Month: December 2023

Time Series

My analysis of the “911 Daily Dispatch Count by Agency” dataset is a comprehensive and methodical exploration into the trends and patterns of emergency dispatches in the Boston area. The study delved deep into the data, which covered dispatch counts for major emergency services including the Boston Police Department (BPD), Boston Fire Department (BFD), and Emergency Medical Services (EMS).

One of the key findings from my analysis is the identification of distinct yearly and monthly trends in dispatch counts for each agency. The BPD, in particular, showed a significant increase in dispatches from 2010 to 2013, followed by a decline in 2014, suggesting shifts in community dynamics or operational strategies. In contrast, BFD and EMS displayed more consistent dispatch patterns over the years. This variation in trends between agencies underscores the diverse nature of emergencies they respond to and highlights the importance of tailored resource allocation and preparedness strategies.

My study also made insightful observations about potential seasonal variations, especially with higher dispatch counts for BPD in specific months like March and December. The application of time series analysis techniques, including the Augmented Dickey-Fuller (ADF) test and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test, revealed non-stationarity in the data, indicating the presence of underlying trends or cyclic behaviors. This aspect of the analysis is crucial for emergency services planning and resource management, as it aids in anticipating periods of high demand.

Furthermore, the implementation and evaluation of an ARIMA (1,1,1) model provided a methodological approach to forecast future dispatch needs. While the model offered valuable insights, my analysis also pointed out the need for refinement, especially in addressing outliers and improving the distribution of residuals. The diagnostic plots from the ARIMA model were instrumental in identifying areas for improvement, emphasizing the model’s limitations and the necessity for additional data to enhance its predictive accuracy.

The comprehensive use of visual tools like correlograms and histograms not only facilitated a deeper understanding of the complex data patterns but also aided in the interpretation of the model’s diagnostics. These visualizations played a significant role in conveying intricate analytical findings in an intuitive manner, making them accessible for decision-making processes.

Overall, my analysis stands out as a robust and nuanced approach to understanding emergency dispatch trends, offering valuable insights that could significantly impact resource distribution, emergency response strategies, and policy-making in the realm of public safety and emergency management.

The SARIMAX model summary provides a comprehensive overview of the time series analysis performed on the variable ‘Total’, with 1268 observations from November 1, 2010, to April 21, 2014. The model used is an ARIMA(1, 1, 1), a type of autoregressive integrated moving average model, which is often used for forecasting time series data.

The results show the following key statistics and diagnostics:

Log Likelihood: The model has a log likelihood of -8626.941, which measures the likelihood of the data given the model.
Akaike Information Criterion (AIC): With a value of 17259.882, the AIC suggests the relative quality of the model, with lower values indicating a better model.
Bayesian Information Criterion (BIC): The BIC value is 17275.315, another criterion for model selection, similar to AIC but with a higher penalty for models with more parameters.
Hannan-Quinn Information Criterion (HQIC): This is another measure for model selection, with a value of 17265.680 in this case.

The coefficients for the autoregressive term (ar.L1) and the moving average term (ma.L1) are 0.3624 and -0.9463, respectively, with the latter showing a strong negative effect. Both coefficients are highly significant, as indicated by their P-values of 0.000.

The variance of the model’s residuals (sigma2) is approximately 48,010, indicating the average squared deviation from the predicted value.

Other diagnostics include:

Ljung-Box Test: With a Q statistic of 4.61 and a P-value of 0.03, this test checks for lack of fit in the model, suggesting that the model fits the data well.
Jarque-Bera Test: The JB statistic is 651.24 with a P-value of 0.00, indicating that the residuals may not follow a normal distribution, given the significant skewness and kurtosis.
Heteroskedasticity: The H statistic is 0.87 with a P-value of 0.16, suggesting that there is no significant heteroskedasticity in the model residuals.

The warning about the covariance matrix being calculated using the outer product of gradients suggests that the standard errors and confidence intervals might not be fully reliable.

Additionally, a future warning indicates that an argument used in the `pd.date_range` function is deprecated, suggesting that the code may need updating to maintain compatibility with future versions of pandas.

These notes provide a concise understanding of the SARIMAX model’s results and its implications for the accuracy and reliability of the time series forecasting.

i worked on the following tests and plots:

1.Standardized Residual Plot:

1. • This plot displays the standardized residuals of the model over time.
   • Residuals are the differences between observed and predicted values.
   • Ideally, residuals should fluctuate randomly around zero, without any discernible pattern.
   • In our plot, we observed a fairly random scatter of residuals, although there are some instances of potential outliers.
2. Histogram and Estimated Density Plot:
   • The histogram bins the standardized residuals to show their distribution.
   • An overlaid kernel density estimate (KDE) shows a smoothed version of this distribution.
   • A standard normal distribution (N(0,1)) is plotted for comparison.
   • A good-fitting model would have residuals that closely follow a normal distribution. The histogram should resemble the bell shape of the normal distribution curve.
3. Normal Q-Q Plot:
   • The quantile-quantile plot compares the quantiles of the residuals with the quantiles of a normal distribution.
   • If the residuals are normally distributed, the points should fall approximately along the red line.
   • In the plot, the points largely follow the line, suggesting normality, but deviations at the ends may indicate heavier tails than the normal distribution.
4. Correlogram (ACF Plot):
   • The correlogram, or autocorrelation function plot, shows the correlation of the residuals with themselves at different lags.
   • We look for correlations that are significantly different from zero at various lag intervals.
   • In a well-fitted model, we expect that there will be no significant autocorrelation in the residuals. Here, most autocorrelations are within the confidence band, indicating no significant autocorrelation.

I attempted to fit an ARIMA model to the ‘Total’ column of the dataset using the `ARIMA` function from the `statsmodels` library.
– The order of the ARIMA model was set to (1, 1, 1), indicating a first-order autoregressive model with a first-order differencing and a first-order moving average component.

Encountered a Warning:
– Upon executing the model fitting code, a `ValueWarning` was generated by the `statsmodels` library.
– This warning indicated that the library had to infer the frequency of the time series data. It assumed a daily frequency, denoted as ‘D’.

Understanding the Warning:
– The warning was not an error but a notification. It suggested that the library wasn’t explicitly provided with the frequency of the data and had to make an assumption.
– If the dataset indeed represents daily observations and there are no gaps in the dates, this assumption by the library is appropriate.

Implications of the Warning:
– If the data genuinely has daily measurements and is consistent without missing days, the inference made by the library aligns with the dataset’s structure.
– However, if the dataset follows a different frequency or has irregular intervals, I would need to set the frequency explicitly to match the actual data pattern.