thesis.pdf - StatResp

One of the most enriching experiences at Vanderbilt was my interaction with fellow students. The most natural measure of response quality is response time, and considerable research has been devoted to this (Zografos et al.

Predicting incident arrival

Temporal Dynamics

The exponential distribution, widely used to model arrival times, is the most natural choice for our model. Grids can experience multiple events, so we consider that each spatial grid introduces a new "subject" every time an event occurs.

Spatial Dynamics

We iteratively merge the two meshes that are most similar, with mesh-to-mesh similarity measured as the distance between the connected wis and wjs. Initially, we maintain a high probability tolerance level to encourage exploration of the solution space and lower it as the algorithm progresses.

Predicting Incident Severity

Features affecting crime

Temporal Features

Police presence: The last class of features that is particularly applicable to our optimization problem involves the effect of police presence on crimes. Specifically, it is often hypothesized that police presence at or near a location will influence future crime at that location Koper (1995).

Spatial and Spatio-Temporal Features

We attempt to capture this relationship by including a feature in the model that corresponds to the number of police vehicles that have passed within the grid, as well as its immediate neighboring grid cells, over the previous two hours.

Features affecting accidents

Each accident is accompanied by its time of occurrence, the time at which the first responding vehicle reached the scene and the time at which the last responding vehicle was back in service, which refers to the completion of the service of an incident. To predict incidents, we extracted highway and street intersections from Open Street Maps (Haklay and Weber, 2008) and weather data were collected at the county level.

Predicting Crimes

Prediction Effectiveness Comparison with DSDA

To make a comparative analysis, we took into account a natural adaptation of the HPA-STAC model, which allows us to compare probabilities. Finally, we assume that the total probability is proportional to the product of the spatial and temporal probabilities.

Prediction Effectiveness Comparison with the Dynamic Bayes-Network

In fact, our model outperforms DSDA in both the temporal and the spatial predictions by a large margin (overall, the improvement in log-likelihood is 25–30%). We also make comparisons by varying the number of grids, shown in Figure III.2 (b), starting around downtown Nashville and gradually moving outward.

Runtime Comparison with DSDA and DBN

Specifically, we divide time into 8-h intervals (same temporal discretization as used by Zhang et al.), and derive the conditional probability of observed crime as follows. To make the fairest comparison with DBN, we use an evaluation metric proposed by Zhang et al.

Predicting Accidents

An important contribution is also the model's ability to capture the effect of police presence both temporally and spatially to predict crime. To learn the former, we proposed a new hierarchical clustering approach to extend the use of survival analysis to predict events by learning from the data model's spatial granularity.

Table III.1: Severity Prediction Accuracy Test

Spatial-Temporal Prediction Model

We consider a set of grids G of the same size, covering the entire spatial area under consideration. Consequently, it suffices to mark the spatial location of each incident with one of the grids.

Adversarial Intervention

Attacker Model

For any gridgi∈G, we use Ni to describe the set of neighboring grids that an attacker could move to in order to commit a crime (note that this includes the original grid, since the attacker could choose not to move). Finally, we capture the attacker's decision to move the incident to the network with the variable yij, which is a.

Figure IV.1: Adversarial Movement across grids

Robust Survival Analysis

The integrity constraints ensure the natural limit that the attacker can move one incident to one specific grid. However, the integral solution still remains optimal, since the attacker's best response is a decision problem.

Robust Survival Analysis with Linear Attack (RSALA)

Further, we useSolve(φk) to denote the solution of problem IV.7 under constraintsφk, and useAttack(θi) to denote the generation of the attacker's best response against θi. We then update the constraint by computing the attacker's response to θk+1 (see step 6).

Adversary Based Gradient Descent (AdGraD)

This process continues until the attacker's gain between successive iterations is within an exogenously specified parameter. Although RSALA is guaranteed to converge to the optimal solution in a finite time, the attacker's strategy space can be extremely large, and solving the optimization problem IV.7 at each iteration of RSALA is computationally slow.

Setup

Results

Finally, we aim to understand the effect of the attacker's geographic limitation (budget) on the robustness of the models. Before we introduce the structure of the problem, it is important to note how police patrols occur.

Figure IV.2: Predicted incident density for assault crimes plotted according to a varying attacker budget

Minimizing Response Time for a Fixed Set of Crime Incidents

The arc capacities are adjusted in our situation, as described later in the problem formulation. In the problem setting we describe, Gi is the set of all nodes in the network.

Stochastic Programming and Sample Average Approximation for Police Place-

We introduce double variablesλsi, ..,λsk for V.3c,δis constraints, ..,δsm for V.3d constraints, fis, ..,fns for V.3e constraints and cs11,cs12..,csnn for V.3f constraints . Consequently, there is always an optimal dual solution that is one of the (finite number of) extreme points of the polyhedron consisting of constraints V.5b and V.6c for the corresponding problems.

Iterative Stochastic Programming

There should be more police coverage in areas that see more crime on average, and. The second term penalizes the placement of too many police vehicles in a grid and therefore forces the model to distribute police between grids.

Experiment Setup

Effectiveness of the Response Time Optimization Method

What makes these problems difficult is that the dependence of not only on how, when and where incidents happen, but also on the severity of the incidents. We provide an outline of the derivation for the waiting time of our queuing model here.

Figure V.2: Response Times (lower is better): (a) using simulated crimes, (b) observed crimes.

Calculating Waiting Time

We make the simplifying assumption that although different severities follow different arrival distributions and have different service time constraints in the optimization model, they follow the same service distribution. This is an assumption that is realistic in many real-life applications, as severities often represent the urgency with which to respond to an incident and are not an indication of actual service time.

Adaptive Random Search for Responder Optimization

In the local search phase, described in Algorithm 6, we repeatedly look at each repository from the current solution and distribute all the pairs assigned to it. We distribute these responses iteratively to other potential repositories not in the current solution deployed in proportion to their demand rates.

Data

First, we evaluate the effectiveness of our approach in predicting incidents, and second, we evaluate the performance of our responder optimization methods.

Incident Prediction

Response Optimization

We appreciate the importance of the local search phase operating in the random candidate list phase. Seeing the importance of the local search phase, we turn to tuning the γ parameter in our model.

Figure VI.1: Phases of GRASP: Objective Value Comparison

Optimal Static Dispatch

We validate our findings by comparing our approach with existing, state-of-the-art approaches (Mukhopadhyay et al., 2017), using real traffic data from a major US metropolis, as well as simulated data. Our results show that our principled approach to dynamic dispatch significantly outperforms the state-of-the-art alternative.

Semi-Markov Decision Processes

Dynamic Bayes Networks

The dynamics of the world are primarily governed by two events that enable decision-making: the occurrence of traffic incidents and the completion of servicing. For example, when an event occurs and the action of dispatching a responder is performed, the occurrence of the next state depends on the time it takes for that responder to travel to the location and deal with the event.

States

We refer to the entire spatial and temporal process of event arrival as well as the status of all responders and repositories as our world. We next describe each of the elements of the SMDP as it captures the relief problem, while explaining the particular structure of this problem that is used to both represent and solve it.

Actions

Transitions

As in standard AFT models, we model the arrival rate for the exponential distribution as a log-linear model in terms of characteristic w. The time between these two states is dt1t2v+ts, where dt1t2 is the distance between the respective networks in states st1 and st2, and v is the (known) speed of the transponders.

Figure VII.1: Transition Times when s j ∈ S c

Rewards

For a compact representation of the state transition distribution p, we use a Dynamic Bayes Network (DBN) to exploit the conditional independence relations between the state variables. While the structure of the DBN is self-explanatory, we point out some of the key insights that use the structure of the response delivery problem.

Dispatch Process and Decision Problem

When a responder is dispatched in statesi to attend to an event that leads to a state completion statej, we denote the reward asdmax−dij, where max is the largest possible distance in the given area under consideration and dij is the distance between the relevant grids. In the first step, the values of states are estimated under a fixed policy from the last iteration.

Discretization

We now present an approach for computing an optimal policy from the formulated SMDP model of dynamic emergency response. Letg(t) represents the density function of the random variable, and f(ts) represents the density of the service time distribution, as mentioned earlier.

SimTrans : Simulate and Transform

If ˜p≤ω, the algorithm accepts the estimates; otherwise, it simulates the world to get an estimate. Second, we have tested our policy iteration algorithm with heuristic policies, which are designed based on previous work (Mukhopadhyay et al., 2017) and domain expertise.

Data

We also use UpdateState(s,a,c) to get the next state of the system given the current state, action and chainc. As soon as a responder becomes available, he or she will be assigned to the incident at the front of the queue.

Data and Methodology

To reduce the search time needed to find vehicle speeds between arbitrary locations, we have stored travel times between locations for different times of the day, with time discretized every 30 minutes. To actualize the cache, we used the same grid system as in the incident model, with each location in the city discretized at the center of its grid.

Experimental Setup

Caching router results: Although we recommend using arouterin in real-time using the exact locations of responders and incidents while making decisions, this is not feasible for our experiments. Our experiments show that travel times in Nashville do not change significantly over this interval (ranging from 2-7 mph).

Results and Discussion

Streaming Survival Analysis

To illustrate this, we present the computation time of the streaming model (for each flow) in Figure VIII.3 and note that it usually takes less than 2 seconds to execute an update, which justifies the use of such models in practice. We see that the ensemble model learns the model poorly since it has access to the updated dataset only in the form of features; transmission model on the other hand,.

Figure VIII.2: Negative Log Likelihood comparison (lower is better) between Batch and Streaming Survival Models.

Responder Dispatch

It is also indicated which emergency responder should be sent to the scene of the incident. Algorithms for the assignment and transport problems. Journal of the Association for Industrial and Applied Mathematics.