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create plots with a dependence

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Niclas
2026-03-11 14:14:10 +01:00
parent 14b4425570
commit 76f982069e
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scripts/.gitignore vendored
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*.html
*plots_dimensions_files/
*plots_a_dependence_files/

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scripts/plots_a_dependence.qmd Normal file
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---
title: "plots of a dependence"
author: "Niclas"
format: html
editor: visual
execute:
echo: true
working-directory: ../
---
# Plots of the dimensions
## Setup
We consider the matrix $QQ^\top$ and look at the smallest eigenvalue, i.e. the
smallest non-zero singular value of $Q$.
The matrix $Q$ is given by
$$
Q_{ik} = \int_{\frac{k}{K}}^{\frac{k+1}{K}} p_a(u| X_i) \, du
$$
with
$$
p_a(u|X) = \frac{f_v(F_a^{-1}(u) - a^\top X)}{f_a(F_a^{-1}(u))}
$$
In this document we plot different the smallest eigenvalue in dependence of the
parameter $a$ with different "ratios" of the parameters $n$ and $k$.
```{r loading libraries}
#| cache: true
#| echo: false
#| collapse: true
# load local files
source(here::here("R", "singular_values.R"))
source(here::here("R", "graphon_distribution.R"))
source(here::here("R","singular_value_plot.R"))
# load libaries for data handling
library(ggplot2)
library(dplyr)
library(latex2exp)
```
## Hyperparameter with n vs. k = log(n)
```{r n / log(n) hyperparameters data generation}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 1000, 100)
Ks <- floor(log(ns))
as <- seq(0, 20, 2)
set.seed(100)
results <- data.frame(dim_n = integer(),
dim_k = integer(),
param_a = double(),
ssv = double())
for (a in as) {
for (i in 1:length(ns)) {
n <- ns[i]
K <- Ks[i]
# use the default seed 1L
out <- smallest_sv_sequence(
a = a,
n = n,
maxK = K,
sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
guard=1e-12,
plot=FALSE,
fv = function(x) {dnorm(x, mean=0, sd=1)},
Fv = function(x) {pnorm(x, mean=0, sd=1)}
)
current_res <- data.frame(dim_n = rep(n, K), dim_k = out$K, param_a = rep(a, K), ssv = out$sv)
results <- rbind(results, current_res)
}
}
```
```{r hyperparameter n vs log(n) plotting}
results |>
mutate(dim_n = as.factor(dim_n)) |>
group_by(dim_n) |>
filter(dim_k == max(dim_k) & param_a > 0) |>
ggplot(aes(param_a, ssv, col=dim_n)) +
geom_point(size=1.5) +
geom_line() +
#scale_y_log10() +
theme_bw() +
labs(x=latex2exp::TeX("$a$"),
y=latex2exp::TeX("Smallest singular value of $Q$"),
title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
subtitle = latex2exp::TeX(("Hyperparameter $k = \\lfloor\\log(n) \\rfloor$")),
colour=latex2exp::TeX("$n$"),
shape=latex2exp::TeX("$a$"))
```
Here we have relatively large values of the smallest singular value (ssv) of $Q$ since
the values for $k$ are relatively small. We have already observed, that with
larger $k$ the ssv shrinks rapidly towards zero. However the largest value for $k$
is at most $k = `r floor(log(1000))`$ for $n = 1000$.
We omitted the value $a = 0$, as this results in a ssv of $10^{-61}$.
Note that this is only one sample and it could produce sighlty different results
for other seeds.
## Hyperparameter $n vs k = n^alpha$
```{r n / log(n) hyperparameters data generation}
#| cache: true
#| echo: false
#| collapse: true
ns <- seq(100, 1000, 100)
as <- seq(0, 20, 2)
alphas <- seq(0.1, 0.5, 0.1)
set.seed(100)
results <- data.frame(dim_n = integer(),
dim_k = integer(),
param_a = double(),
param_alpha = double(),
ssv = double())
for (a in as) {
for (i in 1:length(ns)) {
for (j in 1:length(alphas)) {
n <- ns[i]
K <- floor(n^alphas[j])
if (!K > 0) next # skip if K is equal to zero
# use the default seed 1L
Q <- compute_matrix(seed=1L,
a= a,
n = n,
K = K,
sample_X_fn = function(n) {matrix(rnorm(n), ncol = 1L)},
fv = function(x) {dnorm(x, mean=0, sd=1)},
Fv = function(x) {pnorm(x, mean=0, sd=1)},
guard = 1e-12)
ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
current_res <- data.frame(dim_n = n, dim_k = K, param_a = a, param_alpha=alphas[j], ssv =ssv)
results <- rbind(results, current_res)
}
}
}
```
```{r hyperparameter n / k^alpha = const plotting}
results |>
filter(dim_n %in% c(100, 500, 1000)) |>
mutate(dim_n = as.factor(dim_n),
param_alpha = as.factor(param_alpha)) |>
group_by(dim_n, param_alpha) |>
ggplot(aes(param_a, ssv, col=dim_n, shape=param_alpha)) +
geom_point(size=1.5) +
geom_line() +
#scale_y_log10() +
theme_bw() +
labs(x=latex2exp::TeX("$a$"),
y=latex2exp::TeX("Smallest singular value of $Q$"),
title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $a$."),
subtitle = latex2exp::TeX(("Hyperparameter $k = n^{\\alpha}$")),
colour=latex2exp::TeX("$n$"),
shape=latex2exp::TeX("$\\alpha$"))
```
Here we use $K = \lfloor n^\alpha\rfloor$ as hyperparameter. Why don't we see
any change w.r.t. $a$ for $\alpha = 0.1$?
## Two dimensional example
Here consider $p = 2$ covariate variables for each node. In order to make the
parameter $a$ invariant to the direction, we sample 5 different vectors for
each $a$ and rescale them. We use $K = 5$ as hyperparameter.
```{r data generation for two d example}
#| cache: true
#| echo: false
#| collapse: true
set.seed(10)
ns <- seq(100, 1000, 100)
as <- matrix(rnorm(8), ncol=4)
as_norm <- seq(1, 20, 2)
for (i in 1:ncol(as)){
as[, i] <- as[, i] / sqrt(sum(as[, i]^2))
}
results <- data.frame(dim_n = integer(),
dim_k = integer(),
param_a = integer(),
param_a_norm = double(),
ssv = double())
for (a_norm in as_norm) {
for (i in 1:length(ns)) {
for (j in 1:ncol(as)) {
n <- ns[i]
K <- 5 # floor(sqrt(n))
if (!K > 0) next # skip if K is equal to zero
# use the default seed 1L
Q <- compute_matrix(seed=1L,
a= as.vector(t(as[, j])),
n = n,
K = K,
sample_X_fn = function(n) {matrix(rnorm(2 * n), ncol = 2L)},
fv = function(x) {dnorm(x, mean=0, sd=1)},
Fv = function(x) {pnorm(x, mean=0, sd=1)},
guard = 1e-12)
ssv <- compute_minmax_sv(Q)[["smallest_singular_value"]]
current_res <- data.frame(dim_n = n, dim_k = K, param_a = j, param_a_norm=a_norm, ssv =ssv)
results <- rbind(results, current_res)
}
}
}
```
```{r 2d a parameter plotting}
results |>
filter(dim_n %in% c(100, 500, 1000)) |>
mutate(dim_n = as.factor(dim_n),
param_a = as.factor(param_a)) |>
group_by(dim_n, param_a) |>
ggplot(aes(param_a_norm, ssv, col=dim_n, shape=param_a, interaction(dim_n, param_a))) +
geom_point(size=1.5) +
geom_line() +
#scale_y_log10() +
theme_bw() +
labs(x=latex2exp::TeX("$\\|a\\|$"),
y=latex2exp::TeX("Smallest singular value of $Q$"),
title=latex2exp::TeX("Smallest singular value of $Q$ with respect to the norm of $a$."),
subtitle = latex2exp::TeX(("Hyperparameter $k = 5$")),
colour=latex2exp::TeX("$n$"),
shape=latex2exp::TeX("$\\|\\alpha\\|$"))
```
```{r plot the vectors}
plot(0, 0, xlim=c(-1.5, 1.5), ylim=c(-1.5, 1.5), type="n", xlab="x", ylab="y", asp=1)
arrows(0, 0, as[1, ], as[2, ], col=1:4, lwd=2)
title(main="Vectors for a rescaled to norm one.")
```
It seems, that the direction of the parameter $a$ has a small influence in the
smallest singular value of the matrix $Q$, but not the norm of it??

25
scripts/plots_dimensions.qmd
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@@ -44,7 +44,7 @@ library(latex2exp)
#| cache: true
#| echo: false
#| collapse: true
ns <- c(100, 200, 300, 400, 500)
ns <- c(100, 200, 300, 400, 500, 600, 700, 800)
Ks <- floor(sqrt(ns))
as <- c(0.5, 1.0, 1.5, 2.0)
@@ -83,6 +83,7 @@ for (a in as) {
#| collapse: true
#| fig-cap: "Simulation of the smallest singular values w.r.t. a, n and k"
results |>
filter(dim_n <= 400) |>
mutate(param_a = as.factor(param_a),
dim_n = as.factor(dim_n)) |>
group_by(param_a, dim_n) |>
@@ -119,4 +120,26 @@ summary(model1)
plot(model1)
```
## Plot of n vs. ssv
```{r plot n vs ssv}
results |>
filter(dim_k %in% c(2, 6, 10)) |>
mutate(param_a = as.factor(param_a),
dim_k = as.factor(dim_k)) |>
group_by(param_a, dim_k) |>
ggplot(aes(dim_n, ssv, col=dim_k, shape=param_a, interaction(dim_k, param_a))) +
geom_point(size=1.5) +
geom_line() +
scale_y_log10() +
theme_bw() +
labs(x=latex2exp::TeX("$n$"),
y=latex2exp::TeX("Smallest singular value of $Q$"),
title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $n$, $k$, and $a$."),
colour=latex2exp::TeX("$k$"),
shape=latex2exp::TeX("$a$"))
```
```{r}
Q <- compute_matrix(1, a=0.5, n=10, K = 3, function(n) matrix(rnorm(n), ncol = 1L), fv=dnorm, Fv=pnorm)
```