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experiments with the variance

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This commit is contained in:
Niclas
2026-02-11 19:00:56 +01:00
parent c2c759bb04
commit 9db48a9a33
4 changed files with 364 additions and 24 deletions
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32
R/singular_value_plot.R
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@@ -59,6 +59,7 @@ expr_to_label <- function(expr) {
#' @param curve_col Colour of the reference curve (default = `"red"`).
#' @param curve_lwd Line width of the reference curve (default = 2).
#' @param log_plot If True, then the y-axis is on a log scale.
#' @param main_title Main title for the plot
#' @return A list with the following components
#' \item{K}{Integer vector `1:maxK`.}
#' \item{sv}{Numeric vector of the smallest singular values for each `K`.}
@@ -98,7 +99,8 @@ smallest_sv_sequence <- function(
curve_to = NULL,
curve_col = "red",
curve_lwd = 2,
log_plot = FALSE
log_plot = FALSE,
main_title = "Smallest singular value vs. K"
) {
## 1. Input validation =======================================================
if (!is.numeric(a) || length(a) == 0) {
@@ -129,6 +131,9 @@ smallest_sv_sequence <- function(
if (!inherits(curve_expr, "call") && !is.character(curve_expr)) {
stop("`curve_expr` must be a call (e.g., quote(20/sqrt(x))) or a character string.")
}
if (!is.character(main_title)){
stop("`main_title` must be a character vector.")
}
## 2. Prepare storage ========================================================
K_vec <- seq_len(maxK)
@@ -147,6 +152,8 @@ smallest_sv_sequence <- function(
guard = guard
)
Q <- 1 /sqrt(n) * Q
sv_res <- compute_minmax_sv(Q)
if (!is.list(sv_res) || is.null(sv_res$smallest_singular_value)) {
stop("`compute_minmax_sv()` must return a list containing `$smallest_singular_value`.")
@@ -157,6 +164,7 @@ smallest_sv_sequence <- function(
## 4. Plotting (optional) ====================================================
if (plot) {
## Basic scatter/line plot of the singular values
par(mar = c(5, 4, 4, 8)) # extra space on the right for the legend
plot_args <- list(
x = K_vec,
y = smallest_sv,
@@ -165,17 +173,21 @@ smallest_sv_sequence <- function(
col = "steelblue",
xlab = "K subdivisions",
ylab = "Smallest singular value of Q",
main = "Smallest singular value vs. K"
main = main_title
)
if (log_plot) plot_args$log <- "y"
do.call(graphics::plot, plot_args)
# graphics::plot(
# K_vec, smallest_sv,
# type = "b", pch = 19, col = "steelblue",
# xlab = "K subdivisions", ylab = "Smallest singular value of Q",
# main = "Smallest singular value vs. K"
# )
# add legend. The par(xpd = ...) allows drawing outside of the plot region.
par(xpd = TRUE)
legend("topright",
inset=c(-0.2,0),
legend=c("SV of Q"),
col="steelblue",
title="Legend",
pch = 16,
bty = "n")
par(xpd = FALSE)
## Add the reference curve if requested
if (add_curve) {
## Determine sensible defaults for the curve limits
@@ -197,8 +209,8 @@ smallest_sv_sequence <- function(
# add label with the curve expression
label_txt <- expr_to_label(curve_expr)
x_pos <- curve_from + 0.9 * (curve_to - curve_from)
y_pos <- 0.9 * max(smallest_sv)
x_pos <- curve_from + 0.8 * (curve_to - curve_from)
y_pos <- 0.85 * max(smallest_sv)
graphics::text(
x = x_pos, y = y_pos,
labels = label_txt,

3
R/singular_values.R
View File

@@ -192,6 +192,9 @@ compute_matrix <- function(
#' @export
compute_minmax_sv <- function(M) {
s <- svd(M, nu=0, nv=0)$d
# just a check if we compute the right thing
# s <- sqrt(eigen(M %*% t(M), symmetric = TRUE, only.value=TRUE)$values)
list(
largest_singular_value = max(s),

6
scripts/plot_densities.R
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@@ -50,14 +50,14 @@ title(
)
## 3.3 Plot one dimensional X_i ================================================
p3 <- create_cond_density(a1, dnorm, pnorm, X1)
givenX <- c(-0.5)
p3 <- create_cond_density(2, dnorm, pnorm, X1)
givenX <- c(-3.0)
p3_plot <- \(x) p3(x, givenX)
plot.function(p3_plot, xlab="u", ylab="p(u |x) ")
title(main=expression("Conditional density for" ~ v %~% N(0, 1)), line=2)
title(
main = bquote(
a == (.(paste(a1, collapse = ", "))) ~ "," ~
a == (.(paste(c(2), collapse = ", "))) ~ "," ~
N == .(N) ~ "," ~
x == (.(paste(givenX, collapse = ", ")))
),

347
scripts/plot_sv.R
View File

@@ -3,20 +3,104 @@ source(here::here("R", "singular_values.R"))
source(here::here("R", "graphon_distribution.R"))
source(here::here("R", "singular_value_plot.R"))
# https://stackoverflow.com/a/5790430
resetPar <- function() {
dev.new()
op <- par(no.readonly = TRUE)
dev.off()
op
}
calc_conv_rate <- function(x,y) {
if (!is.numeric(x) || length(x) == 0) {
stop("`x` must be a non‑empty numeric vector.")
}
if (!is.numeric(y) || length(y) == 0) {
stop("`y` must be a non‑empty numeric vector.")
}
if (length(x) != length(y)) {
stop("`x` and `y` must have the same length.")
}
df <- data.frame("x" = x, "y" = y)
lm_model <- lm(log(y) ~ log(x), data=df)
C <- exp(coefficients(lm_model)[[1]])
alpha <- coefficients(lm_model)[[2]]
df[, "y_pred"] <- C * df[, "x"]^alpha
df[, "residual"] <- df[, "y"] - df[, "y_pred"]
out <- list(
"C" = C,
"alpha" = alpha,
"obs" = df
)
out
}
calc_exp_conv_rate <- function(x,y) {
if (!is.numeric(x) || length(x) == 0) {
stop("`x` must be a non‑empty numeric vector.")
}
if (!is.numeric(y) || length(y) == 0) {
stop("`y` must be a non‑empty numeric vector.")
}
if (length(x) != length(y)) {
stop("`x` and `y` must have the same length.")
}
df <- data.frame("x" = x, "y" = y)
fit <- nls(y ~ C * exp(r * x^m),
start = list(C = min(y), r= -0.5, m = 1))
}
# Nearly match with sample function --------------------------------------------
# v ~ N(0,1) and X ~ discrete Uniform on [1:n]
smallest_sv_sequence(
out <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK= 20,
n = 9,
maxK= 3,
sampler_fn = sample,
guard=1e-12,
plot=TRUE,
curve_expr = quote(20 / sqrt(x))
log_plot = TRUE,
curve_expr = quote(1 / x^0.545)
)
conv_rate <- calc_conv_rate(out$K, out$sv)
# Normally distributed X ~ N(0,1) and v ~ N(0,1) -------------------------------
smallest_sv_sequence(
out <- smallest_sv_sequence(
a = c(0.5),
n = 1200,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
curve_expr = quote(1.5 * exp(-0.95 * x^1.34))
#curve_expr = quote( 1/exp(x^1.32))
)
# convergence rate does not work here, probably because the underlying model
# does not work well
conv_rate <- calc_conv_rate(out$K[1:20], out$sv[1:20])
# Uniform distributed X ~ U[0,1] and v ~ N(0,1) --------------------------------
out <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(runif(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
curve_expr = quote(1* exp(-1.1 * x^1.5))
)
# here the optimal fit does not work too, probably other model
calc_conv_rate(out$K[1:9], out$sv[1:9])
# Compare of parameters of Normal distribution ----------------------------------
#
out_sd0_5 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
@@ -24,16 +108,257 @@ smallest_sv_sequence(
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
curve_expr = quote(20 / exp(x^1.32))
fv = function(x) {dnorm(x, mean=0, sd=0.5)},
Fv = function(x) {pnorm(x, mean=0, sd=0.5)},
main_title="Smallest SV of v~ N(0,0.5^2) distribution"
)
# Uniform distributed X ~ U[0,1] and v ~ N(0,1) --------------------------------
smallest_sv_sequence(
out_sd1 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(runif(n), ncol = 1L),
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
curve_expr = quote(20 / x^2)
)
log_plot = TRUE,
fv = function(x) {dnorm(x, mean=0, sd=1)},
Fv = function(x) {pnorm(x, mean=0, sd=1)},
main_title="Smallest SV of v~ N(0,1) distribution"
)
out_sd2 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
fv = function(x) {dnorm(x, mean=0, sd=2)},
Fv = function(x) {pnorm(x, mean=0, sd=2)},
main_title="Smallest SV of v~ N(0,2^2) distribution"
)
out_sd4 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
fv = function(x) {dnorm(x, mean=0, sd=4)},
Fv = function(x) {pnorm(x, mean=0, sd=4)},
main_title="Smallest SV of v~ N(0,4^2) distribution"
)
par(mar = c(5, 4, 4, 8))
plot(out_sd0_5$K, out_sd0_5$sv,
type = "b",
pch = 19,
col = "#D3BA68FF",
xlab = "K subdivisions",
ylab = "Smallest singular value of Q",
main="smallest SV for different variances of a normal distribution",
sub = "n = 400, a = 0.5",
log="y")
lines(out_sd1$K, out_sd1$sv,
type="b", pch=19, col="#D5695DFF", add=TRUE)
lines(out_sd2$K, out_sd2$sv,
type = "b", pch=19, col="#5D8CA8FF", add=TRUE)
lines(out_sd4$K, out_sd4$sv,
type = "b", pch=19, col="#65A479FF", add=TRUE)
par(xpd = TRUE)
legend("topright",
inset=c(-0.2,0),
legend=c("sd=0.5", "sd=1", "sd=2", "sd=4"),
col=c("#D3BA68FF", "#D5695DFF","#5D8CA8FF", "#65A479FF" ),
title="Legend",
pch = 16,
bty = "n")
# Break phenomena of Exp-distribution ------------------------------------------
out_exp <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
fv = function(x) {dexp(x, rate=1)},
Fv = function(x) {pexp(x, rate=1)},
main_title="Smallest SV of v~ Exp(1) distribution"
)
par(resetPar)
plot(out_exp$K, out_exp$sv,
log="y",
xlab="K subdivsions",
ylab="Smallest singular value of Q",
col="steelblue",
type="b",
main="Smallest singular value for v ~ Exp(1)",
sub="a = 0.5, n = 400")
arrows(8, 1e-5, 6.5, 1e-7, angle=20, lty = 1, lwd=2)
text(8.5, 1e-5, "Break only seen for exp-distribution", pos=4)
## Observations of the break point depending the rate --------------------------
#
# Note: this also depends on the the parameter n of samples
out_exp_1 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 80,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=FALSE,
fv = function(x) {dexp(x, rate=1)},
Fv = function(x) {pexp(x, rate=1)}
)
out_exp_2 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 80,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=FALSE,
fv = function(x) {dexp(x, rate=2)},
Fv = function(x) {pexp(x, rate=2)}
)
out_exp_3 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 80,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=FALSE,
fv = function(x) {dexp(x, rate=3)},
Fv = function(x) {pexp(x, rate=3)}
)
out_exp_4 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 80,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=FALSE,
fv = function(x) {dexp(x, rate=4)},
Fv = function(x) {pexp(x, rate=4)}
)
par(mar = c(5, 4, 4, 8))
plot(out_exp_1$K, out_exp_1$sv,
type="b", col="#D3BA68FF", log="y",
main="Smallest SV of Q for different rates of Exp-distribution",
ylab="Smallest singular value of Q",
xlab="K subdivisions",
sub="a = 0.5, n = 400, depending also on n")
lines(out_exp_2$K, out_exp_2$sv, type="b", col="#D5695DFF")
lines(out_exp_3$K, out_exp_3$sv, type="b", col="#5D8CA8FF")
lines(out_exp_4$K, out_exp_4$sv, type="b", col="#65A479FF")
par(xpd=TRUE)
legend("topright",
inset=c(-0.2,0),
legend=c(expression(lambda == 1), expression(lambda == 2), expression(lambda == 3), expression(lambda == 4)),
col=c("#D3BA68FF", "#D5695DFF","#5D8CA8FF", "#65A479FF" ),
title="Rate",
pch = 16,
bty = "n")
# Use of the gamma distribution ------------------------------------------------
# Fix the parameters, such that the mean stays the same and the variance is
# changing.
# From the documentation
# Note that for smallish values of shape (and moderate scale) a large parts of
# the mass of the Gamma distribution is on values of x
# so near zero that they will be represented as zero in computer arithmetic.
# So rgamma may well return values which will be represented as zero.
# (This will also happen for very large values of scale since the actual generation is done for scale = 1.)
#
# Take E(X) = 5, so sigma = 5 / alpha, and with this we have
# Var(X) = sigma^2 * alpha = 25 / alpha.
# -> Increasing alpha yields lower variance
# alpha = 1
alpha <- 1.0
out_gamma_1 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
fv = function(x) {dgamma(x, shape = alpha, rate = 5 / alpha)},
Fv = function(x) {pexp(x, rate=1)},
main_title="Smallest SV of v~ Gamma(1, 5) distribution"
)
# alpha = 2
alpha <- 2.0
out_gamma_2 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
fv = function(x) {dgamma(x, shape = alpha, rate = 5 / alpha)},
Fv = function(x) {pexp(x, rate=1)},
main_title="Smallest SV of v~ Gamma(2, 2.5) distribution"
)
# alpha = 3
alpha <- 3.0
out_gamma_3 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
fv = function(x) {dgamma(x, shape = alpha, rate = 5 / alpha)},
Fv = function(x) {pexp(x, rate=1)},
main_title="Smallest SV of v~ Gamma(3, 5/3) distribution"
)
# alpha = 4
alpha <- 4.0
out_gamma_4 <- smallest_sv_sequence(
a = c(0.5),
n = 400,
maxK = 20,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=TRUE,
log_plot = TRUE,
fv = function(x) {dgamma(x, shape = alpha, rate = 5 / alpha)},
Fv = function(x) {pexp(x, rate=1)},
main_title="Smallest SV of v~ Gamma(3, 5/3) distribution"
)
par(mar = c(5, 4, 4, 8))
plot(out_gamma_1$K, out_gamma_1$sv,
type="b", col="#D3BA68FF", log="y",
main="Smallest SV of Q for variance of the Gamma distribution",
ylab="Smallest singular value of Q",
xlab="K subdivisions",
sub="a = 0.5, n = 400")
lines(out_gamma_2$K, out_gamma_2$sv, type="b", col="#D5695DFF")
lines(out_gamma_3$K, out_gamma_3$sv, type="b", col="#5D8CA8FF")
lines(out_gamma_4$K, out_gamma_4$sv, type="b", col="#65A479FF")
par(xpd=TRUE)
legend("topright",
inset=c(-0.2,0),
legend=c(expression(alpha == 1), expression(alpha == 2), expression(alpha == 3), expression(alpha == 4)),
col=c("#D3BA68FF", "#D5695DFF","#5D8CA8FF", "#65A479FF" ),
title="Rate",
pch = 16,
bty = "n")