ML Interview Q Series: Under what circumstances might you opt for normalization over standardization in linear regression, and vice versa?
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Comprehensive Explanation
Normalization usually rescales your data into a fixed range, often between 0 and 1. On the other hand, standardization transforms the data so that it has mean 0 and standard deviation 1. Both can be crucial when fitting a linear regression model because they help maintain numerical stability and improve the behavior of optimization algorithms, especially in gradient-based methods.
Normalization can be very useful when:
You want your features strictly within a bounded scale.
You suspect large outliers might not be too frequent or would otherwise distort a standardization approach.
You expect the distribution of your data to be somewhat uniform across the range, or you aim to preserve the relative distances on a 0–1 scale for interpretability.
Standardization is especially helpful when:
You want features centered around 0, which can stabilize gradient descent and ensure a more uniform convergence pace across parameters.
Your features are approximately normally distributed or do not have strict minimum and maximum bounds.
You have outliers, but you still want to keep the data’s variation visible (unlike 0–1 normalization, which may collapse outliers to 1).
Normalization Formula
Here, x is a data value in a specific column, x_min is the minimum value in that column, and x_max is the maximum value in that column. The result is that x_norm will lie in the interval from 0 to 1. This approach can be extended to a different target range, like [a, b], by adjusting the numerator and denominator accordingly.
This method is usually sensitive to any change in x_min or x_max because if the range shifts, your normalized scale changes too. Therefore, you want to compute x_min and x_max on your training set and apply the same values to the test set to avoid data leakage.
Standardization Formula
Here, x is a data value in a given column, µ is the mean of that column, and σ is the standard deviation of that column. This will transform x into z, which has mean 0 and standard deviation 1. It is often robust against moderate amounts of outliers, as the effect of an extreme value is moderated by the standard deviation. However, very large outliers can still skew µ and σ.
Use in Linear Regression
In linear regression, especially when using gradient-based optimizers (e.g., in iterative solvers or regularized regression like Ridge or Lasso), features on drastically different scales can slow down or destabilize convergence. Scaling all features to a similar range (whether 0–1 through normalization or approximately standard normal distribution) can improve training stability and sometimes produce more interpretable coefficient magnitudes.
If your data set features are not strictly bounded and you want them to have zero mean for interpretative reasons (like analyzing which features are most influential around the average behavior), standardization often is the natural choice. If your data is definitely bounded and you prefer a strict 0–1 range or a custom range, normalization is more intuitive.
Practical Implementation Example
import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import MinMaxScaler, StandardScaler
from sklearn.model_selection import train_test_split
# Synthetic data
X = np.array([[10, 1000],
[12, 1200],
[18, 1900],
[20, 2100],
[30, 3200],
[40, 4000]], dtype=float)
y = np.array([1, 1.5, 2.0, 2.2, 3.0, 3.8])
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Normalization (0-1 scale)
norm_scaler = MinMaxScaler()
X_train_norm = norm_scaler.fit_transform(X_train)
X_test_norm = norm_scaler.transform(X_test)
reg_norm = LinearRegression()
reg_norm.fit(X_train_norm, y_train)
print("Coefficients after Normalization:", reg_norm.coef_)
print("Intercept after Normalization:", reg_norm.intercept_)
# Standardization (mean=0, std=1)
std_scaler = StandardScaler()
X_train_std = std_scaler.fit_transform(X_train)
X_test_std = std_scaler.transform(X_test)
reg_std = LinearRegression()
reg_std.fit(X_train_std, y_train)
print("Coefficients after Standardization:", reg_std.coef_)
print("Intercept after Standardization:", reg_std.intercept_)
The above example shows how to use MinMaxScaler for normalization and StandardScaler for standardization. Depending on which approach you choose, the linear regression coefficients and intercept may appear quite different, but their ultimate performance and interpretability might be enhanced in different ways.
Potential Follow-Up Questions
How does the presence of outliers influence the choice of scaling?
Outliers can significantly shift mean and standard deviation, thus affecting standardization. If the outliers are genuine but extreme, the standard deviation might become large, leading to a smaller scaled representation of typical data points. Normalization’s min and max values might be even more affected if those outliers happen to be the new minimum or maximum. If you suspect outliers that may not be representative, you might investigate robust scalers (like the robust version of standardization using the median and the interquartile range) or outlier mitigation before applying your chosen scaling.
Do these scalers need to be re-fitted after deployment if new data arrives?
The usual practice is to fit the scalers on the training data set once and then use the same parameters (x_min, x_max, mean, standard deviation) on all future data, including validation and test data. This prevents data leakage, where information about the test distribution influences how you scale. If your data distribution shifts significantly over time, you may do periodic re-fitting on new data, but only after carefully ensuring you preserve training/test separation to avoid leakage.
If some features are already in a small range or are binary, should they still be scaled?
Binary or 0–1 features might not require additional scaling unless you need a strict 0 mean and unit variance approach. Even then, standardizing binary variables may or may not be beneficial depending on how you want them to influence the model. If the rest of your features are large numerical values, ignoring scaling for binary features might still allow them to have an outsized influence. Often, you might scale all numeric features for consistency.
What happens if the distribution of features is heavily skewed?
In that case, a log transform or other nonlinear transformations might be more appropriate prior to normalizing or standardizing. For example, if you have a heavily right-skewed distribution, applying log(x+1) can sometimes yield a more normal-like shape, making standardization more meaningful. After such transformations, normalizing or standardizing again can further stabilize training and improve model performance.
Are there advantages to skipping any scaling entirely?
If all your features are on similar scales and your optimization algorithm converges quickly with stable results, you might not need scaling. For straightforward ordinary least squares solved analytically (via matrix inversions for smaller datasets), scaling might not drastically change results. However, it is still a common best practice for regularized regression or iterative methods to ensure more consistent and robust convergence.
These considerations all help you decide whether to pick normalization or standardization when applying linear regression.