1. Introducción

Este cuaderno ilustra dos técnicas de interpolación espacial: distancia ponderada inversa (IDW) y Kriging ordinario (OK). IDW es una técnica determinista. OK es probabilístico. Ambas técnicas se utilizan aquí para obtener una superficie continua de SOC a 15-30 cm a partir de muestras obtenidas de SoilGrids 250 m.

2. Configuración

Primero, limpia la memoria:

rm(list=ls())

Cargamos las librerias:

library(sp)
library(terra)
## terra 1.7.55
library(sf)
## Linking to GEOS 3.9.3, GDAL 3.5.2, PROJ 8.2.1; sf_use_s2() is TRUE
library(stars)
## Loading required package: abind
library(gstat)
library(automap)
library(leaflet)
library(leafem)
library(ggplot2)
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:terra':
## 
##     intersect, union
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(tmap)
## Breaking News: tmap 3.x is retiring. Please test v4, e.g. with
## remotes::install_github('r-tmap/tmap')
library(curl)
## Using libcurl 8.3.0 with Schannel
h <- new_handle()
handle_setopt(h, http_version = 2)

3. Leer los datos de entrada

Necesitamos leer un conjunto de datos para imitar datos del mundo real. Por lo tanto, leamos la capa SOC que descargamos de ISRIC) usando la biblioteca terra:

archivo <- ("soc_igh_15_30.tif")
(soc <- rast(archivo))
## class       : SpatRaster 
## dimensions  : 1087, 712, 1  (nrow, ncol, nlyr)
## resolution  : 250, 250  (x, y)
## extent      : -8476000, -8298000, 319750, 591500  (xmin, xmax, ymin, ymax)
## coord. ref. : Interrupted_Goode_Homolosine 
## source      : soc_igh_15_30.tif 
## name        : soc_igh_15_30

Ahora, convierta los datos del SOC en porcentaje. Revise el factor de escala de SOC en el sitio web de SoilGrids y anótelo aquí.

soc.perc <-  soc/10

¿Cuál es el CRS de los datos del mundo real?

Parece que necesitamos una transformación de dicho CRS al conocido CRS WGS84:

geog ="+proj=longlat +datum=WGS84"
(geog.soc = project(soc.perc, geog))
## class       : SpatRaster 
## dimensions  : 1101, 757, 1  (nrow, ncol, nlyr)
## resolution  : 0.002216344, 0.002216344  (x, y)
## extent      : -76.11119, -74.43342, 2.87334, 5.313535  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : soc_igh_15_30 
## min value   :       10.9917 
## max value   :      120.7206

Convirtamos la capa SpatRaster en un objeto de estrellas:

stars.soc = st_as_stars(geog.soc)
m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      stars.soc,
      opacity = 0.8,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 8:130)
    ) 
m

4. Muestreo del mundo

Consigamos una muestra de aprox. 500 sitios a partir de datos del mundo real utilizando una muestra ubicada aleatoriamente:

set.seed(123456)

(samples <- spatSample(geog.soc, 500, "random", as.points=TRUE))
##  class       : SpatVector 
##  geometry    : points 
##  dimensions  : 500, 1  (geometries, attributes)
##  extent      : -76.11008, -74.43674, 2.88553, 5.307994  (xmin, xmax, ymin, ymax)
##  coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
##  names       : soc_igh_15_30
##  type        :         <num>
##  values      :            NA
##                        53.37
##                        21.19

Describa las características principales del objeto de muestra.

Las caracteristicas de esta muestra son: el tipo de clase que es Spatvector (vector), la geometria que es de puntos, las dimensiones que toma son de 500 puntos de 1 solo atributo, se muestra la extension que en este caso seria la del departamento del Tolima y ademas se usa el sistema de coordenadas WSG84.

Ahora, necesitamos convertir el objeto spatVector en un objeto de característica simple:

(muestras <- sf::st_as_sf(samples))
nmuestras <- na.omit(muestras)

Visualicemos las muestras:

longit <- st_coordinates(muestras)[,1]
latit <- st_coordinates(muestras)[,2]
soc <- muestras$soc_igh_15_30
id <- seq(1,500,1) 
(sitios <- data.frame(id, longit, latit, soc))

Eliminemos los valores de NA:

sitios <- na.omit(sitios)
head(sitios)

Ahora, visualisemos las muestras:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      stars.soc,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 8:130)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions())
m

Ahora estamos listos para realizar las tareas de interpolación.

5. Interpolación

5.1 Creación del objeto gstat

Para interpolar, primero necesitamos crear un objeto de clase gstat, usando una función del mismo nombre: gstat.

Un objeto gstat contiene toda la información necesaria para realizar la interpolación espacial, a saber:

-La definición del modelo. -Los datos de calibración

Según sus argumentos, la función gstat “entiende” qué tipo de modelo de interpolación queremos usar:

-Sin modelo de variograma → IDW -Modelo de variograma, sin covariables → Kriging ordinario

Vamos a utilizar tres parámetros de la función gstat:

-fórmula: la “fórmula” de predicción que especifica las variables dependientes e independientes (también conocidas como covariables) -datos: los datos de calibración (también conocidos como los datos del tren) -modelo: el modelo de variograma

5.2 interpolación IDW

Para interpolar SOC usando el método IDW creamos el siguiente objeto gstat, especificando solo la fórmula y los datos:

g1 = gstat(formula = soc_igh_15_30 ~ 1, data = nmuestras)

Ahora que nuestro modelo de interpolación g1 está definido, podemos usar la función de predicción para interpolar, es decir, estimar los valores de precipitación.

La función de predicción acepta:

-Un objeto ráster: estrellas, como dem -Un modelo: objeto gstat, como g1

El ráster sirve para dos propósitos:

-Especificar las ubicaciones donde queremos hacer predicciones (en todos los métodos) -Especificación de valores de covariables (solo en Universal Kriging)

Creamos un objeto ráster con valores de celda iguales a 1:

rrr = aggregate(geog.soc, 4)

Que es rrr?:

rrr
## class       : SpatRaster 
## dimensions  : 276, 190, 1  (nrow, ncol, nlyr)
## resolution  : 0.008865376, 0.008865376  (x, y)
## extent      : -76.11119, -74.42677, 2.866691, 5.313535  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : soc_igh_15_30 
## min value   :      12.02751 
## max value   :     103.87885

Definir nuevos valores:

values(rrr) <-1

Definir nuevos nombres:

names(rrr) <- "valor"

Ahora que es rrr?:

rrr
## class       : SpatRaster 
## dimensions  : 276, 190, 1  (nrow, ncol, nlyr)
## resolution  : 0.008865376, 0.008865376  (x, y)
## extent      : -76.11119, -74.42677, 2.866691, 5.313535  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=longlat +datum=WGS84 +no_defs 
## source(s)   : memory
## name        : valor 
## min value   :     1 
## max value   :     1
stars.rrr = st_as_stars(rrr)

Por ejemplo, la siguiente expresión interpola los valores SOC según el modelo definido en g1 y la plantilla ráster definida en stars.rrr:

z1 = predict(g1, stars.rrr)
## [inverse distance weighted interpolation]

Que es z1?:

z1
## stars object with 2 dimensions and 2 attributes
## attribute(s):
##                Min. 1st Qu.   Median     Mean  3rd Qu.     Max.  NA's
## var1.pred  12.85815 32.4155 40.86539 41.46911 50.38098 96.75321     0
## var1.var         NA      NA       NA      NaN       NA       NA 52440
## dimension(s):
##   from  to offset     delta                       refsys x/y
## x    1 190 -76.11  0.008865 +proj=longlat +datum=WGS8... [x]
## y    1 276  5.314 -0.008865 +proj=longlat +datum=WGS8... [y]

Tome nota de los nombres de los dos atributos incluidos dentro del objeto z1.

Podemos crear un subconjunto solo del primer atributo y cambiarle el nombre a “soc”:

z1 = z1["var1.pred",,]
names(z1) = "soc"

Necesitamos una paleta de colores:

paleta <- colorNumeric(palette = c("orange", "yellow", "cyan", "green"), domain = 10:100, na.color = "transparent")

El ráster SOC interpolado, utilizando IDW, se muestra en la siguiente figura:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      z1,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 11:55)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions()) %>%
    addLegend("bottomright", pal=paleta, values= z1$soc,
    title = "IDW SOC interpolation [%]"
    )
m

5.3 Interpolación correcta

Los métodos de Kriging requieren un modelo de variograma. El modelo de variograma es una forma objetiva de cuantificar el patrón de autocorrelación en los datos y asignar ponderaciones en consecuencia al realizar predicciones.

Como primer paso, podemos calcular y examinar el variograma empírico utilizando la función de variograma.

La función requiere dos argumentos:

-fórmula: especifica la variable dependiente y las covariables, como en gstat -datos: la capa de puntos con la variable dependiente y covariables como atributos de puntos

Por ejemplo, la siguiente expresión calcula el variograma empírico de muestras, sin covariables:

v_emp_ok = variogram(soc_igh_15_30 ~ 1, data=nmuestras)

Tracemos el variograma:

plot(v_emp_ok)

Hay varias formas de ajustar un modelo de variograma a un variograma empírico. Usaremos el más simple: ajuste automático usando la función autofitVariogram del paquete automap:

v_mod_ok = autofitVariogram(soc_igh_15_30 ~ 1, as(nmuestras, "Spatial"))

La función elige el tipo de modelo que mejor se adapta y también ajusta sus parámetros. Puede utilizar show.vgms() para mostrar los tipos de modelos de variograma.

Tenga en cuenta que la función autofitVariogram no funciona en objetos sf, por lo que convertimos el objeto en un SpatialPointsDataFrame (paquete sp).

El modelo ajustado se puede trazar con plot:

plot(v_mod_ok)

El objeto resultante es en realidad una lista con varios componentes, incluido el variograma empírico y el modelo de variograma ajustado. El componente $var_model del objeto resultante contiene el modelo real:

v_mod_ok$var_model

En tu cuaderno, explica el significado de cada elemento del modelo anterior.

-Range: Distancia a la cual se alcanza el sill.

-Sill: Maxima Variabilidad aespacial.

-Nug: Variabilidad a la menos distancia posible.

-Ste: Varaiablidad a la mayor distancia posible.

Ahora, el modelo de variograma se puede pasar a la función gstat y podemos continuar con la interpolación Kriging ordinaria:

g2 = gstat(formula = soc_igh_15_30 ~ 1, model = v_mod_ok$var_model, data = nmuestras)
z2= predict(g2, stars.rrr)
## [using ordinary kriging]

Nuevamente, crearemos un subconjunto del atributo de valores predichos y le cambiaremos el nombre:

z2 = z2["var1.pred",,]
names(z2) = "soc"

Las predicciones de Kriging ordinario se muestran en la siguiente figura:

m <- leaflet() %>%
  addTiles() %>%  
  leafem:::addGeoRaster(
      z2,
      opacity = 0.7,                
      colorOptions = colorOptions(palette = c("orange", "yellow", "cyan", "green"), 
                                  domain = 11:55)
    ) %>%
  addMarkers(lng=sitios$longit,lat=sitios$latit, popup=sitios$soc, clusterOptions = markerClusterOptions()) %>%
    addLegend("bottomright", pal = paleta, values= z2$soc,
    title = "OK SOC interpolation [%]"
    )
m

6. Evaluación de resultados

6.1 Evaluación cualitativa

Otra vista de las tres salidas de interpolación:

colores <- colorOptions(palette = c("orange", "yellow", "cyan", "green"), domain = 10:100, na.color = "transparent")
m <- leaflet() %>%
  addTiles() %>%  
  addGeoRaster(stars.soc, opacity = 0.8, colorOptions = colores, group="RealWorld") %>%
  addGeoRaster(z1, colorOptions = colores, opacity = 0.8, group= "IDW")  %>%
  addGeoRaster(z2, colorOptions = colores, opacity = 0.8, group= "OK")  %>%
  # Add layers controls
  addLayersControl(
    overlayGroups = c("RealWorld", "IDW", "OK"),
    options = layersControlOptions(collapsed = FALSE)
  ) %>% 
    addLegend("bottomright", pal = paleta, values= z1$soc,
    title = "Soil organic carbon [%]"
)
m

6.2 Validación cruzada

Hemos estimado las superficies climáticas utilizando dos métodos diferentes: IDW y Kriging Ordinario. Aunque es útil examinar y comparar los resultados gráficamente, también necesitamos una forma objetiva de evaluar la precisión de la interpolación. De esa manera, podemos elegir objetivamente el método más preciso entre los métodos de interpolación disponibles.

En la validación cruzada Leave-One-Out nosotros:

-Saque un punto de los datos de calibración. -Haz una predicción para ese punto. -Repita para todos los puntos. Al final, lo que obtenemos es una tabla con un valor observado y un valor predicho para todos los puntos.

Podemos ejecutar la validación cruzada Leave-One-Out usando la función gstat.cv, que acepta un objeto gstat.

Al escribir el siguiente fragmento, oculte el mensaje y los resultados.

cv1 = gstat.cv(g1)
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## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]
## [inverse distance weighted interpolation]

El resultado de gstat.cv tiene los siguientes atributos:

-var1.pred: valor previsto -var1.var: varianza (solo para Kriging) -observado: valor observado -residual: observado-predicho -zscore: puntuación Z (solo para Kriging) -plegar: ID de validación cruzada

cv1 = na.omit(cv1)
cv1
## class       : SpatialPointsDataFrame 
## features    : 476 
## extent      : -76.06576, -74.44339, 2.88553, 5.307994  (xmin, xmax, ymin, ymax)
## crs         : +proj=longlat +datum=WGS84 +no_defs 
## variables   : 6
## names       :        var1.pred, var1.var,         observed,          residual, zscore, fold 
## min values  : 19.6068594224905,       NA, 12.7732934951782, -20.2674750595145,     NA,    1 
## max values  : 70.2081924593506,       NA, 97.7614059448242,  45.5275049032166,     NA,  476

Convirtamos el objeto cv1:

cv1 = st_as_sf(cv1)

Ahora, grafiquemos los residuos:

sp::bubble(as(cv1[, "residual"], "Spatial"))

Ahora, calculamos índices de precisión de predicción, como el error cuadrático medio (RMSE):

sqrt(sum((cv1$var1.pred - cv1$observed)^2) / nrow(cv1))
## [1] 11.10777

Ahora, repita el proceso con los resultados correctos:

cv2 = gstat.cv(g2)
## [using ordinary kriging]
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## [using ordinary kriging]
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## [using ordinary kriging]
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## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
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## [using ordinary kriging]
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## [using ordinary kriging]
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## [using ordinary kriging]
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## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
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## [using ordinary kriging]
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## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
## [using ordinary kriging]
cv2 = na.omit(cv2)
cv2
## class       : SpatialPointsDataFrame 
## features    : 476 
## extent      : -76.06576, -74.44339, 2.88553, 5.307994  (xmin, xmax, ymin, ymax)
## crs         : +proj=longlat +datum=WGS84 +no_defs 
## variables   : 6
## names       :        var1.pred,         var1.var,         observed,          residual,            zscore, fold 
## min values  : 15.2338001796977, 41.5095520719797, 12.7732934951782, -24.9741925554444, -3.02844651082469,    1 
## max values  : 75.7055196383225, 168.889503939612, 97.7614059448242,  38.3400100998308,  3.92886622692256,  476

Tiempo de conversión:

cv2 = st_as_sf(cv2)
sp::bubble(as(cv1[, "residual"], "Spatial"))

Calcule RSME para obtener resultados correctos:

sqrt(sum((cv2$var1.pred - cv2$observed)^2) / nrow(cv2))
## [1] 8.973377

Escribe un párrafo que resuma tus resultados. Según ellos, qué técnica de interpolación parece más precisa. Explicar.

-En este caso el mejor metodo y el mas preciso, nos damos cuena en el Error Cuadratico Medio (RMSE - Root Mean Square Error), el mejor metodo es el Ok (Ordinary Kriging), teniendo un valor bajo de 8.973377, con los datos que usamos se llega a la conclusión de que con este metodo se llega mas a los valores reales.

7. Referencias

Cuaderno sacado de: Lizarazo, I., 2023. Interpolación espacial del carbono orgánico del suelo. Disponible en https://rpubs.com/ials2un/soc_interp.

sessionInfo()
## R version 4.2.3 (2023-03-15 ucrt)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 19045)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=Spanish_Colombia.utf8  LC_CTYPE=Spanish_Colombia.utf8   
## [3] LC_MONETARY=Spanish_Colombia.utf8 LC_NUMERIC=C                     
## [5] LC_TIME=Spanish_Colombia.utf8    
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] curl_5.1.0    tmap_3.3-4    dplyr_1.1.4   ggplot2_3.4.4 leafem_0.2.3 
##  [6] leaflet_2.2.1 automap_1.1-9 gstat_2.1-1   stars_0.6-4   abind_1.4-5  
## [11] sf_1.0-14     terra_1.7-55  sp_2.1-1     
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.11             lattice_0.20-45         FNN_1.1.3.2            
##  [4] leaflet.providers_2.0.0 png_0.1-8               class_7.3-21           
##  [7] zoo_1.8-12              digest_0.6.33           utf8_1.2.4             
## [10] R6_2.5.1                plyr_1.8.9              evaluate_0.23          
## [13] e1071_1.7-13            highr_0.10              pillar_1.9.0           
## [16] rlang_1.1.2             rstudioapi_0.15.0       raster_3.6-26          
## [19] jquerylib_0.1.4         rmarkdown_2.25          htmlwidgets_1.6.2      
## [22] munsell_0.5.0           proxy_0.4-27            compiler_4.2.3         
## [25] xfun_0.41               tmaptools_3.1-1         pkgconfig_2.0.3        
## [28] base64enc_0.1-3         htmltools_0.5.7         tidyselect_1.2.0       
## [31] tibble_3.2.1            intervals_0.15.4        codetools_0.2-19       
## [34] XML_3.99-0.15           viridisLite_0.4.2       reshape_0.8.9          
## [37] fansi_1.0.5             spacetime_1.3-0         withr_2.5.2            
## [40] grid_4.2.3              lwgeom_0.2-13           jsonlite_1.8.7         
## [43] gtable_0.3.4            lifecycle_1.0.4         DBI_1.1.3              
## [46] magrittr_2.0.3          units_0.8-4             scales_1.2.1           
## [49] KernSmooth_2.23-20      cli_3.6.1               cachem_1.0.8           
## [52] farver_2.1.1            leafsync_0.1.0          bslib_0.5.1            
## [55] ellipsis_0.3.2          xts_0.13.1              generics_0.1.3         
## [58] vctrs_0.6.4             RColorBrewer_1.1-3      tools_4.2.3            
## [61] dichromat_2.0-0.1       glue_1.6.2              crosstalk_1.2.0        
## [64] parallel_4.2.3          fastmap_1.1.1           yaml_2.3.7             
## [67] colorspace_2.1-0        classInt_0.4-10         knitr_1.45             
## [70] sass_0.4.7