SYS 6018 | Spring 2024 | University of Virginia

Homework #9: Density Estimation

Author

Your Name Here

Published

April 4, 2024

Required R packages and Directories

data_dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
library(ks)        # functions for KDE
library(tidyverse) # functions for data manipulation

Problem 1 Geographic Profiling

Geographic profiling, a method developed in criminology, can be used to estimate the home location (roost) of animals based on a collection of sightings. The approach requires an estimate of the distribution the animal will travel from their roost to forage for food.

A sample of \(283\) distances that pipistrelle bats traveled (in meters) from their roost can be found at:

One probability model for the distance these bats will travel is: \[\begin{align*} f(x; \theta) = \frac{x}{\theta} \exp \left( - \frac{x^2}{2 \theta} \right) \end{align*}\] where the parameter \(\theta > 0\) controls how far they are willing to travel.

a. Derive a closed-form expression for the MLE for \(\theta\) (i.e., show the math).

Solution

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b. Estimate \(\theta\) for the bat data using MLE?

Calculate using the solution to part a, or use computational methods.

Solution

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c. Plot the estimated density

Using the MLE value of \(\theta\) from part b, calculate the estimated density at a set of evaluation points between 0 and 8 meters. Plot the estimated density.

  • The x-axis should be distance and y-axis should be density (pdf).
Solution

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d. Estimate the density using KDE.

Report the bandwidth you selected and produce a plot of the estimated density.

Solution

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e. Which model do you prefer, the parametric or KDE?

Solution

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Problem 2: Interstate Crash Density

Interstate 64 (I-64) is a major east-west road that passes just south of Charlottesville. Where and when are the most dangerous places/times to be on I-64? The crash data (link below) gives the mile marker and fractional time-of-week for crashes that occurred on I-64 between mile marker 87 and 136 in 2016. The time-of-week data takes a numeric value of <dow>.<hour/24>, where the dow starts at 0 for Sunday (6 for Sat) and the decimal gives the time of day information. Thus time=0.0417 corresponds to Sun at 1am and time=6.5 corresponds to Sat at noon.

a. Crash Data

Extract the crashes and make a scatter plot with mile marker on x-axis and time on y-axis.

Solution

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b. Use KDE to estimate the mile marker density.

  • Report the bandwidth.
  • Plot the density estimate.
Solution

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c. Use KDE to estimate the temporal time-of-week density.

  • Report the bandwidth.
  • Plot the density estimate.
Solution

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d. Use KDE to estimate the bivariate mile-time density.

  • Report the bandwidth parameters.
  • Plot the bivariate density estimate.
Solution

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e. Crash Hotspot

Based on the estimated density, approximate the most dangerous place and time to drive on this stretch of road. Identify the mile marker and time-of-week pair (within a few miles and hours).

Solution

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