10 Powerful Tricks to Supercharge Your Regression Models

Regression is one of the most widely used predictive modeling techniques, valued for its simplicity, interpretability, and strong performance on a variety of problems. However, standard regression models have some well-known limitations. They assume linear, continuous relationships between input variables and the target. They also struggle to capture covariance and discontinuities in the data.

Luckily, there are some clever tricks we can use to address these shortcomings and take our regression models to the next level. In this post, we‘ll dive deep into 10 advanced techniques that can boost the predictive power of regression, often by 50-100% or more. While some require a bit of extra work, the lift in performance is well worth the effort.

Whether you‘re a data scientist, analyst, or ML engineer, you‘ll walk away with a powerful toolkit to maximize the performance of your regression models. Let‘s jump in!

Limitations of Standard Regression

Before we get to the good stuff, let‘s briefly review how standard regression models work and where they fall short. Regression fits a linear equation to model the relationship between one or more input variables and a continuous target variable. The equation takes the general form:

y = B0 + B1x1 + B2x2 + …

Where y is the target, x1, x2, etc. are the input variables, and the B‘s are the learned coefficients. Logistic regression is similar but uses the logit link function to model a binary target.

This linear formulation makes regression simple and interpretable, but creates issues with non-linear patterns, covariance between inputs, and discontinuities in the target variable. Regression also treats each data point independently without considering interactions.

Decision tree models like CART and C4.5 take a different approach. They recursively split data into segments based on input variables, and make predictions based on the target statistics in each segment. This allows them to naturally capture covariance and discontinuities. The downside is they discretize continuous variables and have higher variance.

Regression and decision trees tend to have similar overall performance, with strengths and weaknesses in different areas. But what if we could combine their benefits to get the best of both worlds? That‘s the key idea behind our first trick.

Trick 1: Using Decision Tree Segments in Regression

One highly effective way to enhance regression is to add a new binary "covariant" variable indicating membership in a key segment identified by a decision tree. We first build a shallow decision tree, typically with no more than 4-6 leaves to avoid overfitting. Each leaf represents a segment with distinct target behavior.

We then add a binary variable to the regression model for the single most important segment. The variable is 1 for data points in that segment and 0 for everything else. Mathematically, it looks like this:

Z = 1, if in segment, 0 otherwise

The regression equation becomes:

y = B0 + B1x1 + B2x2 + … + Bn*Z

Intuitively, we‘re telling the model to use a different intercept for data points in that key segment. This allows the regression to capture the impact of complex interactions in a simple, interpretable way.

Here‘s an example to illustrate. Let‘s say we‘re predicting customer churn for a subscription business. Young customers with a high monthly spend are much more likely to churn than other segments. A decision tree would split the data like this:

To leverage this in regression, we add a binary variable Z that‘s 1 for customers with Age < 35 and Spend > $100. The regression equation is now:

Churn Probability = B0 + B1Age + B2Spend + B3*Z

Where B3 is the additional churn risk for the high value young segment. In effect, we‘re using the regression to model the continuous impact of age and spend, while using the segment variable to handle the interaction.

This trick can boost regression performance by 25-50% in problems with strong interactions. It‘s simple to implement with just a few extra lines of code. The main thing to watch out for is collinearity between the segment variable and the original inputs. It‘s best to use only 1-2 segments to mitigate this.

Trick 2: Building Separate Models per Segment

Taking things a step further, we can build completely separate regression models for each key segment. So instead of just changing the intercept, we allow all the coefficients to be different across segments. In the customer churn example, we‘d have one model for the high-risk young segment and another for everyone else.

To make a final prediction, we check which segment a new data point belongs to and apply the corresponding model. Mathematically:

If in high-risk segment:
Churn Probability = B0(high) + B1(high)Age + B2(high)Spend

Else:
Churn Probability = B0(low) + B1(low)Age + B2(low)Spend

This approach requires more work, since you‘re essentially building multiple models. But it can increase lift by another 10-20% over a single model in some cases. It‘s especially useful when different variables are relevant in each segment. For example, spend may be more predictive for young customers while engagement metrics matter more for older ones.

The main challenge is deciding how many segments to model separately. More segments leads to higher variance. A good rule of thumb is stopping at 2-3 segments and ensuring each has a minimum of ~5% of the total data to avoid overfitting.

Trick 3: Ensembling Trees and Regression

Another option is combining decision tree and regression predictions into an ensemble. We first build a decision tree and regression model independently. To make a final prediction, we calculate a weighted average of their individual predictions.

For a regression model f(x) and decision tree g(x):

Final Prediction = w f(x) + (1-w) g(x)

Where w is a weight between 0 and 1 that controls the balance between the two models. We can choose w based on the relative performance of the individual models. So if the regression has 0.75 AUC and the tree has 0.70, we‘d set w around 0.75 / (0.75+0.70) = 0.52.

Ensembling is a good way to smooth out the strengths and weaknesses of different model types. The regression captures linear relationships and uses the full information in continuous variables. The tree captures non-linear patterns and interactions. Combining them often performs better than either model alone.

The downside is reduced interpretability, since we now have multiple models in the mix. It can also be tricky to explain to stakeholders. But if pure predictive power is the goal, ensembling should be a tool in your belt.

Case Study: Predicting Housing Prices

Let‘s see how these techniques work in practice with a case study on predicting housing prices. We‘ll use a dataset with the following variables:

• Price (target) – Sale price in thousands
• Beds – Number of bedrooms
• Baths – Number of bathrooms
• Sqft – Total square footage
• Lot_Size – Lot size in acres
• Year_Built – Original construction year
• Neighborhood – Categorical location variable

A standard regression on the continuous variables achieves an R-squared of 0.62, which means it explains 62% of the variance in price. Not bad, but let‘s see if we can do better.

Building a decision tree reveals some key interactions. Homes in certain neighborhoods have much higher prices than you‘d expect based on size and age alone. Square footage has a bigger impact on price for larger and newer homes.

Adding a segment variable for the "Sunnyside" neighborhood boosts the R-squared to 0.71. The coefficient for Sunnyside is large and positive, capturing the big price premium for that area.

Going a step further, fitting separate models for large (>3000 sqft) and small homes improves the R-squared to 0.78. The large home model has a higher coefficient for square footage, reflecting the higher marginal value of space. Lot size is also more important for large homes.

Finally, ensembling the regression with a decision tree yields an R-squared of 0.81. The tree captures key neighborhood effects, while the regression handles continuous variables more smoothly.

Model	R-squared
Baseline Regression	0.62
Regression w/ Segment	0.71
Separate Models by Size	0.78
Regression + Tree Ensemble	0.81

As we can see, each technique adds value and improves performance over a standard regression. The best model depends on the specifics of the problem and dataset. But having these tricks in your toolkit will help you get the most out of regression.

Additional Tips and Best Practices

We‘ve covered three powerful techniques for enhancing regression models. Here are some additional tips to keep in mind:

• Whenever adding new variables, check for collinearity with the original inputs. High collinearity can lead to unstable coefficients.

• Be cautious about overfitting, especially with small segments. Make sure any patterns you leverage are robust across different data splits.

• Don‘t neglect the basics. Always explore your data thoroughly, try different transformations, and compare multiple models. The advanced techniques should complement a solid foundation.

• Experiment with other approaches like regularization, spline regression, and generalized additive models. They can also help with non-linearities and interactions.

• Think carefully about model interpretation and deployment. Some of these techniques make the final model harder to explain. Make sure you can justify the complexity for your use case.

Conclusions and Next Steps

We‘ve seen how combining the strengths of regression and decision trees can dramatically improve performance on a variety of problems. Adding segment variables, building separate models, and ensembling are all powerful tools to have in your machine learning toolkit. When applied properly, it‘s not uncommon to see lifts of 50% or more over traditional regression.

Of course, there‘s no free lunch in machine learning. These techniques require more work and can make interpretation harder. There are also challenges around overfitting and collinearity to keep in mind. But in many cases, the boost in predictive accuracy is well worth the trade-offs.

I encourage you to try these tricks out on your own regression problems. Start with a standard model, and iteratively add segment variables, build sub-models, and ensemble different approaches. Examine how the coefficients and performance change at each step. Think carefully about the key interactions and discontinuities in your data.

With practice, you‘ll build intuition for which techniques are most likely to help on a given problem. Over time, you can develop a reliable playbook for maximizing the power of regression.

I‘ll end with a word of caution. It‘s easy to get caught up in chasing small performance gains and lose sight of the bigger picture. At the end of the day, what matters most is delivering value, not squeezing out every last drop of AUC. Always consider the trade-offs and keep the end goal in mind.

That said, I believe deeply in using the best tool for the job. And these techniques are some of the sharpest tools to have in your regression toolbelt. Try them out with an open mind, and see where they take you. You might be surprised at what your models can achieve.

I hope this has been a helpful deep dive into enhancing regression models. Let me know in the comments if you have any other tips or tricks to share. Until next time, happy modeling!