relationships – paulvanderlaken.com

Update March, 2021: My R package for the predictive power score (ppsr) is live on CRAN!
Try install.packages("ppsr") in your R terminal to get the latest version.

A few months ago, I wrote about the Predictive Power Score (PPS): a handy metric to quickly explore and quantify the relationships in a dataset.

As a social scientist, I was taught to use a correlation matrix to describe the relationships in a dataset. Yet, in my opinion, the PPS provides three handy advantages:

PPS works for any type of data, also nominal/categorical variables
PPS quantifies non-linear relationships between variables
PPS acknowledges the asymmetry of those relationships

Florian Wetschoreck came up with the PPS idea, wrote the original blog, and programmed a Python implementation of it (called ppscore).

Yet, I work mostly in R and I was very keen on incorporating this powertool into my general data science workflow.

So, over the holiday period, I did something I have never done before: I wrote an R package!

It’s called ppsr and you can find the code here on github.

Installation

# You can get the official version from CRAN:
install.packages("ppsr")

## Or you can get the development version from GitHub:
# install.packages('devtools')
# devtools::install_github('https://github.com/paulvanderlaken/ppsr')

Usage

The ppsr package has three main functions that compute PPS:

score() – which computes an x-y PPS
score_predictors() – which computes X-y PPS
score_matrix() – which computes X-Y PPS

Visualizing PPS

Subsequently, there are two main functions that wrap around these computational functions to help you visualize your PPS using ggplot2:

visualize_predictors() – producing a barplot of all X-y PPS

visualize_matrix() – producing a heatmap of all X-Y PPS

Note that Species is a nominal/categorical variable, with three character/text options.

A correlation matrix would not be able to show us that the type of iris Species can be predicted extremely well by the petal length and width, and somewhat by the sepal length and width. Yet, particularly sepal width is not easily predicted by the type of species.

Exploring mtcars

It takes about 10 seconds to run 121 decision trees with visualize_matrix(mtcars). Yet, the output is much more informative than the correlation matrix:

cyl can be much better predicted by mpg than the other way around
the classification of vs can be done well using nearly all variables as predictors, except for am
yet, it’s hard to predict anything based on the vs classification
a cars’ am can’t be predicted at all using these variables

The correlation matrix does provides insights that are not provided by the PPS matrix. Most importantly, the sign and strength of any linear relationship that may exist. For instance, we can deduce that mpg relates strongly negatively with cyl.

Yet, even though half of the matrix does not provide any additional information (due to the symmetry), I still find it hard to derive the most important relations and insights at a first glance.

Moreover, the rows and columns for vs and am are not very informative in this correlation matrix as it contains pearson correlations coefficients by default, whereas vs and am are binary variables. The same can be said for cyl, gear and carb, which contain ordinal categories / integer data, so you can discuss the value of these coefficients depicted here.

Exploring trees

In R, there are many datasets built in via the datasets package. Let’s explore some using the ppsr::visualize_matrix() function.

datasets::trees has data on 31 trees’ girth, height and volume.

visualize_matrix(datasets::trees) shows that both girth and volume can be used to predict the other quite well, but not perfectly.

Let’s have a look at the correlation matrix.

The scores here seem quite higher in general. A near perfect correlation between volume and girth.

Is it near perfect though? Let’s have a look at the underlying data and fit a linear model to it.

You will still be pretty far off the real values when you use a linear model based on Girth to predict Volume. This is what the original PPS of 0.65 tried to convey.

Actually, I’ve run the math for this linaer model and the RMSE is still 4.11. Using just the mean Volume as a prediction of Volume will result in 16.17 RMSE. If we map these RMSE values on a linear scale from 0 to 1, we would get the PPS of our linear model, which is about 0.75.

So, actually, the linear model is a better predictor than the decision tree that is used as a default in the ppsr package. That was used to generate the PPS matrix above.

Yet, the linear model definitely does not provide a perfect prediction, even though the correlation may be near perfect.

Conclusion

In sum, I feel using the general idea behind PPS can be very useful for data exploration.

Particularly in more data science / machine learning type of projects. The PPS can provide a quick survey of which targets can be predicted using which features, potentially with more complex than just linear patterns.

Yet, the old-school correlation matrix also still provides unique and valuable insights that the PPS matrix does not. So I do not consider the PPS so much an ~~alternative~~, as much as a complement in the toolkit of the data scientist & researcher.

Enjoy the R package, or the Python module for that matter, and let me know if you see any improvements!

Update March, 2021: My R package for the predictive power score (ppsr) is live on CRAN!
Try install.packages("ppsr") in your R terminal to get the latest version.

Last week, I shared this Medium blog on PPS — or Predictive Power Score — on my LinkedIn and got so many enthousiastic responses, that I had to share it with here too.

Basically, the predictive power score is a normalized metric (values range from 0 to 1) that shows you to what extent you can use a variable X (say age) to predict a variable Y (say weight in kgs).

A PPS high score of, for instance, 0.85, would show that weight can be predicted pretty good using age.

A low PPS score, of say 0.10, would imply that weight is hard to predict using age.

The PPS acts a bit like a correlation coefficient we’re used too, but it is also different in many ways that are useful to data scientists:

PPS also detects and summarizes non-linear relationships
PPS is assymetric, so that it models Y ~ X, but not necessarily X ~ Y
PPS can summarize predictive value of / among categorical variables and nominal data

However, you may argue that the PPS is harder to interpret than the common correlation coefficent:

PPS can reflect quite complex and very different patterns
Therefore, PPS are hard to compare: a 0.5 may reflect a linear relationship but also many other relationships
PPS are highly dependent on the used algorithm: you can use any algorithm from OLS to CART to full-blown NN or XGBoost. Your algorithm hihgly depends the patterns you’ll detect and thus your scores
PPS are highly dependent on the the evaluation metric (RMSE, MAE, etc).

Here’s an example picture from the original blog, showing a case in which PSS shows the relevant predictive value of Y ~ X, whereas a correlation coefficient would show no relationship whatsoever:

https://towardsdatascience.com/rip-correlation-introducing-the-predictive-power-score-3d90808b9598

Here’s two more pictures from the original blog showing the differences with a standard correlation matrix on the Titanic data:

I highly suggest you read the original blog for more details and information, and that you check out the associated Python package ppscore:

Installing the package:

pip install ppscore

Calculating the PPS for a given pandas dataframe:

import ppscore as pps
pps.score(df, "feature_column", "target_column")

You can also calculate the whole PPS matrix:

pps.matrix(df)

There’s no R package yet, but it should not be hard to implement this general logic.

Florian Wetschoreck — the author — already noted that there may be several use cases where he’d think PPS may add value:

Find patterns in the data [red: data exploration]: The PPS finds every relationship that the correlation finds — and more. Thus, you can use the PPS matrix as an alternative to the correlation matrix to detect and understand linear or nonlinear patterns in your data. This is possible across data types using a single score that always ranges from 0 to 1.
Feature selection: In addition to your usual feature selection mechanism, you can use the predictive power score to find good predictors for your target column. Also, you can eliminate features that just add random noise. Those features sometimes still score high in feature importance metrics. In addition, you can eliminate features that can be predicted by other features because they don’t add new information. Besides, you can identify pairs of mutually predictive features in the PPS matrix — this includes strongly correlated features but will also detect non-linear relationships.
Detect information leakage: Use the PPS matrix to detect information leakage between variables — even if the information leakage is mediated via other variables.
Data Normalization: Find entity structures in the data via interpreting the PPS matrix as a directed graph. This might be surprising when the data contains latent structures that were previously unknown. For example: the TicketID in the Titanic dataset is often an indicator for a family.
https://towardsdatascience.com/rip-correlation-introducing-the-predictive-power-score-3d90808b9598