---
title: "6. Spatial Analysis"
author: "Tobias Stephan"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{6. Spatial Analysis}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  markdown: 
    wrap: 72
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

This vignette demonstrates some additional spatially interpolated statistics of 
a stress field.

```{r setup, message=FALSE}
library(tectonicr)
library(ggplot2) # load ggplot library
```

```{r load_data}
data("san_andreas")

data("cpm_models")
por <- cpm_models[["NNR-MORVEL56"]] |>
  equivalent_rotation("na", "pa")

plate_boundary <- subset(plates, plates$pair == "na-pa")
```

`circular_summary()` yields several statistics estimates for a given vector of 
angles, such as mean, median, standard deviation, quasi-quantiles, mode, 95% 
confidence angle, as wells as the moments (, i.e. 2nd moment = variance, 
3rd = skewness, 4th = kurtosis): 

```{r}
circular_summary(san_andreas$azi, w = 1 / san_andreas$unc)
```


## Spatial analysis
Spatial analysis and interpolation of stress data using `stress2grid_stats()` 
or `PoR_stress2grid_stats()` (analysis in the PoR coordinate system) uses a 
moving window with a user defined cell-size (im km) and calculates the summary 
statistics within each cell:

```{r interpolation,message=FALSE,warning=FALSE}
spatial_stats_R <- PoR_stress2grid_stats(san_andreas, PoR = por, gridsize = 1, R_range = 100)
subset(spatial_stats_R, !is.na(mean)) |> head()
```


One can also specify a range of cell-sizes for a wavelength analysis:

```{r interpolation2,message=FALSE,warning=FALSE}
spatial_stats <- PoR_stress2grid_stats(san_andreas, PoR = por, gridsize = 1, R_range = seq(50, 350, 100), mode = TRUE)
```

The mean azimuth for each grid cell:  
```{r plot, warning=FALSE, message=FALSE}
trajectories <- eulerpole_loxodromes(x = por, n = 40, cw = FALSE)

ggplot(spatial_stats) +
  geom_sf(data = plate_boundary, color = "red") +
  geom_sf(data = trajectories, lty = 2) +
  geom_spoke(data = san_andreas, aes(lon, lat, angle = deg2rad(90 - azi)), radius = .3, linewisth = .5, color = "grey30", position = "center_spoke") +
  geom_spoke(aes(lon, lat, angle = deg2rad(90 - mean), alpha = sd, color = mdr), radius = .75, position = "center_spoke", lwd = 1) +
  coord_sf(xlim = range(san_andreas$lon), ylim = range(san_andreas$lat)) +
  scale_alpha(name = "Standard deviation", range = c(1, .25)) +
  scale_color_viridis_c(
    "Wavelength\n(R-normalized mean distance)",
    limits = c(0, 1),
    breaks = seq(0, 1, .25)
  ) +
  facet_wrap(~R)
```

To filter the range of search windows to only keep the shortest wavelength (R) 
with the least variance for each grid cell, use compact_grid2().

```{r}
spatial_stats_comp <- spatial_stats |>
  compact_grid2(var)
``` 

Interpolated median stress field color-coded by the skewness within each search 
window:
```{r}
ggplot(spatial_stats_comp) +
  geom_sf(data = plate_boundary, color = "red") +
  geom_sf(data = trajectories, lty = 2) +
  geom_spoke(data = san_andreas, aes(lon, lat, angle = deg2rad(90 - azi)), radius = .3, color = "grey30", position = "center_spoke") +
  geom_spoke(aes(lon, lat, angle = deg2rad(90 - median), alpha = CI, color = skewness), radius = .5, position = "center_spoke", lwd = 1) +
  coord_sf(xlim = range(san_andreas$lon), ylim = range(san_andreas$lat)) +
  scale_alpha(name = "95% CI", range = c(1, .25)) +
  scale_color_viridis_c(
    "Skewness"
  )
```

Interpolated mode of the stress field color-coded by the absolute kurtosis 
within each search window:

```{r}
ggplot(spatial_stats_comp) +
  geom_sf(data = plate_boundary, color = "red") +
  geom_sf(data = trajectories, lty = 2) +
  geom_spoke(data = san_andreas, aes(lon, lat, angle = deg2rad(90 - azi)), radius = .3, color = "grey30", position = "center_spoke") +
  geom_spoke(aes(lon, lat, angle = deg2rad(90 - mode), alpha = CI, color = abs(kurtosis)), radius = .5, position = "center_spoke", lwd = 1) +
  coord_sf(xlim = range(san_andreas$lon), ylim = range(san_andreas$lat)) +
  scale_alpha(name = "95% CI", range = c(1, .25)) +
  scale_color_viridis_c(
    "|Kurtosis|"
  )
```

## Heat maps for the spatial statistics

`PoR_stress2grid_stats()` and `stress2grid_stats()` allow to create heatmaps 
showing the spatial patterns of any desired statistical estimate (from 
`circular_summary()`). Some examples:

### Spatial central moments

#### Spatial variance
```{r variance, warning=FALSE, message=FALSE}
ggplot(spatial_stats_comp) +
  ggforce::geom_voronoi_tile(
    aes(lon, lat, fill = var),
    max.radius = .7, normalize = FALSE
  ) +
  scale_fill_viridis_c(limits = c(0, 1)) +
  geom_sf(data = plate_boundary, color = "red") +
  geom_sf(data = trajectories, lty = 2) +
  geom_spoke(
    aes(lon, lat, angle = deg2rad(90 - mean)),
    radius = .5, position = "center_spoke", lwd = .2, colour = "white"
  ) +
  coord_sf(xlim = range(san_andreas$lon), ylim = range(san_andreas$lat))
```

#### Skewness:
Skewness is a measure for the asymmetry of the probability distribution. It can 
be either counterclockwise or clockwise skewed, hence values can range between 
negative and positive numbers, respectively. This can be best visualized in a 
diverging color-sequence:

```{r skew, warning=FALSE, message=FALSE}
ggplot(spatial_stats_comp) +
  ggforce::geom_voronoi_tile(
    aes(lon, lat, fill = skewness),
    max.radius = .7, normalize = FALSE
  ) +
  scale_fill_gradient2("|Skewness|", low = "#001260", mid = "#EBE5E0", high = "#590007") +
  geom_sf(data = plate_boundary, color = "red") +
  geom_sf(data = trajectories, lty = 2) +
  geom_spoke(
    aes(lon, lat, angle = deg2rad(90 - median), alpha = var),
    radius = .5, position = "center_spoke", lwd = .2, colour = "white"
  ) +
  scale_alpha("variance", range = c(1, 0)) +
  coord_sf(xlim = range(san_andreas$lon), ylim = range(san_andreas$lat))
```

#### Kurtosis
Kurtosis is a measure of the "tailedness" of the probability distribution. 
Here, colors are in a square-root scale:
```{r kurtosis, warning=FALSE, message=FALSE}
ggplot(spatial_stats_comp) +
  ggforce::geom_voronoi_tile(
    aes(lon, lat, fill = abs(kurtosis)),
    max.radius = .7, normalize = FALSE
  ) +
  scale_fill_viridis_c("|Kurtosis|", transform = "sqrt") +
  geom_sf(data = plate_boundary, color = "red") +
  geom_sf(data = trajectories, lty = 2) +
  geom_spoke(
    aes(lon, lat, angle = deg2rad(90 - mode), alpha = var),
    radius = .5, position = "center_spoke", lwd = .2, colour = "white"
  ) +
  scale_alpha("variance", range = c(1, 0)) +
  coord_sf(xlim = range(san_andreas$lon), ylim = range(san_andreas$lat))
```


### Kernel dispersion

Another way to analyse spatial misfits is the kernel dispersion, i.e. the
local dispersion within a user-defined window (kernel). The kernel´s half width 
can be a single number (km) or a range of widths. The latter requires to compact
the grid result (`x`) to find the smallest kernel size containing the the least 
dispersion (`compact_grid(x, 'dispersion')`).

> It is recommended to calculate the kernel dispersion on PoR transformed data to 
avoid angle distortions due to projections.

```{r kernel_disp}
san_andreas_por <- san_andreas
san_andreas_por$azi <- PoR_shmax(san_andreas, por, "right")$azi.PoR # transform to PoR azimuth
san_andreas_por$prd <- 135 # test direction
san_andreas_kdisp <- kernel_dispersion(san_andreas_por, gridsize = 1, R_range = seq(50, 350, 100))
san_andreas_kdisp <- compact_grid(san_andreas_kdisp, "dispersion")

ggplot(san_andreas_kdisp) +
  ggforce::geom_voronoi_tile(
    aes(lon, lat, fill = stat),
    max.radius = .7, normalize = FALSE
  ) +
  scale_fill_viridis_c("Dispersion", limits = c(0, 1)) +
  geom_sf(data = trajectories, lty = 2) +
  geom_spoke(
    data = san_andreas,
    aes(lon, lat, angle = deg2rad(90 - azi), alpha = unc),
    radius = .5, position = "center_spoke", lwd = .2, colour = "white"
  ) +
  scale_alpha("Standard deviation", range = c(1, .25)) +
  coord_sf(xlim = range(san_andreas$lon), ylim = range(san_andreas$lat))
```
