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03-methods.qmd
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03-methods.qmd
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# Methodology {#methodology}
```{r setup, file = "R/chapter_start.R", include = FALSE, cache = FALSE}
# a number of commands need to run at the beginning of each chapter. This
# includes loading libraries that I always use, as well as options for
# displaying numbers and text.
library(readxl)
```
In this section we describe a model framework designed to evaluate resilience
using a logit-based choice metric. This framework is heavily based on tools and
methods in existing statewide travel models, with a few necessary extensions. We
then describe the implementation of this model framework to a prioritization
exercise on the Utah Statewide Travel Model (USTM).
## Model Design
The overall model framework is presented in @fig-framework, and is designed to
capture the utility-based accessibility for a particular origin zone $i$ and
trip purpose $m$. The model begins with a travel time skim procedure, to
determine the congested travel time from zone $i$ to zone $j$ by auto as well as
the shortest network distance for non-motorized modes. The transit travel time
skim is fixed, assuming that transit infrastructure would not be affected by
changes to the highway network. Throughout this section, lower-cased index
variables $k$ belong to a set of all indices described by the corresponding
capital letter $K$.
```{r framework, out.width="100%", fig.align="center", fig.cap = "Model framework.", echo = FALSE}
#| label: fig-framework
#| fig-cap: Model framework with feedback cycle. Blue boxes are calculated after the second feedback loop.
DiagrammeR::grViz("
digraph boxes_and_circles {
# a 'graph' statement
graph [overlap = true, fontsize = 10]
# several 'node' statements
node [shape = box,
fontname = Helvetica]
'Time and Distance Skim';
'Mode Choice Logsums'; 'Destination Choice Logsums'; 'Trips'; 'Congested Network'
node [shape = box,
fontname = Helvetica, color = '#D14124']
'Time and Distance Costs'; 'Logsum Costs' [color='#0072CE']
node [shape = oval,
width = 0.9,
color = black] // sets as circles
'USTM Network'; 'Trip Productions'; 'Parameters'; 'Transit and Non-Motorized Skims';
# several 'edge' statements
'USTM Network' -> 'Time and Distance Skim' [label ='Broken link?', fontname = Helvetica]
'Time and Distance Skim' -> 'Mode Choice Logsums'
'Transit and Non-Motorized Skims' -> 'Mode Choice Logsums'
'Trip Productions' -> Trips
'Parameters' -> 'Mode Choice Logsums'
'Mode Choice Logsums' -> 'Destination Choice Logsums'
'Parameters' -> 'Destination Choice Logsums'
'Mode Choice Logsums' -> 'Trips'
'Destination Choice Logsums' -> Trips
Trips -> 'Congested Network'
'Congested Network' -> 'Time and Distance Skim' [color = '#D14124', label = 'Feedback',fontname = Helvetica, fontcolor='#D14124']
Trips -> 'Time and Distance Costs' [color = '#D14124']
'Time and Distance Skim' -> 'Time and Distance Costs' [color = '#D14124']
'Destination Choice Logsums' -> 'Logsum Costs' [color = '#0072CE']
'Trip Productions' -> 'Logsum Costs' [color = '#0072CE']
}
")
```
With the travel time $t_{ijk}$ for all modes $k \in K$, the model computes mode
choice utility values. The multinomial logit mode choice model describes the
probability of a person at origin $i$ choosing mode $k$ for a trip to
destination $j$: $$
\mathcal{P}_{ijm}(k) = \frac{\exp(f(\beta_{m}, t_{ijk}))}{\sum_{K}\exp(f(\beta_{m}, t_{ijk}))}
$$ {#eq-mcp} The log of the denominator of the this equation is called the mode
choice logsum, $MCLS_{ijm}$ and is a measure of the travel cost by all modes,
weighted by utility parameters $\beta_m$ that may vary by trip purpose.
The $MCLS$ is then used as a travel impedance term in the multinomial logit
destination choice model, where the probability of a person at origin $i$
choosing destination $j \in J$ is $$
\mathcal{P}_{im}(j) = \frac{\exp(f(\gamma_{m}, MCLS_{ijm}, A_j))}{\sum_{J}\exp(f(\gamma_{m}, MCLS_{ijm}, A_j))}
$$ {#eq-dcp} where $A_j$ is the attractiveness --- represented in terms of
socioeconomic activity --- of zone $j$. As with mode choice, the log of the
denominator of this model is the destination choice logsum, $DCLS_{im}$. This
quantity represents the value access to all destinations by all modes of travel,
and varies by trip purpose.
The $DCLS_{im}$ measure is relative, but can be compared across scenarios. The
difference between the measures of two scenarios $$
\Delta_{im} = DCLS_{im}^{\mathrm{Base}} - DCLS_{im}^{\mathrm{Scenario}}
$$ {#eq-deltas} provides an estimate of the accessibility lost when
$t_{ij\mathrm{drive}}$ changes due to a damaged highway link. This accessibility
change is *per trip*, meaning that the total lost accessibility is
$P_{im} * \Delta_{im}$ where $P$ is the number of trip productions at zone $i$
for purpose $m$. This measure is given in units of dimensionless utility, but
the mode choice cost coefficient $\beta_{\mathrm{cost}}$ provides a conversion
factor between utility and cost. The total financial cost of a damaged link for
the entire region for all trip purposes is $$
\mathrm{Cost} = \sum_{I}\sum_{M} -1 / \beta_{\mathrm{cost},m} * P_{im} \Delta_{im}
$$ {#eq-totalcost}
For comparison to a common method that only includes the increased
travel time between origins and destinations (and not the cost and opportunities
of changing modes and destinations), we compute the change in congested
travel time between $\delta t_{ij}$ and multiply the number of trips by this
change and a value of time coefficient derived from the cost and vehicle time
coefficients of the mode choice model, $$
\mathrm{Cost}' = \sum_I \sum_J \sum_M \frac{\beta_{\mathrm{time}, m} }{\beta_{\mathrm{cost}, m}} T_{ijm} \delta t_{ijm}
$$ {#eq-ttmethod}
## Model Implementation in Utah
The Utah Department of Transportation (UDOT) manages an extensive highway
network consisting of interstate freeways (I-15, I-80, I-70, and I-84),
intraurban expressways along the Wasatch Front, and rural highways throughout
the state. The rugged mountain and canyon topography throughout the state places
severe constraints on possible redundant paths in the highway network. A
landslide or rock fall in any single canyon may isolate a community or force a
redirection of traffic that could be several hours longer than the preferred
route; understanding which of these many possible choke points is most critical
is a key and ongoing objective of the agency.
Several data elements for the model described above were obtained from the Utah
Statewide Travel Model (USTM). USTM is a trip-based statewide model that is
focused exclusively on long-distance and rural trips: intraurban trips within
existing Metropolitan Planning Organization (MPO) model regions are pre-loaded
onto the USTM highway network. This means that USTM as currently constituted can
be used for infrastructure planning purposes, but would be inadequate to
evaluate the systemic resiliency of the highway network given the disparate
methodologies of the MPO models. USTM can, however, provide the following data
elements
1. *Highway Network*: including free flow and congested travel speeds, link
length, link capacity estimates, etc.
2. *Zonal Productions* $P_{im}$: available for all zones by purpose, including
those in the MPO region areas.
3. *Zonal Socioeconomic Data*: the destination choice model described in
@eq-dcp calculates attractions $A_{jm}$ from the USTM zonal socioeconomic
data based on the utility coefficients in @tbl-coeffs.
4. *Calibration Targets*: USTM base scenario estimates of mode split and trip
length were used to calibrate the utility coefficients as described below.
Among MPO models in Utah, only the model jointly operated by the Wasatch Front
Regional Council (WFRC, Salt Lake area MPO) and the Mountainland Association of
Governments (MAG, Provo area MPO) model includes a substantive transit
forecasting component. The transit travel time skim from the WFRC / MAG model
was used for the mode choice model in @eq-mcp; the zonal travel time between the
smaller WFRC / MAG model zones was averaged to the larger USTM zones, and the
minimum time among the several modes available (commuter rail, light rail, bus
rapid transit, local bus) was taken as the travel time for a single transit mode
in this implementation.
```{r coeffs}
#| label: tbl-coeffs
#| tbl-cap: Choice Model Coefficients
tar_load(coefficient_table)
kbl(coefficient_table, booktabs = TRUE,
col.names = c("Model", "Variable", "HBW", "HBO", "NHB"), digits = 4) %>%
kable_styling() %>%
collapse_rows(1:2, row_group_label_position = 'stack', latex_hline = "major")
```
The utility coefficients for the destination and mode choice models are
presented in @tbl-coeffs. The mode choice coefficients were adapted from USTM
and supplemented with coefficients from the Roanoke (Virginia) Valley
Transportation Planning Organization (RVTPO) travel model [@wang2016]. This
model was selected for its simplicity and analogous data elements to the
proposed model. The alternative-specific constants were calibrated to regional
mode choice targets developed from the 2012 Utah Household Travel Survey (UHTS)
using methods described by @koppelman2006.
The destination choice utility equation consists of three parts: a size term, a
travel impedance term, and a calibration polynomial. Coefficients for the size
term and travel impedance terms were adapted from the Oregon Statewide
Integrated Model (SWIM) [@donnelly2017] for all purposes except HBW, which coefficients
were adapted from the RVTPO model as the SWIM uses a different methodology for
selecting work locations. The distance polynomial coefficients were
calibrated to targets developed from the 2012 UHTS.
### Vulnerable Link Identification
To develop evaluation scenarios on which to apply the model, we used information
contained in the UDOT Risk Priority Analysis online map [@UDOT2020]. This map
considers the probability of various events that could impact road performance
including rock falls, avalanches, landslides, and other similar occurrences.
Using this tool, combined with information gathered from the research team and
UDOT officials, 41 locations of interest were identified for analysis, at
the locations shown in @fig-linksmap. Each link
was identified due to its location in relation to population centers, remote
geographic location, and proximity to other highway facilities, or because the
link was known to be at risk due to geologic or geographic features, or because
it was a suspected choke point in the network.