Making aoristic density maps in R

I saw Jerry the other day made/updated an R package to do aoristic analysis. A nice part of this is that it returns the weights breakdown for individual cases, which you can then make maps of. My goto hot spot map for data visualization, kernel density maps, are a bit tough to work with weighted data though in R (tough is maybe not the right word, to use ggplot it takes a bit of work leveraging other packages). So here are some notes on that.

I have provided the data/code here. It is burglaries in Dallas, specifically I filter out just for business burglaries.

R Code Snippet

First, for my front end I load the libraries I will be using, and change the working directory to where my data is located.

############################
library(aoristic) #aoristic analysis 
library(rgdal)    #importing spatial data
library(spatstat) #weighted kde
library(raster)   #manipulate raster object
library(ggplot2)  #for contour graphs
library(sf)       #easier to plot sf objects

my_dir <- "D:\\Dropbox\\Dropbox\\Documents\\BLOG\\aoristic_maps_R\\data_analysis"
setwd(my_dir)
############################

Next I just have one user defined function, this takes an input polygon (the polygon that defines the borders of Dallas here), and returns a raster grid covering the bounding box. It also have an extra data field, to say whether the grid cell is inside/outside of the boundary. (This is mostly convenient when creating an RTM style dataset to make all the features conform to the same grid cells.)

###########################
#Data Manipulation Functions

#B is border, g is size of grid cell on one side
BaseRaster <- function(b,g){
    base_raster <- raster(ext = extent(b), res=g)
    projection(base_raster) <- crs(b)
    mask_raster <- rasterize(b, base_raster, getCover=TRUE) #percentage of cover, 0 is outside
    return(mask_raster)
}
###########################

The next part I grab the datasets I will be using, a boundary file for Dallas (in which I chopped off the Lochs, so will not be doing an analysis of boat house burglaries today), and then the crime data. R I believe you always have to convert date-times when reading from a CSV (it never smartly infers that a column is date/time). And then I do some other data fiddling – Jerry has a nice function to check and make sure the date/times are all in order, and then I get rid of points outside of Dallas using the sp over function. Finally the dataset is for both residential/commercial, but I just look at the commercial burglaries here.

###########################
#Get the datasets

#Geo data
boundary <- readOGR(dsn="Dallas_MainArea_Proj.shp",layer="Dallas_MainArea_Proj")
base_Dallas <- BaseRaster(b=boundary,g=200) 
base_df <- as.data.frame(base_Dallas,long=TRUE,xy=TRUE)

#Crime Data
crime_dat <- read.csv('Burglary_Dallas.csv', stringsAsFactors=FALSE)
#prepping time fields
crime_dat$Beg <- as.POSIXct(crime_dat$StartingDateTime, format="%m/%d/%Y %H:%M:%OS")
crime_dat$End <- as.POSIXct(crime_dat$EndingDateTime, format="%m/%d/%Y %H:%M:%OS")

#cleaning up data
aor_check <- aoristic.datacheck(crime_dat, 'XCoordinate', 'YCoordinate', 'Beg', 'End')
coordinates(crime_dat) <- crime_dat[,c('XCoordinate', 'YCoordinate')]
crs(crime_dat) <- crs(boundary)
over_check <- over(crime_dat, boundary)
keep_rows <- (aor_check$aoristic_datacheck == 0) & (!is.na(over_check$city))
crime_dat_clean <- crime_dat[keep_rows,]

#only look at business burgs to make it go abit faster
busi_burgs <- crime_dat_clean[ crime_dat_clean$UCROffense == 'BURGLARY-BUSINESS', ]
###########################

The next part preps the aoristic weights. First, the aoristic.df function is from Jerry’s aoristic package. It returns the weights broken down by 168 hours per day of the week. Here I then just collapse across the weekdays into the same hour, which to do that is simple, just add up the weights.

After that it is some more geographic data munging using the spatstat package to do the heavy lifting for the weighted kernel density estimate, and then stuffing the result back into another data frame. My bandwidth here, 3000 feet, is a bit large but makes nicer looking maps. If you do this smaller you will have a more bumpy and localized hot spots in the kernel density estimate.

###########################
#aoristic weights

#This takes like a minute
res_weights <- aoristic.df(busi_burgs@data, 'XCoordinate', 'YCoordinate', 'Beg', 'End')

#Binning into same hourly bins
for (i in 1:24){
    cols <- (0:6*24)+i+5
    lab <- paste0("Hour",i)
    res_weights[,c(lab)] <- rowSums(res_weights[,cols])
}

#Prepping the spatstat junk I need
peval <- rasterToPoints(base_Dallas)[,1:2]
spWin <- as.owin(as.data.frame(peval))
sp_ppp <- as.ppp(res_weights[,c('x_lon','y_lat')],W=spWin) #spp point pattern object

#Creating a dataframe with all of the weighted KDE
Hour_Labs <- paste0("Hour",1:24)

for (h in Hour_Labs){
  sp_den <- density.ppp(sp_ppp,weights=res_weights[,c(h)],
                        sigma=3000,
                        edge=FALSE,warnings=FALSE)
  sp_dat <- as.data.frame(sp_den)
  kd_raster <- rasterFromXYZ(sp_dat,res=res(base_Dallas),crs=crs(base_Dallas))
  base_df[,c(h)] <- as.data.frame(kd_raster,long=TRUE)$value
}
###########################

If you are following along, you may be wondering why all the hassle? It is partly because I want to use ggplot to make maps, but for its geom_contour it does not except weights, so I need to do the data manipulation myself to supply ggplot the weighted data in the proper format.

First I turn my Dallas boundary into a simple feature sf object, then I create my filled contour graph, supplying the regular grid X/Y and the Z values for the first Hour of the day (so between midnight and 1 am).

###########################
#now making contour graphs

dallas_sf <- st_as_sf(boundary)

#A plot for one hour of the day
hour1 <- ggplot() + 
  geom_contour_filled(data=base_df, aes(x, y, z = Hour1), bins=9) +
  geom_sf(data=dallas_sf, fill=NA, color='black') +
  scale_fill_brewer(palette="Greens") +
  ggtitle('       Hour [0-1)') + 
  theme_void() + theme(legend.position = "none")
hour1

png('Hour1.png', height=5, width=5, units="in", res=1000, type="cairo") 
hour1
dev.off()
###########################

Nice right! I have in the code my attempt to make a super snazzy small multiple plot, but that was not working out so well for me. But you can then go ahead and make up other slices if you want. Here is an example of taking an extended lunchtime time period.

###########################
#Plot for the afternoon time period
base_df$Afternoon <- rowSums(base_df[,paste0("Hour",10:17)])

afternoon <- ggplot() + 
  geom_contour_filled(data=base_df, aes(x, y, z = Afternoon), bins=9) +
  geom_sf(data=dallas_sf, fill=NA, color='black') +
  scale_fill_brewer(palette="Greens") +
  ggtitle('       Hour [9:00-17:00)') + 
  theme_void() + theme(legend.position = "none")
afternoon
###########################

So you can see that the patterns only slightly changed compared to the midnight prior graph.

Note that these plots will have different breaks, but you could set them to be equal by simply specifying a breaks argument in the geom_contour_filled call.

I will leave it up so someone who is more adept at R code than me to make a cool animated viz over time from this. But that is a way to mash up the temporal weights in a map.

RTM Deep Learning Style

In my quest to better understand deep learning, I have attempted to replicate some basic models I am familiar with in criminology, just typical OLS and the more complicated group based trajectory models. Another example I will illustrate is doing a variant of Risk Terrain Modeling.

The typical way RTM is done is:

Data Prep Part:

  1. create a set of independent variables for crime generators (e.g. bars, subway stops, liquor stores, etc.) that are either the distance to the nearest or the kernel density estimate
  2. Turn these continuous estimates into dummy variables, e.g. if within 100 meters = 1, else = 0. For kernel density they typically z-score and if a z-score > 2 the dummy variable equals 1.
  3. Do 2 for varying distance/bandwidth selections, e.g. 100 meters, 200 meters, etc. So you end up with a collection of distance variables, e.g. Bars_100, Bars_200, Bars_400, etc.

Modeling Part

  1. Fit a Lasso regression predicting your crime outcome constraining all of the variables to be positive. (So RTM will never say a crime generator has a negative effect.)
  2. For the variables that passed this Lasso stage, then do a variable selection routine. So instead of the final model having Bars_100 and Bars_400, it will only choose one of those variables.

For the modeling part, we can replicate various parts of these in a deep learning environment. For the constrain the coefficients to be positive, when you see folks refer to a “RelU” or the rectified linear unit function, all this means is that the coefficients are constrained to be positive. For the variable selection part, I needed to hack my own – it ends up being a combo of a custom dropout scheme and then pruning in deep learning lingo.

Although RTM is typically done on raster grid cells for the spatial unit of analysis, this is not a requirement. You can do all these steps on vector (e.g. street segments) or other areal spatial units of analysis.

Here I illustrate using street units (intersections and street segments) from DC. The crime generator data I take from my dissertation (and I have a few pubs in Crime & Delinquency based on that work). The crime data I illustrate using 2011 violent Part 1 UCR (homicide, agg assault, robbery, no rape in the public data).

The crime dataset is over time, and I describe in an analysis (with Billy Zakrzewski) on examining pre/post crime around DC medical marijuana dispensaries.

The data and code to replicate can be downloaded here. It is python, and for the deep learning model I used pytorch.

RTM Example in Python

So I will walk through briefly my second script, 01_DeepLearningRTM.py. The first script, 00_DataPrep.py, does the data prep, so this data file already has the crime generator variables prepared in the manner RTM typically creates them. (The rtm_dl_funcs.py has the functions to do the feature extraction and create the deep learning model, to do distance/density in sci-kit is very slick, only a few lines of code.)

So first I just define the libraries I will be using, and import my custom rtm functions I created.

######################################################
import numpy as np
import pandas as pd
import torch
device = torch.device("cuda:0")
import os
import sys

my_dir = r'C:\Users\andre\OneDrive\Desktop\RTM_DeepLearning'
os.chdir(my_dir)
sys.path.append(my_dir)
import rtm_dl_funcs
######################################################

The next set of code grabs the crime data, and then defines my variable sets. I have plenty more crime generator data from my dissertation, but to make it easier on myself I just focus on distance to metro entrances, the density of 311 calls (a measure of disorder), and the distance and density of alcohol outlets (this includes bars/liquor stores/gas stations that sell beer, etc.).

Among these variable sets, the final selected model will only choose one within each set. But I have also included the ability for the model to incorporate other variables that will just enter in no-matter what (and are not constrained to be positive). This is mostly to incorporate an intercept into the regression equation, but here I also include the percent of sidewalk encompassing one of my street units (based on the Voronoi tessellation), and a dummy variable for whether the street unit is an intersection. (I also planned on included the area of the tessalation, but it ended up being an explosive effect, my dissertation shows its effect is highly non-linear, so didn’t want to worry about splines here for simplicity.)

######################################################
#Get the Prepped Data
crime_data = pd.read_csv('Prepped_Crime.csv')

#Variable sets for each
db = [50, 100, 200, 300, 400, 500, 600, 700, 800]
metro_set = ['met_dis_' + str(i) for i in db]
alc_set = ['alc_dis_' + str(i) for i in db]
alc_set += ['alc_kde_' + str(i) for i in db]
c311_set = ['c31_kde_' + str(i) for i in db]

#Creating a few other generic variables
crime_data['PercSidewalk'] = crime_data['SidewalkArea'] / crime_data['AreaMinWat']
crime_data['Const'] = 1
const_li = ['Const','Intersection','PercSidewalk']
full_set = const_li + alc_set + metro_set + c311_set
######################################################

The next set of code turns my data into a set of torch tensors, then I grab the size of my independent variable sets, which I will end up needing when initializing my pytorch model.

Then I set the seed (to be able to reproduce the results), create the model, and set the loss function and optimizer. I use a Poisson loss function (will need to figure out negative binomial another day).

######################################################
#Now creating the torch tensors
x_ten = torch.tensor(crime_data[full_set].to_numpy(), dtype=float)
y_ten = torch.tensor(crime_data['Viol_2011'].to_numpy(), dtype=float)
out_ten = torch.tensor(crime_data['Viol_2012'].to_numpy(), dtype=float)

#These I need to initialize the deep learning model
gen_lens = [len(alc_set), len(metro_set), len(c311_set)]
    
#Creating the model 
torch.manual_seed(10)

model = rtm_dl_funcs.RTM_torch(const=len(const_li), 
                               gen_list=gen_lens)
criterion = torch.nn.PoissonNLLLoss(log_input=True, reduction='mean')
optimizer = torch.optim.Adam(model.parameters(), lr=0.001) #1e-4
print( model )
######################################################

If you look at the printed out model, it gives a nice summary of the different layers. We have our one layer for the fixed coefficients, and another three sets for our alcohol outlets, 311 calls, and metro entrances. We then have a final cancel layer. The idea behind the final cancel layer is that the variable selection routine in RTM can still end up not selecting any variables for a set. I ended up not using it here though, as it was too aggressive in this example. (So will need to tinker with that some more!)

The variable selection routine is very volatile – so if you have very correlated inputs, you can essentially swap one for the other and get near equivalent predictions. I often see folks who do RTM analyses say something along the lines of, “OK this RTM selected A, and this RTM selected B, so they are different effects in these two samples” (sometimes pre/post, other times comparing different areas, and other times different crime outcomes). I think this is probably wrong though to make that inference, as there is quite a bit of noise in the variable selection process (and the variable selection process itself precludes making inferences on the coefficients themselves).

My deep learning example inherited the same problems. So if you change the initialized weights, it may end up selecting totally different inputs in the end. To get the variable selection routine to at least select the same crime generator variables in my tests, I do a burn in period in which I implement a random dropout scheme. So instead of the typical dropout, for every forward pass it does a random dropout to only select one variable randomly out of each crime generator set. After that converges, I then use a pruning layer to only pick the coefficient that has the largest effect, and again do a large set of iterations to make sure the results converged. So different means but same ends to the typical RTM steps 4 and 5 above. I also have like I said a ReLU transformation after each layer, so the crime generator variables are always positive, any negative effects will be pruned out.

One thing that is nice about deep learning is that it can be quite fast. Here each of these 10,000 iteration sets take less than a minute on my desktop with a GPU. (I’ve been prototyping models with more parameters and more observations at work on my laptop with just a CPU that only take like 10 to 20 minutes).

######################################################
#Burn in part, random dropout
for t in range(10000):
    #Forward pass
    y_pred = model(x=x_ten)
    #Loss
    loss_insample = criterion(y_pred, y_ten)
    optimizer.zero_grad()
    loss_insample.backward(retain_graph=True)
    optimizer.step()
    if t % 1000 == 0:
        print(f'loss: {loss_insample.item()}' )

#Switching to pruning all but the largest effects
model.l1_prune()

for t in range(10000):
    #Forward pass
    y_pred = model(x=x_ten, mask_type=None, cancel=False)
    #Loss
    loss_insample = criterion(y_pred, y_ten)
    optimizer.zero_grad()
    loss_insample.backward(retain_graph=True)
    optimizer.step()
    if t % 1000 == 0:
        print(f'loss: {loss_insample.item()}' )

print( model.coef_df(nm_li=full_set, cancel=False) )
######################################################

And this prints out the results (as incident rate ratios), so you can see it selected 50 meters alcohol kernel density, 50 meters distance to the nearest metro station, and kernel density for 311 calls with an 800 meter bandwidth.

I have in the code another example model when using a different seed. So testing out on around 5 different seeds it always selected these same distance/density variables, but the coefficients are slightly different each time. Here is an example from setting the seed to 12.

These models are nothing to brag about, using the typical z-score the predictions and set the threshold to above 2, the PAI is only around 3 (both for in-sample 2011 and out of sample 2012 is slightly lower). It is a tough prediction task – the mean number of violent crimes per street unit per year is only 0.3. Violent crime is fortunately very rare!

But with only three different risk variables, we can do a quick conjunctive analysis, and look at the areas of overlap.

######################################################
#Adding model 1 predictions back into the dataset
pred_mod1 = pd.Series(model(x=x_ten, mask_type=None, cancel=False).exp().detach().numpy())
crime_data['Pred_M1'] = pred_mod1

#Check out the areas of overlapping risk
mod1_coef = model.coef_df(nm_li=full_set, cancel=False)
risk_vars = list(set(mod1_coef['Variable']) - set(const_li))
conj_set = crime_data.groupby(risk_vars, as_index=False)['Const','Pred_M1','Viol_2012'].sum()
print(conj_set)
######################################################

In this table Const is the total number of street units selected, Pred_M1 is the expected number of crimes via Model 1, and then I show how well it conforms to the predictions out of sample 2012. So you can see in the aggregate the predictions are not too far off. There only ends up being one street unit that overlaps for all three risk factors in the study area.

I believe the predictions would be better if I included more crime generator variables. But ultimately the nature of how RTM works it trades off accuracy for simple models. Which is fair – it helps to ease the nature of how a police department (or some other entity) responds to the predictions.

But this trade off results in predictions that don’t fare as well compared with more complicated models. For example I show (with Wouter Steenbeek) that random forests do much better than RTM. To make those models more interpretable we did local decompositions for hot spots, so say this hot spot is 30% alcohol outlets, 20% nearby apartments, etc.

So there is no doubt more extensions for RTM you could do in a deep learning framework, but they will likely always result in more complicated and less interpretable models. Also here I don’t think this code will be better than the traditional RTM folks, the only major benefit of this code is it will run faster – minutes instead of overnight for most jobs.

Creating high crime sub-tours

I was nerdsniped a bit by this paper, Targeting Knife-Enabled Homicides For Preventive Policing: A Stratified Resource Allocation Model by Vincent Hariman and Larry Sherman (HS from here on).

It in, HS attempt to define a touring schedule based on knife crime risk at the lower super output area in London. So here are the identified high risk areas:

And here are HS’s suggested hot spot tours schedule.

This is ad-hoc, but an admirable attempt to figure out a reasonable schedule. As you can see in their tables, the ‘high’ knife crime risk areas still only have a handful of homicides, so if reducing homicides is the objective, this program is a bit dead in the water (I’ve written about the lack of predictive ability of the model here).

I don’t think defining tours to visit everywhere makes sense, but I do think a somewhat smaller in scope question, how to figure out geographically informed tours for hot spot areas does. So instead of the single grid cell target ala PredPol, pick out multiple areas to visit for hot spots. (I don’t imagine the 41 LSOA areas are geographically contiguous either, e.g. it would make more sense to pick a tour for areas connected than for areas very far apart.)

Officers don’t tend to like single tiny areas either really, and I think it makes more sense to widen the scope a bit. So here is my attempt to figure those reasonable tours out.

Defining the Problem

The way I think about that problem is like this, look at the hypothetical diagram below. We have two choices for the hot spot location we are targeting, where the crime counts for locations are noted in the text label.

In the select the top hot spot (e.g. PredPol) approach, you would select the singlet grid cell in the top left, it is the highest intensity. We have another choice though, the more spread out hot spot in the lower right. Even though it is a lower density, it ends up capturing more crime overall.

I subsequently formulated an integer linear program to try to tackle the problem of finding good sub-tours through the graph that cumulatively capture more crime. So with the above graph, if I select two subtours, I get the results as (where nodes are identified by their (x,y) position):

  • ['Begin', (1, 4), 'End']
  • ['Begin', (4, 0), (4, 1), (3, 1), (3, 0), (2, 0), 'End']

So it can select singlet areas if they are islands (the (1,4) area in the top left), but will grow to wind through areas. Also note that the way I have programmed this network, it doesn’t skip the zero area (4,1) (it needs to go through at least one in the bottom right unless it doubles back on itself).

I will explain the meaning of the begin and end nodes below in my description of the linear program. It ends up being sort of a mash-up of traveling salesman type vehicle routing and min cost max flow type problems.

The Linear Program

The way I think about this problem formulation is like this: we have a directed graph, in which you can say, OK I start from location A, then can go to B, than go to C. In my set of decision variables, I have choices that look like this, where the first subscript denotes the from node, and the second subscript denotes the to node.

D_ab := node a -> node b
D_bc := node b -> node c

etc. In our subsequent linear program, the destination node is the node that we calculate our cumulative crime density statistics. So if node B had 10 crimes and 0.1 square kilometers, we would have a density of 100 crimes per square kilometer.

Now to make this formulation work, we need to add in a set of special nodes into our usual location network. These nodes I call ‘Begin’ and ‘End’ nodes (you may also call them source/sink nodes though). The begin nodes all look like this:

D_{begin},a
D_{begin},b
D_{begin},c

So you do that for every node in your network. Then you have End nodes as well, e.g.

D_a,{end}
D_b,{end}
D_c,{end}

In this formulation, since we are only concerned about the crime stats for the to node, not the from node, the Begin nodes just inherit the crime density stats from the original node data. For the end nodes though, you just set their objective value stats to zero (they are only relevant to define the constraints).

Now here is my linear program formulation:

Maximize 
  Sum [ D_ij ( CrimeDensity_j - DensityPenalty_j ) ]

Subject To:

 1. Sum( D_in for each neighbor of n ) <= 1, 
      for each original node n
 2. Sum( D_in for each neighbor of n ) =  Sum( D_ni for each neighbor of n ), 
      for each original node n
 3. Sum( D_bi for each begin node ) = k routes
 4. Sum( D_ie for each end node ) = k routes
 5. Sum( D_ij + D_ji ) <= 1, for each unique i,j pair
 6. D_ij is an element of {0,1}

Constraint 1 is a flow constraint. If a node has an incoming edge set to one, it cannot have any other incoming edge set to one (so a location can only be chosen once).

Constraint 2 is a constraint that says if an incoming node is selected, one of the outgoing edges needs to be selected.

Constraints 3 & 4 determine the number of k tours/routes to choose in the end. Since the begin/end nodes are special we have k routes going out of the begin nodes, and k routes going into the end nodes.

With just these constraints, you can still get micro-cycles I found. So something like, X -> Z -> X. Constraint 5 (for only the undirected edges) prevents this from happening.

Constraint 6 is just setting the decision variables to binary 0/1. So it is a mixed integer linear program.

The final thing to note is the objective function, I have CrimeDensity_j - DensityPenalty_j, so what exactly is DensityPenalty? This is a value that penalizes visiting areas that are below this threshold. Basically the way that this works is that, the density penalty sets an approximate threshold for the minimum density a tour should contain.

I suggest a default of a predictive accuracy index of 10. Where do I get 10 you ask? Weisburd’s law of crime concentration suggests 5% of the areas should contain 50% of the crime, which is a PAI of 0.5/0.05 = 10. In my example with DC data then I just calculate the actual density of crime per unit area that corresponds to a PAI of 10.

You can adjust this though, if you prefer smaller tours of higher crime density you would up the value. If you prefer longer tours decrease it.

This is the best way I could figure out how to trade off the idea of spreading out the targeted hot spot vs selecting the best areas. If you spread out you will ultimately have a lower density. This turns it into a soft objective penalty to try to keep the selected tours at a particular density threshold (and will scoop up better tours if they are available). For awhile I tried to figure out if I could maximize the PAI metric, but it is the case with larger areas the PAI will always go down, so you need to define the objective some other way.

This formulation I only consider linked nodes (unlike the usual traveling salesman in which it is a completely linked distance graph). That makes it much more manageable. In this formulation, if you have N as the number of nodes/areas, and E is the number of directed edges between those areas, we will then have:

  • 2*N + E decision variables
  • 2 + 2*N + E/2 constraints

Generally if you are doing directly connected areas in geographic networks (i.e. contiguity connections), you will have less than 8 (typically more like an average of 6) neighbors per each area. So in the case of the 4k London lower super output areas, if I chose tours I would guess it would end up being fewer than 2*4,000 + 8*4,000 = 40,000 decision variables, and then fewer than that constraints.

Since that is puny (and I would suggest doing this at a smaller geographic resolution) I tested it out on a harder network. I used the data from my dissertation, a network of 21,506 street units (both street segments and intersections) in Washington, D.C. The contiguity I use for these micro units is based on the Voronoi tessellation, so tends to have more neighbors than you would with a strictly road based network connectivity. Still in the end it ends up being a shade fewer than 200k decision variables and 110k constraints. So is a better test for in the wild whether the problem can be feasibly solved I think.

Example with DC Data

Here I have posted the python code and data used for this analysis, I end up having a nice function that you just submit your network with the appropriate attributes and out pops the different tours.

So I end up doing examples of 4 and 8 subtours based on 2011 violent UCR crime data (agg assaults, robberies, and homicides, no rapes in the public data). I use for the penalty that PAI = 10 threshold, so it should limit tours to approximately that value. It only ends up taking 2 minutes for the model to converge for the 4 tours and less than 2.5 minutes for the 8 tours on my desktop. So it should be not a big problem to up the decision variables to more sub-areas and still be solvable in real life applications.

The area estimates are in square meters, hence the high numbers. But on the right you can see that each sub-tour has a PAI above 10.

Here is an interactive map for you to zoom into each 4 subtour example. Below is a screenshot of one of the subtours. You can see that since I have defined my connected areas in terms of Voronoi tessalations, they don’t exactly follow the street network.

For the 8 tour example, it ends up returning several zero tours, so it is not possible in this data to generate 8 sub-tours that meet that PAI >= 10 threshold.

You can see that it ends up being the tours have higher PAI values, but lower overall crime counts.

You may think, why does it not pick at least singlet areas with at least one crime? It ends up being that I weight areas here by their area (this formulation would be better with grid cells of equal area, so my objective function is technically Sum [ D_ij * w_j *( CrimeDensity_j - DensityPenalty_j ) ], where w_j is the percent of the total area (so the denominator in the PAI calculation). So it ends up picking areas that are the tiniest areas, as they result in the smallest penalty to the objective function (w_j is tiny). I think this is OK though in the end – I rather know that some of the tours are worthless.

You can also see I get one subtour that is just under the PAI 10 threshold. Again possible here, but should be only slightly below in the worst case scenario. The way the objective function works is that it is pretty tricky to pick out subtours below that PAI value but still have a positive contribution to the overall objective function.

Future Directions

The main thing I wish I could do with the current algorithm (but can’t the way the linear program is set up), is to have minimum and maximum tour area/length constraints. I think I can maybe do this by adapting this code (I’m not sure how to do the penalties/objectives though). So if others have ideas let me know!

I admit that this may be overkill, and maybe just doing more typical crime clustering algorithms may be sufficient. E.g. doing DBSCAN hot spots like I did here.

But this is my best attempt shake at the problem for now!

Using Steiner trees to select a subgraph of interest

This is just a quick blog post. A crime analyst friend the other day posed a network problem to me. They had a social network in which they had particular individuals of interest, and wanted to show just a subset of that graph that connected those key individuals. The motivation was for plotting – if you show the entire hairball it can become really difficult to uncover any relationships.

Here is an example gang network from this paper. I randomly chose 10 nodes to highlight (larger red circles), and you can see it is quite hairy. You often want to label the nodes for these types of graphs, but that becomes impossible with so many intertwined nodes.

One solution to select out a subgraph of the connected bits is to use a Steiner tree. Here is that graph after running the approximate Steiner tree algorithm in networkx (in python).

Much simpler! And much more space to put additional labels.

I’ve posted the code and data to replicate here. Initially I debated on solving this via setting up the problem as a min-cost-flow, where one of the highlighted nodes had the supply, and the other highlighted nodes had the demand. But this approximate algorithm in my few tests looks really good in selecting tiny subsets, so not much need.

A few things to note about this. It is likely for many dense networks there will be many alternative subsets that are the same size, but different nodes (e.g. you can swap out a node and have the same looking network). A better approach to see connections between interesting nodes may be a betweenness centrality metric, where you only consider the flows between the highlighted nodes.

A partial solution to that problem is to add nodes/edges back in after the Steiner tree subset. Here is an example where I add back in all first degree nodes to the red nodes of interest:

So it is still a tiny enough network to plot. This just provides a way to identify higher order nodes of interest that aren’t directly connected to those red nodes.

Resources of interest for criminologists and crime analysts

I tend to get about one email per week asking for help. Majority of folks are either students asking for general research advice, or individuals who came across my webpage asking for advice about code.

This is great, and everyone should always feel open to send me an email. The utility of me answering these questions (for everyone) are likely greater than spending time working on a paper, so I do not mind at all. I can currently keep up with the questions given the volume (but not by much, and is dependent on how busy I am with other work/family things). Worst case I will send an email response that says sorry I cannot respond to this anytime soon.

Many times there are other forums though for people to post questions that are ultimately better. One, I participate in many of these, so it is not like sending an email just to me, it is like sending an email to me + 40 other people who can answer your question. Also from my perspective it is better to answer a question once in one of these forums, than repeat the same answer a dozen different times. (Many times I write a blog post if I get the same question multiple times.)

While the two groups overlap a bit, I separate out resources aimed at criminologists (as typically more interested in research and are current master/PhD students), whereas crime analysts are embedded in a criminal justice agency.

For Criminologists

For resources on where to ask questions, Jacob Kaplan recently created a slack channel, crimhelp.slack.com. It has been joined by a variety of criminologists, folks in think tanks/research institutes, current graduate students, and some working crime analysts. It is new, but you can go and peruse the topics so far, they are pretty wide in scope.

So that forum you can really ask about anything crim related, the remaining resources are more devoted towards programming/statistical analysis.

If you are interested in statistical or programming questions, I used to participate in StackOverflow, Cross Validated (the stats site), and the GIS site. They are good places to check out prior answers, and are worth a shot asking a question on occasion. For tricky python or R coding questions that are small in scope, StackOverflow is excellent. Anything more complicated it is more hit or miss.

Many programming languages have their own question boards. Stata and SPSS are ones I am familiar with and tend to receive good responses (I still actively participate in the SPSS board). If I’m interested in learning some new command/library in Stata, I often just search the forum for posts related to it to check it out in the wild.

For programming questions, it is often useful to create a minimal reproducible example to describe the problem, show what the input data looks like and how you want the output data to look like. (In fact on the forums I link to you will almost always be asked explicitly to do that.)

For Crime Analysts

In similar spirit to the crim slack channel, Police Rewired has a Discord group for crime analysts (not 100% sure who started it, Andreas Varotsis is one of the people involved though). So that was founded by some UK analysts, but there are US analysts participating as well (and the problems folks deal with are very similar, so no real point in making a distinction between US/UK).

For crime analysts in the US, you should likely join either the IACA or a local crime analyst network. Many of the local ones come bundled, so if you join the Texas analyst network TXLEAN you also automatically get an IACA membership. To join is cheap (especially for current students). IACA has also started a user question forum as well.

For folks looking to get an entry level gig, the IACA has a job board that is really good. So it is worth the $10 just for that. They have various other intro resources though as well. For current BA/Masters students who are looking to get a job, I also suggest applying to private sector analyst jobs as well. They are mostly exchangeable with a crime analyst role. (Think more excel jockey than writing detailed statistic programming.)

How I learn to code

What prompted this blog post is that I’ve gotten asked by maybe 5 different people in the past month or so asking for resources to learn about statistical programming. And honestly I do not have a good answer, I’ve never really sat down with a book and learned a statistical software (tried on a few occasions and failed). I’m always just project focused.

So I wanted to do an example conjunctive analysis, or deep learning with pytorch, or using conformal prediction intervals to generate synthetic control estimates, etc. So I just sat down and figured out how to do those specific projects using various resources around the internet. One of my next personal projects is going to estimate prediction intervals for logistic multilevel models using Julia (based on this very nice set of intros to the language). I also need to get a working familiarity with Tableau. (Both are related to projects I am doing at work.) So expect to see a Tableau dashboard on the blog sometime in the near future!

Also many statistical programming languages are pretty much exchangeable for the vast majority of tasks people do. You can see that I have example blog posts for Excel, Access/SQL, R, SPSS, Stata, python, and ArcGIS. Just pick one and figure it for a particular project.

For criminologists, I have posted my Phd research course materials, and for Crime Analysts I have posted my GIS Class and my Crime Analysis course materials (although the GIS course is already out of date, it uses Arc Desktop instead of ArcPro). I don’t suggest you sit down and go through the courses though page-by-page. What I more suggest is look at the table of contents, see if anything strikes your fancy, read that particular lecture/code, and if you want to apply it to your own projects try to work it out. (At least that is how I go about learning coding.)

If you want more traditional learning materials for learning how to do code (e.g. textbooks or online courses), I suggest you ask folks on those forums I mentioned. They will likely be able to provide much better advice than I would.

To end it is totally normal to want to ask questions, get advice, or get feedback. Both my experience in Academia and in Crime Analysis it can be very lonely (I was in a small department, so was the only analyst). Folks on these forums are happy to help and connect.

Using Association Rules to Conduct Conjunctive Analysis

I’ve suggested to folks a few times in the past that a popular analysis in CJ, called conjunctive analysis (Drawve et al., 2019; Miethe et al., 2008; Hart & Miethe, 2015), could be automated in a fashion using a popular machine learning technique called association rules. So I figured a blog post illustrating it would be good.

I was motivated by some recent work by Nix et al. (2019) examining officer involved injuries in NIBRS data. So I will be doing a relevant analysis (although not as detailed as Justin’s) to illustrate the technique.

This ended up being quite a bit of work. NIBRS is complicated, and I had to do some rewrites of finding frequent itemsets to not run out of memory. I’ve posted the python code on GitHub here. So this blog post will be just a bit of a nicer walkthrough. I also have a book chapter illustrating geospatial association rules in SPSS (Wheeler, 2017).

A Brief Description of Conjunctive Analysis

Conjunctive analysis is more of an exploratory technique examining high cardinality categorical sets. Or in other words, you search though a database of cases that have many categories to find “interesting” patterns. It is probably easier to see an example than for me to describe it. Here is an example from Miethe et al. (2008):

You can see that here they are looking at characteristics of drug offenders, and then trying to identify particular sets of characteristics that influence the probability of a prison sentence. So this is easy to do in one dimension, it gets very difficult in multiple dimensions though.

Association rules were created for a very different type of problem – identifying common sets of items that shoppers buy together at the same time. But you can borrow that work to aid in conducting conjunctive analysis.

Data Prep for NIBRS

So here I am using 2012 NIBRS data to conduct analysis. Like I mentioned, I was motivated by the Nix and company paper examining officer injuries. They were interested in specifically examining officer involved injuries, and whether the perception that domestic violence cases were more dangerous for officers was justified.

For brevity I only ended up examining five different variable sets in NIBRS (Justin has quite a few more in his paper):

  • assault (or injury) type V4023
  • victim/off relationship V4032
  • ucr type V2006
  • drug use V2009 (also includes computer use!)
  • weapon V2017

All of these variables have three different item sets in the NIBRS codes, and many categories. You will have to dig into the python code, 00_AssocRules.py in the GitHub page to see how I recoded these variables.

Also maybe of interest I have some functions to do one-hot encoding of wide data. So a benefit of NIBRS is that you can have multiple crimes in one incident. So e.g. you can have one incident in which an assault and a burglary occurs. I do the analysis in a way that if you have common co-crimes they would pop out.

Don’t take this as very formal though. Justin’s paper which used 2016 NIBRS data only had 1 million observations, whereas here I have over 5 million (so somewhere along the way me and Justin are using different units of analysis). Also Justin’s incorporates dozens of other different variables into the analysis I don’t here.

It ends up being that with just these four variables (and the reduced sets of codes I created), there still end up being 34 different categories in the data.

Analysis of Frequent Item Sets

The first part of conjunctive analysis (or association rules) is to identify common item sets. So the work of Hart/Miethe is always pretty vague about how you do this. Association rules has the simple approach that you find any item sets, categories in which a particular itemset meets an arbitrary threshold.

So the way you represent the data is exactly how the prior Miethe et al. (2008) data showed, you create a series of dummy 0/1 variables. Then in association rules you look for sets in which for different cases all of the dummy variables take the value of 1.

The code 01_AssocRules.py on GitHub shows this going from the already created dummy variable data. I ended up writing my own function to do this, as I kept getting out of memory errors using the mlextend library. (I don’t know if this is due to my data is large N but smaller number of columns.) You can see my freq_sets function to do this.

Typically in association rules you identify item sets that meet a particular support threshold. Support here just means the proportion of cases that those items co-occur. E.g. if 20% of cases of assault also have a weapon of fists listed. Instead though I wrote the code to have a minimum N, which I choose here to be 1000 cases. (So out of 5 million cases, this is a support of 1/5000.)

I end up finding a total of 411 frequent item sets in the data that have at least 1000 cases (out of the over 5 million). Here are a few examples, with the frequencies to the left. So there are over 2000 cases in the 2012 NIBRS data that had a known relationship between victim/offender, resulted in assault, the weapon used was fists (or kicking), and involved computer use in some way. I only end up finding two itemsets that have 5 categories and that is it, there are no higher sets of categories that have at least 1000 cases in this dataset.

3509    {'rel_Known', 'ucr_Assault', 'weap_Fists', 'ucr_Drug'}
2660    {'rel_Known', 'ucr_Assault', 'weap_Firearm', 'ucr_WeaponViol'}
2321    {'rel_Known', 'ucr_Assault', 'weap_Fists', 'drug_ComputerUse'}
1132    {'rel_Known', 'ucr_Assault', 'weap_Fists', 'weap_Knife'}
1127    {'ucr_Assault', 'weap_Firearm', 'weap_Fists', 'ucr_WeaponViol'}
1332    {'rel_Known', 'ass_Argument', 'rel_Family', 'ucr_Assault', 'weap_Fists'}
1416    {'rel_Known', 'rel_Family', 'ucr_Assault', 'weap_Fists', 'ucr_Vandalism'}

Like I said I was interested in using NIBRS because of the Nix example. One way we can then examine what variables are potentially related to officer involved injuries during a commission of a crime would be to just pull out any itemsets which include the variable of interest, here ass_LEO_Assault.

4039    {'ass_LEO_Assault'}
1232    {'rel_Known', 'ass_LEO_Assault'}
4029    {'ucr_Assault', 'ass_LEO_Assault'}
1856    {'ass_LEO_Assault', 'weap_Fists'}
1231    {'rel_Known', 'ucr_Assault', 'ass_LEO_Assault'}
1856    {'ucr_Assault', 'ass_LEO_Assault', 'weap_Fists'}

So we see there are a total of just over 4000 officer assaults in the dataset. Unsurprisingly almost all of these also had an UCR offense of assault listed (4029 out of 4039).

Analysis of Association Rules

Sometimes just identifying the common item sets is what is of main interest in conjunctive analysis (see Hart & Miethe, 2015 for an example of examining the geographic characteristics of crime events).

But the apriori algorithm is one way to find particular rules that are of the form if A occurs then B occurs quite often, but swap out more complicated itemsets in the antecedent (A) and consequent (B) in the prior statement, and different ways of quantifying ‘quite often’.

I prefer conditional probability notation to the more typical association rule one, but for typical rules we have (here I use A for antecedent and B for consequent):

  • confidence: P(A & B) / P(B). So if the itemset of just B occurs 20% of the time, and the itemset of A and B together occurs 10% of the time, the confidence would be 50%. (Or more simply the probability of B conditional on A, P(B | A)).
  • lift: confidence(A,B) / P(B). This is a ratio of the baseline a category occurs for the denominator, and the numerator is the prior confidence category. So if you have a baseline B occurring 25% of the time, and the confidence of A & B is 50%, you would then have a lift of 2.

There are other rules as well that folks use, but those are the most common two I am interested in.

So for example in this data if I draw out rules that have a lift of over 2, I find rules like {'ucr_Vandalism', 'rel_Family'} -> {'ass_Argument'} produces a lift of over 6. (I can use the mlextend implementation here in this code, it was only the frequent itemsets code that was giving me problems.) So it ends up being arguments are listed in the injury codes around 1.6% of the time, but when you have a ucr crime of vandalism, and the relationship between victim/offender are family members, injury type of argument happens around 10.5% of the time (so 10.5/1.6 ~= 6).

The original use case for this is recommender systems/market analysis (so say if you see someone buy A, give them a coupon for B). So this ends up being not so interesting in this NIBRS example when you have you have more clear cause-effect type relationships criminologists would be interested in. But I describe in the next section some further potential machine learning models that may be more relevant, or how I might in the future amend the apriori algorithm for examining specific outcomes.

Further Notes

If you have a particular outcome you are interested in a specific outcome from the get go (so not so much totally exploratory analysis as here), there are a few different options that may make more sense than association rules.

One is the RuleFit algorithm, which basically just uses a regularized regression to find simple models and low order interactions. An example of this idea using police stop data is in Goel et al. (2016). These are very similar in the end to simple decision trees, you can also have continuous covariates in the analysis and it splits them into binary above/below rules. So you could say do RTM distance analysis, and still have it output a rule if < 1000 ft predict high risk. But they are fit in a way that tend to behave better out of sample than doing simple decision trees.

Another is fitting a more complicated model, say random forests, and then having reduced form summaries to describe those models. I have some examples of using shapely values for spatial crime prediction in Wheeler & Steenbeek (2020), but for a more if-then type sets of rules you could look at Scoped Rules.

I may need to dig into the association rules code some more though, and try to update the code to take the sample sizes and statistical significance into account for a particular outcome variable. So if you find higher lift in a four set predicting a particular outcome, you search the tree for more sets with a smaller support in the distribution. (I should probably also work on some cool network viz. to look at all the different rules.)

References

 

An intro to linear programming for criminologists

Erik Alda made the point the other day on twitter that we are one of the few crim folks that do anything related to linear programming. I think it is crazy useful – much more so than say teaching myself some new regression technique or a programming language.

I don’t quite remember the motivation to learn it. I think I kept seeing repeated applications in papers I read, but was also totally baffled by it; I did not understand peoples notation for it at all. In retrospect that was because it is not statistics. You are optimizing a function by estimating some parameters (there is nothing stochastic about it, so there is no statistical inference). So it is more like finding the min/max of a function in calculus.

I think the best way to think about linear programming is in terms of decision analysis. We have a set of options among which we need to choose some action. So we make the choices that either maximize or minimize some objective, but also take into account constraints on the decisions we can make.

For social scientists here is an example that hopefully illustrates the difference between statistics and linear programming. Say we are interested in conducting a hot spots policing randomized experiment. So we define our top 20 crime hot spots in the city, and randomly assign 10 of them to receive the hot spots treatment. Linear programming is basically the obverse of this, given our 20 hot spot areas, which are the best 10 locations to choose for our intervention.

This problem as stated you might be thinking is trivial – just rank each of the 20 hot spots by the total number of crimes, and then choose the top 10. Where linear programming really helps though is if you have constraints on the final choices you make. Say you did not want to choose hot spots that are within 1 mile of each other (to spread out the hot spot interventions throughout the city). There is no simple way to sort your hot spots to obey that constraint, but you can encode that in the linear program and have the computer solve it quite easily.

There is no shortage of ways you could expand the complexity of this example hot spot decision analysis. Say you had two different types of hot spot treatments, and that they had different efficacy in different areas (one was good for property crime, and the other was better for violent crime). You might think of this as doing two separate decision analyses, where a constraint is that an area can only be assigned one of the two interventions.

Here I will provide some code examples in python using the pulp library to illustrate some more examples using data you can see in action, as well as different ways to think about linear programming problems in practice. (Technically the examples I give are all mixed integer linear programs, as the decision variables are binary 0/1.)

Formulating Objectives and Constraints

For this example I will be simulating data, but imagine a case you are an analyst for the IRS, and you want to determine which business tax returns to audit. We want to audit cases that both have a high probability of being fraudulent, as well as cases in which the total amount of the underpayment is large. (If you want a more typical criminology example, imagine assigning criminal cases to detectives, some cases have more costs, e.g. homicide vs burglary, and some cases have different probabilities of being solvable. This type of decision problem is very common in my experience – pretty much any time you have to make a binary choice, and those choices have variable costs/benefits.)

First I start off by simulating some data (the only libraries we need are numpy and pulp). So I simulate 1000 business tax returns, which have an estimate of the probability they are fraud, prob_fraud, and an estimate of the amount they underpayed, underpay_est.

import numpy as np
import pulp

###########################################################
#Simulate data for costs and probabilities

np.random.seed(10)
total_cases = 1000
underpay_est = np.random.uniform(1000,100000,total_cases)
prob_fraud = np.random.uniform(0,1,total_cases)
exp_return = prob_fraud*underpay_est

###########################################################

The objective we will be maximizing then is the expected return of auditing a tax return, exp_return, which is simply the multiplication of the probability of fraud multiplied by the amount of the underpayment. For a simple example, say we have a case where fraud is estimated to be 50%, and the estimate of the underpayment amount is $10,000. So our expected return for auditing that case is $5,000.

We need these two estimates external to our linear programming problem, and they themselves can be informed by predictive models (or simpler estimates, e.g. underpayment is proportional ~30% of deductions or something like that).

Now we have all we need to set up our linear programming problem. I am going to choose 100 cases out of these 1000 to audit. Hopefully that code is documented enough to see creating the decision variables (each tax return either gets a 1 if it is chosen, or a 0 if it is not), the original objective function that we are maximizing, and the results.

#Setting up the problem
case_index = list(range(total_cases))
tot_audit = 100

####################################
#Basic Problem
P = pulp.LpProblem("Choosing Cases to Audit", pulp.LpMaximize)
D = pulp.LpVariable.dicts("Decision Variable", [i for i in case_index], lowBound=0, upBound=1, cat=pulp.LpInteger)
#Objective Function
P += pulp.lpSum( D[i]*exp_return[i] for i in case_index)
#Constraint on total number of cases audited
P += pulp.lpSum( D[i] for i in case_index ) == tot_audit
#Solve the problem
P.solve()
#Get the decision variables
dec_list = [D[i].varValue for i in case_index]
dec_np = np.asarray(dec_list)
#Expected return
print( (dec_np * exp_return).sum() )
#Should be the same
print( pulp.value(P.objective) )
#Hit rate of cases
print( (dec_np * prob_fraud).sum()/tot_audit )
####################################

If you are following along in python, you will see that the total expected return is 7,287,915, and the estimated hit rate (or clearance rate) of the audits is around 0.88.

This example would be no different if we just chose the top 100 cases based on the expected return. Say that you thought the hit rate though of 88% was too low. You will choose cases that are big dollar amounts, but not necessarily a very high probability. So you may say I want my clearance rate to be over 90% overall. That is a simple constraint to add into the above model.

####################################
#Updating the problem to constrain on the hit rate
#Above a particular threshold
hit_rate = 0.9
P += pulp.lpSum( D[i]*prob_fraud[i] for i in case_index ) >= hit_rate*tot_audit
P.solve()
#Get the decision variables
dec_list = [D[i].varValue for i in case_index]
dec_np = np.asarray(dec_list)
#Expected return is slightly lower than before
print( pulp.value(P.objective) )
#Hit rate of cases
print( (dec_np * prob_fraud).sum()/tot_audit )
####################################

So now the total expected return is lower than without the constraint, 7,229,140 (so a reduction of about $60k), but our expected hit rate is just above 90%.

You may be thinking, “why not just eliminate cases with a probability of lower than 90%”, and then amongst those left over select the highest expected return. That meets your constraints, but has a lower expected return than this program! Think of this program as more tit-for-tat. High expected return / lower probability audits can still be selected with this model, but you need to balance them out with some high probability cases in response to tip the scales to meet the overall hit rate objective.

Trade-Offs and the Frontier Curve

So you may be thinking, ok the trade-off to get a 90% clearance was not too bad in terms of total extra taxes collected. So why not set the constraint to 95%. When you create constraints, they always lower the objective function (lower or equal to be more precise). The question then becomes quantifying that trade off.

You can subsequently vary the hit rate constraint, and see how much it changes the total expected return. Here is an example of doing that, each model only takes around a second to complete.

###########################################################
#Drawing the trade-off in hit rate vs expected return

hit_rate = np.linspace(0.85, 0.95, 30)
total_return = []

#Function to estimate the model
def const_hit_rate(er, prob, tot_n, hr):
    c_index = range(len(er))
    Prob = pulp.LpProblem("Choosing Cases to Audit", pulp.LpMaximize)
    Dec = pulp.LpVariable.dicts("Decision Variable", [i for i in c_index], lowBound=0, upBound=1, cat=pulp.LpInteger)
    Prob += pulp.lpSum( Dec[i]*er[i] for i in c_index)
    Prob += pulp.lpSum( Dec[i] for i in c_index ) == tot_n
    Prob += pulp.lpSum( Dec[i]*prob[i] for i in c_index ) >= hr*tot_n
    Prob.solve()
    dec_li = [Dec[i].varValue for i in c_index]
    dec_np = np.asarray(dec_li)
    return pulp.value(Prob.objective), dec_np

for h in hit_rate:
    print(f'Estimating hit rate {h}')
    obj, dec_res = const_hit_rate(exp_return, prob_fraud, 100, h)
    total_return.append(obj)

###########################################################

For this simulated data example, there end up being pretty severe trade-offs in the total return after you get above 91% hit rates, so from this it may not be worth the trade-off to get a much higher hit rate in practice. Just depends on how much you are willing to trade-off one for the other.

There are other ways to formulate this trade off (via bi-objective functions/soft-constraints, or weighted ranking schemes), but the blog post is long enough as is!

Other Potential Applications

So in terms of my work, I have examples of using linear programs to make spatial location decisions, encode fairness constraints into predictive policing, and allocate treatment assignment with network spillovers.

Erik Alda and Joseph Ferrandino have conducted frontier analysis of different criminal justice organizations, which is based on estimating the frontier curve above from data (instead of a pre-specified objective function).

That is about it for criminologists that I know of, but there are plenty of applications towards criminal justice topics using linear programming (or related concepts). It is most popular among operations researchers, of which Laura Albert is one of my favorites. (Criminal Justice as a field might not exist for Albert Blumstein, who was also a very influential operations researcher.)

One of the things that makes this different from more traditional quantitative work in the social sciences is that again it is not statistics – we are not testing hypotheses. The contribution is simply formulating the decision problem in a tractable way that can be solved, and the drawing of the trade-offs I showed above.

It is one of the ways I really like it though – unlike saying how your regression model can be used to inform decisions, this much more explicitly shows the utility of the results of those models in some practice.

Conjoint Analysis of Crime Rankings

So part of my recent research mapping crime harm spots uses cost of crime estimates relevant to police departments (Wheeler & Reuter, 2020). But a limitation of this is that cost of crime estimates are always somewhat arbitrary.

For a simple example, those cost estimates are based mostly on people time by the PD to respond to crimes and devote investigative resources. Many big city PDs entirely triage crimes like breaking into vehicles though. So based on PD response the cost of those crimes are basically $0 (especially if PDs have an online reporting system).

But I don’t think the public would agree with that sentiment! So in an act of cognitive dissonance with my prior post, I think asking the public is likely necessary for police to be able to ultimately serve the publics interest when doing valuations. For some ethical trade-offs (like targeting hot spots vs increasing disproportionate minority contact, Wheeler, 2019) I am not sure there is any other reasonable approach than simply getting a bunch of peoples opinions.

But that being said, I suspected that these different metrics would provide pretty similar rankings for crime severity overall. So while it is criminology 101 that official crime and normative perceptions of deviance are not a perfect 1 to 1 mapping, most folks (across time and space) have largely similar agreement on the severity of different crimes, e.g. that assault is worse than theft.

So what I did was grab some survey ranking of crime data from the original source of crime ranking that I know of, Marvin Wolfgang’s supplement to the national crime victimization survey (Wolfgang et al., 2006). I have placed all the code in this github folder to replicate. And in particular check out this Jupyter notebook with the main analysis.

Conjoint Analysis of Crime Ranks

This analysis is often referred to as conjoint analysis. There are a bunch of different ways to conduct conjoint analysis – some ask folks to create a ranked list of items, others ask folks to choose between a list of a few items, and others ask folks to rank problems on a Likert item 1-5 scale. I would maybe guess Likert items are the most common in our field, see for example Spelman (2004) using surveys of asking people about disorder problems (and that data is available to, Taylor, 2008).

The Wolfgang survey I use here is crazy complicated, see the codebook, but in a nutshell they had an anchoring question where they assigned stealing a bike to a value of 10, and then asked folks to give a numeric score relative to that theft for a series of 24 other crime questions. Here I only analyze one version of the questionnaire, and after eliminating missing data there are still over 4,000 responses (in 1977!).

So you could do analyze those metric scores directly, but I am doing the lazy route and just doing a rank ordering (where ties are the average rank) within person. Then conjoint analysis is simply a regression predicting the rank. See the notebook for a more detailed walkthrough, so this just produces the same analysis as looking at the means of the ranks.

About the only thing I do different here than typical conjoint analysis is that I rescale the frequency weights (just changes the degrees of freedom for standard error estimates) to account for the repeated nature of the observations (e.g. I treat it like a sample of 4000 some observations, not 4000*25 observations). (I don’t worry about the survey weights here.)

To test my assertion of whether these different ranking systems will be largely in agreement, I take Jerry’s crime harm paper (Ratcliffe, 2015), which is based on sentencing guidelines, and map them as best I could to the Wolfgang questions (you could argue with me some though on those assements – and some questions don’t have any analog, like a company dumping waste). I rescaled the Wolfgang rankings to be in a range of 1-14, same as Jerry’s, instead of 1-25.

Doing a more deep dive into the Wolfgang questions, there are definately different levels in the nature of the questions you can tease out. Folks clearly take into account both harm to the victim and total damages/theft amounts. But overall the two systems are fairly correlated. So if an analyst wants to make crime harm spots now, I think it is reasonable to use one of these ranking systems, and then worry about getting the public perspective later on down the line.

The Wolfgang survey is really incredible. In this regression framework you can either adjust for other characteristics (e.g. it asks about all the usual demographics) or look at interactions (do folks who were recently victimized up their scores). So this is really just scratching the surface. I imagine if someone redid it with current data many of the metrics would be similar as well, although if I needed to do this I don’t think I would devise something as complicated as this, and would ask people to rank a smaller set of items directly.

References

  • Ratcliffe, J.H. (2015). Towards an index for harm-focused policing. Policing: A Journal of Policy and Practice, 9(2), 164-182.
  • Spelman, W. (2004). Optimal targeting of incivility-reduction strategies. Journal of Quantitative Criminology, 20(1), 63-88.
  • Taylor, R.B. (2008). Impacts of Specific Incivilities on Responses to Crime and Local Commitment, 1979-1994: [Atlanta, Baltimore, Chicago, Minneapolis-St. Paul, and Seattle]. https://doi.org/10.3886/ICPSR02520.v1
  • Wheeler, A.P., & Reuter, S. (2020). Redrawing hot spots of crime in Dallas, Texas. https://doi.org/10.31235/osf.io/nmq8r
  • Wheeler, A.P. (2019). Allocating police resources while limiting racial inequality. Justice Quarterly, Online First.
  • Wolfgang, M.E., Figlio, R.M., Tracy, P.E., and Singer, S.I. (2006). National Crime Surveys: Index of Crime Severity, 1977. https://doi.org/10.3886/ICPSR08295.v1

Nearby Analysis Example (Excel)

The other day on Twitter I made a comment to Joel Caplan about how I would solve analysis with multiple buffers and not counting overlaps. A typical GIS workflow would go:

  • take your points of interest and create buffers
  • join the points to the buffer polygons, and get a count of the crimes of interest

I often do the analysis in different way though – I do a spatial join of the location of interest to the point features, in which you get a field that is the distance to the nearest feature, and then subsequently do analysis on that distance field. In that workflow, it makes it much easier to change the size of the buffer for sensitivity analysis, or conduct analysis on different subsets of data.

To start I am going to be working with a set of robberies in Dallas (from the open data, not quite 16k), and DART stations (n = 74). (DART is the Dallas above ground train.) You can access the Excel file I am doing analysis with here. Using excel as I often suggest it for undergrads/masters for projects who aren’t up to speed with programming – so this is a good illustration of that buffer analysis workflow.

Distance to Nearest

To start, I would typically use a GIS system (or R/python/SQL) to calculate the distance to a nearest object. But I don’t have access to Arc anymore, so I am going to show you a way to do this right in Excel. This only works for projected data (not latitude/longitude), and calculating distance from point-to-point.

So first, to figure out the distance between two points in Euclidean space, we can just use the Pythagorean theorem that you learned in grade school, Distance = sqrt( (x1 - x2)^2 + (y1 - y2)^2 ). Because we are doing this in an Excel spreadsheet and want to find the nearest Dart station to the robbery, we will use a little array formula magic. I named my table of Dart locations Dart, and so the array formula to find the nearest distance in Excel is:

=MIN( SQRT( (B2 - Dart[X])^2  + (C2 - Dart[Y])^2))

When you enter this formula, hit Ctrl + Shift + Enter, else it just returns the distance to the first Dart station. If you did this right, you will see the formula have {} brackets around it in the formula bar.

Distance will be defined in whatever the projected units are in – here they are in feet. But by using MIN with the array, it returns the distance to the nearest station. To get the ID of the associated station, we need to do a similar formula (and this only works with numeric ID fields). You can basically do an array IF formula, and the only station this is true for will be the MAX of that array. (Again hit Ctrl + Shift + Enter when finishing off this cell calculation instead of just Enter.)

=MAX(IF(F2=SQRT((B2 - Dart[X])^2  + (C2 - Dart[Y])^2), Dart[DartID],0))

User beware – this runs super fast on my machine (surprisingly) but it is quite a few computations under the hood. For much larger data again use a GIS/database/Stat program to do these calculations.

Using Pivot Tables to do Buffer Analysis

So now that we have those distance fields, it is easy to do a formula along the lines of you want to count up the robberies within 1000 feet. You can do another IF formula that is something like IF([@Distance] < 1000, 1, 0).

And then go ahead and make a pivot table, and put the DartID as the rows, and the Within distance field you just made as the values (to sum in the pivot table).

Then bam, you have your buffer analysis. Here I sorted the pivot table so you can see the highest crime Dart is 12. (I haven’t looked up which one this is, you can use Excel though to map them out).

So say you wanted to change the buffer size? It is as simple as changing out the 1000 in the prior formula to a different value. One thing I like to do though is to make a lookup table to define different bins. You can see I named that table BuffTable (naming the tables makes it easier to refer to them later in array formulas, also I shifted down the pivot table to not accidently overwrite it later).

And now I use a combination of MATCH to find what row it falls into for this table, and INDEX to return the row label I want. So first I have =MATCH([@Distance],BuffTable[Within Bounds],1). This is very similar to VLOOKUP, and will match to the row that the distance is less than.

This just returns the row number of the match though – I want to pipe in those nicer labels I made. To do that, I nest the match results within index, =INDEX(BuffTable, MATCH([@Distance],BuffTable[Within Bounds],1)+1, 2). And voila, I get my binned data.

Now we can do our pivot table so the columns are the new field we just made (make sure to refresh the pivot table).

And we can do our buffer analysis and varying buffers. Just update the tables to however you want the buffers, hit refresh, and everything will be updated. (I should have done the labels so they are ordered a bit more nicely in the pivot table.)

I like this approach for students, as it is easy to pivot/filter on other characteristics as well. Want to get arrest rates around areas? Want to see changes in crimes nearby different DART stations over time? It is just a few formulas/filters and a pivot table away in this format.

Distance to Nearest Analysis for DART stations

Another analysis I think is useful is to look at the cumulative distance analysis. I got this idea from a paper of Jerry Ratcliffe’s.

So what you can do is to round the distance data, e.g. using a formula like this will round the data to every 300 feet.

=ROUND([@Distance]/300,0)*300

And then you can make a pivot table of the rounded counts. Here I also did additional stats to calculate the spatial density of the points, and show the distance decay curve.

Jerry’s paper I linked to looks for change points – I think that idea is somewhat misleading though. It does look like a change point in Jerry’s graphs, but that is a function of the binning I think (see this Xu/Griffiths paper, same method, finer bins, and shows a more smooth decay).

So here I tied the round function to a cell, and again I can just update the value to a different bin size, and everything get auto-updated in the spreadsheet. Here is a bin size of 100 feet, which introduces some volatility in the very nearby locations, but you can see still pretty much follows that smooth distance decay effect.

Actually the Xu/Griffiths paper looks at the street network distance, which I think makes more sense. (And again need a GIS to do that analysis.) The buffer areas can behave funny, and won’t have a direct relationship to the street length exposure, so I think the typical Euclidean analysis can be misleading in some cases. I will need to save that for another blog post though!

The Failed Idea Bin: Temporal Aggregation and the Crime/Stop Relationship

A recent paper by the Hipp/Kim/Wo trio analyzing robbery at very fine temporal scales in NYC reminded me on a failed project I never quite worked out to completion. This project was about temporal aggregation bias. We talk about spatial aggregation bias quite a bit, which I actually don’t think is that big of deal for many projects (for reasons discussed in my dissertation).

I think it is actually a bigger deal though when dealing with temporal relationships, especially when we are considering endogenous relationships between crime and police action in response to crime. This is because they are a countervailing endogenous relationship – most endogenous relationships are positively correlated, but here we think police do more stuff (like arrests and stops) in areas with more crime, and that crime falls in response.

I remember the first time I thought about the topic was when I was working with the now late Dennis Smith and Robert Purtell as a consultant for the SQF litigation in NYC. Jeff Fagan had some models predicting the number of stops in an area, conditional on crime and demographic factors at the quarterly level. Dennis and Bob critiqued this as not being at the right temporal aggregation – police respond to crime patterns much faster than at the quarterly level. So Jeff redid his models at the monthly level and found the exact same thing as he did at the quarterly level. This however just begs the question of whether monthly is the appropriate temporal resolution.

So to try to tackle the problem I took the same approach as I did for my dissertation – I pretend I know what the micro level equation looks like, and then aggregate it up, and see what happens. So I start with two endogenous equations:

crime_t1 = -0.5*(stops_t0) + e_c
stops_t1 =  0.5*(crime_t0) + e_s

And then aggregation is just a sum of the micro level units:

Crime_T = (crime_t1 + crime_t0)
Stops_T = (stops_t1 + stops_t0)

And then what happens when we look at the aggregate relationship?

Crime_T = Beta*(Stops_T)

Intuitively here you may see where this is going. Since crime and stops have the exact same countervailing effects on each other, they cancel out if you aggregate up one step. I however show in the paper if you aggregate up more than two temporal units in this situation the positive effect wins. The reason is that back substitution for prior negative time series relationships oscillates (so a negative covariance at t-1 is a positive covariance at t-2). And in the aggregate the positive swamps the negative relationship. Even estimating Crime_T = Beta*(Stops_T-1) does not solve the problem. These endogenous auto-regressive relationships actually turn into an integrated series quite quickly (a point that cannot be credited to me, Clive Granger did a bunch of related work).

So this presented a few hypotheses. One, since I think short run effects for stops and crime are more realistic (think the crackdown literature), the covariance between them at higher resolutions (say monthly) should be positive. You should only be able to recover the deterrent effect of stops at very short temporal aggregations I think. Also crime and stops should be co-integrated at large temporal aggregations of a month or more.

Real life was not so convenient for me though. Here I have the project data and code saved. I have the rough draft of the theoretical aggregation junk here for those interested. Part of the reason this is in the failed idea bin is that neither of my hypotheses appears to be true with the actual crime and stop data. For the NYC citywide data I broke up stops into radio-runs and not-radio-runs (less discretion for radio runs, but still should have similar deterrent effects), and crimes as Part 1 Violent, Part 1 Non-Violent, and Part 2. More recently I handed it off to Zach Powell, and he ran various vector auto-regression models at the monthly/weekly/daily/hourly levels. IIRC it was pretty weak sauce evidence that stops at the lower temporal aggregations showed greater evidence of reducing crime.

There of course is a lot going on that could explain the results. Others have found deterrent effects using instrumental variable approaches (such as David Greenberg’s work using Arellano-Bond, or Wooditch/Weisburd using Bartik instruments). So maybe my idea that spatial aggregation does not matter is wrong.

Also there is plenty of stuff going on specifically in NYC. We had the dramatic drop in stops due to the same litigation. Further work by MacDonald/Fagan/Geller have shown stops that met a higher reasonable suspicion standard based on the reported data have greater effects than others (essentially using Impact zones as an instrument there).

So it was a question I was never able to figure out – how to correctly identify the right temporal unit to examine crime and deterrence from police action.