Previously we had seen how we were trying to fit Poisson Distribution to the traffic data on our blog. Link to the Post Today, we study why our analysis could be wrong.By looking at this plot, we suspect three things: 1. The data has outliers. A bunch of points around May 2021. 2. The data has some local trend. Upward trend till Jan 2022 and then downward trend till Mar 2023. 3. Because of the above two points, we feel: there is constant rate (average) for the events occurring within a specific time interval.
Can we use poisson distribution when we suspect there are outliers in the data?
The Poisson distribution has its limitations when dealing with outliers. Here's why:
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Underlying Assumption: The Poisson distribution assumes a constant rate of events across the observed interval. Outliers, by definition, deviate significantly from this expected pattern.
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Impact on Results: If outliers are present, the Poisson model might not accurately reflect the actual probability of observing certain counts. This can lead to misleading interpretations of the data.
However, there are still ways to approach this situation:
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Outlier Detection: You can use statistical methods to identify potential outliers in your data. Techniques like Grubbs' test or examining standardized residuals can help [3].
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Alternative Models: Depending on the nature of the outliers, alternative models like the negative binomial distribution might be a better fit. This model accounts for overdispersion, which is a higher variance than expected by the Poisson distribution, often caused by outliers [1].
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Data Transformation: In some cases, transforming the data (like using the Anscombe transform) before applying the Poisson model can help reduce the influence of outliers [3].
Ultimately, the decision of using the Poisson distribution with outliers depends on the severity of the outliers and the purpose of your analysis.
Here are some additional resources you might find helpful:
- Outliers in long-tailed discrete data: https://web-archive.lshtm.ac.uk/csm.lshtm.ac.uk/wp-content/uploads/sites/6/2016/04/Mario-Cortina-Borja-16-11-2012.pdf
And can we use poisson distribution when we suspect there is a trend in the data?
No, the standard Poisson distribution is not ideal for data with a suspected trend. Here's why:
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Constant Rate Assumption: The Poisson distribution assumes a constant rate (average) for the events occurring within a specific time interval. A trend implies a non-constant rate, where the frequency of events increases or decreases over time.
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Misleading Results: Using the Poisson model with a trend can lead to inaccurate estimates of event probabilities and unreliable hypothesis tests.
However, there are ways to analyze count data with trends:
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Poisson Regression: This statistical technique incorporates a trend term into the Poisson model. You can use a linear trend or even more complex functions to capture the non-constant rate [2].
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Segmented Poisson Regression: This approach divides the data into segments with potentially different rates. It's useful when the trend has breaks or shifts at specific points [2].
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Change-Point Detection: Methods exist to statistically identify the time points where the event rate changes, allowing for further analysis of segments with constant rates [1].
In conclusion, the standard Poisson distribution is not suitable for data with trends. Consider using Poisson regression or other techniques that account for the non-constant event rate.
Here are some additional resources you might find helpful:
- Trend analysis with count data: https://pubmed.ncbi.nlm.nih.gov/29775435/
- Poisson regression for trends: https://pubmed.ncbi.nlm.nih.gov/7977282/
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