Are You Misinterpreting Correlation for Predictiveness?

Correlation measures how two features vary with one another linearly (or monotonically).

This makes correlation symmetric: corr(A, B) = corr(B, A).

Yet, associations are often asymmetric.

For instance, given a date, it is easy to tell the month. But given a month, you can never tell the date.

Correlation, being symmetric, entirely ignores this notion.

What’s more, it is not meant to quantify how well a feature can predict the outcome, as demonstrated below:

Yet, at times, it is misinterpreted as a measure of “predictiveness”.

Lastly, correlation is mostly limited to numerical data. But categorical data is equally important for predictive models.

The Predictive Power Score (PPS) addresses each of these limitations.

As the name suggests, it measures the predictive power of a feature.

PPS(a → b) is calculated as follows:

• If the target (b) is numeric:

  • Train a Decision Tree Regressor that predicts b using a.

  • Find PPS by comparing its MAE to the MAE of a baseline model (median prediction).

• If the target (b) is categorical:

  • Train a Decision Tree Classifier that predicts b using a.

  • Find PPS by comparing its F1 to the F1 of a baseline model (random or most frequent prediction).

Thus, PPS:

• is asymmetric, meaning PPS(a, b) != PPS(b, a).

• can be used on categorical targets (b).

• can be used to measure the predictive power of categorical features (a).

• works well for linear and non-linear relationships.

• works well for monotonic and non-monotonic relationships.

Its effectiveness is evident from the image below.

For all three datasets:

• Correlation is low.

• PPS (x → y) is high.

• PPS (y → x) is zero.

That being said, it is important to note that correlation has its place.

When selecting between PPS and correlation, first set a clear objective about what you wish to learn about the data:

• Do you want to know the general monotonic trend between two variables? Correlation will help.

• Do you want to know the predictiveness of a feature? PPS will help.

👉 Over to you: What other points will you add here about PPS vs. Correlation?

Get started with PPS: GitHub.

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