146 lines
3.8 KiB
Plaintext
146 lines
3.8 KiB
Plaintext
---
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title: "Plots of n vs. k"
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author: "Niclas"
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format: html
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editor: visual
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execute:
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echo: true
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working-directory: ../
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---
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# Plots of the dimensions
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## Setup
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We consider the matrix $QQ^\top$ and look at the smallest eigenvalue, i.e. the
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smallest non-zero singular value of $Q$.
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The matrix $Q$ is given by
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$$
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Q_{ik} = \int_{\frac{k}{K}}^{\frac{k+1}{K}} p_a(u| X_i) \, du
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$$
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with
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$$
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p_a(u|X) = \frac{f_v(F_a^{-1}(u) - a^\top X)}{f_a(F_a^{-1}(u))}
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$$
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## Plots of n vs. k
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- The $v$'s are normally distributed with $v \sim \mathcal N(0,1)$
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- Plot $n = 100, 200, 300, 400$ and $k = 1, \dots, K$ with $K = \sqrt n$.
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```{r Load Libraries}
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# load local files
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source(here::here("R", "singular_values.R"))
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source(here::here("R", "graphon_distribution.R"))
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source(here::here("R","singular_value_plot.R"))
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# load libaries for data handling
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library(ggplot2)
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library(dplyr)
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library(latex2exp)
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```
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```{r Compute the data}
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#| cache: true
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#| echo: false
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#| collapse: true
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ns <- c(100, 200, 300, 400, 500, 600, 700, 800)
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Ks <- floor(sqrt(ns))
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as <- c(0.5, 1.0, 1.5, 2.0)
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# set a global seed
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set.seed(42)
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results <- data.frame(dim_n = integer(),
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dim_k = integer(),
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param_a = double(),
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ssv = double())
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for (a in as) {
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for (i in 1:length(ns)) {
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n <- ns[i]
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K <- Ks[i]
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# use the default seed 1L
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out <- smallest_sv_sequence(
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a = a,
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n = n,
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maxK = K,
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sampler_fn =function(n) matrix(rnorm(n), ncol = 1L),
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guard=1e-12,
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plot=FALSE,
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fv = function(x) {dnorm(x, mean=0, sd=1)},
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Fv = function(x) {pnorm(x, mean=0, sd=1)}
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)
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current_res <- data.frame(dim_n = rep(n, K), dim_k = out$K, param_a = rep(a, K), ssv = out$sv)
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results <- rbind(results, current_res)
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}
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}
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```
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```{r plot the results}
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#| cache: true
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#| echo: false
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#| collapse: true
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#| fig-cap: "Simulation of the smallest singular values w.r.t. a, n and k"
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results |>
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filter(dim_n <= 400) |>
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mutate(param_a = as.factor(param_a),
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dim_n = as.factor(dim_n)) |>
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group_by(param_a, dim_n) |>
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ggplot(aes(dim_k, ssv, col=dim_n, shape=param_a, interaction(dim_n, param_a))) +
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geom_point(size=1.5) +
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geom_line() +
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scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$k$"),
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y=latex2exp::TeX("Smallest singular value of $Q$"),
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title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $n$, $k$, and $a$."),
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colour=latex2exp::TeX("$n$"),
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shape=latex2exp::TeX("$a$"))
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```
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The data for $n = 100$ is covered by the data for $n = 200$.
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## Analysis of the convergence
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We assume that the smallest singular value $\sigma$ can be approximated by:
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$$
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\sigma = C \cdot n^\eta \cdot k^\kappa \cdot a^\alpha
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$$
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to estimate the coefficients we make a log-transform and perform a linear regression, i.e.
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$$
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\log(\sigma) = \log (C) + \eta \log(n) + \kappa\log(k) + \alpha \log(a).
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$$
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```{r estimate coeffs}
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model1 <- results |>
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filter(ssv > 1e-15) |> # exclude to small values
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lm(formula = log(ssv) ~ log(dim_n) + log(dim_k) + log(param_a))
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summary(model1)
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plot(model1)
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```
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## Plot of n vs. ssv
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```{r plot n vs ssv}
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results |>
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filter(dim_k %in% c(2, 6, 10)) |>
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mutate(param_a = as.factor(param_a),
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dim_k = as.factor(dim_k)) |>
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group_by(param_a, dim_k) |>
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ggplot(aes(dim_n, ssv, col=dim_k, shape=param_a, interaction(dim_k, param_a))) +
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geom_point(size=1.5) +
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geom_line() +
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scale_y_log10() +
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theme_bw() +
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labs(x=latex2exp::TeX("$n$"),
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y=latex2exp::TeX("Smallest singular value of $Q$"),
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title=latex2exp::TeX("Smallest singular value of $Q$ with respect to $n$, $k$, and $a$."),
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colour=latex2exp::TeX("$k$"),
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shape=latex2exp::TeX("$a$"))
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```
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```{r}
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Q <- compute_matrix(1, a=0.5, n=10, K = 3, function(n) matrix(rnorm(n), ncol = 1L), fv=dnorm, Fv=pnorm)
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```
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