commit 5c94df30e2a2e8b14384093a0cdf0fcc8e892652
Author: Maxime Wack <maximewack@free.Fr>
Date:   Mon Jan 8 23:31:36 2018 +0100

    Initial commit

diff --git a/cours.pptx b/cours.pptx
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diff --git a/exemples.R b/exemples.R
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+library(tidyverse)
+
+# Un temps ----
+
+## Data
+data(iris)
+data(mtcars)
+
+mtcars %>%
+  mutate_at(vars(am, cyl, vs, gear), factor) ->
+mtcars
+
+## Deux groupes
+
+### Plots
+
+boxplot(mpg ~ am, data = mtcars)
+
+ggplot(mtcars) +
+  aes(x = am, y = mpg) +
+  geom_boxplot() +
+  geom_point(alpha = .5)
+
+### Paramétrique
+t.test(mpg ~ am, data = mtcars, var.equal = T)
+oneway.test(mpg ~ am, data = mtcars, var.equal = T)
+
+### Non-paramétrique
+wilcox.test(mpg ~ am, data = mtcars)
+
+## Plusieurs groupes
+
+### Plots
+iris %>%
+  ggplot() +
+  aes(x = Species, y = Sepal.Width) +
+  geom_violin() +
+  geom_point(position = "jitter", alpha = .5)
+
+iris %>%
+  ggplot() +
+  aes(x = Petal.Length) +
+  geom_histogram(data = iris %>% select(-Species)) +
+  geom_histogram(aes(fill = Species)) +
+  facet_grid(~Species)
+
+### Paramétrique
+oneway.test(Sepal.Width ~ Species, data = iris)
+
+### Non-paramétrique
+kruskal.test(Sepal.Width ~ Species, data = iris)
+
+## Multivarié
+
+### Régression linéaire
+
+# Y = a₁X₁ + a₂X₂ + _ + anXn + ε
+# Y = Σ aiXi + ε, ε -> N
+plot(mtcars)
+
+fit <- lm(mpg ~ am, data = mtcars)
+summary(fit)
+plot(fit)
+
+fit <- lm(mpg ~ cyl + disp + hp + drat + wt + vs + am, data = mtcars)
+summary(fit)
+plot(fit)
+
+### Régression logistique
+fit <- glm(am ~ mpg + drat + wt, data = mtcars, family = "binomial")
+summary(fit)
+plot(fit)
+
+# Deux temps ----
+
+## Data
+data(ChickWeight)
+
+### Test apparié
+
+#### Plots
+ChickWeight %>%
+  filter(Time %in% c(0, 2)) %>%
+  ggplot() +
+  aes(x = Time, y = weight, group = Chick, color = Diet) +
+  geom_point() +
+  geom_line()
+
+#### Test
+ChickWeight %>%
+  group_by(Chick) %>%
+  summarise(maxTime = max(Time)) %>%
+  summarise(minTime = min(maxTime))
+
+t.test(ChickWeight$weight[ChickWeight$Time == 0], ChickWeight$weight[ChickWeight$Time == 2], paired = T)
+
+### Test apparié à la main
+ChickWeight %>%
+  filter(Time %in% c(0, 2)) %>%
+  group_by(Chick) %>%
+  summarise(weightDiff = last(weight) - first(weight),
+            Diet = first(Diet)) -> deuxTemps
+
+#### Plots
+deuxTemps %>%
+  ggplot() +
+  aes(x = weightDiff, fill = Diet) +
+  geom_density(alpha = .25) #, position = "fill") # position = "stack" "fill"
+
+oneway.test(weightDiff ~ Diet, data = deuxTemps)
+kruskal.test(weightDiff ~ Diet, data = deuxTemps)
+
+# Survie ----
+library(survival)
+
+## Data
+data(lung)
+
+## Bivarié
+lung$Surv <- Surv(lung$time, lung$status)
+survfit(Surv ~ 1, data = lung) -> fit
+
+plot(fit)
+
+survfit(Surv ~ sex, data = lung) -> fit
+plot(fit, conf.int = T, col = c("blue", "red"))
+legend("topright", legend = c("Hommes", "Femmes"), col = c("blue", "red"), lty = 1)
+
+survdiff(Surv ~ sex, data = lung)
+
+## Cox
+coxph(Surv ~ sex + age + ph.ecog + ph.karno + meal.cal + wt.loss, data = lung) -> fit
+cox.zph(fit) %>% plot
+
+# N temps ----
+library(lme4)
+library(lmerTest)
+
+## Data
+data(ChickWeight)
+
+## Plots
+ChickWeight %>%
+  ggplot() +
+  aes(x = Time, y = weight, group = Chick, color = Diet) %>%
+  geom_line()#alpha = .5) +
+  geom_smooth(aes(x = Time, y = weight, color = Diet))
+
+## Test
+lmer(weight ~ Diet + (1|Time) + (1|Chick), data = ChickWeight) -> fit
+
+summary(fit)
+plot(fit)
+
+# Trajectoires ----
+library(TraMineR)
+library(cluster)
+
+## Data
+data(mvad)
+
+## Description
+mvad.seq <- seqdef(mvad, 17:86, xtstep = 12)
+plot(mvad.seq, 1:20)
+seqdplot(mvad.seq)
+
+## Clustering
+mvad.om <- seqdist(mvad.seq, method = "OM", sm = "TRATE")
+
+clusterward <- agnes(mvad.om, diss = T, method = "ward")
+mvad.cl <- cutree(clusterward, k = 4)
+cl.lab <- factor(mvad.cl, labels = paste("Cluster", 1:4))
+seqdplot(mvad.seq, group = cl.lab)
+seqfplot(mvad.seq, group = cl.lab)
+seqmsplot(mvad.seq, group = cl.lab)
+
+## Analyse
+mvad.ent <- seqient(mvad.seq)
+lm.ent <- lm(mvad.ent ~ male + funemp + gcse5eq, data = mvad)
+summary(lm.ent)
+
+# Séries temporelles ----
+library(astsa)
+library(Rssa)
+library(zoo)
+library(TSA)
+
+## Data
+data(cmort)
+
+## Description des données
+is.ts(cmort) #on vérifie qu'il s'agit bien d'un objet série temporelle
+start(cmort) #Première observation
+end(cmort) #Dernière observation
+frequency(cmort) #Fréquance des observations
+deltat(cmort) #intervalle de temps entre deux observations
+
+length(cmort)
+head(cmort) #visualisation des premier éléments de l'objet cmort
+
+tsplot(cmort) #visualisation graphique de la série
+
+
+## Méthode de décomposition 
+MannKendall(cmort) #test de tendance : significatif donc il existe bien une tendance
+
+### Modèle additif
+D_additif1 = decompose(cmort, "additive")
+plot(D_additif1)
+
+### Modèle additif(autre méthode)
+D_additif2 = stl(cmort, "periodic")
+plot(D_additif2)
+
+### Modèle multiplicatif
+D_multiplicatif = decompose(cmort, "multiplicative")
+plot(D_multiplicatif)
+
+## Plots
+
+#Jaune : signal recomposé
+#Noir pointillé signal initial 
+graph = function(var){
+  plot(var, lwd = 2, col = "#FFC000", main = substitute(var))
+  lines(cmort,lty = 2)
+}
+
+#Reconstitution signal avec décomposition additive et bruit
+cmort_ad1 = D_additif1$trend + D_additif1$seasonal  + D_additif1$random
+graph(cmort_ad1)
+#Reconstitution signal avec décomposition additive sans bruit
+cmort_ad1_ssrandom = D_additif1$trend + D_additif1$seasonal  
+graph(cmort_ad1_ssrandom)
+
+
+#Reconstitution signal avec décomposition additive et bruit
+cmort_ad2 = D_additif2$time.series[,1] + D_additif2$time.series[,2]  + D_additif2$time.series[,3]
+graph(cmort_ad2)
+#Reconstitution signal avec décomposition additive sans bruit
+cmort_ad2_ssrandom = D_additif2$time.series[,1] + D_additif2$time.series[,2]  
+graph(cmort_ad2_ssrandom)
+
+#Reconstitution signal avec décomposition multiplicative et bruit
+cmort_mult = D_multiplicatif$trend*D_multiplicatif$seasonal*D_multiplicatif$random
+graph(cmort_mult)
+#Reconstitution signal avec décomposition multiplicative sans bruit
+cmort_mult_ssrandom = D_multiplicatif$trend*D_multiplicatif$seasonal
+graph(cmort_mult_ssrandom)
+
+
+### Autocorrélation et Modélisation 
+
+#autocorrélation
+acf(cmort)
+
+#autocorrélation partielle
+pacf(cmort)
+
+
+### Analyse spectrale 
+
+#périodogramme 
+p = periodogram(cmort)
+#Trouver la période : 
+# Définir la fréquence où le spectre est maximum
+# Période : 1/Freq
+frequence = p$freq[which.max(p$spec)]
+Periode = 1/frequence
+
+#Lissage par moyenne mobile 
+plot(cmort, ylab = "cmort", main = "Lissage par moyenne glissante")
+lines(filter(cmort, rep(1/20, 20), side = 1), col = "#FFC000", lwd = 2)
+lines(filter(cmort, rep(1/12, 12), side = 1), col = "red", lwd = 2)
+lines(filter(cmort, rep(1/50, 50), side = 1), col = "indianred4", lwd = 2)
+legend("topright", col = c("red", "#FFC000", "indianred4"), legend = c("1/12", "1/20", "1/100"), lwd = c(2,2,2))
+
+#Lissage par noyaux
+#Calcul de 3 noyaux
+k1 <- kernel("daniell", 10)
+k2 <- kernel("daniell", 5)
+k3 <- kernel("daniell", 30)
+#Application des 3 noyaux aux données
+x1 <- kernapply(cmort, k1)
+x2 <- kernapply(cmort, k2)
+x3 <- kernapply(cmort, k3)
+#Graphiques
+plot(cmort, main = "Lissage par noyaux")
+lines(x1, col = "#FFC000", lwd = 2)
+lines(x2, col = "red", lwd = 2)
+lines(x3, col = "indianred4", lwd = 2)
+legend("topright", col = c("red", "#FFC000", "indianred4"), legend = c("5", "10", "30"), lwd = c(2,2,2))
+
+
+### Série temporelle multiple 
+
+#Graphique 
+serie = cbind(lap[,10],cmort)
+colnames(serie)[1] = "ozone"
+plot(serie)
+abline(v = 1970:1980, col ="#FFC000", lty = 2, lwd = 2)
+
+#Test de causalité de Granger
+grangertest(cmort ~ lap[,4]) #Température
+grangertest(cmort ~ lap[,5]) #Humidité relative
+grangertest(cmort ~ lap[,6]) #Monoxyde de carbone
+grangertest(cmort ~ lap[,7]) #Dioxyde de soufre
+grangertest(cmort ~ lap[,8]) #Dioxyde d'azote
+grangertest(cmort ~ lap[,9]) #Hydrocarbure
+grangertest(cmort ~ lap[,10]) #Ozone
+grangertest(cmort ~ lap[,11]) #Particule
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diff --git a/illustrations.R b/illustrations.R
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+library(tidyverse)
+library(patchwork)
+
+
+# Graphique test statistique ----
+
+seq(100, 0) %>%
+  map(function(x){
+        ci <- function(m, s, n, z = 1.34)
+        {
+          dev <- z * (s / sqrt(n))
+          c(m - dev, m + dev)
+        }
+
+        ztest <- function(m1, s1, n1, m2, s2, n2)
+        {
+          zscore <- abs(m2 - m1) / sqrt((s1^2 / n1) + (s2^2 / n2))
+          2 * (1 - pnorm(zscore))
+        }
+
+        tibble(m1 = x/10,
+               m2 = - x/10,
+               s = 10,
+               n = 20) -> df
+
+        df %>%
+          ggplot() +
+          aes(x = m1) +
+          stat_function(fun = dnorm, args = list(mean = df$m1, sd = df$s), color = "blue", size = 1.5, n = 1000) +
+          stat_function(fun = dnorm, args = list(mean = df$m2, sd = df$s), color = "red", size = 1.5, n = 1000) +
+          geom_vline(xintercept = ci(df$m1, df$s, df$n), color = "blue") +
+          geom_vline(xintercept = ci(df$m2, df$s, df$n), color = "red") +
+          geom_text(x = -30, hjust = 0, y = .05, vjust = 1, label = str_c("p = ", ztest(df$m1, df$s, df$n, df$m2, df$s, df$n)), size = 4) +
+          scale_x_continuous(limits = c(-30 , 30)) +
+          scale_y_continuous(limits = c(0, .05)) +
+          theme_classic() +
+          xlab("x") -> p1
+
+        df %>%
+          ggplot() +
+          aes(x = m1) +
+          stat_function(fun = dnorm, args = list(mean = abs(df$m2 - df$m1), sd = sqrt(df$s^2 / df$n + df$s^2 / df$n))) +
+          geom_vline(xintercept = 0) +
+          geom_vline(xintercept = ci(abs(df$m2 - df$m1), sqrt(df$s^2 / df$n + df$s^2 / df$n), 1, 1.96)) +
+          scale_x_continuous(limits = c(-10, 40)) +
+          theme_classic() +
+          xlab("x") -> p2
+
+        p1 + p2 + plot_layout(ncol = 1, heights = c(8, 2))
+}) -> plots
+
+1:101 %>%
+  map(~ ggsave(plot = plots[[.]],
+               filename = str_c("anim/", str_pad(., width = 3, pad = "0"), ".png"),
+               unit = "cm",
+               dpi = 200,
+               width = 16,
+               height = 9))
+# Un temps ----
+
+one <- tibble(color = rep(LETTERS[1:3], each = 10),
+              y = map((-1:1), ~rnorm(10, ., .1)) %>% unlist,
+              x = rnorm(30, 0, .1))
+
+one %>%
+  ggplot() +
+  geom_segment(x = -1, y = 0, xend = 1, yend = 0, color = "darkgrey", arrow = arrow(), size = 2) +
+  geom_segment(jx = 0, y = -.1, xend = 0, yend = .1, color = "darkgrey", size = 2) +
+  aes(x = x, y = y, color = color) +
+  geom_point(size = 2) +
+  scale_x_continuous(limits = c(-1, 1)) +
+  scale_y_continuous(limits = c(-2, 2)) +
+  theme_void() +
+  theme(legend.position = "none",
+        plot.background = element_rect(fill = "transparent", colour = "transparent")) ->
+oneplot
+
+ggsave(plot = oneplot, filename = "one.png", bg = "transparent")
+
+# Deux temps ----
+
+two <- tibble(x = map(c(-.5, .5), ~rnorm(10, ., .1)) %>% unlist,
+              y = map(c(-1, 1), ~rnorm(10, ., .1)) %>% unlist,
+              colour = rep("A", 20))
+
+two %>%
+  ggplot() +
+  geom_segment(x = -1, y = 0, xend = 1, yend = 0, color = "darkgrey", arrow = arrow(), size = 2) +
+  geom_segment(x = -.5, y = -.1, xend = -.5, yend = .1, color = "darkgrey", size = 2) +
+  geom_segment(x = .5, y = -.1, xend = .5, yend = .1, color = "darkgrey", size = 2) +
+  aes(x = x, y = y, colour = colour) +
+  geom_point(size = 2) +
+  scale_x_continuous(limits = c(-1, 1)) +
+  scale_y_continuous(limits = c(-2, 2)) +
+  theme_void() +
+  theme(legend.position = "none",
+        plot.background = element_rect(fill = "transparent", colour = "transparent")) ->
+twoplot
+
+ggsave(plot = twoplot, filename = "two.png", bg = "transparent")
+
+# N temps ----
+
+many <- tibble(x = map(-3:3, ~rnorm(10, ., .1)) %>% unlist,
+               y = runif(7, -1.5, 1.5) %>% map(~rnorm(10, ., .1)) %>% unlist,
+               colour = rep("A", 70))
+
+many %>%
+  ggplot() +
+  geom_segment(x = -4, y = 0, xend = 4, yend = 0, color = "darkgrey", arrow = arrow(), size = 2) +
+  geom_segment(x = -3, y = -.1, xend = -3, yend = .1, color = "darkgrey", size = 2) +
+  geom_segment(x = -2, y = -.1, xend = -2, yend = .1, color = "darkgrey", size = 2) +
+  geom_segment(x = -1, y = -.1, xend = -1, yend = .1, color = "darkgrey", size = 2) +
+  geom_segment(x = 0, y = -.1, xend = 0, yend = .1, color = "darkgrey", size = 2) +
+  geom_segment(x = 1, y = -.1, xend = 1, yend = .1, color = "darkgrey", size = 2) +
+  geom_segment(x = 2, y = -.1, xend = 2, yend = .1, color = "darkgrey", size = 2) +
+  geom_segment(x = 3, y = -.1, xend = 3, yend = .1, color = "darkgrey", size = 2) +
+  aes(x = x, y = y, colour = colour) +
+  geom_point(size = 2) +
+  scale_x_continuous(limits = c(-4, 4)) +
+  scale_y_continuous(limits = c(-2, 2)) +
+  theme_void() +
+  theme(legend.position = "none",
+        plot.background = element_rect(fill = "transparent", colour = "transparent")) ->
+manyplot
+
+ggsave(plot = manyplot, filename = "many.png", bg = "transparent")
+
+# Série temporelle ----
+
+## Figure introduction
+library(astsa)
+data(flu) #chargement de la base "flu" du package astsa
+
+## Représentation graphique de la série temporelle
+tsplot(flu,xlab = "Time", ylab = "Deaths per 10,000 people",
+       main="Monthly pneumonia and influenza deaths per 10,000 people \n in the United States for 11 years, 1968 to 1978")
+
+## Simulation série
+
+### Simulation des paramêtres temps, tendance, saisonalité, bruit
+t = time(cmort)
+m = 0.2*t+2.1
+s = cos(5*t)
+epsilon = rnorm(length(t),0,0.5)
+
+### Simulation des séries temporelles comme modèle additif/multiplicatif/Hybride
+simu.ts.additif = ts(m+s+epsilon,frequency = 52, start = c(1970, 1) )
+tsplot(simu.ts.additif)
+
+simu.ts.multiplictif = ts(m*s*epsilon,frequency = 52, start = c(1970, 1) )
+tsplot(simu.ts.multiplictif)
+
+simu.ts.hybride = ts(m*s+epsilon*100,frequency = 52, start = c(1970, 1) )
+tsplot(simu.ts.hybride)
+
+#### Courbes
+plot(t,m, type = "l", main = "Tendance", xlab = "Temps", ylab = "")
+plot(t,s, type = "l", main = "Saisonnalité", xlab = "Temps", ylab = "")
+plot(t,epsilon, type = "l", main = "Bruit", xlab = "Temps", ylab = "")
+
+### Décomposition des signaux simulés
+plot(decompose(simu.ts.additif, type="additive")) #décomposition trend/seasonnality/noise, modèle additif
+plot(stl(simu.ts.additif, "periodic"))
+plot(decompose(decompose(simu.ts.additif, type="additive")$random, type="additive"))
+
+## Simulation modèles AR et MA
+
+### Simulation AR(1) : X_t = 0.6X_{t???1} + ??_t
+X=arima.sim(n = 2400, list(ar = 0.6),sd = 1)
+plot(acf(X),lwd=2, main = "Acf AR(1)")
+plot(pacf(X),lwd=2, main = "Pacf AR(1)")
+
+### Simulation AR(2) : Xt = 0.6X_{t???1} - 0.4X_{t???2} + ??t
+X=arima.sim(n = 2400, list(ar = c(0.6,-0.4)),sd = 1)
+plot(acf(X),lwd=2, main = "Acf AR(2)")
+plot(pacf(X),lwd=2, main = "Pacf AR(2)")
+
+### Simulation MA(1) X_t =  ??_t + 0.7??_{t???1}
+X=arima.sim(n = 2400, list(ma = .7),sd = 1)
+plot(acf(X),lwd=2, main = "Acf MA(1)")
+plot(pacf(X),lwd=2, main = "Pacf MA(1)")
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