MONTHLY CRIME DECOMPOSITION
National trends and city departures
Every city in the RTCI national sample. Rates are annualized from monthly counts and shown per 100,000 residents.
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Global model decomposition
Observed is the population-weighted monthly sample rate, and sample fitted aggregates the city-specific structural curves using population weights. Global trend and global plus season describe a reference city whose random effects equal zero; they need not match the sample level because cities have different intercepts and the logit transformation is nonlinear.
Seasonal component
Seasonal effect is the change in annualized rate associated with the three-harmonic Fourier curve, relative to the long-run trend.
Shared time period effect
This is the fitted (1 | time_period) effect shared by all cities in a month, transformed to an annualized rate change. It is distinct from the city-by-month effect. The horizontal zero line is the reference.
City detail
Observed is this city's reported rate. Global baseline contains the crime-specific intercept, trend, and season. City fitted additionally contains the city random intercept and the city's trend and seasonal deviations.
City and US-wide trends
The city and US-wide spline trends are centered separately on the logit scale. Centering removes the city intercept so differences between the curves represent differences in long-run trajectory rather than average crime level. The band is an approximate 95% pointwise conditional interval for the city deviation.
City and US-wide seasonal patterns
The curves compare the city's fitted Fourier seasonal component with the component shared across the RTCI sample. Intercepts, long-run trends, shared time-period effects, and city-month effects are excluded. The band is an approximate 95% pointwise conditional interval for the city deviation.
City and US-wide residuals
The US-wide residual is the shared (1 | time_period) effect. The city residual adds this city's (1 | cell_id) observation-level effect, matching the residual comparison in the paper. The band is an approximate 95% pointwise conditional interval based on the city-month residual's SE(e).
Monthly observed and expected outcomes
Expected values include the city's fitted level, trend, season, and the shared time-period effect, but exclude the city-month residual e. Counts are monthly; rates are annualized per 100,000. Highlighted rows have |e| greater than two conditional standard errors.
City-specific trends across the sample
Each grey line is a centered city-specific spline trend; the blue line is the centered global trend. The shaded band contains the middle 80% of city curves at each time point. Hover over a city line to identify it. Centering removes city intercepts so this view compares trend shapes.
City-specific seasonal patterns across the sample
Each grey line is a city's centered Fourier seasonal curve; the blue line is the global seasonal curve. The shaded band contains the middle 80% of city curves. Hover over a city line to identify it. Trend, intercept, and residual terms are excluded.
City-specific monthly residuals across the sample
Each grey line is one city's fitted observation-level residual eit; the blue line marks zero. The shaded band contains the middle 80% of city residuals in each month. Hover over a line to identify the city. The smooth trends, seasonal terms, city intercepts, and shared time-period effect are excluded.