Landsat Data in R
This tutorial covers the basic acquisition, import, and visualization of Landsat data in R.
Landsat is the name of a joint NASA / USGS program that provides repetitive, calibrated, satellite-based observations of the earth at a spatial resolution that enables analysis of man's interaction with the environment. (NASA 2021).
The Landsat 1 satellite was launched on 23 July 1972 and subsequent satellites have provided continuous satellite imagery of the Earth. This is arguably one of the most important scientific enterprises of our time, and if you work with remote sensing, you will probably use Landsat data on multiple occasions.
Landsat 7 (launched 1999) and Landsat 8 (launched 2013) are operational as of this writing. Landsat 9 is scheduled for launch in 2021 and will replace Landsat 7 with a satellite that is functionally similar to Landsat 8 (NASA 2020).
Landsat satellites complete just over 14 orbits a day, covering the entire earth every 16 days. Landsat satellites travel in a near-polar, sun-synchronous orbit so they spend as much time as possible looking at sunlit area.
Acquire the Data
The USGS makes recent and historic Landsat data freely available via the EarthExplorer website.
- Scenes: Landsat data is publicly available as scenes (images) that cover an area around 100 miles square. Scenes are designated by a path number (scanning path west to east) and row number (scene location on the path south to north).
- Collections: Landsat data has been released in two different collections, numbered one and two. Collection 2 incorporates the most recent processing enhancements to provide the best accuracy (USGS 2021).
- Levels: Landsat data is also released at two different quality levels. Level-1 scenes are higher quality than Level-2 scenes.
- Sensors: While historic scenes dating back to Landsat 1 (1972) are available from the multispectral scanner system carried on Landsats 1 to 4, this sensor had lower spatial resolution (68 m X 83 m) and lower spectral resolution (four bands) than the thematic mapper sensors (30 m resolution and seven bands) that were launched on Landsat 4 in 1982 and Landsat 5 in 1984. Therefore, unless you specifically need scenes from the 1970s, the oldest data you will probably want to use will be Landsat 4 or 5.
- Clouds: Because Landsat only passes over any given location on the surface of the earth every 16 days, if that happens to be a cloudy day at your point of interest on the day of interest, you will be out of luck. Therefore finding a clear scene will often be a challenge, especially during winter in cloudy northern climates.
In order to download data, you need to register (for free). Click the Login button and select Create New Account.
For the most recent scenes, you should use Landsat 8 data. In this example we download both a summer and a winter scene.
- Log in
- To get data for a specific location, search with a Feature Name and State and select a location that you want to explore.
- Select the appropriate collection and sensor.
- If you want the most recent scenes, use: Landsat -> Landsat Collection 2 Level-1 -> Landsat 8 OLTI/TIRS C1 Level-1.
- If you want historical scenes, use: Landsat -> Landsat Collection 2 Level-1 -> Landsat 4 - 5 TM C1 Level-1.
- View scene footprints that are in the time frame of interest, that cover your area of interest, and are not covered in clouds. Be aware that for vegetation analysis, summer scenes are better.
- Click Download and you will be given multiple options.
- If you just need visible light imagery, the Full Resulotion Browse (Natural Color) GeoTIFF is the simplest (single file) option.
- If you are doing some kind of multi-band analysis (such as NDVI analysis), dowload the full (and large) Landsat Collection 2 Level-1 Product Bundle.
- Create a new working directory and
uncompress the data files into that directory.
- The Natural Color image files are made available directly as GeoTIFF (.tif) files that require no additional processing.
- The Product Bundle data is a .tar archive contianing GeoTIFF files for individual bands that can be extracted using a program like 7-zip.
Individual Landsat bands for a scene are placed in separate GeoTIFF files that are distinguished by the ends of their names. For example, for Landsat 8 level 1 data (LC08_L1TP), path 23 and row 32 (023032) on 6 June 2021 (20210622), the following files are data files for the 11 different bands
- LC08_L1TP_023032_20210622_20210629_02_T1_B1.TIF (coastal aerosol - ultraviolet)
- LC08_L1TP_023032_20210622_20210629_02_T1_B2.TIF (blue)
- LC08_L1TP_023032_20210622_20210629_02_T1_B3.TIF (green)
- LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF (red)
- LC08_L1TP_023032_20210622_20210629_02_T1_B5.TIF (near infrared)
- LC08_L1TP_023032_20210622_20210629_02_T1_B6.TIF (shortwave infrared 1)
- LC08_L1TP_023032_20210622_20210629_02_T1_B7.TIF (shortwave infrared 2)
- LC08_L1TP_023032_20210622_20210629_02_T1_B8.TIF (panchromatic)
- LC08_L1TP_023032_20210622_20210629_02_T1_B9.TIF (cirrus)
- LC08_L1TP_023032_20210622_20210629_02_T1_B10.TIF (thermal infrared 1)
- LC08_L1TP_023032_20210622_20210629_02_T1_B11.TIF (thermal infrared 2)
Additional GeoTIFF files provide more-detailed information about the data and data processing.
- LC08_L1TP_023032_20210622_20210629_02_T1_QA_PIXEL.TIF: Pixel quality adjustment band
- LC08_L1TP_023032_20210622_20210629_02_T1_QA_RADSAT.TIF: Radiometric saturation and terrain occlusion QA band
- LC08_L1TP_023032_20210622_20210629_02_T1_SAA.TIF: Solar azimuth angle
- LC08_L1TP_023032_20210622_20210629_02_T1_SZA.TIF: Solar zenith angle
- LC08_L1TP_023032_20210622_20210629_02_T1_VAA.TIF: Sensor azimuth angle
- LC08_L1TP_023032_20210622_20210629_02_T1_VZA.TIF: Sensor zenith angle
Memory and Bandwidth Considerations
Note that rasters are often large files that take a long time to load and consume considerable disk space, memory, and connection bandwith.
If your machine has inadequate memory to keep all the data in memory, your machine may begin thrashing data back and forth between RAM and your hard disk (virtual memory). This will result in very slow performance from all apps running on your machine, and you will likely need to kill R to regain control of your computer.
If you have inadequate memory, you will need to find a machine with more RAM (such as a lab machine on UIUC AnyWare), you will need to add memory to your machine, or you may need to acquire a more powerful machine.
An unrelated source of performance degradation can occur if you are working on a lab machine that stores data on a computer network. Having to constantly transfer your data across a network can consume a considerable amount of unnecessary time. This can be especially serious if you are connected to the internet through a poor Wi-Fi connection rather than a fast wired connection.
If you are storing your data on a network drive, you should consider downloading and performing your work on your local hard drive, and then saving your work at completion to the network drive.
Landsat 8 RGB Visualization
The raster Package
Raster data in R is commonly handled with the raster package, which you can install with install.packages() before you use it the first time.
The raster package provides three different raster structures that you choose from depending on how many bands you need for a scene, and how many files they come from:
- raster() = one file, one band
- brick() = one file, multiple bands. Commonly used with RGB images
- stack() = multiple files, multiple bands (Landsat GeoTIFFs)
To create a RasterStack for a visible light RGB scene:
- Create three raster() for the red, green, and blue band files.
- Combine those into a single raster stack().
- Use the plotRGB() to display the stack
- The scale=65535 parameter is needed because Landsat 8 data is published with 16-bit values that range from 0 to 65535.
Because the Landsat sensors have a wide dynamic range to be able to handle a variety of different terrains, the values for a scene may be in the low range of potential values, which will result in a very dark visualization.
library(raster) blue = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B2.TIF") green = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B3.TIF") red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") rgb = stack(red, green, blue) plotRGB(rgb, scale=65535)
Landsat scenes cover a land area of over 200 square kilometers, which, along with the dark rendering, can make it difficult to distinguish what locations are being displayed.
If your focus is on a fairly small area within the scene, cropping the scene will both make it easier to see where you are, and will reduce the memory needed to process the scene further.
- Create an empty raster() with the top left (ymx and xmn) and bottom right (ymn and xmx) lat/long coordinates.
- projectExtent() that raster into the projected coordinate system of the source rasterStack.
- crop() the source rasterStack to the projected extent.
- plotRGB() the cropped rasterStack.
boundary = raster(ymx=40.17, xmn=-88.38, ymn=40.03, xmx=-88.10) boundary = projectExtent(boundary, rgb@crs) champaign = crop(rgb, boundary) plotRGB(champaign)
In some cases, the stretch option on plotRGB() can improve the rendering of the scene. lin stretches the values evenly from the minimum to the minimum, and hist distributes them more evenly across the range.
In both cases with this particular scene, the results make features more visible, but the visual contrast is harsh and, arguably, unappealing.
Using hist() to show a histogram of the distributions of values across the different bands shows the heavy skewing of the distributions so that most values are in the (dark) low end of the distribution in all bands. This explains why the default plotRGB() is so dark.
This function reads all values from the raster files and may take a few seconds to complete.
Given that knowledge of the distribution, you can use the zlim and scale parameters to more clearly define range of values that will be displayed. The values that look best will depend on the content of your specific scene.
- The champaign - 6000 darkens the overall scene by subtracting 6000 from all pixels so that most of the lowest values are clipped off.
- The ext parameter defines the range of values that can be processed.
- The scale parameter defines the highest value in the data. This should probably be the same as the maximum ext value unless you specifically want to darken the scene by making the highest displayed value less than the highest value that the display can render.
plotRGB(champaign - 5000, scale=12000, zlim=c(0, 12000))
The custom stretch performs the same operation on all values, which can imbalance the colors and gives the scene a slight tinting and or haze. The channel rasters can be individually adjusted for leveling and gamma shift.
- The crop() should be performed before the math to reduce the amount of memory needed for the modified raster data.
- The subtraction and division serves the same purpose as the xlim minimum and maximum scaling above.
- The reset of negative values to zero is needed to avoid invalid values when exponents are used.
- The exponents perform a gamma stretch by shifting the lower values up to lighten the dark areas with saturated greens in this largely agricultural area. Values below one increase the curve shift and values above one decrease the shift.
- This scaling and shifting moves the values to a range of zero to one, which should be reflected in the plotRGB() parameters.
blue = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B2.TIF") green = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B3.TIF") red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") blue = crop(blue, boundary) green = crop(green, boundary) red = crop(red, boundary) blue = ((blue - 7000) / 10000) green = ((green - 7000) / 10000) red = ((red - 7000) / 10000) blue[blue < 0] = 0 green[green < 0] = 0 red[red < 0] = 0 blue = blue^0.95 green = green^0.5 red = red^0.4 rgb = stack(red, green, blue) plotRGB(rgb, scale=1, zlim=c(0, 1))
Landsat 5 RGB Visualization
Because Landsat data has been collected for almost half a century, it can be used to detect broad land use changes. The process of visualizing Landsat 5 RGB data is largely the same as with Landsat 8, although the differences in band numbering and bit depth necessitate different parameters.
This Landsat 5 scene is from 28 August 1984. Note that the Landsat 5 Thematic Mapper band numbering is different from the newer Landsat 8 band numbering.
library(raster) blue = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B1.TIF") green = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B2.TIF") red = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B3.TIF") rgb = stack(red, green, blue) boundary = raster(ymx=40.17, xmn=-88.38, ymn=40.03, xmx=-88.10) boundary = projectExtent(boundary, rgb@crs) champaign = crop(rgb, boundary) plotRGB(champaign, stretch="lin")
As with the Landsat 8 scene, linear stretching creates harsh contrast that can be scaled and stretched to give a more pleasing visualization.
blue = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B1.TIF") green = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B2.TIF") red = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B3.TIF") boundary = raster(ymx=40.17, xmn=-88.38, ymn=40.03, xmx=-88.10) boundary = projectExtent(boundary, red@crs) blue = crop(blue, boundary) green = crop(green, boundary) red = crop(red, boundary) blue = ((blue - 30) / 120) green = ((green - 30) / 70) red = ((red - 30) / 70) blue[blue < 0] = 0 green[green < 0] = 0 red[red < 0] = 0 blue = blue^1.7 green = green^0.3 red = red^0.2 rgb = stack(red, green, blue) plotRGB(rgb, scale=1, zlim=c(0, 1))
Natural Color Images
USGS also publishes preprocessed three-band composite RGB files that can be downloaded directly without having to combined band files or perform additional correction. These are the Full-Resolution Browse (Natural Color) GeoTIFF option on the download dialog in EarthExplorer. These are actually false-color images constructed with the infrared and red bands, but in they often render well in RGB and require much less work to use than stacks constructed from band files.
Because the natural color scenes are in single composite files, use the brick() function to load the image into memory and the plotRGB() function to display the brick as a single image. Optionally, you can also crop the image as described above.
rgb = brick("LC08_L1TP_023031_20200822_20200905_02_T1_refl.tif") boundary = raster(ymx=41.922, xmn=-87.72, ymn=41.838, xmx=-87.552) boundary = projectExtent(boundary, rgb@crs) rgb = crop(rgb, boundary) plotRGB(rgb)
Other Band Combinations
Although RGB is the most obvious band combination for Landsat data, the comparatively low resolution (30 meters) severely limits visible light image quality compared to other contemporary high-resolution sources, especially when focusing on localities.
However, data from the eight other infrared and ultraviolet bands can be combined with the visible light bands in a variety of ways to create useful visualizations for biogeography and agriculture.
This visualization colors near-infrared (band 5) as red, red (band 4) as green, and green (band 3) as blue.
Healthy photosynthetic plants reflect infrared light, so helthy vegetation is red, bare or urbanized areas are white, and dark areas are water.
green = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B3.TIF") red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") near.infrared = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B5.TIF") rgb = stack(near.infrared, red, green) boundary = raster(ymx=40.17, xmn=-88.38, ymn=40.03, xmx=-88.10) boundary = projectExtent(boundary, rgb@crs) rgb = crop(rgb, boundary) plotRGB(rgb, scale=65535, stretch="lin")
This alternative vegetation visualization colors short-wave infrared 2 (band 7) as red, short-wave infrared 1 (band 6) as green, and red(band 4) as blue.
Density of vegetation is indicated by the darkness of the shades of green. Urban areas are blue and soil is brown.
red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") sw.infrared.1 = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B6.TIF") sw.infrared.2 = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B7.TIF") rgb = stack(sw.infrared.2, sw.infrared.1, red) boundary = raster(ymx=40.17, xmn=-88.38, ymn=40.03, xmx=-88.10) boundary = projectExtent(boundary, rgb@crs) rgb = crop(rgb, boundary) plotRGB(rgb, scale=65535, stretch="lin")
The Landsat multispectral data is of considerable value for the remote sensing of the environment.
One commonly used technique is the calculation of Normalized Difference Vegetation Index (NDVI), which can be used to detect the presence of and changes in photosynthetic vegetation.
NDVI = (infrared - red) / (infrared + red)
NDVI is based on the observation that green plants reflect infrared light (to avoid overheating), but absorb visible light (to power photosynthesis). This gives green plants a light reflection signature that can distinguish them from objects that absorb all spectra of light or different spectra of light from plants.
While you could use the level of green light to detect green (photosynthetic) plants, the level of green light depends on the intensity and angle of the sun relative to the terrain. The use of this combination of red and near-infrared gives measurements that are normalized to be the consistent across time and space regardless of terrain or level of lighting.
Creating an NDVI Raster
Rasters with the same extents and pixel dimensions can be used in mathematical operations like vectors or matrices. This is called raster algebra.
In this example:
- The raster() calls load the red and near.infrared band files.
- raster() is also used to create an extent for cropping and projectExtent() reprojects that extent to the projection and resolution used by the Landsat data.
- The crop() function crops the red and near infrared rasters to the extent defined for the desired boundary.
- The NDVI formula is used in raster algebra to create a new NDVI raster.
- The colorRampPalette() is used to define 255 colors that are then passed with the raster to plot() a visualization of the NDVI raster.
red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") near.infrared = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B5.TIF") boundary = raster(ymx=40.17, xmn=-88.38, ymn=40.03, xmx=-88.10) boundary = projectExtent(boundary, red@crs) red = crop(red, boundary) near.infrared = crop(near.infrared, boundary) ndvi = (near.infrared - red) / (near.infrared + red) colors = colorRampPalette(c("red3", "white", "darkcyan"))(255) plot(ndvi, zlim=c(0, 0.6), col=colors)
The range of NDVI is -1 to +1, although NDVI values will generally be in a small area in the middle of the full potential range of -1 to +1. You can use this information to adjust the zlim parameter to distribute the color values and make them legible.
Blending with a Base Map For Context
Since the objective of NDVI analysis is usually to know what is going on at a specific location or set of locations, it can be helpful to plot NDVI with a base map that gives geographic context to the data:
- A raster() call converts the OpenStreetMap tiles to an RGB RasterStack.
- projectRaster() crops and reprojects the red and near infrared rasters so they are in the same format as the base map.
- The red and near infrared are converted to NDVI using the formula given above.
- The RGB() function converts the single-band NDVI raster to a three-band RGB rasterStack using the 255 colors defined with colorRampPalette().
- The base map and colored NDVI stacks are divided by 255 (so their range is 0 to 1) and multplied to blend the colors together.
- Finally, the blended stack is plotted to output device using plotRGB(). The scale=1 reflects that the values are 0 to 1, and the interpolate=T parameter smooths the pixels together for display.
library(OpenStreetMap) upperLeft = c(40.1171, -88.26) lowerRight = c(40.0817, -88.19) basemap = openmap(upperLeft, lowerRight, zoom=13) basemap = raster(basemap) red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") near.infrared = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B5.TIF") red = projectRaster(red, basemap) near.infrared = projectRaster(near.infrared, basemap) ndvi = (near.infrared - red) / (near.infrared + red) colors = colorRampPalette(c("red3", "white", "darkcyan"))(255) ndvi = RGB(ndvi, col=colors) blended = (basemap / 255.0) * (ndvi / 255.0) plotRGB(blended, scale=1, interpolate=1)
Analyzing NDVI Change
The same type of raster algebra used to calculate NDVI can be used to calculate changes in NDVI between scenes.
When looking for a scene for for comparison a different date, it is important that you use an image from the same path and row. Although Landsat scenes overlap, using the same path and row will maximize the amount of common area and, therefore the amount of area covered in the analysis.
This example displays changes in NDVI in Champaign-Urbana, IL between the 1984 Landsat 5 scene and the 2021 Landsat 8 scene.
While the urbanized areas seem largely unchanged, the surrounding area has noticeably lower NDVI, which reflects both the construction of residential developments on the edge of the city (sprawl), and may reflect decreased agricultural intensity in the areas around Champaign-Urbana, which is noticeable by comparing the visible light visualizations above. However, this may also reflect differences in the sensor hardware.
- The raster() and crop() function calls create the 2021 rasters and crop them to the desired area for viewing.
- The second set of raster() calls creates the 1984 rasters, and the projectRaster() calls crop and align them with the 2021 rasters.
- A raster with difference between the 2021 and 1984 NDVI rasters can be created by subtracting the 1984 raster from the 2021 raster.
red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") near.infrared = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B5.TIF") boundary = raster(ymx=40.17, xmn=-88.38, ymn=40.03, xmx=-88.10) boundary = projectExtent(boundary, red@crs) red = crop(red, boundary) near.infrared = crop(near.infrared, boundary) ndvi.2021 = (near.infrared - red) / (near.infrared + red) red = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B3.TIF") near.infrared = raster("LT05_L1TP_022032_19840828_20200918_02_T1_B4.TIF") red = projectRaster(red, ndvi.2021) near.infrared = projectRaster(near.infrared, ndvi.2021) ndvi.1984 = (near.infrared - red) / (near.infrared + red) ndvi.difference = ndvi.2021 - ndvi.1984 colors = colorRampPalette(c("red3", "white", "darkcyan"))(255) plot(ndvi.difference, col=colors, zlim=c(-1, 1))
Interactive Display with Leaflet
When exploring data, it can be helpful to have an interactive visualization that you can zoom and pan. You can either export your data to a web service like ArcGIS Online or use an library like Leaflet for R that provides a similar level of interactivity on your desktop.
If you haven't already, you can install the Leaflet for R library with install.packages().
The following script will create an interactive map using the NDVI change raster and open it in a new browser window. Note that unless you publish this map to a web server, it will only be available to you on your personal machine, and cannot be embedded into a notebook.
- leaflet() creates the map.
- addTiles() adds a base map to the map using OpenStreetMap tiles.
- colorNumeric() creates a color palette that Leaflet can use.
- addRasterImage() add the raster layer to the map along with the palette used to color it. The opacity=0.7 parameter makes the map semitransparent so you can see the base map features for context.
- addLegend() adds a legend so you know what numeric values the colors represent.
library(leaflet) map = leaflet() map = addTiles(map) palette = colorBin(c("red3", "white", "darkcyan"), values(ndvi.difference)) map = addRasterImage(map, ndvi.difference, palette, opacity=0.7) map = addLegend(map, pal=palette, values=values(ndvi.difference), title="NDVI") map
NDVI vs. Demographics
Vegetation (especially trees) is often associated with residential areas that are both more attractive and more healthful places to live (Visram 2021). Tree cover reduces summer street temperatures and promotes the presence of birds (Rubiano 2020).
There is a commonly cited relationship between neighborhood demographics and levels of vegetation that has been referred to as a luxury effect (Leong, Dunn, and Trautwein 2018) or tree gap (Visram 2021). Residential tree cover is considered a matter of environmental justice (Leahy and Serkez 2021), and American Forests' Tree Equity Score tracks the tree cover as critical infrastructure.
While NDVI does not clearly distinguish trees from grasses, levels of different types of vegetation tend to correlate in space, so NDVI can be used to assess the relationship between neighborhood greenness and demographics.
Load County Census Tracts
For this example, we will use demographic data from the US Census Bureau's 2015-2019 American Community Survey five-year estimates. A GeoJSON file of that data for census tracts in Illinois is available here.
- That GeoJSON file can be loaded into R with the st_read() file from the Simple Features (sf) package, and then displayed with the plot() function.
- This analysis will focus on Champaign County, IL. The subsetting line selects only tracts with a FIPS code that begins with the FIPS code for Champaign County (17019). The grepl() function subsets rows by returning TRUE for every FIPS field value that begins with the county FIPS code.
library(sf) tracts = st_read("2015-2019-acs-il_tracts.geojson", stringsAsFactors=F, quiet=T) tracts = tracts[grepl("^17019", tracts$FIPS),] plot(tracts["Median.Household.Income"], breaks="jenks", pal=colorRampPalette(c("red3", "white", "darkcyan")))
To filter other counties in Illinois, below are the FIPS codes for the top 14 Illinois counties by population.
- Cook (Chicago): 17031
- DuPage (Naperville): 17043
- Lake (Waukegan): 17097
- Will (Joliet): 17197
- Kane (Aurora): 17089
- McHenry (Crystal Lake): 17111
- Winnebago (Rockford): 17201
- Madison (Granite City):17119
- St. Clair (Belleville): 17163
- Champaign (Champaign): 17019
- Sangamon (Springfield): 17167
- Peoria (Peoria): 17143
- McLean (Bloomington): 17113
- Rock Island (Moline): 17161
Calculate County NDVI
NDVI is calculated using the same technique described above. For this example, we crop the rasters to the area occupied by all Champaign County tracts.
library(raster) red = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B4.TIF") near.infrared = raster("LC08_L1TP_023032_20210622_20210629_02_T1_B5.TIF") boundary = raster(as_Spatial(tracts)) boundary = projectExtent(boundary, red@crs) red = crop(red, boundary) near.infrared = crop(near.infrared, boundary) ndvi = (near.infrared - red) / (near.infrared + red) tracts = st_transform(tracts, st_crs(red)) colors = colorRampPalette(c("red3", "white", "darkcyan"))(255) plot(ndvi, zlim=c(0, 0.6), col=colors, interpolate=T, axes=F, box=F) plot(st_geometry(tracts), add=T)
Extract Mean NDVI for Each Tract
When used with a set of polygons like the tract polygons, the extract() function from the raster package finds all raster values covered by each polygon and combines them using the function specified with the fun parameter. For this example we will take the mean of all NDVI values within each tract.
tracts$NDVI = extract(ndvi, as_Spatial(tracts), fun=mean) plot(tracts["NDVI"], breaks="jenks", pal=colorRampPalette(c("red3", "white", "darkcyan")))
Save Tract Data
In order to preserve your processed data for future analysis, you may want to save it to a new GeoJSON file.
st_write(tracts, "2021-minn-champaign-ndvi.geojson", delete_dsn=T)
Correlate NDVI and Median Household Income
Plotting income against NDVI, we see a fairly strong luxury effect in Champaign County, IL.
plot(tracts$NDVI, tracts$Median.Household.Income) model = lm(Median.Household.Income ~ NDVI, weights=Total.Population, data=tracts) abline(model, lwd=3, col="navy") summary(model) Weighted Residuals: Min 1Q Median 3Q Max -3131216 -956475 -120820 905910 3644378 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -31924 14138 -2.258 0.0293 * NDVI 257878 40223 6.411 1.13e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1427000 on 41 degrees of freedom Multiple R-squared: 0.5006, Adjusted R-squared: 0.4885 F-statistic: 41.1 on 1 and 41 DF, p-value: 1.125e-07
To get geographic context for the low NDVI tract, we order() the tracts by mean NDVI, take the first feature, create an OpenStreetMap base map for the area covered by the bounding box, and overlay the outline of the tract.
This low NDVI tract is the barren student ghetto (Campustown) to the west of the University of Illinois campus.
tracts = tracts[order(tracts$NDVI),] low.tract = tracts[1,] library(OpenStreetMap) bbox = st_bbox(st_buffer(st_transform(low.tract, 4326), 0.001)) upperLeft = bbox[c("ymax", "xmin")] lowerRight = bbox[c("ymin", "xmax")] basemap = openmap(upperLeft, lowerRight) plot(basemap) plot(st_geometry(st_transform(low.tract, 3857)), lwd=3, add=T)
The high NDVI tract can be visualized by reversing the sort. In this case, we use an aerial imagery from Bing as the basemap.
As might be expected, this is a sparsely-populated rural agricultural area in the southeast corner of the county.
tracts = tracts[order(tracts$NDVI, decreasing=T),] high.tract = tracts[1,] library(OpenStreetMap) bbox = st_bbox(st_buffer(st_transform(high.tract, 4326), 0.001)) upperLeft = bbox[c("ymax", "xmin")] lowerRight = bbox[c("ymin", "xmax")] basemap = openmap(upperLeft, lowerRight, type="bing") plot(basemap) plot(st_geometry(st_transform(high.tract, 3857)), lwd=3, border="red", add=T)
A question we might ask would be, what mechanism ties income and neighborhood vegetation together?
- Is vegetation an amenity that attracts wealthy residents?
- Do wealthier people actively plant more vegetation?
- Are wealthier people able to afford more vegetation?
- Are places with older populations both more wealthy and filled with older houses, which would be more likely to be surrounded by mature trees, compared to places where less wealthy young families live in newer homes?
- Is vegetation simply a function of population density, where wealthier people can live on larger lots, which will have more vegetation?
To examine these questions, we can create a multiple regression model which considers additional variables that we hypothesize may be significant.
The results show that when population density (Pop.per.Square.Mile) is considered, it becomes significant, while median household income falls to the wayside. The percentage of homeowners is also presumably related since home ownership in this part of the state usually means a single family home on a lot with a lawn and trees, compared to renters (especially students) who live in dense, multi-family housing units with less surrounding vegetation.
This hints that the correlation between income and NDVI (at least in this part of the state) is indirect. The more money you have, the more space you can afford to occupy, and the more space you occupy, the more you will be surrounded by vegetation. However, this relationship is confounded in a largely low density area like Champaign County, poor people can also afford homes, which makes the income/vegetation relationship less clear than it might be in a more densely populated area.
model = lm(NDVI ~ Median.Household.Income + Pop.per.Square.Mile + Median.Age + Percent.Pre.War.Units + Percent.Homeowners + Percent.Bachelors.Degree, weights=Total.Population, data=tracts) summary(model) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.678e-01 4.127e-02 6.488 1.55e-07 *** Median.Household.Income -8.987e-07 6.107e-07 -1.472 0.149795 Pop.per.Square.Mile -4.081e-06 8.961e-07 -4.555 5.80e-05 *** Median.Age -3.266e-04 1.413e-03 -0.231 0.818572 Percent.Pre.War.Units -1.750e-04 4.854e-04 -0.360 0.720592 Percent.Homeowners 2.324e-03 6.482e-04 3.585 0.000992 *** Percent.Bachelors.Degree 1.775e-03 1.147e-03 1.548 0.130356 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.767 on 36 degrees of freedom Multiple R-squared: 0.7811, Adjusted R-squared: 0.7446 F-statistic: 21.41 on 6 and 36 DF, p-value: 1.592e-10
An issue with the aforementioned interpretation of the model results is that multiple regression assumes the independent variables are, indeed, independent and not correlated. The presence of correlation in explanatory variables is called multicollinearity.
While multicollinearity will not affect model fit, it can make model parameters ambiguous and cause coefficients to be marked as statistically insignificant even though their presence improves model fit (Ghosh 2017).
The imcdiag() function from the mctest library can be used to perform multiple tests on the model to identify multicollinearity.
library(mctest) model = lm(NDVI ~ Median.Household.Income + Pop.per.Square.Mile + Median.Age + Percent.Pre.War.Units + Percent.Homeowners + Percent.Bachelors.Degree, weights=Total.Population, data=tracts) imcdiag(model) All Individual Multicollinearity Diagnostics Result VIF TOL Wi Fi Leamer CVIF Klein Median.Household.Income 7.7600 0.1289 50.0239 64.2198 0.3590 -1.6275 1 Pop.per.Square.Mile 2.1322 0.4690 8.3781 10.7557 0.6848 -0.4472 0 Median.Age 3.9510 0.2531 21.8375 28.0347 0.5031 -0.8286 0 Percent.Pre.War.Units 1.1029 0.9067 0.7618 0.9779 0.9522 -0.2313 0 Percent.Homeowners 8.8169 0.1134 57.8449 74.2603 0.3368 -1.8491 1 Percent.Bachelors.Degree 1.9967 0.5008 7.3758 9.4689 0.7077 -0.4188 0 IND1 IND2 Median.Household.Income 0.0174 1.4406 Pop.per.Square.Mile 0.0634 0.8781 Median.Age 0.0342 1.2352 Percent.Pre.War.Units 0.1225 0.1543 Percent.Homeowners 0.0153 1.4662 Percent.Bachelors.Degree 0.0677 0.8255 1 --> COLLINEARITY is detected by the test 0 --> COLLINEARITY is not detected by the test Median.Household.Income , Median.Age , Percent.Pre.War.Units , Percent.Bachelors.Degree , coefficient(s) are non-significant may be due to multicollinearity
If you do not know how to interpret the different test statistics, the easiest approach is to look for a one in the Klein's rule column (Imdadullah, Aslam, and Altaf 2016; Klein 1962).
In this case Median.Household.Income and Percent.Homeowners are flagged as collinear. While more complex techniques like factor analysis can be used to mitigate the multicollinearity, the simplest approach is to remove the flagged variables.
model = lm(NDVI ~ Pop.per.Square.Mile + Median.Age + Percent.Pre.War.Units + Percent.Bachelors.Degree, weights=Total.Population, data=tracts) imcdiag(model) All Individual Multicollinearity Diagnostics Result VIF TOL Wi Fi Leamer CVIF Klein Pop.per.Square.Mile 1.5860 0.6305 7.6174 11.7191 0.7941 -10.7487 0 Median.Age 1.3766 0.7264 4.8957 7.5319 0.8523 -9.3298 0 Percent.Pre.War.Units 1.0204 0.9800 0.2648 0.4074 0.9900 -6.9155 0 Percent.Bachelors.Degree 1.1878 0.8419 2.4409 3.7552 0.9176 -8.0500 0 IND1 IND2 Pop.per.Square.Mile 0.0485 1.7999 Median.Age 0.0559 1.3327 Percent.Pre.War.Units 0.0754 0.0972 Percent.Bachelors.Degree 0.0648 0.7701 1 --> COLLINEARITY is detected by the test 0 --> COLLINEARITY is not detected by the test Percent.Pre.War.Units , Percent.Bachelors.Degree , coefficient(s) are non-significant may be due to multicollinearity R-square of y on all x: 0.6755 summary(model) Weighted Residuals: Min 1Q Median 3Q Max -5.8128 -2.4011 0.2137 2.3086 5.3079 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.292e-01 4.401e-02 5.207 6.93e-06 *** Pop.per.Square.Mile -4.680e-06 8.759e-07 -5.343 4.53e-06 *** Median.Age 3.382e-03 1.012e-03 3.342 0.00188 ** Percent.Pre.War.Units 1.931e-04 5.380e-04 0.359 0.72159 Percent.Bachelors.Degree 9.499e-04 1.036e-03 0.917 0.36511 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 3.216 on 38 degrees of freedom Multiple R-squared: 0.6879, Adjusted R-squared: 0.655 F-statistic: 20.94 on 4 and 38 DF, p-value: 3.468e-09
Removing the statistically insigificant variables leaves only population density and median age in the model, which still has an R2 indicating a fairly good fit.
model = lm(NDVI ~ Pop.per.Square.Mile + Median.Age, weights=Total.Population, data=tracts) imcdiag(model) All Individual Multicollinearity Diagnostics Result VIF TOL Wi Fi Leamer CVIF Klein IND1 Pop.per.Square.Mile 1.3646 0.7328 14.9482 Inf 0.856 -137.1094 0 0.0179 Median.Age 1.3646 0.7328 14.9482 Inf 0.856 -137.1094 0 0.0179 IND2 Pop.per.Square.Mile 1 Median.Age 1 1 --> COLLINEARITY is detected by the test 0 --> COLLINEARITY is not detected by the test * all coefficients have significant t-ratios R-square of y on all x: 0.6686
Spatial Lag Regression
A second issue is that multiple regression assumes that variables are not autocorrelated with themselves. Because all of these variables are likely spatially autocorrelated (clustered in space), running the same variables in an spatial lag model can consider whether spatial autocorrelation affects the relative importance of the relationships.
- The lagsarlm() function from the spdep library performs regression that attempts to compensate for spatial autocorrelation.
- The poly2nb() function determines which of the polygons are neighbors (adjacent to each other) and the nb2listw() function creates a weighting matrix that is used by lagsarlm().
In the model output, Rho is the parameter for autoregression. The p-values for the Liklihood Ratio (LR) test and Asymptotic standard error t-test indicate the significance of this parameter, with values under 0.05 indicating statistical significance. The low p-values indicate that autocorrelation is significant consideration with this data. (Anselin 2003).
However, the most notable change is that median age no longer significant in the spatial lag model. This leaves only population density as a significant predictor of average NDVI in a census tract.
library(spdep) neighbors = poly2nb(as_Spatial(tracts)) weights = nb2listw(neighbors, zero.policy=T) model = lagsarlm(NDVI ~ Pop.per.Square.Mile + Median.Age, data=tracts, listw=weights, tol.solve=1.0e-30, zero.policy=T) summary(model) Coefficients: (asymptotic standard errors) Estimate Std. Error z value Pr(>|z|) (Intercept) 8.1738e-02 4.2607e-02 1.9184 0.0550577 Pop.per.Square.Mile -2.6416e-06 7.0301e-07 -3.7576 0.0001716 Median.Age 1.3295e-03 7.4975e-04 1.7732 0.0761953 Rho: 0.66608, LR test value: 16.651, p-value: 4.4935e-05 Asymptotic standard error: 0.1169 z-value: 5.698, p-value: 1.212e-08 Wald statistic: 32.468, p-value: 1.212e-08 Log likelihood: 80.25075 for lag model ML residual variance (sigma squared): 0.0012555, (sigma: 0.035433) Number of observations: 43 Number of parameters estimated: 5 AIC: -150.5, (AIC for lm: -135.85) LM test for residual autocorrelation test value: 0.15444, p-value: 0.69432
Principal Component Analysis
In dealing with multicollinearity above, our approach was to simply remove correlated variables from our model. However, in eliminating variables, we may have also removed important information. We also cannot know whether variables we never chose not to include could have been significant.
One technique for addressing those issues is principal component analysis (PCA), which uses linear algebra to reduce a matrix of all the possible variables to a smaller set of principal component variables. When used as part of factor analysis, PCA is used to find relationships between observed, correlated variables that point to unobserved variables (called factors or latent variables) that explain the broader relationships (Majumdar 2018).
This reduction in the number of variables (dimensions) needed for analysis is called dimensionality reduction. Because this technique makes a determination of the importance of different variables without direct human intervention or judgement, PCA can also be considered a form of unsupervised machine learning.
For this example, we use a set of census tract demographic variables that we hypothesize may affect NDVI, but which we have observed above as being correlated. These variables are passed as the right side of a formula to the prcomp() function.
The print() of the output of the prcomp() function lists the calculated principal components along the top (PC1 - PC6) along with the percent of each variable that contributes to that component. These can be interpreted to infer patterns of relationships that hint at latent variables.
- PC1 draws strongly on income, age, and home ownership, which might broadly be considered a wealth component.
- PC2 draws strongly on percent of residents with bachelor's degrees, which may make it an education component.
- PC3 draws strongly on the percent of pre-war (built before 1940) building, which may make it a building age component.
- PC4 seems to be a population density component.
- PC5 seems to be an age component.
- PC6 draws on income and the inverse of home ownership, which may represent a high income renters component.
pca = prcomp(~ Median.Household.Income + Pop.per.Square.Mile + Median.Age + Percent.Pre.War.Units + Percent.Homeowners + Percent.Bachelors.Degree, data=tracts, scale=T) print(pca) Standard deviations (1, .., p=6):  1.7817884 1.1120335 0.9529559 0.6684303 0.4106193 0.2551065 Rotation (n x k) = (6 x 6): PC1 PC2 PC3 PC4 Median.Household.Income 0.50453008 -0.29820975 0.11966764 -0.09439929 Pop.per.Square.Mile -0.42854455 -0.27840409 0.16474412 0.79941428 Median.Age 0.50489392 -0.12534918 0.03310023 0.42237713 Percent.Pre.War.Units 0.03034971 0.53816646 0.83838676 0.02159398 Percent.Homeowners 0.53538904 -0.04247929 0.07006712 0.24128032 Percent.Bachelors.Degree -0.13899124 -0.72554949 0.49964152 -0.33902899 PC5 PC6 Median.Household.Income -0.478345209 0.63598320 Pop.per.Square.Mile -0.244642283 0.11307993 Median.Age 0.727370616 0.14423463 Percent.Pre.War.Units 0.005540119 0.07788716 Percent.Homeowners -0.381645934 -0.70906620 Percent.Bachelors.Degree 0.191268633 -0.23042000
We can test the statistical significance of these principal components as predictors of NDVI by creating a multiple regression model using the lm() function.
The listing of coefficients indicates that the wealth component is the strongest, followed by high income renters, and population density.
pca.data = as.data.frame(predict(pca, st_drop_geometry(tracts))) pca.data$NDVI = tracts$NDVI model = lm(NDVI ~ ., data=pca.data) summary(model) Residuals: Min 1Q Median 3Q Max -0.074877 -0.026456 -0.002282 0.024347 0.107067 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.3385591 0.0063652 53.189 < 2e-16 *** PC1 0.0363484 0.0036147 10.056 5.35e-12 *** PC2 0.0049857 0.0057917 0.861 0.39502 PC3 -0.0002518 0.0067585 -0.037 0.97048 PC4 -0.0241218 0.0096353 -2.503 0.01697 * PC5 0.0014034 0.0156850 0.089 0.92920 PC6 -0.0781579 0.0252465 -3.096 0.00379 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.04174 on 36 degrees of freedom Multiple R-squared: 0.7658, Adjusted R-squared: 0.7268 F-statistic: 19.62 on 6 and 36 DF, p-value: 5.166e-10
Since the principal components still represent spatial data, we should try spatial lag regression to verify that those variables are still significant when compensating for spatial autocorrelation.
Although the p-values for the Liklihood Ratio (LR) test and Asymptotic standard error t-test indicate autocorrelation is significant (both below 0.05), the p-values on the components largely match the significance observed above.
library(spdep) neighbors = poly2nb(as_Spatial(tracts)) weights = nb2listw(neighbors, zero.policy=T) model = lagsarlm(NDVI ~ ., data=pca.data, listw=weights, tol.solve=1.0e-30, zero.policy=T) summary(model) Type: lag Coefficients: (asymptotic standard errors) Estimate Std. Error z value Pr(<|z|) (Intercept) 0.1431644 0.0441071 3.2458 0.001171 PC1 0.0207942 0.0039552 5.2575 1.461e-07 PC2 0.0055409 0.0044463 1.2462 0.212693 PC3 0.0026783 0.0052533 0.5098 0.610175 PC4 -0.0172517 0.0074813 -2.3060 0.021111 PC5 0.0052327 0.0120898 0.4328 0.665145 PC6 -0.0593586 0.0193892 -3.0614 0.002203 Rho: 0.57636, LR test value: 11.936, p-value: 0.00055063 Asymptotic standard error: 0.12802 z-value: 4.5021, p-value: 6.7284e-06 Wald statistic: 20.269, p-value: 6.7284e-06 Log likelihood: 85.35492 for lag model ML residual variance (sigma squared): 0.001023, (sigma: 0.031984) Number of observations: 43 Number of parameters estimated: 9 AIC: -152.71, (AIC for lm: -142.77) LM test for residual autocorrelation test value: 10.882, p-value: 0.00097091