12 Interpolation Paris : Corrigé

Author

Dr. Elisabetta Pietrostefani

Published

March 22, 2026

Corrigé — à consulter après avoir essayé par vous-mêmes

Ce document contient les réponses à l’exercice Paris posé dans Lissage et Interpolation Spatiale.

library(sf)
library(gstat)
library(ggplot2)
library(dplyr)
library(scales)
library(patchwork)

12.1 Partie A — IDW sur le prix Airbnb

12.1.1 Charger et préparer les données

PA <- read_sf("data/Paris/Polygons/arrondiss_lambert.shp") |>
  st_transform(2154)

listings_sf <- read.csv("data/Paris/airbnb/listings.csv") |>
  st_as_sf(coords = c(8, 7)) |>
  st_set_crs(4326) |>
  st_transform(st_crs(PA))

listings_filtered <- listings_sf |> filter(price < 200)

cat("Nb annonces filtrées :", nrow(listings_filtered), "\n")

Nb annonces filtrées : 46157

ggplot() +
  geom_sf(data = PA, fill = "grey92", color = "white", linewidth = 0.4) +
  geom_sf(data = listings_filtered, aes(color = price),
          size = 0.3, alpha = 0.4) +
  scale_color_viridis_c(
    option    = "plasma", direction = -1,
    name      = "Prix (€/nuit)",
    labels    = label_dollar(prefix = "€")
  ) +
  labs(
    title    = "Annonces Airbnb < 200 €/nuit — Paris",
    subtitle = paste0(format(nrow(listings_filtered), big.mark = " "), " annonces"),
    caption  = "Source : Inside Airbnb | Cours R-Carto"
  ) +
  theme_void(base_size = 12) +
  theme(
    plot.title    = element_text(face = "bold", hjust = 0.5),
    plot.subtitle = element_text(color = "grey40", hjust = 0.5)
  )

Figure 12.1: Annonces Airbnb < 200 €/nuit — points colorés par prix

12.1.2 Appliquer l’IDW

listings_sp <- sf::as_Spatial(listings_filtered)

gs <- gstat(
  formula = price ~ 1,
  data    = listings_sp,
  nmax    = 50,
  set     = list(idp = 2)
)

grid_sf <- st_make_grid(PA, cellsize = 250, what = "centers") |>
  st_as_sf() |>
  st_filter(PA)

idw_sf <- predict(gs, sf::as_Spatial(grid_sf)) |>
  st_as_sf() |>
  rename(pred = var1.pred)

[inverse distance weighted interpolation]

ggplot() +
  geom_sf(data = idw_sf, aes(color = pred), size = 2.2, shape = 15) +
  scale_color_viridis_c(
    option    = "plasma", direction = -1,
    name      = "Prix interpolé\n(€/nuit)",
    labels    = label_dollar(prefix = "€")
  ) +
  geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.5) +
  labs(
    title    = "IDW — Prix Airbnb à Paris",
    subtitle = "N voisins = 50 | puissance p = 2 | résolution 250 m",
    caption  = "Source : Inside Airbnb | Cours R-Carto"
  ) +
  theme_void(base_size = 12) +
  theme(
    plot.title        = element_text(face = "bold", hjust = 0.5),
    plot.subtitle     = element_text(color = "grey40", hjust = 0.5),
    legend.key.height = unit(1.5, "cm")
  )

Figure 12.2: Surface de prix Airbnb interpolée par IDW — Paris (résolution 250 m)

Effet des paramètres

nmax petit (ex. 10) → surface très localisée, forte influence des points isolés
nmax grand (ex. 100) → surface plus lisse, influence étendue
idp grand (ex. 4) → décroissance rapide, les points lointains n’influencent presque pas
cellsize petite (ex. 100 m) → image plus fine mais calcul plus lent

12.2 Partie B — Heatmap de densité Airbnb

12.2.1 Densité de toutes les annonces

coords_all <- st_coordinates(listings_filtered) |>
  as.data.frame() |>
  setNames(c("x", "y"))

ggplot() +
  geom_sf(data = PA, fill = "grey15", color = "grey35", linewidth = 0.4) +
  stat_density_2d(
    data        = coords_all,
    aes(x = x, y = y, fill = after_stat(density)),
    geom        = "raster",
    contour     = FALSE,
    interpolate = TRUE,
    alpha       = 0.85
  ) +
  scale_fill_viridis_c(option = "inferno", name = "Densité", guide = "none") +
  geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.4) +
  labs(
    title    = "Densité des annonces Airbnb — Paris",
    subtitle = "Annonces < 200 €/nuit",
    caption  = "Source : Inside Airbnb | Cours R-Carto"
  ) +
  theme_void(base_size = 12) +
  theme(
    plot.title      = element_text(face = "bold", hjust = 0.5, color = "white"),
    plot.subtitle   = element_text(color = "grey60", hjust = 0.5),
    plot.caption    = element_text(color = "grey50", size = 9),
    plot.background = element_rect(fill = "grey15", color = NA)
  )

Figure 12.3: Densité des annonces Airbnb — heatmap par noyau gaussien

12.2.2 Comparaison : toutes annonces vs annonces > 150 €/nuit

coords_premium <- listings_filtered |>
  filter(price > 150) |>
  st_coordinates() |>
  as.data.frame() |>
  setNames(c("x", "y"))

p1 <- ggplot() +
  geom_sf(data = PA, fill = "grey15", color = "grey35", linewidth = 0.3) +
  stat_density_2d(
    data = coords_all,
    aes(x = x, y = y, fill = after_stat(density)),
    geom = "raster", contour = FALSE, interpolate = TRUE, alpha = 0.85
  ) +
  scale_fill_viridis_c(option = "inferno", guide = "none") +
  geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.3) +
  labs(title = "Toutes annonces (< 200 €)") +
  theme_void(base_size = 11) +
  theme(plot.title      = element_text(face = "bold", hjust = 0.5, color = "white"),
        plot.background = element_rect(fill = "grey15", color = NA))

p2 <- ggplot() +
  geom_sf(data = PA, fill = "grey15", color = "grey35", linewidth = 0.3) +
  stat_density_2d(
    data = coords_premium,
    aes(x = x, y = y, fill = after_stat(density)),
    geom = "raster", contour = FALSE, interpolate = TRUE, alpha = 0.85
  ) +
  scale_fill_viridis_c(option = "magma", guide = "none") +
  geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.3) +
  labs(title = "Annonces premium (> 150 €)") +
  theme_void(base_size = 11) +
  theme(plot.title      = element_text(face = "bold", hjust = 0.5, color = "white"),
        plot.background = element_rect(fill = "grey15", color = NA))

p1 + p2 +
  plot_annotation(
    caption = "Source : Inside Airbnb | Cours R-Carto",
    theme   = theme(
      plot.background = element_rect(fill = "grey15", color = NA),
      plot.caption    = element_text(color = "grey50", size = 9)
    )
  )

Figure 12.4: Densité globale vs annonces premium (> 150 €/nuit)

12.3 IDW vs Heatmap — quelle méthode pour quelle question ?

	IDW	Heatmap
Question	Quel prix prédit-on ici ?	Combien d’annonces ici ?
Variable	Prix (attribut des points)	Présence des points uniquement
Résultat	Surface de valeurs continues	Surface de densité
Usage	Estimer une variable manquante	Identifier des zones de concentration

← Retour au cours : Lissage et Interpolation

Dr. Elisabetta Pietrostefani — Directrice Adjointe, Geographic Data Science Lab — Université de Liverpool | Co-Directrice, Imago : Data Service for Imagery | Chercheuse Associée, London School of Economics

--- title: "Interpolation Paris : Corrigé" author: "Dr. Elisabetta Pietrostefani" date: today format: html: toc: true toc-depth: 3 toc-title: "Table des matières" number-sections: true code-fold: false code-tools: true theme: cosmo highlight-style: github embed-resources: true fig-align: center fig-width: 8 fig-height: 6 execute: warning: false message: false echo: true --- ::: callout-important ## Corrigé — à consulter après avoir essayé par vous-mêmes Ce document contient les réponses à l'exercice Paris posé dans [Lissage et Interpolation Spatiale](interpolation.qmd). ::: ```{r} #| label: setup #| message: false library(sf) library(gstat) library(ggplot2) library(dplyr) library(scales) library(patchwork) ``` ------------------------------------------------------------------------ ## Partie A — IDW sur le prix Airbnb {#sec-idw} ### Charger et préparer les données ```{r} #| label: load-data PA <- read_sf("data/Paris/Polygons/arrondiss_lambert.shp") |> st_transform(2154) listings_sf <- read.csv("data/Paris/airbnb/listings.csv") |> st_as_sf(coords = c(8, 7)) |> st_set_crs(4326) |> st_transform(st_crs(PA)) listings_filtered <- listings_sf |> filter(price < 200) cat("Nb annonces filtrées :", nrow(listings_filtered), "\n") ``` ```{r} #| label: fig-points #| fig-cap: "Annonces Airbnb < 200 €/nuit — points colorés par prix" ggplot() + geom_sf(data = PA, fill = "grey92", color = "white", linewidth = 0.4) + geom_sf(data = listings_filtered, aes(color = price), size = 0.3, alpha = 0.4) + scale_color_viridis_c( option = "plasma", direction = -1, name = "Prix (€/nuit)", labels = label_dollar(prefix = "€") ) + labs( title = "Annonces Airbnb < 200 €/nuit — Paris", subtitle = paste0(format(nrow(listings_filtered), big.mark = " "), " annonces"), caption = "Source : Inside Airbnb | Cours R-Carto" ) + theme_void(base_size = 12) + theme( plot.title = element_text(face = "bold", hjust = 0.5), plot.subtitle = element_text(color = "grey40", hjust = 0.5) ) ``` ### Appliquer l'IDW ```{r} #| label: idw-paris listings_sp <- sf::as_Spatial(listings_filtered) gs <- gstat( formula = price ~ 1, data = listings_sp, nmax = 50, set = list(idp = 2) ) grid_sf <- st_make_grid(PA, cellsize = 250, what = "centers") |> st_as_sf() |> st_filter(PA) idw_sf <- predict(gs, sf::as_Spatial(grid_sf)) |> st_as_sf() |> rename(pred = var1.pred) ``` ```{r} #| label: fig-idw-paris #| fig-cap: "Surface de prix Airbnb interpolée par IDW — Paris (résolution 250 m)" ggplot() + geom_sf(data = idw_sf, aes(color = pred), size = 2.2, shape = 15) + scale_color_viridis_c( option = "plasma", direction = -1, name = "Prix interpolé\n(€/nuit)", labels = label_dollar(prefix = "€") ) + geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.5) + labs( title = "IDW — Prix Airbnb à Paris", subtitle = "N voisins = 50 | puissance p = 2 | résolution 250 m", caption = "Source : Inside Airbnb | Cours R-Carto" ) + theme_void(base_size = 12) + theme( plot.title = element_text(face = "bold", hjust = 0.5), plot.subtitle = element_text(color = "grey40", hjust = 0.5), legend.key.height = unit(1.5, "cm") ) ``` ::: callout-tip ## Effet des paramètres - **`nmax` petit** (ex. 10) → surface très localisée, forte influence des points isolés - **`nmax` grand** (ex. 100) → surface plus lisse, influence étendue - **`idp` grand** (ex. 4) → décroissance rapide, les points lointains n'influencent presque pas - **`cellsize` petite** (ex. 100 m) → image plus fine mais calcul plus lent ::: ------------------------------------------------------------------------ ## Partie B — Heatmap de densité Airbnb {#sec-heatmap} ### Densité de toutes les annonces ```{r} #| label: fig-heatmap-all #| fig-cap: "Densité des annonces Airbnb — heatmap par noyau gaussien" coords_all <- st_coordinates(listings_filtered) |> as.data.frame() |> setNames(c("x", "y")) ggplot() + geom_sf(data = PA, fill = "grey15", color = "grey35", linewidth = 0.4) + stat_density_2d( data = coords_all, aes(x = x, y = y, fill = after_stat(density)), geom = "raster", contour = FALSE, interpolate = TRUE, alpha = 0.85 ) + scale_fill_viridis_c(option = "inferno", name = "Densité", guide = "none") + geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.4) + labs( title = "Densité des annonces Airbnb — Paris", subtitle = "Annonces < 200 €/nuit", caption = "Source : Inside Airbnb | Cours R-Carto" ) + theme_void(base_size = 12) + theme( plot.title = element_text(face = "bold", hjust = 0.5, color = "white"), plot.subtitle = element_text(color = "grey60", hjust = 0.5), plot.caption = element_text(color = "grey50", size = 9), plot.background = element_rect(fill = "grey15", color = NA) ) ``` ### Comparaison : toutes annonces vs annonces \> 150 €/nuit ```{r} #| label: fig-heatmap-compare #| fig-cap: "Densité globale vs annonces premium (> 150 €/nuit)" #| fig-height: 7 coords_premium <- listings_filtered |> filter(price > 150) |> st_coordinates() |> as.data.frame() |> setNames(c("x", "y")) p1 <- ggplot() + geom_sf(data = PA, fill = "grey15", color = "grey35", linewidth = 0.3) + stat_density_2d( data = coords_all, aes(x = x, y = y, fill = after_stat(density)), geom = "raster", contour = FALSE, interpolate = TRUE, alpha = 0.85 ) + scale_fill_viridis_c(option = "inferno", guide = "none") + geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.3) + labs(title = "Toutes annonces (< 200 €)") + theme_void(base_size = 11) + theme(plot.title = element_text(face = "bold", hjust = 0.5, color = "white"), plot.background = element_rect(fill = "grey15", color = NA)) p2 <- ggplot() + geom_sf(data = PA, fill = "grey15", color = "grey35", linewidth = 0.3) + stat_density_2d( data = coords_premium, aes(x = x, y = y, fill = after_stat(density)), geom = "raster", contour = FALSE, interpolate = TRUE, alpha = 0.85 ) + scale_fill_viridis_c(option = "magma", guide = "none") + geom_sf(data = PA, fill = NA, color = "white", linewidth = 0.3) + labs(title = "Annonces premium (> 150 €)") + theme_void(base_size = 11) + theme(plot.title = element_text(face = "bold", hjust = 0.5, color = "white"), plot.background = element_rect(fill = "grey15", color = NA)) p1 + p2 + plot_annotation( caption = "Source : Inside Airbnb | Cours R-Carto", theme = theme( plot.background = element_rect(fill = "grey15", color = NA), plot.caption = element_text(color = "grey50", size = 9) ) ) ``` ------------------------------------------------------------------------ ## IDW vs Heatmap — quelle méthode pour quelle question ? {#sec-compare} | | IDW | Heatmap | |----|----|----| | **Question** | Quel prix prédit-on ici ? | Combien d'annonces ici ? | | **Variable** | Prix (attribut des points) | Présence des points uniquement | | **Résultat** | Surface de valeurs continues | Surface de densité | | **Usage** | Estimer une variable manquante | Identifier des zones de concentration | ------------------------------------------------------------------------ > **← Retour au cours : [Lissage et Interpolation](interpolation.qmd)** ------------------------------------------------------------------------ *Dr. Elisabetta Pietrostefani — Directrice Adjointe, Geographic Data Science Lab — Université de Liverpool \| Co-Directrice, [Imago : Data Service for Imagery](https://imago.ac.uk/) \| Chercheuse Associée, London School of Economics*