CVEN9407 Chap.8 Time Series Models for Monthly Crash Counts
Time Series Models for Monthly Crash Counts
The time series step turns your area's crashes into a monthly or quarterly series and asks what its past says about its future. This chapter shows how to read a series for trend, seasonality and memory, why stationarity matters, and how differencing achieves it. It defines autoregressive and moving average terms and the ARIMA order, and works an AR(1) forecast to show how predictions decay towards the long-run mean.
It then sets out what the time series part of the second report should contain.
The second deliverable pairs this time series work with the count models of the next chapter, so the two analyses use the same aggregated crashes from different angles. The time series view asks how crashes evolve and what comes next; the count view asks which conditions raise crash frequency.
A report that connects the two, for example by checking whether a seasonal peak matches wet months, reads as one piece of analysis.
What this chapter covers
- 01
Trend, seasonality and memory
- 02
The autocorrelation function
- 03
Stationarity and differencing
- 04
Autoregressive and moving average terms
- 05
ARIMA order and model checking
Worked example · free
Forecasting from an AR(1) after a bad month
- 1Check stationarity: the coefficient 0.6 has absolute value below 1, so a long-run mean exists.
- 1One-month forecast: 8 + 0.6 x 25 = 23 crashes.
- 1Long-run mean: 8 / (1 - 0.6) = 20 crashes a month, the level forecasts approach as the horizon grows.
Key terms
- Autocorrelation function
- The autocorrelation function gives the correlation between a series and itself shifted by each number of periods.
- Differencing
- Differencing replaces each value of a series by its change from the previous period, usually removing a trend.
- ARIMA model
- An ARIMA model combines autoregressive and moving average terms on a series differenced enough times to be stationary.
- Seasonality
- Seasonality is a pattern that repeats at a fixed period, such as a twelve-month cycle in monthly crash counts.
- White noise
- White noise is a series of uncorrelated random shocks with zero mean and constant variance, the target for model residuals.
Time Series Models for Monthly Crash Counts FAQ
Why must a series be stationary before fitting ARIMA terms?
Autoregressive and moving average terms assume stable mean, variance and correlations. Fitted to a trending series, they can find spurious relationships created only by the shared trend. Differencing first, then modelling the stationary result, avoids that trap.
How do I choose the ARIMA order?
Look at the autocorrelation and partial autocorrelation plots of the stationary series for a starting guess, compare candidate orders with an information criterion, and accept a model only when its residuals look like white noise.
Should I use monthly or quarterly counts?
The brief allows either. Monthly counts give more observations and show seasonality clearly, but small areas may have many low or zero months; quarterly totals are steadier but leave fewer points to fit and test.
What is the difference between an AR and an MA term?
An autoregressive term makes the current value depend on past values of the series itself. A moving average term makes it depend on past random shocks. AR effects fade gradually, while an MA(1) shock affects only the next period, which is why their autocorrelation plots look different.
How far ahead can an ARIMA model forecast crashes?
Forecasts from a stationary model drift back to the long-run mean within a few periods, and their intervals widen with the horizon. A few years of monthly data support short-term forecasts; longer horizons are better treated as scenarios than predictions.
Assessment move
Plot your area's monthly counts before any modelling and write three sentences on trend, season and memory. Difference the series and plot it again. Fit two or three candidate models, keep the one with clean residuals and a sensible forecast, and explain any break you see in terms of something that happened on the roads or in the records.
Keep one plot of the raw series and one of the forecast with its interval for the presentation.
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