experiments with the variance
This commit is contained in:
Niclas
@@ -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 = ", ")))
|
||||
),
|
||||
|
||||
@@ -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")
|
||||
|
||||
Reference in New Issue
Block a user