##3D Kernel Density Estimation and Bandwidth Selection in Agricultural Data Analysis ##### Three-Dimensional Density Estimation

For a more advanced visualization, we can create a three-dimensional plot of the conditional density.

library(ggplot2)
library(MASS)
library(boot)

# Generate synthetic data for 3D plot
set.seed(123)
n <- 300
waiting_time <- sort(runif(n, min=40, max=100))
duration_time <- 6 - 2*exp(-waiting_time/10) + rnorm(n, mean=0, sd=0.3)

# 3D Kernel density estimation
library(plot3D)
kde3d <- kde2d(waiting_time, duration_time, n=50)
persp3D(z=kde3d$z, x=kde3d$x, y=kde3d$y, theta=30, phi=20, expand=0.6, 
        col=heat.colors(50), xlab="Waiting time", ylab="Duration time", 
        zlab="Density", main="3D Conditional Density Estimation")

### Bayesian Nonparametric Models

Bandwidth Selection Methods for Conditional Density

Bandwidth selection is a critical component in kernel conditional density estimation. Proper bandwidth selection can significantly improve the accuracy of density estimates, which is crucial for interpreting the relationships between variables in agricultural economics. This chapter explores various bandwidth selection methods and their applications, supported by detailed examples and visualizations.

Introduction to Conditional Density Estimation

Conditional density estimation involves estimating the probability density function of a response variable conditioned on one or more explanatory variables. This approach extends standard kernel density estimation by accounting for covariates, providing deeper insights into the data.

The kernel estimator for the conditional density of \(Y\) given \(X = x\) is:

\[ \hat{f}(y | x) = \frac{\sum_{i=1}^{n} K\left(\frac{x - X_i}{h_x}\right) K\left(\frac{y - Y_i}{h_y}\right)}{h_x \sum_{i=1}^{n} K\left(\frac{x - X_i}{h_x}\right)} \]

where \(K\) is the kernel function, and \(h_x\) and \(h_y\) are the bandwidths for the explanatory and response variables, respectively.

Bias, Variance, and AMISE

Bias and variance are key considerations in kernel density estimation. Bias refers to the systematic deviation of the estimated density from the true density, while variance measures the variability of the estimate from sample to sample.

The Asymptotic Mean Integrated Squared Error (AMISE) combines bias and variance into a single performance measure:

\[ \text{AMISE}(h) = \frac{R(K)}{nh} + \frac{h^4}{4} \mu_2(K)^2 R(f'') \]

where: - \(R(K)\) is the roughness of the kernel function. - \(\mu_2(K)\) is the second moment of the kernel. - \(R(f'')\) is the roughness of the second derivative of the true density function.

The optimal bandwidth minimizes the AMISE, balancing bias and variance trade-offs.

The Curse of Dimensionality

The curse of dimensionality refers to the exponential increase in data sparsity as the number of dimensions grows. This phenomenon complicates density estimation, as higher dimensions require more data to maintain estimation accuracy.

To address these challenges, dimensionality reduction techniques like Principal Component Analysis (PCA) can be employed, or more advanced density estimation methods suited for high-dimensional data can be used.

Bandwidth Selection Methods

Rule-of-Thumb Bandwidth Selection

Rule-of-thumb methods offer quick bandwidth estimates based on assumed data distributions. For univariate density estimation, Silverman’s rule of thumb is:

\[ h = \left(\frac{4 \hat{\sigma}^5}{3n}\right)^{\frac{1}{5}} \]

where \(\hat{\sigma}\) is the data’s standard deviation and \(n\) is the sample size. For conditional density estimation, this rule can be adapted as:

\[ h_x = \left(\frac{4 \hat{\sigma}_x^5}{3n}\right)^{\frac{1}{5}}, \quad h_y = \left(\frac{4 \hat{\sigma}_y^5}{3n}\right)^{\frac{1}{5}} \]

3.4.4.2 Cross-Validation Bandwidth Selection

Cross-validation is a robust method that minimizes prediction error. The leave-one-out cross-validation score for bandwidth \(h\) is:

\[ \text{CV}(h) = \frac{1}{n} \sum_{i=1}^{n} \left( \hat{f}_{-i}(x_i) - f(x_i) \right)^2 \]

where \(\hat{f}_{-i}(x_i)\) is the density estimate excluding the \(i\)-th observation.

Bootstrap Bandwidth Selection

Bootstrap methods involve resampling data with replacement to create multiple samples. Each sample’s bandwidth minimizes the integrated mean squared error (IMSE). The average bandwidth is then selected:

\[ \hat{h} = \frac{1}{B} \sum_{b=1}^{B} h_b \]

where \(B\) is the number of bootstrap samples and \(h_b\) is the selected bandwidth for the \(b\)-th sample.

Examples in Agricultural Economics

To illustrate these concepts, we generate data and create visualizations similar to those shown in the images provided. These examples demonstrate bandwidth selection methods for conditional density estimation in agricultural economics.

Crop Yield Prediction
Plug-In Bandwidth Selection for Conditional Density Estimation

The images provided illustrate various aspects of kernel density estimation and bandwidth selection:

  1. 3D Conditional Density Estimation Plot: This plot shows the conditional density of a response variable given two explanatory variables using a 3D surface plot.
  2. Contour Plots for Different Densities: These plots visualize the density functions for different bandwidths and kernel functions, demonstrating the impact of bandwidth selection on the density estimation.
  3. Plug-In Bandwidth Selection Visualization: This image illustrates the process and result of applying the plug-in method for bandwidth selection in kernel density estimation.
  4. Comparative Plots of Conditional Densities: These visualizations compare the conditional density estimates under various conditions and bandwidth selections.

Summary

Bandwidth selection is a crucial step in kernel conditional density estimation, affecting the accuracy and reliability of the estimates. This chapter explored various bandwidth selection methods, including rule-of-thumb, cross-validation, and bootstrap methods, and demonstrated their application in agricultural economics. By understanding the trade-offs between bias and variance and employing appropriate bandwidth selection techniques, researchers can obtain more accurate and insightful density estimates.

References

3.5 Examples in Agricultural Economics

To illustrate these concepts, we use the provided data on soil quality, rain, pesticide usage, fertilizer application, labor, and crop yield. The correlation matrix visualizes the relationships between these variables.

Example: Correlation Matrix for Agricultural Data

The correlation matrix provides insights into the relationships between different variables. High correlations between explanatory variables and the response variable (crop yield) suggest strong linear relationships, which can inform the bandwidth selection for conditional density estimation.

# Load necessary libraries

library(ggplot2)
library(boot)
library(corrplot)
setwd("C:/Users/ASUS/Downloads/agriculture")


# Load the provided data
data <- read.csv('agricultural_data.csv')

# Calculate the correlation matrix
corr_matrix <- cor(data)
corr_matrix <- cor(data[, sapply(data, is.numeric)]) 
corrplot(corr_matrix, method = "color", col = colorRampPalette(c("blue", "white", "red"))(200), title = "Correlation Matrix of Agricultural Data Variables", addCoef.col = "black", number.cex = 0.7, tl.cex = 0.8)