K-nearest neighbour adaptive bandwidth
Jeremy Gelb
-2024-05-19
+2024-10-20
Source:vignettes/web_vignettes/AdaptiveBW.Rmd
AdaptiveBW.RmdFinding the optimal bandwidth
+
Abramson, Ian S. 1982. “On Bandwidth Variation in Kernel
Estimates-a Square Root Law.” The Annals of Statistics,
diff --git a/docs/articles/web_vignettes/Isochrones.html b/docs/articles/web_vignettes/Isochrones.html
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+
+Calculating isochrones • spNetwork
+
+
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+
+ Calculating isochrones
+ Jeremy
+Gelb
+
+ 2024-10-20
+
+ Source: vignettes/web_vignettes/Isochrones.Rmd
+ Isochrones.Rmd
+
+
+
+
+
+
+Introduction
+
+An isochrone is a geometrical representation of the area that can be
+reached for a starting on a network within a given distance (in time or
+space units).
+There are already several libraries to build isochrones with R
+(opentripplanner, osrm, rmapzen,
+etc.), but they are all based on OpenStreetMap data or on external API.
+We propose here a simple implementation that can be used on a
+geographical network read as SpatialLinesDataFrame.
+We give here a simple example with the Montreal network dataset and
+calculate several isochrones from the centre of the dataset.
+
+
+calculating the isochrones
+
+The first step is to load (or create) a geographical network and to
+have some starting points.
+
+# first load data and packages
+library(sf)
+library(spNetwork)
+library(tmap)
+
+data(mtl_network)
+
+# calculating the length of each segment
+mtl_network$length <- as.numeric(st_length(mtl_network))
+
+# extracting the coordinates in one matrix
+coords <- st_coordinates(mtl_network)
+center <- colMeans(coords)
+df_center <- data.frame("OID" = 1,
+ "x" = center[1],
+ "y" = center[2])
+df_center <- st_as_sf(df_center, coords = c("x","y"), crs = st_crs(mtl_network))
+
+# then plotting the data
+tm_shape(mtl_network) +
+ tm_lines("black") +
+ tm_shape(df_center) +
+ tm_dots("red", size = 0.2)
+
+The second step is to calculate the isochrones with the function
+calc_isochrones. In this example, we use the length of a
+segment as the weight but other values could be used. We also consider
+that all the roads can be used in both directions (see parameter
+direction)
+
+iso_results <- calc_isochrones(lines = mtl_network,
+ start_points = df_center,
+ dists = c(500,1000,2000),
+ weight = "length"
+ )
+The results are reported as lines, which can be mapped.
+
+# creating a factor and changing order for better vizualisation
+iso_results$fac_dist <- as.factor(iso_results$distance)
+iso_results <- iso_results[order(-1*iso_results$distance),]
+
+tm_shape(mtl_network) +
+ tm_lines(col = "grey") +
+ tm_shape(iso_results) +
+ tm_lines(col = "fac_dist",title.col = "distance (m)",
+ palette = c("500"="#005f73", "1000"="#ca6702", "2000"="#9b2226"))+
+ tm_layout(legend.outside = TRUE) +
+ tm_shape(df_center) +
+ tm_dots(col = "black", size = 0.1)
+
+
+
+From lines to polygons
+
+The lines are the most exact form of an isochrone because they depict
+exactly what parts of the network can be reached. However, isochrones
+are represented as polygons most of the time. We propose here some
+approaches to build some meaningful polygons for isochrones.
+The best starting point is probably to calculate for each set of
+points a concave hull. This is different from the traditional convex
+hull and is more adapted to the irregular geometry of an isochrone. To
+do so, we use the library concaveman
+
+library(concaveman)
+
+# identifying each isochrone
+iso_results$iso_oid <- paste(iso_results$point_id,
+ iso_results$distance,
+ sep = "_")
+
+# creating the polygons for each isochrone
+polygons <- lapply(unique(iso_results$iso_oid), function(oid){
+
+ # subseting the required lines
+ lines <- subset(iso_results, iso_results$iso_oid == oid)
+
+ # extracting the coordinates of the lines
+ coords <- st_coordinates(lines)
+ poly_coords <- concaveman(points = coords, concavity = 3)
+ poly <- st_polygon(list(poly_coords[,1:2]))
+ return(poly)
+})
+
+# creating a SpatialPolygonsDataFrame
+iso_sp <- st_sf(
+ iso_oid = unique(iso_results$iso_oid),
+ distance = unique(iso_results$distance),
+ geometry = polygons,
+ crs = st_crs(iso_results)
+)
+We can now map the obtained polygons.
+
+# creating a factor and changing order for better vizualisation
+iso_sp$fac_dist <- as.factor(iso_sp$distance)
+iso_sp <- iso_sp[order(-1*iso_sp$distance),]
+
+tm_shape(iso_results) +
+ tm_lines(col = "black")+
+ tm_shape(iso_sp) +
+ tm_polygons(col = "fac_dist",title.col = "distance (m)",
+ palette = c("500"="#005f73", "1000"="#ca6702", "2000"="#9b2226"),
+ alpha = 0.6, border.col = "white") +
+ tm_shape(df_center) +
+ tm_dots(col = "black", size = 0.1)
+
+This is more similar to classical isochrone maps. Another possible
+enhancement would be to simplify and smooth the limits of the polygons.
+To do so, we will use the functions st_simplify from
+sf and smooth from smoothr.
+
+library(smoothr)
+
+simple_polygons <- st_simplify(iso_sp, dTolerance = 50)
+smooth_iso <- smooth(simple_polygons, method = "chaikin",
+ refinements = 5)
+
+tm_shape(iso_results) +
+ tm_lines(col = "black")+
+ tm_shape(smooth_iso) +
+ tm_polygons(col = "fac_dist",
+ title.col = "distance (m)",
+ palette = c("500"="#005f73", "1000"="#ca6702", "2000"="#9b2226"),
+ border.col = "white",
+ alpha = 0.6) +
+ tm_shape(df_center) +
+ tm_dots(col = "black", size = 0.1) +
+ tm_layout(legend.outside = TRUE)
+
+The shapes are cleaner for mapping but the results are not exact
+anymore.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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diff --git a/docs/articles/web_vignettes/NKDEdetailed.html b/docs/articles/web_vignettes/NKDEdetailed.html
index 6208e7d2..483afc99 100644
--- a/docs/articles/web_vignettes/NKDEdetailed.html
+++ b/docs/articles/web_vignettes/NKDEdetailed.html
@@ -33,7 +33,7 @@
@@ -53,7 +53,7 @@
K-nearest neighbour adaptive bandwidth
- Calculating isochrones
+ Calculating isochrones
Network k Functions
@@ -110,7 +110,7 @@ Details about NKDE
Jeremy
Gelb
- 2024-05-19
+ 2024-10-20
Source: vignettes/web_vignettes/NKDEdetailed.Rmd
NKDEdetailed.Rmd
@@ -194,8 +194,8 @@ The simple method
d3_plot_situation(all_lines, event, samples_pts, simple_kernel, scales = c(1,1,3))
-
-(The function used to display the 3d plot is available in the source
+
+(The function used to display the 3d plot is available in the source
code of the vignette)
As one can see, NKDE is only determined by the distance between the
sampling points and the event. This is a problem, because at
@@ -249,8 +249,8 @@
The discontinuous method verbose = FALSE)
d3_plot_situation(all_lines, event, samples_pts, discontinuous_kernel, scales = c(1,1,3))
-
-As one can see, the values of the NKDE are split at intersections to
+
+As one can see, the values of the NKDE are split at intersections to
avoid the multiplication of the mass observed in the simple version.
However, this NKDE is discontinuous, which is counter-intuitive, and
leads to sharp differences between density values in the network, and
@@ -300,8 +300,8 @@
The continuous method verbose = FALSE)
d3_plot_situation(all_lines, event, samples_pts, continuous_kernel, scales = c(1,1,3))
-
-As one can see, the values of the NKDE are continuous, and the
+
+As one can see, the values of the NKDE are continuous, and the
density values close to the events have been adjusted. This method
produces smoother results than the discontinuous method.
what to retain about the discontinuous method
@@ -338,8 +338,8 @@ The continuous method verbose = FALSE)
d3_plot_situation(all_lines, events, samples_pts, continuous_kernel, scales = c(1,1,3))
-
-The larger the bandwidth, the more common links the events share in
+
+The larger the bandwidth, the more common links the events share in
their range.
@@ -351,13 +351,13 @@ The continuous method verbose = FALSE)
d3_plot_situation(all_lines, events, samples_pts, continuous_kernel, scales = c(1,1,3))
-
-
+
+
References
-
+
Okabe, Atsuyuki, and Kokichi Sugihara. 2012. Spatial Analysis Along
Networks: Statistical and Computational Methods. John Wiley &
diff --git a/docs/articles/web_vignettes/SpaceTimeDBscan.html b/docs/articles/web_vignettes/SpaceTimeDBscan.html
index d72c5f58..6202ae79 100644
--- a/docs/articles/web_vignettes/SpaceTimeDBscan.html
+++ b/docs/articles/web_vignettes/SpaceTimeDBscan.html
@@ -33,7 +33,7 @@
@@ -53,7 +53,7 @@
K-nearest neighbour adaptive bandwidth
- Calculating isochrones
+ Calculating isochrones
Network k Functions
@@ -114,7 +114,7 @@ Spatio-temporal dbscan with network
Jeremy
Gelb
- 2024-05-19
+ 2024-10-20
Source: vignettes/web_vignettes/SpaceTimeDBscan.Rmd
SpaceTimeDBscan.Rmd
diff --git a/docs/articles/web_vignettes/SpeedVSprecision.html b/docs/articles/web_vignettes/SpeedVSprecision.html
index d7e7970e..a840e7f5 100644
--- a/docs/articles/web_vignettes/SpeedVSprecision.html
+++ b/docs/articles/web_vignettes/SpeedVSprecision.html
@@ -33,7 +33,7 @@
@@ -53,7 +53,7 @@
K-nearest neighbour adaptive bandwidth
- Calculating isochrones
+ Calculating isochrones
Network k Functions
@@ -110,7 +110,7 @@ Analysing the impact of max_dept parameter for
Jeremy
Gelb
- 2024-05-19
+ 2024-10-20
Source: vignettes/web_vignettes/SpeedVSprecision.Rmd
SpeedVSprecision.Rmd
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diff --git a/docs/authors.html b/docs/authors.html
index ba3af048..ad14ff3a 100644
--- a/docs/authors.html
+++ b/docs/authors.html
@@ -17,7 +17,7 @@
@@ -35,7 +35,7 @@
K-nearest neighbour adaptive bandwidth
- Calculating isochrones
+ Calculating isochrones
Network k Functions
diff --git a/docs/index.html b/docs/index.html
index 2bc239b7..c97597f3 100644
--- a/docs/index.html
+++ b/docs/index.html
@@ -33,7 +33,7 @@
@@ -53,7 +53,7 @@
K-nearest neighbour adaptive bandwidth
Calculating isochrones
+Jeremy +Gelb
+ +2024-10-20
+ + Source:vignettes/web_vignettes/Isochrones.Rmd
+ Isochrones.RmdIntroduction +
+An isochrone is a geometrical representation of the area that can be +reached for a starting on a network within a given distance (in time or +space units).
+There are already several libraries to build isochrones with R
+(opentripplanner, osrm, rmapzen,
+etc.), but they are all based on OpenStreetMap data or on external API.
+We propose here a simple implementation that can be used on a
+geographical network read as SpatialLinesDataFrame.
We give here a simple example with the Montreal network dataset and +calculate several isochrones from the centre of the dataset.
+calculating the isochrones +
+The first step is to load (or create) a geographical network and to +have some starting points.
+
+# first load data and packages
+library(sf)
+library(spNetwork)
+library(tmap)
+
+data(mtl_network)
+
+# calculating the length of each segment
+mtl_network$length <- as.numeric(st_length(mtl_network))
+
+# extracting the coordinates in one matrix
+coords <- st_coordinates(mtl_network)
+center <- colMeans(coords)
+df_center <- data.frame("OID" = 1,
+ "x" = center[1],
+ "y" = center[2])
+df_center <- st_as_sf(df_center, coords = c("x","y"), crs = st_crs(mtl_network))
+
+# then plotting the data
+tm_shape(mtl_network) +
+ tm_lines("black") +
+ tm_shape(df_center) +
+ tm_dots("red", size = 0.2)
The second step is to calculate the isochrones with the function
+calc_isochrones. In this example, we use the length of a
+segment as the weight but other values could be used. We also consider
+that all the roads can be used in both directions (see parameter
+direction)
+iso_results <- calc_isochrones(lines = mtl_network,
+ start_points = df_center,
+ dists = c(500,1000,2000),
+ weight = "length"
+ )The results are reported as lines, which can be mapped.
+
+# creating a factor and changing order for better vizualisation
+iso_results$fac_dist <- as.factor(iso_results$distance)
+iso_results <- iso_results[order(-1*iso_results$distance),]
+
+tm_shape(mtl_network) +
+ tm_lines(col = "grey") +
+ tm_shape(iso_results) +
+ tm_lines(col = "fac_dist",title.col = "distance (m)",
+ palette = c("500"="#005f73", "1000"="#ca6702", "2000"="#9b2226"))+
+ tm_layout(legend.outside = TRUE) +
+ tm_shape(df_center) +
+ tm_dots(col = "black", size = 0.1)
From lines to polygons +
+The lines are the most exact form of an isochrone because they depict +exactly what parts of the network can be reached. However, isochrones +are represented as polygons most of the time. We propose here some +approaches to build some meaningful polygons for isochrones.
+The best starting point is probably to calculate for each set of
+points a concave hull. This is different from the traditional convex
+hull and is more adapted to the irregular geometry of an isochrone. To
+do so, we use the library concaveman
+library(concaveman)
+
+# identifying each isochrone
+iso_results$iso_oid <- paste(iso_results$point_id,
+ iso_results$distance,
+ sep = "_")
+
+# creating the polygons for each isochrone
+polygons <- lapply(unique(iso_results$iso_oid), function(oid){
+
+ # subseting the required lines
+ lines <- subset(iso_results, iso_results$iso_oid == oid)
+
+ # extracting the coordinates of the lines
+ coords <- st_coordinates(lines)
+ poly_coords <- concaveman(points = coords, concavity = 3)
+ poly <- st_polygon(list(poly_coords[,1:2]))
+ return(poly)
+})
+
+# creating a SpatialPolygonsDataFrame
+iso_sp <- st_sf(
+ iso_oid = unique(iso_results$iso_oid),
+ distance = unique(iso_results$distance),
+ geometry = polygons,
+ crs = st_crs(iso_results)
+)We can now map the obtained polygons.
+
+# creating a factor and changing order for better vizualisation
+iso_sp$fac_dist <- as.factor(iso_sp$distance)
+iso_sp <- iso_sp[order(-1*iso_sp$distance),]
+
+tm_shape(iso_results) +
+ tm_lines(col = "black")+
+ tm_shape(iso_sp) +
+ tm_polygons(col = "fac_dist",title.col = "distance (m)",
+ palette = c("500"="#005f73", "1000"="#ca6702", "2000"="#9b2226"),
+ alpha = 0.6, border.col = "white") +
+ tm_shape(df_center) +
+ tm_dots(col = "black", size = 0.1)
This is more similar to classical isochrone maps. Another possible
+enhancement would be to simplify and smooth the limits of the polygons.
+To do so, we will use the functions st_simplify from
+sf and smooth from smoothr.
+library(smoothr)
+
+simple_polygons <- st_simplify(iso_sp, dTolerance = 50)
+smooth_iso <- smooth(simple_polygons, method = "chaikin",
+ refinements = 5)
+
+tm_shape(iso_results) +
+ tm_lines(col = "black")+
+ tm_shape(smooth_iso) +
+ tm_polygons(col = "fac_dist",
+ title.col = "distance (m)",
+ palette = c("500"="#005f73", "1000"="#ca6702", "2000"="#9b2226"),
+ border.col = "white",
+ alpha = 0.6) +
+ tm_shape(df_center) +
+ tm_dots(col = "black", size = 0.1) +
+ tm_layout(legend.outside = TRUE)
The shapes are cleaner for mapping but the results are not exact +anymore.
+Details about NKDE
Jeremy Gelb
-2024-05-19
+2024-10-20
Source:vignettes/web_vignettes/NKDEdetailed.Rmd
NKDEdetailed.RmdThe simple method
d3_plot_situation(all_lines, event, samples_pts, simple_kernel, scales = c(1,1,3))
d3_plot_situation(all_lines, event, samples_pts, simple_kernel, scales = c(1,1,3))
d3_plot_situation(all_lines, event, samples_pts, discontinuous_kernel, scales = c(1,1,3))As one can see, the values of the NKDE are split at intersections to +
+As one can see, the values of the NKDE are split at intersections to avoid the multiplication of the mass observed in the simple version. However, this NKDE is discontinuous, which is counter-intuitive, and leads to sharp differences between density values in the network, and @@ -300,8 +300,8 @@
The continuous method verbose = FALSE)
d3_plot_situation(all_lines, event, samples_pts, continuous_kernel, scales = c(1,1,3))
-
-As one can see, the values of the NKDE are continuous, and the
+
+As one can see, the values of the NKDE are continuous, and the
density values close to the events have been adjusted. This method
produces smoother results than the discontinuous method.
what to retain about the discontinuous method
@@ -338,8 +338,8 @@ The continuous method verbose = FALSE)
d3_plot_situation(all_lines, events, samples_pts, continuous_kernel, scales = c(1,1,3))
-
-The larger the bandwidth, the more common links the events share in
+
+The larger the bandwidth, the more common links the events share in
their range.
@@ -351,13 +351,13 @@ The continuous method verbose = FALSE)
d3_plot_situation(all_lines, events, samples_pts, continuous_kernel, scales = c(1,1,3))
-
-
+
+
References
-
+
Okabe, Atsuyuki, and Kokichi Sugihara. 2012. Spatial Analysis Along
Networks: Statistical and Computational Methods. John Wiley &
diff --git a/docs/articles/web_vignettes/SpaceTimeDBscan.html b/docs/articles/web_vignettes/SpaceTimeDBscan.html
index d72c5f58..6202ae79 100644
--- a/docs/articles/web_vignettes/SpaceTimeDBscan.html
+++ b/docs/articles/web_vignettes/SpaceTimeDBscan.html
@@ -33,7 +33,7 @@
@@ -53,7 +53,7 @@
K-nearest neighbour adaptive bandwidth
- Calculating isochrones
+ Calculating isochrones
Network k Functions
@@ -114,7 +114,7 @@ Spatio-temporal dbscan with network
Jeremy
Gelb
- 2024-05-19
+ 2024-10-20
Source: vignettes/web_vignettes/SpaceTimeDBscan.Rmd
SpaceTimeDBscan.Rmd
diff --git a/docs/articles/web_vignettes/SpeedVSprecision.html b/docs/articles/web_vignettes/SpeedVSprecision.html
index d7e7970e..a840e7f5 100644
--- a/docs/articles/web_vignettes/SpeedVSprecision.html
+++ b/docs/articles/web_vignettes/SpeedVSprecision.html
@@ -33,7 +33,7 @@
@@ -53,7 +53,7 @@
K-nearest neighbour adaptive bandwidth
- Calculating isochrones
+ Calculating isochrones
Network k Functions
@@ -110,7 +110,7 @@ Analysing the impact of max_dept parameter for
Jeremy
Gelb
- 2024-05-19
+ 2024-10-20
Source: vignettes/web_vignettes/SpeedVSprecision.Rmd
SpeedVSprecision.Rmd
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K-nearest neighbour adaptive bandwidth
- Calculating isochrones
+ Calculating isochrones
Network k Functions
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K-nearest neighbour adaptive bandwidth
d3_plot_situation(all_lines, event, samples_pts, continuous_kernel, scales = c(1,1,3))
d3_plot_situation(all_lines, events, samples_pts, continuous_kernel, scales = c(1,1,3))The larger the bandwidth, the more common links the events share in +
+The larger the bandwidth, the more common links the events share in their range.
@@ -351,13 +351,13 @@ The continuous method verbose = FALSE)
d3_plot_situation(all_lines, events, samples_pts, continuous_kernel, scales = c(1,1,3))
References
-Spatio-temporal dbscan with network
Jeremy
Gelb
- 2024-05-19
+ 2024-10-20
Source: vignettes/web_vignettes/SpaceTimeDBscan.Rmd
SpaceTimeDBscan.Rmd
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@@ -33,7 +33,7 @@
SpaceTimeDBscan.RmdAnalysing the impact of max_dept parameter for
Jeremy
Gelb
- 2024-05-19
+ 2024-10-20
Source: vignettes/web_vignettes/SpeedVSprecision.Rmd
SpeedVSprecision.Rmd
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SpeedVSprecision.Rmd
