# 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.

# Some additional plots to go with Crime Increase Dispersion

So Jerry nerdsniped me again with his Crime Increase Dispersion statistic (Ratcliffe, 2010). Main motivation for this post is that I don’t find that stat very intuitive to be frank. So here are some alternate plots, based on how counts of crime approximately follow a Poisson distribution. These get at the same question though as Jerry’s work, is a crime increase (or decrease) uniform across the city or specific to a few particular sub-areas.

First, in R I am going to simulate some data. This creates a set of data that has a constant increase over 50 areas of 20%, but does the post crime counts as Poisson distributed (so it isn’t always exactly a 20% increase). I then create 3 outliers (two low places and one high place).

###########################################
#Setting up the simulation
set.seed(10)
n <- 50
low <- 10
hig <- 400
inc <- 0.2
c1 <- trunc(runif(n,low,hig))
c2 <- rpois(n,(1+inc)*c1)
#Putting in 2 low outliers and 1 high outlier
c2[5] <- c1[5]*0.5
c2[10] <- c1[10]*0.5
c2[40] <- c1[40]*2
#data frame for ggplot
my_dat <- data.frame(pre=c1,post=c2)
###########################################

The first plot I suggest is a simple scatterplot of the pre-crime counts on the X axis vs the post-crime counts on the Y axis. My make_cont function takes those pre and post crime counts as arguments and creates a set of contour lines to put as a backdrop to the plot. Points within those lines support the hypothesis that the area increased in crime at the same rate as the overall crime increase, taking into account the usual ups and downs you would expect with Poisson data. This is very similar to mine and Jerry’s weighted displacement difference test (Wheeler & Ratcliffe, 2018), and uses a normal based approximation to examine the differences in Poisson data. I default to plus/minus three because crime data tends to be slightly over-dispersed (Wheeler, 2016), so coverage with real data should be alittle better (although here is not necessary).

###########################################
#Scatterplot of pre vs post with uniform
#increase contours

make_cont <- function(pre_crime,post_crime,levels=c(-3,0,3),lr=10,hr=max(pre_crime)*1.05,steps=1000){
#calculating the overall crime increase
ov_inc <- sum(post_crime)/sum(pre_crime)
#Making the sequence on the square root scale
gr <- seq(sqrt(lr),sqrt(hr),length.out=steps)^2
cont_data <- expand.grid(gr,levels)
names(cont_data) <- c('x','levels')
cont_data$inc <- cont_data$x*ov_inc
cont_data$lines <- cont_data$inc + cont_data$levels*sqrt(cont_data$inc)
return(as.data.frame(cont_data))
}

contours <- make_cont(c1,c2)

library(ggplot2)
eq_plot <- ggplot() +
geom_line(data=contours, color="darkgrey", linetype=2,
aes(x=x,y=lines,group=levels)) +
geom_point(data=my_dat, shape = 21, colour = "black", fill = "grey", size=2.5,
alpha=0.8, aes(x=pre,y=post)) +
scale_y_continuous(breaks=seq(0,500,by=100)) +
coord_fixed() +
xlab("Pre Crime Counts") + ylab("Post Crime Counts")
#scale_y_sqrt() + scale_x_sqrt() #not crazy to want square root scale here
eq_plot

#weighted correlation to view the overall change
cov.wt(my_dat[,c('pre','post')], wt = 1/sqrt(my_dat$pre), cor = TRUE)$cor[1,2]
########################################### 

So places that are way outside the norm here should pop out, either for increases or decreases. This will be better than Jerry’s stats for identifying outliers in lower baseline crime places.

I also show how to get an overall index based on a weighted correlation coefficient on the last line (as is can technically return a value within (-1,1), so might square it for a value within (0,1)). But I don’t think the overall metric is very useful – it has no operational utility for a crime department deciding on a strategy. You always need to look at the individual locations, no matter what the overall index metric says. So I think you should just cut out the middle man and go straight to these plots. I’ve had functionally similar discussions with folks about Martin Andresen’s S index metric (Wheeler, Steenbeek, Andresen, 2018), just make your graphs and maps!

An additional plot that basically takes the above scatterplot and turns it on its side is a Poisson version of a Bland-Altman plot. Traditionally this plot is the differences of two measures on the Y axis, and the average of the two measures on the X axis. Here to make the measures have the same variance, I divide the post-pre crime count differences by sqrt(post+pre). This is then like a Poison Z-score, taking into account the null of an equal increase (or decrease) in crime stats among all of the sub-areas. (Here you might also use the Poisson e-test to calculate p-values of the differences, but the normal based approximation works really well for say crime counts of 5+.)

###########################################
#A take on the Bland-Altman plot for Poisson data

ov_total <- sum(my_dat$post)/sum(my_dat$pre)
my_dat$dif <- (my_dat$post - ov_total*my_dat$pre)/sqrt(my_dat$post + my_dat$pre) my_dat$ave <- (my_dat$post + my_dat$pre)/2

ba_plot <- ggplot(data=my_dat, aes(x=ave, y=dif)) +
geom_point(shape = 21, colour = "black", fill = "grey", size=2.5, alpha=0.8) +
scale_y_continuous(breaks=seq(-8,6,by=2)) +
xlab("Average Crime") + ylab("Z-score (Equal Increase)")

ba_plot

#false discovery rate correction
my_dat$p_val <- pnorm(-abs(my_dat$dif))*2 #two-tailed p-value
my_dat$p_adj <- p.adjust(my_dat$p_val,method="BY") #BY correction since can be correlated
my_dat <- my_dat[order(my_dat$p_adj),] my_dat #picks out the 3 cases I adjusted ########################################### So again places with large changes that do not follow the overall trend will pop out here, both for small and large crime count places. I also show here how to do a false-discovery rate correction (same as in Wheeler, Steenbeek, & Andresen, 2018) if you want to actually flag specific locations for further investigation. And if you run this code you will see it picks out my three outliers in the simulation, and all other adjusted p-values are 1. One thing to note about these tests are they are conditional on the observed overall citywide crime increase. If it does happen that only one area increased by alot, it may make more sense to set these hypothesis tests to a null of equal over time. If you see that one area is way above the line and a ton are below the line, this would indicate that scenario. To set the null to no change in these graphs, for the first one just pass in the same pre estimates for both the pre and post arguments in the make_cont function. For the second graph, change ov_total <- 1 would do it. # References • Ratcliffe, J. H. (2010). The spatial dependency of crime increase dispersion. Security Journal, 23(1), 18-36. • Wheeler, A. P. (2016). Tables and graphs for monitoring temporal crime trends: Translating theory into practical crime analysis advice. International Journal of Police Science & Management, 18(3), 159-172. • Wheeler, A. P., & Ratcliffe, J. H. (2018). A simple weighted displacement difference test to evaluate place based crime interventions. Crime Science, 7(1), 11. • Wheeler, A. P., Steenbeek, W., & Andresen, M. A. (2018). Testing for similarity in area‐based spatial patterns: Alternative methods to Andresen’s spatial point pattern test. Transactions in GIS, 22(3), 760-774. # Setting the threshold for bail decisions I am at it again on the bail reform stuff. So we have critics of algorithms in-place of bail say that these bail reforms are letting too many people out (see Chicago and NYC). We also have folks on the other side saying such systems are punishing too many people (the Philly piece is more about probation, but the critique applies the same to pre-trial algorithms). So how can risk assessment algorithms both be too harsh and too lenient at the same time? The answer is that how harsh or lenient is not intrinsic to the algorithm itself, people can choose the threshold to be either. At its most basic, risk assessment algorithms provide an estimate of future risk over some specific time horizon. For example, after scored for a pre-trial risk assessment, the algorithm may say an individual has a 30% probability of committing a crime within the next 3 months. Is 30% low risk, and so they should be released? Or is 30% high risk? The majority of folks do not go on to commit a new offense awaiting for trial, so 30% overall may be really high actually relative to most folks being assessed. So how exactly do you decide the threshold – at what % is it too high and they should be detained pre-trial? For an individual cost-benefit analysis, you consider the costs of pre-trial detainment (the physical costs to detain a person for a specific set of time, as well as the negative externalities of detention for individuals) vs the cost to society of whether a future crime occurs. So for the 30% mark, say that average crime cost to society is$10,000 (this is ignoring that violent crimes cost more than property crime, in practice you can get individual probability estimates for each and combine them together). So the expected risk if we let this person go would then be $10,000*0.3 =$3,000. Whether the 3k risk is worth the pre-trial detention depends on how exactly we valuate detention. Say we have on average 90 days pre-trial detention, and the cost is something like $200 up front fixed costs plus$50 per day. We would then have a cost to detain this person as $200 +$50*90 = $4,700. From that individual perspective, it is not worth to detain that person pre-trial. This is a simplified example, e.g. this ignores any negative externality cost of detaining (e.g. lost wages for individuals detained), but setting the threshold risk for detaining or releasing on recognizance (ROR) should look something like this. One of the things I like about this metric is that it places at the forefront how another aspect of bail reform – the time to trial – impacts costs. So if you reduce the time to trial, for those ROR’d you reduce the time horizon of risk. (Those only out 1 month have less time to recidivate than those that have to wait 6 months.) It also reduces the cost of detaining individuals pre-trial as well (costs less for both the state and the individual to only be in jail 1 month vs 6 months). It may be to make everyone happy, we should reduce the queue so we can get people to trial faster. (But in the short term risk assessment tools I think have a great upside to both increase public safety as well as reduce imprisonment costs.) # Evaluating Overall Crime Effects Part of the reason folks are not happy with the current bail reforms is that they think it is increasing overall crime (see this example in Dallas). Here is an example graph though that folks doing these assessments should be providing, both in terms of up-front establishing the threshold for risk, as well as evaluating the efficacy of the risk assessment tool in practice. I will use the data from my prior posts on false positives to illustrate. For the graph, you place on the X axis the cumulative number of people that are detained, and the Y axis you place the expected number of crimes that your model thinks will be committed by those folks who are ROR’d. So a simplified table may be Person %crime A 0.5 B 0.4 C 0.3 D 0.2 E 0.1 If we let all of these folks go, we would expect they commit a total of 1.5 crimes (the sum of the percent predicted crime column) forecasted per our risk assessment algorithm. If we detained just person A, we have 1 in the detain column, and then a cumulative risk for the remaining folks of 1 (the sum of the predicted crime column for those that are remaining and are not detained). So then we go from the above table to this table by increasing the number of folks detained one-by-one. Detained ExpectedCrimes 0 1.5 1 1.0 2 0.6 3 0.3 4 0.1 5 0 Here is what that graph looks like using the ProPublica data, so if we apply this strategy to the just under 3,000 cases (in my test set from the prior blog post). So you can see that if we decided to detain no-one, we would expect a total of 1,200 extra crimes. And this curve decreases over detaining everyone. So you may say I don’t want more than 200 crimes, which you would need to have detained 1,500 people in the prior example (and happens to result in a risk threshold of 36% in this sample). Using historical data, this is good to establish a overall amount of crime risk you expect to occur from a particular set of bail reform decisions. To apply it to the future threshold decision making, you need to assume the past in terms of the total number of people arrested as well as the risk distribution stays the same (the latter I don’t think is a big issue, and the former you should be able to make reasonable projections if it is not constant). But this sets up the hypothetical, OK if we release this many more people ROR, we expect this many more crimes to occur as an upfront expectation of bail reform. It may be even if the individual cost-benefit calculation above says release, this would result in a total number of extra crimes folks deem unacceptable when applying that decision to everyone. So we can set the threshold to say we only want 10 extra crimes to happen because of bail reform, or 50 extra, or 100 extra, etc. This example again just aggregates all crime together, but you can do the same thing for every individual crime outcome you are interested in. After the assessment is in place, this is actually necessary monitoring folks should do be doing anyway to ensure the model is working as expected. That is, you get an estimate of the number of crimes folks who are released you think would commit per your risk assessment model. If you predict more/less than what you see amongst those released, your model is not well calibrated and needs to be updated. In practice you can’t just estimate a predictive model once and then use it forever, you need to constantly monitor whether it is still working well in real life. (Actually I showed in my prior blog post that the model was not very good, so this is a pretty big over-estimate of the number of crimes in this sample.) This should simultaneously quell complaints about bail reform is causing too many crimes. The lack of this information is causing folks to backlash against these predictive algorithms (although I suspect they do better than human judges, so I suspect they can reduce crime overall if used wisely). Offhand the recent crime increases in Philly, NYC, and Dallas I’m skeptical are tied to these bail reform efforts (they seem too big of increases or too noisy up/downs to reliably pin to just this), but maybe I am underestimating how many people they are letting out and the cumulative overall risk expected from the current models in those places. On the flip-side folks are right to question those Chicago stats, I suspect the risk algorithm should be saying that more crimes are occurring then they observed (ignoring what they should or should not be counting as recidivated). I’d note these metrics I am suggesting here should be pretty banal to produce in practice. It is administrative data already collected and should be available in various electronic databases. So in practice I don’t know why this data is not readily available in various jurisdictions. # What about False Positives? One thing you may notice is that in my prior cost-benefit analysis I didn’t take into consideration false positives. Although my prior post details how you would set this, there is a fundamental problem with monitoring false positives (those detained but who would not go on to recidivate) in practice. In practice, you can’t observe this value (you can only estimate it from historical data). Once you detain an individual, by construction they aren’t given the chance to recidivate. So you don’t get any on-policy feedback about false-positives, only false-negatives (folks who were released and went on to commit a crime pre-trial). This I think puts a pretty big nail in the coffin of using false positive rates as a policy goal for bail reform in practice. Like I said earlier, you can’t just set a model once and expect it to work forever in the future. But, I actually don’t think that should be considered in the cost-benefit calculus anyway. So traditionally people tend to think of setting the threshold for predictive models like this confusion table, where different outcomes in the table have different costs to individuals and to society: In this table those on the bottom row are those detained pre-trial. So in the hypothetical, you may say if we could someone know the false positives, we should calculate extra harm that pre-trial detainment causes to those individuals (lost wages, losing job, health harms, etc.). But for the folks who would have gone on and recidivated, we should just calculate the bare bones cost of detainment. I think this is the wrong way to think about it though. Those harms are basically across the board for everyone – even if the person was likely to recidivate they still bear those subsequent harms of being incarcerated. Whether you think people deserve the harm does not make it go away. The main reason I am harping on bail reform so much (folks who know my work will realize it is outside my specific research area) is that the current bail system is grossly inefficient and unequitable. These are folks that piling on monetary bail costs are the exact wrong way to ensure safety and to promote better outcomes for these folks. It is a hard decision to make on who to detain vs who to let go. But pretending the current state of judges making these decisions on whatever personal whims they have and thinking we are better off than using a cost-benefit approach and algorithmic assessments is just sticking your head in the sand. # Balancing False Positives One area of prediction in criminal justice I think has alot of promise is using predictive algorithms in place of bail decisions. So using a predictive instrument to determine whether someone is detained pre-trial based on risk, or released on recognizance if you are low risk. Risk can be either defined as based on future dangerousness or flight risk. This cuts out the middle man of bail, which doesn’t have much evidence of effectiveness, and has negative externalities of placing economic burdens on folks we really don’t want to pile that onto. It is also the case algorithms can likely do quite a bit better than judges in figuring out future risk. So an area I think they can really do good compared to current status quo in the CJ system. A reasonable critique of such systems though is they can have disparate racial impact. For example, ProPublica had an article on how the Compas risk assessment instrument resulted in more false positives for black than white individuals. Chris Stucchio has a nice breakdown for why this occurs, which is not due to the Compas being intrinsically racist algorithm, but due to the nature of the baseline risks for the two groups. Consider a very simple example to illustrate. Imagine based on our cost-benefit analysis, we determine the probability threshold to flag a individual as high risk is 60%. Now say our once we apply our predictions, for those above the threshold, whites are all predicted to be 90%, and blacks are all 70%. If our model is well calibrated (which is typically the case), the false positive rate for whites will be 10%, and will be 30% for blacks. It is actually a pretty trivial problem though to balance false positive rates between different groups, if that is what you want to do. So I figured I would illustrate here using the same ProPublica data. There are trade-offs though with this, balancing false positives means you lose out on other metrics of fairness. In particular, it means you don’t have equality of treatment – different racial groups will have different thresholds. The full data and code I use to illustrate this can be downloaded here. ## An Example in Python To illustrate how we would balance the false positive rates between groups, I use the same ProPublica risk assessment data. So this isn’t per se for bail decisions, but works fine as an illustration. First in python I load my libraries, and then read in the data – it is a few over 11,000 cases. import pandas as pd import os import numpy as np from sklearn.ensemble import RandomForestClassifier import matplotlib.pyplot as plt my_dir = r'C:\Users\andre\Dropbox\Documents\BLOG\BalanceFalsePos' os.chdir(my_dir) #For notes on data source, check out #https://github.com/apwheele/ResearchDesign/tree/master/Week11_MachineLearning recid = pd.read_csv('PreppedCompas.csv') print( recid.head() ) Next I prepare the dataset for modelling. I am not using all of the variables in the dataset. What I predict here is recidivism post 30 days (there are a bunch of recidivism right away in the dataset, so I am not 100% sure those are prior to screening). I use the three different aggregate compas scores, juvenile felony count, whether they were male, how old they were, and whether the current charge to precipitate screening is a felony or misdemeanor. I include the race variable in the dataset, but I won’t be using it in the predictive model. (That point deserves another blog post, contra to what you might expect, leaving race flags in will often result in better outcomes for that protected class.) #Preparing the variables I want recid_prep = recid[['Recid30','CompScore.1','CompScore.2','CompScore.3', 'juv_fel_count','YearsScreening']] recid_prep['Male'] = 1*(recid['sex'] == "Male") recid_prep['Fel'] = 1*(recid['c_charge_degree'] == "F") recid_prep['Mis'] = 1*(recid['c_charge_degree'] == "M") recid_prep['race'] = recid['race'] print( recid['race'].value_counts() ) #pretty good sample size for both whites/blacks Next I make my testing and training sets of data. In practice I can perfectly balance false positives retrospectively. But having a test set is a better representation of reality, where you need to make some decisions on the historical data and apply it forward. #Now generating train and test set recid_prep['Train'] = np.random.binomial(1,0.75,len(recid_prep)) recid_train = recid_prep[recid_prep['Train'] == 1] recid_test = recid_prep[recid_prep['Train'] == 0] Now the procedure I suggest to balance false-positives doesn’t matter how you generate the predictions, just that we need a predicted probability. Here I use random forests, but you could use whatever machine learning or logistic regression model you want. Second part just generates the predicted probabilities for the training dataset. #Now estimating the model ind_vars = ['CompScore.1','CompScore.2','CompScore.3', 'juv_fel_count','YearsScreening','Male','Fel','Mis'] #no race in model dep_var = 'Recid30' rf_mod = RandomForestClassifier(n_estimators=500, random_state=10) rf_mod.fit(X = recid_train[ind_vars], y = recid_train[dep_var]) #Now getting the predicted probabilities in the training set pred_prob = rf_mod.predict_proba(recid_train[ind_vars] ) recid_train['prob'] = pred_prob[:,1] recid_train['prob_min'] = pred_prob[:,0] Now to balance false positives, I will show a graph. Basically this just sorts the predicted probabilities in descending order for each racial group. Then you can calculate a cumulate false positive rate for different thresholds for each group. #Making a cusum plot within each racial group for the false positives recid_train.sort_values(by=['race','prob'], ascending=False, inplace=True) recid_train['const'] = 1 recid_train['cum_fp'] = recid_train.groupby(['race'])['prob_min'].cumsum() recid_train['cum_n'] = recid_train.groupby(['race'])['const'].cumsum() recid_train['cum_fpm'] = recid_train['cum_fp'] / recid_train['cum_n'] white_rt = recid_train[recid_train['race'] == 'Caucasian'] black_rt = recid_train[recid_train['race'] == 'African-American' ]  And now the fun part (and least in output, not really in writing matplotlib code). #now make the chart for white and black fig, ax = plt.subplots() ax.plot(black_rt['prob'], black_rt['cum_fpm'], drawstyle='steps', color='b', label='Black') ax.plot(white_rt['prob'], white_rt['cum_fpm'], drawstyle='steps', color='r', label='White') ax.set_xlim(1, 0) # decreasing probs plt.xticks(np.arange(1.0,-0.1,-0.1)) ax.set_xlabel('Predicted Probability') ax.set_ylabel('Mean False Positive Rate') ax.grid(True,linestyle='--') ax.legend(facecolor='white', framealpha=1) plt.savefig('FP_Rate.png', dpi=2000, bbox_inches='tight') plt.show() So what this chart shows is that if we set our threshold to a particular predicted probability (X axis), based on the data we would expect a false positive rate (Y axis). Hence if we want to balance false positives, we just figure out the race specific thresholds for each group at a particular Y axis value. Here we can see the white line is actually higher than the black line, so this is reverse ProPublica findings, we would expect whites to have a higher false positive rate than blacks given a consistent predicted probability of high risk threshold. So say we set the threshold at 10% to flag as high risk, we would guess the false positive rate among blacks in this sample should be around 40%, but will be closer to 45% in the white sample. Technically the lines can cross at one or multiple places, and those are places where you get equality of treatment and equality of outcome. It doesn’t make sense to use that though from a safety standpoint – those crossings can happen at a predicted probability of 99% (so too many false negatives) or 0.1% (too many false positives). So say we wanted to equalize false positive rates at 30% for each group. Here this results in a threshold for whites as high risk of 0.256, and for blacks a threshold of 0.22. #Figuring out where the threshold is to limit the mean FP rate to 0.3 #For each racial group white_thresh = white_rt[white_rt['cum_fpm'] > 0.3]['prob'].max() black_thresh = black_rt[black_rt['cum_fpm'] > 0.3]['prob'].max() print( white_thresh, black_thresh ) Now for the real test, lets see if my advice actually worked in a new sample of data to balance the false positive rate. #Now applying out of sample, lets see if this works pred_prob = rf_mod.predict_proba(recid_test[ind_vars] ) recid_test['prob'] = pred_prob[:,1] recid_test['prob_min'] = pred_prob[:,0] white_test = recid_test[recid_test['race'] == 'Caucasian'] black_test = recid_test[recid_test['race'] == 'African-American' ] white_test['Flag'] = 1*(white_test['prob'] > white_thresh) black_test['Flag'] = 1*(black_test['prob'] > black_thresh) white_fp= 1 - white_test[white_test['Flag'] == 1][dep_var].mean() black_fp = 1 - black_test[black_test['Flag'] == 1][dep_var].mean() print( white_fp, black_fp ) And we get a false positive rate of 54% for whites (294/547 false positives), and 42% for blacks (411/986) – yikes (since I wanted a 30% FPR). As typical, when applying your model to out of sample data, your predictions are too optimistic. I need to do some more investigation, but I think a better way to get error bars on such thresholds is to do some k-fold metrics and take the worst case scenario, but I need to investigate that some more. The sample sizes here are decent, but there will ultimately be some noise when deploying this in practice. So basically if you see in practice the false positive rates are within a few percentage points that is about as good as you can get in practice I imagine. (And for smaller sample sizes will be more volatile.) # Knowing when to fold them: A quantitative approach to ending investigations The recent work on investigations in the criminal justice field has my head turning about potential quantitative applications in this area (check out the John Eck & Kim Rossmo podcasts on Jerry’s site first, then check out the recent papers in Criminology and Public Policy on the topic for a start). One particular problem that was presented to me was detective case loads — detectives are humans, so can only handle so many cases at once. Triage is typically taken at the initial crime reporting stage, with inputs such as seriousness of the offense, the overall probability of the case being solved, and future dangerousness of folks involved being examples of what goes into that calculus to assign a case. Here I wanted to focus on a different problem though — how long to keep cases open? There are diminishing returns to keeping cases open indefinitely, and so PDs should be able to right size the backend of detective open cases as well as the front end triaging. Here my suggested solution is to estimate a survival model of the probability of a case being solved, and then you can estimate an expected return on investment given the time you put in. Here is a simplified example. Say the table below shows the (instantaneous) probability of a case being solved per weeks put into the investigation. Week 1 20% Week 2 10% Week 3 5% Week 4 3% Week 5 1% In survival model parlance, this would be the hazard function in discrete time increments. And then we have diminishing probabilities over time, which should also be true (e.g. a higher probability of being solved right away, and gets lower over time). The expected return of investigating this crime at time t is the cumulative probability of the crime being solved at time t, multiplied by whatever value you assign to the case being solved. The costs of investigating will be fixed (based on the detective salary), so is just a multiple of t*invest_costs. So just to fill in some numbers, lets say that it costs the police department$1,000 a week to keep an investigation going. Also say a crime has a return of \$10,000 if it is solved (the latter number will be harder to figure out in practice, as cost of crime estimates are not a perfect fit). So filling in our table, we have below our detective return on investment estimates (note that the cumulative probability of being solved is not simply the sum of the instantaneous probabilities, else it would eventually go over 100%). So return on investment (ROI), at week 1 is 10,000*0.2 = 2,000, at week 2 is 10,000*0.28 = 2,800, etc.

        h(t) solved%  cum-costs   ROI
Week 1  20%    20%     1,000     2,000
Week 2  10%    28%     2,000     2,800
Week 3   5%    32%     3,000     3,200
Week 4   3%    33%     4,000     3,300
Week 5   1%    34%     5,000     3,400

So the cumulative costs outweigh the total detective resources devoted to the crime by Week 4 here. So in practice (in this hypothetical example) you may say to a detective you get 4 weeks to figure it out, if not solved by then it should be closed (but not cleared), and you should move onto other things. In the long run (I think) this strategy will make sure detective resources are balanced against actual cases solved.

This right sizes investigation lengths from a global perspective, but you also might consider whether to close a case on an individual case-by-case basis. In that case you wouldn’t calculate the sunk cost of the investigation so far, it is just the probability of the case being solved going forward relative to future necessary resources. (You do the same table, just start the cum-costs and solved percent columns from scratch whenever you are making that decision.)

In an actual applied setting, you can estimate the survival function however you want (e.g. you may want a cure mixture-model, so not all cases will result in 100% being solved given infinite time). It is also the case that different crimes will not only have different survival curves, but also will have different costs of crime (e.g. a murder has a greater cost to society than a theft) and probably different investigative resources needed (detective costs may also get lower over time, so are not constant). You can bake that all right into this estimate. So you may say the cost of a murder is infinite, and you should forever keep that case open investigating it. A burglary though may be a very short time interval before it should be dropped (but still have some initial investment).

Another neat application of this is that if you can generate reasonable returns to solving crimes, you can right size your overall detective bureau. That is you can make a quantitative argument I need X more detectives, and they will help solve Y more crimes resulting in Z return on investment. It may be we should greatly expand detective bureaus, but have them only keep many cases open a short time period. I’m thinking of the recent officer shortages in Dallas, where very few cases are assigned at all. (Some PDs have patrol officers take initial detective duties on the crime scene as well.)

There are definitely difficulties with applying this approach. One is that getting the cost of solving a crime estimate is going to be tough, and bridges both quantitative cost of crime estimates (although many of them are sunk costs after the crime has been perpetrated, arresting someone does not undo the bullet wound), likelihood of future reoffending, and ethical boundaries as well. If we are thinking about a detective bureau that is over-booked to begin with, we aren’t deciding on assigning individual cases at that point, but will need to consider pre-empting current investigations for new ones (e.g. if you drop case A and pick up case B, we have a better ROI). And that is ignoring the estimating survival part of different cases, which is tricky using observational data as well (selection biases in what cases are currently assigned could certainly make our survival curve estimates too low or too high).

This problem has to have been tackled in different contexts before (either by actuaries or in other business/medical contexts). I don’t know the best terms to google though to figure it out — so let me know in the comments if there is related work I should look into on solving this problem.

# Actively monitoring place based crime interventions

Recently I presented my work (with Jerry Ratcliffe) on the Weighted Displacement Difference test at the New York State GIVE Symposium. My talk fit right in with what the folks at the American Society of Evidence Based Police discussed as well. In particular, Jason Potts and Jeremiah Johnson gave a talk about how officers can conduct their own experiments, and my work provides a simple tool to test if changes over time are significant or just due to chance.

There was one point of contention though between us — ASEBP folks advocate for the failing fast model of evaluation, whereas I advocated for planning more long term experiments. In particular, I suggest this chart to plan your experiments. So say if you have an area with only 10 crimes per month, I would suggest you should do the experiment for at least 4 months, so if what your are doing is 50% effective at reducing crime, you will conclude it has at least weak evidence of effectiveness using my WDD test. If you think 50% is too high of a bar, if you do it for 12 months it only needs to be alittle over 25% effective to tell if it is working.

The ideal behind failing fast and innovating I totally get, but whether or not we can actually see if something is effective in the short run with low baseline crime counts may be a road block to this idea in practice. Note that this is not me bagging on people doing experiments — what is most likely to happen if you do an experiment with low power is you will conclude it is not effective, even if it partially works. So I’m more concerned the BetaGov fail fast model is likely to throw out cost-effective interventions that don’t appear on their face to be effective, as opposed to false positives.1

Am I being too negative though? And also can we create a monitoring tool to give more immediate feedback — so instead of waiting a year and seeing the results, evaluating the efficacy of an intervention over time? To do this I am giving cusum charts a try, and did a little simulation to show how it might look in practice. SPSS Code to replicate the findings here.

So what I did was simulate a baseline control area with 10 crimes per time period, and a treated area that had a 20% reduction (so goes down to 8 crimes on average per time period). Here is the time series of those two areas, black line is the control area, and red line is the treated area. Time periods can be whatever you want (e.g. days, weeks, months), what matters is the overall average and the difference between the two series.

Based on this graph, you can’t really visually tell if the red treated area is doing any better than the black control area — they overlap too much. But we aren’t just interested in the effect for any one time period, but in the cumulative effect over time. To calculate that, you just subtract the black line from the red line, and take the cumulative sum of that difference. The next chart shows that statistic, along with 100 simulated lines showing what happens when you do the same cumulative statistic to data with no changes.

So you can see here that it takes about 13 time periods to show the cumulative effects are outside of the simulation boundaries, but you might conclude there is suggestive evidence of effectiveness after say 8+ time periods. Going further out it still shows the cumulative number of crimes prevented over the life of the intervention, so goes down to around 75 crimes prevented by 25 time periods.

The number of time periods necessary to show divergence is dependent on how effective the intervention is. So if we have the same baseline average of 10 crimes per time period, but the intervention is 50% effective (so reduces to an average of 5 crimes per time period), you can tell it diverges by period 6 in this second simulation example.

If we go back to my power chart I made for the WDD test, you can see that these effective time periods are close to my power chart suggestions for the weak evidence line. So this cusum approach is maybe slightly more diagnostic, and has the benefit you may be able to stop the experiment early if it is really effective.

You should still commit to running the experiment though for a set amount of time. The amount of time should be based on how effective you think the experiment should be, as well as cost-benefit analyses. If something costs alot of money (e.g. overtime) the effectiveness threshold to make it worthwhile in practice is a much higher bar than something that is closer to zero cost (such as shifting around current assignments). Given that ASEBP is advocating for lower level officers to experiment, they are more likely to be the latter type of low cost interventions.

There are still a few issues with using cusum charts like this though. One, this is very dependent on having a control area that is the same level of crime counts. So my WDD test only needs the parallel trends assumption, but this needs both the parallel trends and equal levels. (There are ways to get rid of the equal levels assumption though, one is to take the differences in differences and calculate those cusums over time.)

Another is that you need to reset cusum charts after a particular time period — you can see the simulations are random walks, and so grow much wider over time. I’m not sure at that point though if you should choose a new control area, or just stick with the prior one though. In the first example you can see the red line overestimates the effectiveness — the first chart the true estimate should be -2*time period (estimated -75 versus should be -50 after 25 time periods). For the second the true effect is -5*time period, so the estimate is a slight underestimate (estimated -100 versus should be -125 after 25 time periods).

But this is about the best meet in the middle of actively monitoring place based crime interventions and my advocacy for planning long term interventions that I can drum up for now. It is short term feedback, but you should be committed to running the experiment for a longer period of time. The sequential monitoring allows you to stop early if the intervention is really effective, see this example for A/B tests. But otherwise you are often better off just planning a long term intervention and not peek at the short term results.

Besides the technical stats portion of being able to tell if it diverges from a control area, it may also be behavioral, in that you need a longer period of time to generate deterrence, or for officers to effectively implement the strategy. You notice in these examples if you only did 5 time periods, they meander about 0 and so don’t appear to be effective. It takes longer time periods, even with the 50% effective intervention, to know if the intervention was effective given these low of baseline crime counts.

1. Although with low power you do have issues with what Andrew Gelman calls type S (sign) and type M (magnitude) errors as well. That is even if you do think it is effective, you may think it is way more effective than it actually is in reality based on your noisy estimates. Or you conclude it has iatrogenic effects when it works in practice.

# Optimal treatment assignment with network spillovers

Motivated by a recent piece by Wood and Papachristos (2019), (WP from here on) which finds if you treat an individual at high risk for gun shot victimization, they have positive spillover effects on individuals they are connected to. This creates a tricky problem in identifying the best individuals to intervene with given finite resources. This is because you may not want to just choose the people with the highest risk – the best bang for your buck will be folks who are some function of high risk and connected to others with high risk (as well as those in areas of the network not already treated).

For a simplified example consider the network below, with individuals baseline probabilities of future risk noted in the nodes. Lets say the local treatment effect reduces the probability to 0, and the spillover effect reduces the probability by half, and you can only treat 1 node. Who do you treat?

We could select the person with the highest baseline probability (B), and the reduced effect ends up being 0.5(B) + 0.1(E) = 0.6 (the 0.1 is for the spillover effect for E). We could choose node A, which is a higher baseline probability and has the most connections, and the reduced effect is 0.4(A) + 0.05(C) + 0.05(D) + 0.1(E) = 0.6. But it ends up in this network the optimal node to choose is E, because the spillovers to A and B justify choosing a lower probability individual, 0.2(E) + 0.2(A) + 0.25(B) = 0.65.

Using this idea of a local effect and a spillover effect, I formulated an integer linear program with the same idea of a local treatment effect and a spillover effect:

$\text{Maximize} \{ \sum_{i = 1}^n (L_i\cdot p_{li} + S_i \cdot p_{si}) \}$

Where $p_{li}$ is the reduction in the probability due to the local effect, and $p_{si}$ is the reduction in the probability due to the spillover effect. These probabilities are fixed values you know at the onset, e.g. estimated from some model like in Wheeler, Worden, and Silver (2019) (and Papachristos has related work using the network itself to estimate risk). Each node, i, then gets two decision variables; $L_i$ will equal 1 if that node is treated, and $S_i$ will equal 1 if the node gets a spillover effect (depending on who is treated). Actually the findings in WP show that these effects are not additive (you don’t get extra effects if you are treated and your neighbors are treated, or if you have multiple neighbors treated), and this makes it easier to keep the problem on the probability scale. So we then have our constraints:

1. $L_i , S_i \in \{ 0,1 \}$
2. $\sum L_i = K$
3. $S_i \leq 1 + -1\cdot L_i , \forall \text{ Node}_i$
4. $\sum_{\text{neigh}(i)} L_j \geq S_i , \forall \text{ Node}_i$

Constraint 1 is that these are binary 0/1 decision variables. Constraint 2 is we limit the number of people treated to K (a value that we choose). Constraint 3 ensures that if a local decision variable is set to 1, then the spillover variable has to be set to 0. If the local is 0, it can be either 0 or 1. Constraint 4 looks at the neighbor relations. For Node i, if any of its neighbors local treated decision variable is set to 1, the Spillover decision variable can be set to 1.

So in the end, if the number of nodes is n, we have 2*n decision variables and 2*n + 1 constraints, I find it easier just to look at code sometimes, so here is this simple network and problem formulated in python using networkx and pulp. (Here is a full file of the code and data used in this post.) (Update: I swear I’ve edited this inline code snippet multiple times, if it does not appear I have coded constraints 3 & 4, check out the above linked code file. Maybe it is causing problems being rendered.)

####################################################
import pulp
import networkx

Nodes = ['a','b','c','d','e']
Edges = [('a','c'),
('a','d'),
('a','e'),
('b','e')]

p_l = {'a': 0.4, 'b': 0.5, 'c': 0.1, 'd': 0.1,'e': 0.2}
p_s = {'a': 0.2, 'b': 0.25, 'c': 0.05, 'd': 0.05,'e': 0.1}
K = 1

G = networkx.Graph()

P = pulp.LpProblem("Choosing Network Intervention", pulp.LpMaximize)
L = pulp.LpVariable.dicts("Treated Units", [i for i in Nodes], lowBound=0, upBound=1, cat=pulp.LpInteger)
S = pulp.LpVariable.dicts("Spillover Units", [i for i in Nodes], lowBound=0, upBound=1, cat=pulp.LpInteger)

P += pulp.lpSum( p_l[i]*L[i] + p_s[i]*S[i] for i in Nodes)
P += pulp.lpSum( L[i] for i in Nodes ) == K

for i in Nodes:
P += pulp.lpSum( S[i] ) <= 1 + -1*L[i]
ne = G.neighbors(i)
P += pulp.lpSum( L[j] for j in ne ) >= S[i]

P.solve()

#Should select e for local, and a & b for spillover
print(pulp.value(P.objective))
print(pulp.LpStatus[P.status])

for n in Nodes:
print([n,L[n].varValue,S[n].varValue])
####################################################

And this returns the correct results, that node E is chosen in this example, and A and B have the spillover effects. In the linked code I provided a nicer function to just pipe in your network, your two probability reduction estimates, and the number of treated units, and it will pipe out the results for you.

For an example with a larger network for just proof of concept, I conducted the same analysis, choosing 20 people to treat in a network of 311 nodes I pulled from Rostami and Mondani (2015). I simulated some baseline probabilities to pipe in, and made it so the local treatment effect was a 50% reduction in the probability, and a spillover effect was a 20% reduction. Here red squares are treated, pink circles are the spill-over, and non-treated are grey circles. It did not always choose the locally highest probability (largest nodes), but did tend to choose highly connected folks also with a high probability (but also chose some isolate nodes with a high probability as well).

This problem is solved in an instant. And I think out of the box this will work for even large networks of say over 100,000 nodes (I have let CPLEX churn on problems with near half a million decision variables on my desktop overnight). I need to check myself to make 100% sure though. A simple way to make the problem smaller if needed though is to conduct the analysis on subsets of connected components, and then shuffle the results back together.

Looking at the results, it is very similar to my choosing representatives work (Wheeler et al., 2019), and I think you could get similar results with just piping in 1’s for each of the local and spillover probabilities. One of the things I want to work on going forward though is treatment non-compliance. So if we are talking about giving some of these folks social services, they don’t always take up your offer (this is a problem in choose rep’s for call ins as well). WP actually relied on this to draw control nodes in their analysis. I thought for a bit the problem with treatment non-compliance in this setting was intractable, but another paper on a totally different topic (Bogle et al., 2019) has given me some recent hope that it can be solved.

This same idea is also is related to hot spots policing (think spatial diffusion of benefits). And I have some ideas about that to work on in the future as well (e.g. how wide of net to cast when doing hot spots interventions given geographical constraints).

# References

• Bogle, J., Bhatia, N., Ghobadi, M., Menache, I., Bjørner, N., Valadarsky, A., & Schapira, M. (2019). TEAVAR: striking the right utilization-availability balance in WAN traffic engineering. In Proceedings of the ACM Special Interest Group on Data Communication (pp. 29-43).
• Rostami, A., & Mondani, H. (2015). The complexity of crime network data: A case study of its consequences for crime control and the study of networks. PloS ONE, 10(3), e0119309.
• Wheeler, A. P., McLean, S. J., Becker, K. J., & Worden, R. E. (2019). Choosing Representatives to Deliver the Message in a Group Violence Intervention. Justice Evaluation Journal, Online First.
• Wheeler, A. P., Worden, R. E., & Silver, J. R. (2019). The Accuracy of the Violent Offender Identification Directive Tool to Predict Future Gun Violence. Criminal Justice and Behavior, 46(5), 770-788.
• Wood, G., & Papachristos, A. V. (2019). Reducing gunshot victimization in high-risk social networks through direct and spillover effects. Nature Human Behaviour, 1-7.

# Finding the dominant set in a network (python)

My paper, Choosing representatives to deliver the message in a group violence intervention, is now published online at the Justice Evaluation Journal. For those who don’t have access to that journal, here is a link good for 50 e-prints (for a limited time), and here is a pre-print version, and you can always send me an email for the published copy.

I’ve posted Python code to replicate the analysis, including the original network nodes and edges group data. I figured I would go through a quick example of applying the code for others to use the algorithm.

The main idea is that for a focused deterrence initiative, for the call-ins you want to identify folks to spread the deterrence message around the network. When working with several PDs I figured looking at who was called in would be interesting. Literally the first network graph I drew was below on the left — folks who were called in are the big red squares. This was one of the main problem gangs, and the PD had done several call-ins for over a year at this point. Those are not quite the worst set of four folks to call-in based on the topology of the network, but damn close.

But to criticize the PD I need to come up with a better solution — which is the graph on the right hand side. The larger red squares are my suggested call-ins, and they reach everyone within one step. That means everyone is at most just one link away from someone who attended the call-in. This is called a dominant set of a graph when all of the graph is colored in.

Below I give a quicker example using my code for others to generate the dominant set (instead of going through all of the replication analysis). If you are a PD interested in applying this for your focused deterrence initiative let me know!

So first to set up your python code, I import all of the needed libraries (only non-standard is networkx). Then I import my set of functions, named MyFunctions.py, and then change the working directory.

############################################################
#The libraries I need

import itertools
import networkx as nx
import csv
import sys
import os

#Now importing my own functions I made
locDir = r'C:\Users\axw161530\Dropbox\Documents\BLOG\DominantSet_Python'
sys.path.append(locDir)
from MyFunctions import *

#setting the working directory to this location
os.chdir(locDir)
#print(os.getcwd())
############################################################

The next part I read in the CSV data for City 4 Gang 1, both the nodes and the edges. Then I create a networkx graph simply based on the edges. Technically I do not use the node information at all for this, just the edges that list a source and a target.

############################################################
#Reading in the csv files that have the nodes and the edges
#And turning into a networkX graph

#simple function to read in csv files
tup = []
with open(loc) as f:
for row in z:
tup.append(tuple(row))
return tup

#Turning my csv files into networkx objects

#Turning my csv files into networkx objects
C1G4 = nx.Graph()
############################################################

Now it is quite simple, to get my suggested dominant set it is simple as this function call:

ds_C1G4 = domSet_Whe(C1G4)
print(ds_C1G4)

In my current session this gives the edges ['21', '18', '17', '16', '3', '22', '20', '6']. Which if you look to my original graph is somewhat different, but all are essentially single swaps where the best node to choose is arbitrary.

I have a bunch of other functions in the analysis, one of interest will be given who is under probation/parole who are the best people to call in (see the domSet_WheSub function). Again if you are interested in pursuing this further always feel free to reach out to me.

# David Bayley

David Bayley is most known in my research area, policing interventions to reduce crime, based on this opening paragraph in Police for the future:

The police do not prevent crime. This is one of the best kept secrets of modern life. Experts know it, the police know it, but the public does not know it. Yet the police pretend that they are society’s best defense against crime and continually argue that if they are given more resources, especially personnel, they will be able to protect communities against crime. This is a myth.

This quote is now paraded as backwards thinking, often presented before discussing the overall success of hot spots policing. If you didn’t read the book, you might come to the conclusion that this quote is a parallel to the nothing works mantra in corrections research. That take is not totally off-base: Police for the future was published in 1994, so it was just at the start of the CompStat revolution and hot spots policing. The evidence base was no doubt much thinner at that point and deserving of skepticism.

I don’t take the contents of David’s book as so hardlined on the stance that police cannot reduce crime, at least at the margins, as his opening quote suggests though. He has a chapter devoted to traditional police responses (crackdowns, asset forfeiture, stings, tracking chronic offenders), where he mostly expresses scientific skepticism of their effectiveness given their cost. He also discusses problem oriented approaches to solving crime problems, how to effectively measure police performance (outputs vs outcomes), and promotes evaluation research to see what works. Still all totally relevant twenty plus years later.

The greater context of David’s quote comes from his work examining police forces internationally. David was more concerned about professionalization of police forces. Part of this is better record keeping of crimes, and in the short term crime rates will often increase because of this. In class he mocked metrics used to score international police departments on professionalization that used crime as a measure that went into their final grade. He thought the function of the police was broader than reducing crime to zero.

I was in David’s last class he taught at Albany. The last day he sat on the desk at the front of the room and expressed doubt about whether he accomplished anything tangible in his career. This is the fate of most academics. Very few of us can point to direct changes anyone implemented in response to our work. Whether something works is independent of an evaluation I conduct to show it works. Even if a police department takes my advice about implementing some strategy, I am still only at best indirectly responsible for any crime reductions that follow. Nothing I could write would ever compete with pulling a single person from a burning car.

While David was being humble he was right. If I had to make a guess, I would say David’s greatest impact likely came about through his training of international police forces — which I believe spanned multiple continents and included doing work with the United Nations. (As opposed to saying something he wrote had some greater, tangible impact.) But even there if we went and tried to find direct evidence of David’s impact it would be really hard to put a finger on any specific outcome.

If a police department wanted to hire me, but I would be fired if I did not reduce crimes by a certain number within that first year, I would not take that job. I am confident that I can crunch numbers with the best of them, but given real constraints of police departments I would not take that bet. Despite devoting most of my career to studying policing interventions to reduce crime, even with the benefit of an additional twenty years of research, I’m not sure if David’s quote is as laughable as many of my peers frame it to be.

# Plotting Predictive Crime Curves

Writing some notes on this has been in the bucket list for a bit, how to evaluate crime prediction models. A recent paper on knife homicides in London is a good use case scenario for motivation. In short, when you have continuous model predictions, there are a few different graphs I would typically like to see, in place of accuracy tables.

The linked paper does not provide data, so what I do for a similar illustration is grab the lower super output area crime stats from here, and use the 08-17 data to predict homicides in 18-Feb19. I’ve posted the SPSS code I used to do the data munging and graphs here — all the stats could be done in Excel though as well (just involves sorting, cumulative sums, and division). Note this is not quite a replication of the paper, as it includes all cases in the homicide/murder minor crime category, and not just knife crime. There ends up being a total of 147 homicides/murders from 2018 through Feb-2019, so the nature of the task is very similar though, predicting a pretty rare outcome among almost 5,000 lower super output areas (4,831 to be exact).

So the first plot I like to make goes like this. Use whatever metric you want based on historical data to rank your areas. So here I used assaults from 08-17. Sort the dataset in descending order based on your prediction. And then calculate the cumulative number of homicides. Then calculate two more columns; the total proportion of homicides your ranking captures given the total proportion of areas.

Easier to show than to say. So for reference your data might look something like below (pretend we have 100 homicides and 1000 areas for a simpler looking table):

 PriorAssault  CurrHom CumHom PropHom PropArea
1000          1         1      1/100    1/1000
987          0         1      1/100    2/1000
962          2         4      4/100    3/1000
920          1         5      5/100    4/1000
.          .         .       .        .
.          .         .       .        .
.          .         .       .        .
0          0       100    100/100 1000/1000

You would sort the PriorCrime column, and then calculate CumHom (Cumulative Homicides), PropHom (Proportion of All Homicides) and PropArea (Proportion of All Areas). Then you just plot the PropArea on the X axis, and the PropHom on the Y axis. Here is that plot using the London data.

Paul Ekblom suggests plotting the ROC curve, and I am too lazy now to show it, but it is very similar to the above graph. Basically you can do a weighted ROC curve (so predicting areas with more than 1 homicide get more weight in the graph). (See Mohler and Porter, 2018 for an academic reference to this point.)

Here is the weighted ROC curve that SPSS spits out, I’ve also superimposed the predictions generated via prior homicides. You can see that prior homicides as the predictor is very near the line of equality, suggesting prior homicides are no better than a coin-flip, whereas using all prior assaults does alittle better job, although not great. SPSS gives the area-under-the-curve stat at 0.66 with a standard error of 0.02.

Note that the prediction can be anything, it does not have to be prior crimes. It could be predictions from a regression model (like RTM), see this paper of mine for an example.

So while these do an OK job of showing the overall predictive ability of whatever metric — here they show using assaults are better than random, it isn’t real great evidence that hot spots are the go to strategy. Hot spots policing relies on very targeted enforcement of a small number of areas. The ROC curve shows the entire area. If you need to patrol 1,000 LSOA’s to effectively capture enough crimes to make it worth your while I wouldn’t call that hot spots policing anymore, it is too large.

So another graph you can do is to just plot the cumulative number of crimes you capture versus the total number of areas. Note this is based on the same information as before (using rankings based on assaults), just we are plotting whole numbers instead of proportions. But it drives home the point abit better that you need to go to quite a large number of areas to be able to capture a substantive number of homicides. Here I zoom in the plot to only show the first 800 areas.

So even though the overall curve shows better than random predictive ability, it is unclear to me if a rare homicide event is effectively concentrated enough to justify hot spots policing. Better than random predictions are not necessarily good enough.

A final metric worth making note of is the Predictive Accuracy Index (PAI). The PAI is often used in evaluating forecast accuracy, see some of the work of Spencer Chainey or Grant Drawve for some examples. The PAI is simply % Crime Captured/% Area, which we have already calculated in our prior graphs. So you want a value much higher than 1.

While those cited examples again use tables with simple cut-offs, you can make a graph like this to show the PAI metric under different numbers of areas, same as the above plots.

The saw-tooth ends up looking very much like a precision-recall curve, but I haven’t sat down and figured out the equivalence between the two as of yet. It is pretty noisy, but we might have two regimes based on this — target around 30 areas for a PAI of 3-5, or target 150 areas for a PAI of 3. PAI values that low are not something to brag to your grandma about though.

There are other stats like the predictive efficiency index (PAI vs the best possible PAI) and the recapture-rate index that you could do the same types of plots with. But I don’t want to put everyone to sleep.