In wjones127/thesis: Bike Routes Thesis (Title Case)

knitr::opts_chunk$set(echo = FALSE)

The Data

Ride Report

Ride collection:

• route, timestamp, length collected automatically
• ratings provided by user

Three main issues in data (we are concerned with here):

1. Subjective ratings
2. Many ratings missing
3. Many rides misclassified

Other issues:

• Some rides are split
• No obvious way to model routes

Why study this data?

Ride Report's Goals

• For cities: identify the problematic street segments in the city for urban cyclists
• For cyclists: identify the best routes for commuting by bike

Our Approach

• We attempt to address:
• Issue 1 (subjective ratings) with multilevel models
• Issue 2 (missing ratings) with the expectation maximization algorithm
• We don't use route information; further research needed

Weather Data Sources

We combined the ride data with

• daily weather data from the KPDX weather station [@wunderground]
• hourly rain guage data from the Portland Fire Bureau rain guage [@pdxrain]

A note about reproducibility

This entire analysis is availible as an R package in a GitHub repository

• URL: https://github.com/wjones127/thesis
• But, Ride Report data is not included

Ride Rating Models

Notation

We have $n$ observations of rides.

\begin{equation} y_i = \begin{cases} 1, & \text{if ride } i \text{ was given a negative rating;}\ 0, & \text{otherwise.} \end{cases} \end{equation}

Define predictors,

• $x_i^\text{length}$, log length of ride
• $x_i^\text{rain}$, rainfall during hour of ride
• $x_i^\text{rain4h}$, cumulative rainfall from past four hours
• $x_i^\text{wind}$, mean wind speed that day
• $x_i^\text{gust}$, maximum gust speed
• $x_i^\text{temp}$, mean temperature that day
• $t_i$, time of day of ride

for $i = 1, \ldots, n$. All but the last we represent together as matrix $X$.

Six Models for Ride Rating

• Model 1: Logistic regression model
• Model 2: Add rider random intercepts
• Model 3: Add trigonometric terms for $t$
• Model 4: Additive multilevel model with cyclic cubic spline for $t$
• Model 5: Model 4 with cubic splines for $x^\text{length}$
• Model 6: Model 4 with fixed intercept

How well do they fit?

\begin{table}[htb] \centering \caption{Fit summaries for Models 1--6.\label{tab:modelfits}} \begin{tabular}{lrrr} \toprule \textbf{Model} & \textbf{$\log (\mathcal{L})$} & \textbf{AIC} & \textbf{AUC}$_{\text{CV}}$\footnotemark\ \midrule \rowcolor{red} Model 1 & -4,786 & 9,586 & 0.552\ Model 2 & -3,971 & 7,957 & 0.797\ Model 3 & -3,923 & 7,877 & 0.802\ Model 4 & -3,930 & 7,870 & 0.802\ Model 5 & -3,928 & 7,878 & 0.803\ \rowcolor{red} Model 6 & -4,713 & 9,455 & 0.601\ \bottomrule \end{tabular} \end{table} \footnotetext{Area under ROC curve estimated with 10-fold cross-validation.}

Time of Day Trends

What do the intercepts encode?

Classifying Riders

What features can we use?

For riders $j = 1, \ldots, l$, we have rider-level predictors

• $u^\text{freq}_j$, frequency of rides
• $u^\text{weekend}_j$, proportion of rides on weekends
• $u^\text{med.len}_j$, median length of weekday rides
• $u^\text{med.len.w}_j$, median length of weekend rides
• $u^\text{var.len}_j$, variance of length of weekday rides
• $u^\text{var.len.w}_j$, variance of length of weekend rides
• $u^\text{morning}_j$, proportion of rides in morning
• $u^\text{lunch}_j$, proportion of rides during lunch
• $u^\text{evening}_j$, proportion of rides in evening

Variables were standardized and clustered using $k$-means clustering

What patterns do these riders exhibit?

What good are these as predictors for rider intercepts?

Rider clusters and rider-level predictors

• Model using rider-level predictors to predict intercept found they were poor predictors
• Model using cluster intercepts performed much worse than rider intercepts
• Unsupervised learning methods are an art; maybe there is a better way

Missing Data

Missing Ratings

Of $n = 25,397$ rides, $11,365$ not rated.

• Is it safe to ignore these observations?

Types of Missing Data

Let,

\begin{equation} r_i = \begin{cases} 1, & \text{if ride } i \text{ is missing a rating;}\ 0, & \text{otherwise.} \end{cases} \end{equation}

Rubin classifies missing data into three situations^[@little1987 (page 14)]:

1. Missing Completely at Random (MCAR), where $r$ is independent of $r$ and the predictors $X$. i.e. $\mathbb{P} (r = 1| y, X) = \mathbb{P}(r = 1$)
2. Missing at Random (MAR), where $r$ is independent of $y$, but may depend on $X$, i.e. $\mathbb{P} (r = 1 |y, X) = \mathbb{P} (r = 1 | X )$
3. Nonignorable, or not MCAR nor MAR, where $r$ is dependent on $y$.

We believe the missing ratings are nonignorable.

The EM Algorithm for Missing Data

• The Expectation Maximization (EM) algorithm: procedure for fitting models with latent variables.
• Here, latent variables are missing observations
• We use the weighting method proposed by Ibrahim and Lipsitz^[@ibrahim1996].

The EM Algorithm: Setup

• Have data model: $f(y \;|\; X, \beta)$ and missing data model: $f(r \;|\; X, y, \alpha)$

EM algorithm general procedure^[@little1987]:

1. E-step: Compute expected loglikelihood, \begin{equation} Q(\alpha, \beta | \alpha^{(t)}, \beta^{(t)}) = \int l(\alpha, \beta | y) \cdot f(y_\text{mis} | \: y_\text{obs}, \alpha^{(t)}, \beta^{(t)}) \; dy_\text{mis} \end{equation}
2. M-step: maximize $Q(\alpha, \beta | \alpha^{(t)}, \beta^{(t)})$ to get $(\alpha^{(t + 1)}, \beta^{(t + 1)})$

EM Algorithm: Weighting procedure

1. Get initial estimates of $\alpha$ and $\beta$.
2. Compute weights \begin{equation} w_{i\: y_i}^{(t)} = \frac{f(y_i \;|\; x_i, \beta^{(t)}) f(r_i \;|\; x_i, y_i, \alpha^{(t)})}{ \sum_{y_i \in {0,1}} f(y_i \;|\; x_i, \beta^{(t)}) f(r_i \;|\; x_i, y_i, \alpha^{(t)}) }. \end{equation}

3. Create augmented data:

4. Fit data model and missing data model separately using augmented data

5. Repeat 2--4 until loglikelihood converges

EM Algorithm: Augmented Data

• Allows us to fit models with any packages that supports weighting observations

\begin{figure}[htb] \centering \caption{Creation of augmented data set for the weighted method of the EM algorithm for missing response data. \label{fig:augmented-data}} \begin{tabular}{lcl} \toprule \textbf{Original Data} & & \textbf{Augmented Data}\ \midrule

\begin{tabular}{lll} $y_i$ & $x_i$ & $r_i$\ \midrule 1 & 2.4 & 0\ 0 & 1.3 & 0\ NA & -0.4 & 1\ & & \end{tabular} & $\to$ & \begin{tabular}{llll} $y_i$ & $x_i$ & $r_i$ & $w_i$\ \midrule 1 & 2.4 & 0 & 1\ 0 & 1.3 & 0 & 1\ 1 & -0.4 & 1 & 0.2\ 0 & -0.4 & 1 & 0.8 \end{tabular}\ \bottomrule \end{tabular} \end{figure}

Missing Data Model Results: Data Model

\begin{table}[htb] \centering \begin{tabular}{lrrrr} \toprule \textbf{Parameter} & \textbf{Model 4} & \textbf{EM Model}\ \midrule Log(Length) & -0.147 & 0.205\ & \footnotesize (-0.290, -0.005) & \footnotesize (0.106, 0.304)\ Mean Temperature & 0.142 & 0.100\ & \footnotesize (0.004, 0.281) & \footnotesize (0.005, 0.196)\ Mean Wind Speed & 0.002 & -0.026\ & \footnotesize (-0.054, 0.057) & \footnotesize (-0.069, 0.016)\ Max Gust Speed & -0.005 & 0.020\ & \footnotesize (-0.031, 0.021) & \footnotesize (0.001, 0.039)\ Rainfall & 0.050 & 0.051\ & \footnotesize (-0.017, 0.117) & \footnotesize (0.009, 0.093)\ Rainfall 4-Hour & 0.022 & 0.017\ & \footnotesize (0.003, 0.041) & \footnotesize (0.003, 0.030)\ Intercept & -2.792 & -3.144\ & \footnotesize (-3.334, -2.250) & \footnotesize (-3.604, -2.684)\ \bottomrule \end{tabular} \end{table}

Missing Data Model Results: Nonresponse Model

\begin{table}[htb] \centering \begin{tabular}{lrrrr} \toprule \textbf{Parameter} & \textbf{Basic Model} & \textbf{EM Model}\ \midrule $y$ & 0.730 & 1.035\ & \footnotesize (0.235, 1.224) & \footnotesize (0.493, 1.577) \ Log(Length) & -0.297 & -0.327\ & \footnotesize (-0.362, -0.232) & \footnotesize (-0.393, -0.262)\ Mean Temperature & 0.200 & 0.139\ & \footnotesize (0.139, 0.262) & \footnotesize (0.077, -0.262)\ Mean Wind Speed & 0.032 & 0.031\ & \footnotesize (0.003, 0.060) & \footnotesize (0.001, 0.061) \ Max Gust Speed & -0.003 & -0.007\ & \footnotesize (-0.016, 0.010) & \footnotesize (-0.021, 0.006) \ Rainfall & 0.007 & -0.024\ & \footnotesize (-0.028, 0.041) & \footnotesize (-0.057, 0.009)\ Rainfall 4-Hour & -0.002 & 0.010\ & \footnotesize (-0.012, 0.009) & \footnotesize (-0.001, 0.021) \ Intercept & -0.927 & -0.967\ & \footnotesize (-1.124, -0.729) & \footnotesize (-1.163, -0.771)\ \bottomrule \end{tabular} \end{table}

Should we trust these results?

• Recall issue 4: some rides are misclassified as bike rides
• Perhaps most unrated rides are misclassified?
• $\implies$ need to fix misclassification before missing data methods can be properly applied

Conclusions

What we've learned

• Allowing random intercepts for rider gives huge improvements in model performance
• Modeling missing data may be essential, but data quality of missing data must be fixed

What remains to be researched

• How do we create models that use routes?
• Can riders be clustered in a way that predicts their rider intercept?
• How will rider intercepts change when route is incorporated?

References {.allowframebreaks}

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nocite: | @stan, @lme4, @gamm4, @Rlang, @wunderground, @pdxrain, @ridereport ...

wjones127/thesis documentation built on May 4, 2019, 7:34 a.m.