Groupe 03 Projet DANJOU - DUROUSSEAU
- -Exercise 1 : Negative weighted mixture
-Definition
-Question 1
-The conditions for a function f to be a probability density are :
--
-
- f is defined on \(\mathbb{R}\) -
- f is non-negative, ie \(f(x) \ge -0\), \(\forall x \in -\mathbb{R}\) -
- f is Lebesgue-integrable -
- and \(\int_{\mathbb{R}} f(x) \,dx = -1\) -
The function f, to be a density, needs to be non-negative, ie \(f(|x|) \ge 0\) when \(|x| \rightarrow \infty\)
-We have, when \(|x| \rightarrow -\infty\), \(f_i(|x|) \sim -exp(\frac{-x^2}{\sigma_i^2})\) for \(i -= 1, 2\)
-Then, \(f(|x|) \sim -exp(\frac{-x^2}{\sigma_1^2}) - exp(\frac{-x^2}{\sigma_2^2})\)
-We want \(f(|x|) \ge 0\), ie, \(exp(\frac{-x^2}{\sigma_1^2}) - -exp(\frac{-x^2}{\sigma_2^2}) \ge 0\)
-\(\Leftrightarrow \sigma_1^2 \ge -\sigma_2^2\)
-We can see that \(f_1\) dominates -the tail behavior.
-
-
Question 2
-For given parameters \((\mu_1, -\sigma_1^2)\) and \((\mu_2, -\sigma_2^2)\), we have \(\forall x \in -\mathbb{R}\), \(f(x) \ge 0\)
-\(\Leftrightarrow \frac{1}{\sigma_1} -exp(\frac{-(x-\mu_1)^2}{2 \sigma_1^2}) \ge \frac{a}{\sigma_2} -exp(\frac{-(x-\mu_2)^2}{2 \sigma_2^2})\)
-\(\Leftrightarrow 0 < a \le a^* = -\min_{x \in \mathbb{R}} \frac{f_1(x)}{f_2(x)} = \min_{x \in \mathbb{R}} -\frac{\sigma_2}{\sigma_1} exp(\frac{(x-\mu_2)^2}{2 \sigma_2^2} - -\frac{(x-\mu_1)^2}{2 \sigma_1^2})\)
-To find \(a^*\), we just have to -minimize \(g(x) := \frac{(x-\mu_2)^2}{2 -\sigma_2^2} - \frac{(x-\mu_1)^2}{2 \sigma_1^2}\)
-First we derive \(g\): \(\forall x \in \mathbb{R}, g^{\prime}(x) = \frac{x -- \mu_2}{\sigma_2^2} - \frac{x - \mu_1}{\sigma_1^2}\)
-We search \(x^*\) such that \(g^{\prime}(x^*) = 0\)
-\(\Leftrightarrow x^* = \frac{\mu_2 -\sigma_1^2 - \mu_1 \sigma_2^2}{\sigma_1^2 - \sigma_2^2}\)
-Then, we compute \(a^* = -\frac{f_1(x^*)}{f_2(x^*)}\)
-We call \(C \in \mathbb{R}\) the -normalization constant such that \(f(x) = C -(f_1(x) - af_2(x))\)
-To find \(C\), we know that \(1 = \int_{\mathbb{R}} f(x) \,dx = -\int_{\mathbb{R}} C (f_1(x) - af_2(x)) \,dx = C \int_{\mathbb{R}} f_1(x) -\,dx - Ca \int_{\mathbb{R}} f_2(x) \,dx = C(1-a)\) as \(f_1\) and \(f_2\) are density functions and by -linearity of the integrals.
-\(\Leftrightarrow C = -\frac{1}{1-a}\)
-
-
Question 3
-f <- function(a, mu1, mu2, s1, s2, x) {
- fx <- dnorm(x, mu1, s1) - a * dnorm(x, mu2, s2)
- fx[fx < 0] <- 0
- return(fx / (1 - a))
-}
-
-a_star <- function(mu1, mu2, s1, s2) {
- x_star <- (mu2 * s1^2 - mu1 * s2^2) / (s1^2 - s2^2)
- return(dnorm(x_star, mu1, s1) / dnorm(x_star, mu2, s2))
-}mu1 <- 0
-mu2 <- 1
-s1 <- 3
-s2 <- 1
-
-x <- seq(-10, 10, length.out = 1000)
-as <- a_star(mu1, mu2, s1, s2)
-a_values <- c(0.1, 0.2, 0.3, 0.4, 0.5, as)
-
-plot(x, f(as, mu1, mu2, s1, s2, x),
- type = "l",
- col = "red",
- xlab = "x",
- ylab = "f(a, mu1, mu2, s1, s2, x)",
- main = "Density function of f(a, mu1, mu2, s1, s2, x) for different a"
-)
-for (i in (length(a_values) - 1):1) {
- lines(x, f(a_values[i], mu1, mu2, s1, s2, x), lty = 3, col = "blue")
-}
-legend("topright", legend = c("a = a*", "a != a*"), col = c("red", "blue"), lty = 1)
We observe that for small values of a, the density f is close to the -density of \(f_1 \sim \mathcal{N}(\mu_1, -\sigma_1^2)\). When a increases, the shape of evolves into the -combinaison of two normal distributions. We observe that for \(a = a^*\), the density the largest value of -a for which the density is still a density function, indeed for \(a > a^*\), the function f takes negative -values so it is no longer a density.
-s2_values <- seq(1, 10, length.out = 5)
-a <- 0.2
-
-plot(x, f(a, mu1, mu2, s1, max(s2_values), x),
- type = "l",
- xlab = "x",
- ylab = "f(a, mu1, mu2, s1, s2, x)",
- col = "red",
- main = "Density function of f(a, mu1, mu2, s1, s2, x) for different s2"
-)
-
-for (i in length(s2_values):1) {
- lines(x, f(a, mu1, mu2, s1, s2_values[i], x), lty = 1, col = rainbow(length(s2_values))[i])
-}
-
-legend("topright", legend = paste("s2 =", s2_values), col = rainbow(length(s2_values)), lty = 1)
We observe that when \(\sigma_2^2 = -1\), the density \(f\) has two -peaks and when \(\sigma_2^1 > 1\), -the density \(f\) has only one -peak.
-mu1 <- 0
-mu2 <- 1
-sigma1 <- 3
-sigma2 <- 1
-a <- 0.2
-as <- a_star(mu1, mu2, sigma1, sigma2)
-
-cat(sprintf("a* = %f, a = %f, a <= a* [%s]", as, a, a <= as))## a* = 0.313138, a = 0.200000, a <= a* [TRUE]We have \(\sigma_1^2 \ge -\sigma_2^2\) and \(0 < a \le -a^*\), so the numerical values are compatible with the -constraints defined above.
-Inverse c.d.f Random Variable simulation
-Question 4
-To prove that the cumulative density function F associated with f is -available in closed form, we need to compute \(F(x) = \int_{-\infty}^{x} f(t) \,dt = -\frac{1}{1-a} (\int_{-\infty}^{x} f_1(t)\, dt - a \int_{-\infty}^{x} -f_2(t)\, dt) = \frac{1}{1-a} (F_1(x) - aF_2(x))\) where \(F_1\) and \(F_2\) are the cumulative density functions -of \(f_1 \sim \mathcal{N}(\mu1, -\sigma_1^2)\) and \(f_2 \sim -\mathcal{N}(\mu2, \sigma_2^2)\) respectively.
-Then, \(F\) is a closed-form as a -finite sum of closed forms.
-F <- function(a, mu1, mu2, s1, s2, x) {
- Fx <- pnorm(x, mu1, s1) - a * pnorm(x, mu2, s2)
- return(Fx / (1 - a))
-}To construct an algorithm that returns the value of the inverse -function method as a function of u ∈(0,1), of the parameters \(a, \mu_1, \mu_2, \sigma_1, \sigma_2, x\), -and of an approximation precision \(\epsilon\), we can use the bisection -method.
-We fixe \(\epsilon > 0\).
-We set \(u \in (0, 1)\).
-We define \(L = -10\) and \(U = -L\), the bounds and \(M = \frac{L + U}{2}\), the middle of our
-interval.
-While \(U - L > \epsilon\) :
-
-
- We compute \(F(M) = \frac{1}{1-a} (F_1(M) -- aF_2(M))\) -
- If \(F(M) < u\), we set \(L = M\) -
- Else, we set \(U = M\) -
- We set \(M = \frac{L + U}{2}\) -
End while
-For the generation of random variables from F, we can use the inverse
-transform sampling method.
-
We set \(X\) as an empty array and
-\(n\) the number of random variables we
-want to generate.
-We fixe \(\epsilon > 0\).
-For \(i = 1, \dots, n\) :
-
-
We set \(u \in (0, 1)\).
-
-We define \(L = -10\) and \(U = -L\), the bounds and \(M = \frac{L + U}{2}\), the middle of our -interval.
-
-While \(U - L > \epsilon\) -:
-We compute \(F(M) = \frac{1}{1-a} -(F_1(M) - aF_2(M))\)
-If \(F(M) < u\), we set \(L = M\)
-Else, we set \(U = M\)
-We set \(M = \frac{L + -U}{2}\)
-End while
-We add \(M\) to \(X\)
-
End for We return \(X\)
-Question 5
-inv_cdf <- function(n) {
- X <- numeric(n)
- for (i in 1:n) {
- u <- runif(1)
- L <- -10
- U <- -L
- M <- (L + U) / 2
- while (U - L > 1e-6) {
- FM <- F(a, mu1, mu2, s1, s2, M)
- if (FM < u) {
- L <- M
- } else {
- U <- M
- }
- M <- (L + U) / 2
- }
- X[i] <- M
- }
- return(X)
-}set.seed(123)
-n <- 10000
-X <- inv_cdf(n)
-x <- seq(-10, 10, length.out = 1000)
-
-hist(X, breaks = 100, freq = FALSE, col = "lightblue", main = "Empirical density function", xlab = "x")
-lines(x, f(a, mu1, mu2, s1, s2, x), col = "red")
-legend("topright", legend = c("Theorical density", "Empirical density"), col = c("red", "lightblue"), lty = 1)
plot(ecdf(X), col = "blue", main = "Empirical cumulative distribution function", xlab = "x", ylab = "F(x)")
-lines(x, F(a, mu1, mu2, s1, s2, x), col = "red")
-legend("bottomright", legend = c("Theoretical cdf", "Empirical cdf"), col = c("red", "blue"), lty = 1)
We can see of both graphs that the empirical cumulative distribution -function is close to the theoretical one.
-Accept-Reject Random Variable simulation
-Question 6
-To simulate under f using the accept-reject algorithm, we need to
-find a density function g such that \(f(x) \le
-M g(x)\) for all \(x \in
-\mathbb{R}\), where M is a constant.
-
Then, we generate \(X \sim g\) and
-\(U \sim \mathcal{U}([0,1])\).
-We accept \(Y = X\) if \(U \le \frac{f(X)}{Mg(X)}\). Return \(1\) otherwise.
The probability of acceptance is \(\int_{\mathbb{R}} \frac{f(x)}{Mg(x)} g(x) \,dx = -\frac{1}{M} \int_{\mathbb{R}} f(x) \,dx = \frac{1}{M}\)
-Here we pose \(g = f1\).
-Then we have \(\frac{f(x)}{g(x)} = -\frac{1}{1-a} (1 - a\frac{f_2(x)}{f_1(x)})\)
-We pose earlier that \(a^* = \min_{x \in -\mathbb{R}} \frac{f_1(x)}{f_2(x)} \Rightarrow \frac{1}{a^*} = \max_{x -\in \mathbb{R}} \frac{f_2(x)}{f_1(x)}\).
-We compute in our fist equation : \(\frac{1}{1-a} (1 - a\frac{f_2(x)}{f_1(x)}) -\leq \frac{1}{1-a} (1 - \frac{a}{a^*}) \leq \frac{1}{1-a}\) -because \(a \leq a^* \Rightarrow 1 - -\frac{a}{a^*} \leq 0\)
-To conclude, we have \(M = -\frac{1}{1-a}\) and the probability of acceptance is \(\frac{1}{M} = 1-a\)
-Question 7
-accept_reject <- function(n, a) {
- X <- numeric(0)
- M <- 1 / (1 - a)
-
- while (length(X) < n) {
- Y <- rnorm(1, mu1, s1)
- U <- runif(1)
-
- if (U <= f(a, mu1, mu2, s1, s2, Y) / (M * dnorm(Y, mu1, s1))) {
- X <- append(X, Y)
- }
- }
- return(X)
-}set.seed(123)
-n <- 10000
-a <- 0.2
-X <- accept_reject(n, a)
-x <- seq(-10, 10, length.out = 1000)
-
-hist(X, breaks = 100, freq = FALSE, col = "lightblue", main = "Empirical density function", xlab = "x")
-lines(x, f(a, mu1, mu2, s1, s2, x), col = "red")
-legend("topright", legend = c("Theorical density", "Empirical density"), col = c("red", "lightblue"), lty = 1)
set.seed(123)
-
-acceptance_rate <- function(n, a = 0.2) {
- Y <- rnorm(n, mu1, s1)
- U <- runif(n)
- return(mean(U <= f(a, mu1, mu2, s1, s2, Y) / (M * dnorm(Y, mu1, s1))))
-}
-
-M <- 1 / (1 - a)
-n <- 10000
-cat(sprintf("[M = %.2f] Empirical acceptance rate: %f, Theoretical acceptance rate: %f \n", M, acceptance_rate(n), 1 / M))## [M = 1.25] Empirical acceptance rate: 0.802600, Theoretical acceptance rate: 0.800000Question 8
-set.seed(123)
-a_values <- seq(0.01, 1, length.out = 100)
-acceptance_rates <- numeric(length(a_values))
-as <- a_star(mu1, mu2, s1, s2)
-
-for (i in seq_along(a_values)) {
- acceptance_rates[i] <- acceptance_rate(n, a_values[i])
-}
-
-plot(a_values, acceptance_rates, type = "l", col = "blue", xlab = "a", ylab = "Acceptance rate", main = "Acceptance rate as a function of a")
-points(as, acceptance_rate(n, as), col = "red", pch = 16)
-legend("topright", legend = c("Acceptance rate for all a", "Acceptance rate for a_star"), col = c("lightblue", "red"), lty = c(1, NA), pch = c(NA, 16))