variogram_tutorial1.rmd

---
title: "Semivariograms part 1"
author: "Daniel M. Parker (w/ help from ChatGPT)"
date: "`r Sys.Date()`"
output:
  html_document:
    toc: true
    toc_float: false
    number_sections: false
    code_folding: "hide"
    theme: cosmo
    highlight: textmate
---


# **Introduction to Semivariograms**
A semivariogram describes how data values change with increasing distance. It helps quantify **spatial dependence**.

## **Spatial Dependence and Tobler's Law**
Tobler’s First Law of Geography states:
> "Everything is related to everything else, but near things are more related than distant things."

In semivariograms:
- **Short distances** → Similar values (low variance).
- **Large distances** → Increasing dissimilarity.

---

# **Semivariogram Components**
The key parameters of a semivariogram are:

- **Nugget (`C₀`)**: Variance at distance = 0 (measurement error or microscale variation).
- **Sill (`C`)**: Total variance when the semivariogram levels off.
- **Range (`A`)**: Distance at which spatial autocorrelation disappears.

### **A basic semivariogram**
```{r semivariogram-plot, echo=FALSE}
library(ggplot2)

# Define parameters
distances <- seq(0, 100, by=5)  # Distance values
nugget <- 15  # Nugget above zero
sill <- 160  # Sill where semivariance levels off
range_value <- 30  # Range where spatial dependence disappears

# Spherical model approximation
spherical_variogram <- function(h, nugget, sill, range_value) {
  ifelse(h < range_value, 
         nugget + (sill - nugget) * (1.5 * (h / range_value) - 0.5 * (h / range_value)^3), 
         sill)
}

# Compute semivariance values
semivariance <- sapply(distances, spherical_variogram, nugget = nugget, sill = sill, range_value = range_value)

# Create a data frame
data <- data.frame(Distance = distances, Semivariance = semivariance)

# Plot using ggplot2
library(ggplot2)
ggplot(data, aes(x = Distance, y = Semivariance)) +
  geom_line(color = "blue", linewidth = 1.2) +  # Semivariogram line
  geom_hline(yintercept = sill, linetype = "dashed", color = "red") +  # Sill
  geom_hline(yintercept = nugget, linetype = "dashed", color = "green") +  # Nugget
  geom_vline(xintercept = range_value, linetype = "dashed", color = "purple") +  # Range
  annotate("text", x = 5, y = nugget + 5, label = "Nugget", color = "green", fontface = "bold") +
  annotate("text", x = 80, y = sill - 5, label = "Sill", color = "red", fontface = "bold") +
  annotate("text", x = range_value + 2, y = sill / 2, label = "Range", color = "purple", fontface = "bold", angle = 90) +
  labs(title = "Improved Semivariogram Diagram", x = "Distance", y = "Semivariance") +
  ylim(0, 170) +  # Set y-axis range from 0 to 170
  theme_minimal()
```

---

# **Basic Variogram Statistics**

### **1. Sill - Nugget**
- Measures the structured spatial variance.
- **Large value** → Strong spatial structure.
- **Small value** → Weak spatial dependence.

### **2. Relative Nugget Effect (Nugget/Sill)**
- **High (~1)** → Mostly random noise.
- **Low (~0)** → Strong spatial structure.

### **3. Range-to-Distance Ratio (Range / Max Distance)**
- **Close to 1** → Strong correlation across the study area.
- **Close to 0** → Correlation fades quickly.

### **4. Partial Sill (Sill - Nugget)**
- Measures the variance **explained by spatial dependence**.

### **5. Proportion of Spatially Structured Variation**
- **Formula:** Partial Sill / Sill.
- **Close to 1** → Mostly structured.
- **Close to 0** → Mostly random.

### **6. Mean Variogram Value Over Specific Lag Distances**
- Helps analyze how spatial dependence changes at different distances.

### **7. Relative Range (Range / Study Area Size)**
- **Close to 1** → Correlation spans the entire study area.
- **Much less than 1** → Limited spatial dependence.
- **This will be discussed briefly at the end of the tutorial.**

### **8. Anisotropy**
- Sometimes spatial dependence varies by direction, a phenomenon known as **anisotropy**.
- To measure anisotropy, we can compute variograms along different angles (e.g., **0°, 45°, 90°**).
- If spatial correlation differs by direction, we can quantify it using **anisotropy ratios**.

---

# **Working Through a Real Example: The Meuse Dataset**
```{r empirical-variogram, message=FALSE}
library(sp)
library(gstat)

data(meuse, package = "sp")
coordinates(meuse) <- ~x+y  # Define spatial coordinates

# Compute the Empirical Variogram
variogram_model <- variogram(zinc ~ 1, meuse)
plot(variogram_model, main = "Empirical Semivariogram")

# Fit a Variogram Model
vgm_model_nugget <- fit.variogram(variogram_model, 
                                  vgm(psill = 10000, model = "Nug", range = 0, add.to = 
                                      vgm(psill = 150000, model = "Sph", range = 600)))
print(vgm_model_nugget)
plot(variogram_model, vgm_model_nugget, main = "Fitted Semivariogram Model")
```
# Next Step

For practice with your own CSV data (and side-by-side comparison of correlograms and semivariograms), see:

- [Spatial Dependence Part 2 repo](https://github.com/parker-group/spatial_depend_2)  
- [Live tutorial site](https://parker-group.github.io/spatial_depend_2/)



### **Rendering the RMarkdown File**
Run in R:
```r
rmarkdown::render("variogram_tutorial1.Rmd")
```
This will create a Markdown file that you can push to GitHub.