Traffic Camera Violations in Chicago

Brian Detweiler
April 24, 2017

The Data

Interactive map: http://bdetweiler.github.io/projects/chicago-traffic-cameras.html

ADDRESS CAMERA.ID VIOLATION.DATE VIOLATIONS
7738 S WESTERN CHI065 2014-07-08 65
1111 N HUMBOLDT CHI010 2014-07-16 56
5520 S WESTERN CHI069 2014-07-08 10
1111 N HUMBOLDT CHI010 2014-07-26 101
1111 N HUMBOLDT CHI010 2014-07-27 92

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City-Level Data

  • Aggregate violations by date
  • High level summary view of all violations in Chicago as Time Series

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Stationarity of First Difference

  • \( \nabla Y_t \) results in constant mean over time

  • Variance still quite large, and heteroskedastic in some places

  • Augmented Dickey-Fuller test p-value \( < 0.01 \)

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ACF and PACF of First Difference

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  • ACF is problematic, and appears to have some seasonal influences at \( 7t \)

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  • PACF is also problematic
  • Difficult to pick a model based on these

arma.subsets()

Suggests an ARIMA(7, 1, 10) given by the equation \[ \begin{split} W_t &= \nabla Y_t = Y_t - Y_{t - 1} \\ W_t &= \phi_1 W_{t - 1} + \phi_7 W_{t - 7} + e_t - \theta_{1} e_{1} - \theta_6 e_{t - 6} - \theta_7 e_{t - 7} - \theta_8 e_{t - 8} - \theta_{10} e_{t - 10} \\ \end{split} \]

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Model Diagnostics

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  • AIC: 15246.82

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  • Shapiro-Wilk test p-value << 0.001
  • Ljung-Box test p-value > 0.95
    • Indepenent

Forecast

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auto.arima()

Suggests an ARIMA(1, 1, 3) given by the equation \[ \begin{split} W_t &= \nabla Y_t = Y_t - Y_{t - 1} \\ W_t &= \phi_1 W_{t - 1} + e_t - \theta_{1} e_{1} - \theta_2 e_{t - 2} - \theta_3 e_{t - 3} \\ \end{split} \]

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Model Diagnostics

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  • AIC: 15331.86

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  • Shapiro-Wilk test p-value << 0.001
  • Ljung-Box test p-value > 0.95
    • Indepenent

Forecast

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Conclusion

  • Data difficult to fit due to high day-to-day variance
  • Addative Outliers present possibly due to weather (snow)
  • Both models have i.i.d. normal residuals - Model 2 contains autocorrelation
  • Model 1 has lower AIC
  • Model 2 has more realistic prediction limits