(sim_results) # Plot mean vs variance varmeanplot <- ggplot(st_combined, aes(x = avg, y = sqrt(var))) + geom_point() + geom_vline(aes(xintercept = rank)) + xlab("Mean estimated rank") + ylab("Standard deviation of subspace distance") + ggtitle(title, subtitle = subtitle) + theme_bw() # Plot mean vs variance of coefficients coefplot <- ggplot(st_combined, aes(x = avg, y = sqrt(var.subspace.coef))) + geom_point() + geom_vline(aes(xintercept = rank)) + xlab("Mean estimated rank") + ylab(expression(sqrt("Mean of coefficient ranks"))) + ggtitle(title, subtitle = subtitle) + theme_bw() return(list(rplot = varmeanplot, coefplot = coefplot)) }
p21 <- stability_plots_sim(sim_res21, rank = r, title = "Simulation 2", subtitle = "b = 0.05") p22 <- stability_plots_sim(sim_res22, rank = r, title = "Simulation 3", subtitle = "b = 0.1") p23 <- stability_plots_sim(sim_res23, rank = r, title = "Simulation 4", subtitle = "b = 0.3") p4 <- stability_plots_sim(sim_res4, rank = 0, title = "Simulation 1", subtitle = "b = 0")
# ---- stability-sims-rplots ---- grid.arrange(p4$rplot, p21$rplot, p22$rplot, p23$rplot, nrow = 2) # ---- stability-sims-coefplots ---- grid.arrange(p4$coefplot, p21$coefplot, p22$coefplot, p23$coefplot, nrow = 2) # ---- stability-sims-r0-plots ---- grid.arrange(p4$rplot, p4$coefplot, nrow = 1)
# ---- rank1-sims ---- # Simulations and plots for rank 1 matrices set.seed(9988999) n = 38 # samples q = 44 # number of response variables rho = 0.5 # correlation between Xs rX = 8 # rank of X p = 23 # number of explanatory variables simr1 <- rrr.sim6(n, p, q, r=1, b = 0.3, rho, rX) simr1$s2n sim_annr1 <- rrr2(simr1$Y, simr1$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_resr1 <- stability_ann_sim(simr1$Y, simr1$X, sim_annr1$lambda, subset = 0.7, ntimes = 100) simr12 <- rrr.sim6(n, p, q, r=1, b = 0.05, rho, rX) simr12$s2n sim_annr12 <- rrr2(simr12$Y, simr12$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_resr12 <- stability_ann_sim(simr12$Y, simr12$X, sim_annr12$lambda, subset = 0.7, ntimes = 100) simr13 <- rrr.sim6(n, p, q, r=1, b = 0.1, rho, rX) simr13$s2n sim_annr13 <- rrr2(simr13$Y, simr13$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_resr13 <- stability_ann_sim(simr13$Y, simr13$X, sim_annr13$lambda, subset = 0.7, ntimes = 100) simr14 <- rrr.sim6(n, p, q, r=1, b = 0, rho, rX) simr14$s2n sim_annr14 <- rrr2(simr14$Y, simr14$X, penaltySVD = "ann", modstr = list(gamma = 2)) sim_resr14 <- stability_ann_sim(simr14$Y, simr14$X, sim_annr14$lambda, subset = 0.7, ntimes = 100) pr1 <- stability_plots_sim(sim_resr1, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr1$s2n, 2))) pr12 <- stability_plots_sim(sim_resr12, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr12$s2n, 2))) pr13 <- stability_plots_sim(sim_resr13, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr13$s2n, 2))) pr14 <- stability_plots_sim(sim_resr14, rank = 1, title = "", subtitle = paste("Signal-to-noise ratio: ", round(simr14$s2n, 2)))
grid.arrange(pr1$coefplot, pr12$coefplot, pr13$coefplot, pr14$coefplot, nrow = 2)
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