Poisson Regression: The Robust Extension of Linear Regression

Understanding the modeling limitations of linear regression.

Linear regression comes with its own set of challenges/assumptions.

For instance:

• After modeling, the output can be negative for some inputs.

• But this may not make sense at times — predicting the number of goals scored, number of calls received, etc.

• Thus, it is clear that it cannot model count (or discrete) data.

Furthermore, in linear regression:

• Residuals are expected to be normally distributed around the mean.

• Hence, the outcomes on either side of the mean (m-x, m+x) are equally likely.

Probability density on either side is the same

For instance:

• if the expected number (mean) of calls received is 1...

• ...then, according to linear regression, receiving 3 calls (1+2) is just as likely as receiving -1 (1-2) calls. (This relates to the concept of prediction intervals which I discussed in one of my previous posts here: Prediction intervals.)

• But in this case, a negative prediction does not make any sense.

Thus, if the above assumptions don't hold, linear regression won't help.

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Instead, what you need is Poisson regression (Sklearn docs).

Poisson regression:

• is more suitable if your response (or outcome) is count-based.

• assumes that the response comes from a Poisson distribution.

\(P(x) = \frac{{e^{-\lambda} \lambda ^x }}{{x!}}\)

It is a type of generalized linear model (GLM) that is used to model count data.

It works by estimating a Poisson distribution parameter (λ), which is directly linked to the expected number of events in a given interval.

Contrary to linear regression, in Poisson regression:

• Residuals may follow an asymmetric distribution around the mean (λ).

• Hence, outcomes on either side of the mean (λ-x, λ+x) are NOT equally likely.

Probability on either side is not the same

For instance:

• if the expected number (mean) of calls received is 1...

• ...then, according to Poisson regression, it is possible to receive 3 (1+2) calls, but it is impossible to receive -1 (1-2) calls.

• This is because its outcome is also non-negative.

The regression fit is mathematically defined as follows:

\(ln(E(Y|X)) = w_{1}X_1 + w_{2}X_2 + \cdots + w_{n}X_n\)

The effectiveness of Poisson regression is evident from the image below:

Lineare Regression vs Poisson Regression

👉 Can you tell some limitations of Poisson regression?

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I like to explore, experiment and write about data science concepts and tools. You can read my articles on Medium. Also, you can connect with me on LinkedIn and Twitter.