ML Interview Q Series: If you are using k-Nearest Neighbors, what type of Normalization should be used?

📚 Browse the full ML Interview series here.

Comprehensive Explanation

k-Nearest Neighbors (KNN) is a distance-based algorithm. It compares data points in the feature space by measuring how “far” they are from one another, typically using a distance metric such as Euclidean distance. When features have significantly different scales (for instance, one feature measured in thousands and another in decimals), those with the larger scale dominate the distance calculation. This can mislead the algorithm into giving disproportionate importance to certain features.

To address this, features are generally scaled to a similar range before applying KNN. Common approaches include min-max scaling (often called “normalization”) or standardization (“z-score scaling”). The decision about which scaling method to use often depends on outliers and the distribution of features. Both methods can be valid, but standardization is typically robust when moderate outliers are present, whereas min-max scaling is common when the dataset is known to have bounded values without extreme outliers.

Below is the typical Euclidean distance formula for KNN:

Here x and x' represent two points in d-dimensional space. Each x_i and x'_i is the value of the i-th feature for the points x and x' respectively.

Min-Max Scaling

Min-max scaling transforms each feature such that its minimum value becomes 0 and its maximum value becomes 1. This is useful when you want all features to be confined within a fixed [0,1] interval. It can be sensitive to outliers; if the data has points that are far outside the typical range, the rest of the points might get squashed into a very small range. A common formula for min-max scaling is shown below:

Where x is the original feature value, x_min is the minimum observed value of that feature, and x_max is the maximum observed value of that feature. The transformed value x' then lies between 0 and 1.

Standardization (Z-Score Scaling)

In standardization, data is scaled to have zero mean and unit variance. This method is less affected by outliers than min-max scaling. The common formula for standardization is:

Where x is the original feature value, μ is the mean of that feature across the dataset, and σ is the standard deviation of that feature. After this transformation, the resulting distribution for the feature has mean 0 and standard deviation 1.

In practice, either min-max scaling or standardization can work for KNN, with standardization often being the more typical default when you do not have strong reasons to use a strict [0,1] range or you expect moderate outliers in your data.

When to Use Standardization or Min-Max Scaling

If your data has no severe outliers and you want all features strictly in a certain range [0,1], min-max scaling is an intuitive approach. However, if outliers exist or if features follow approximately a normal distribution, standardization is generally preferable.

Potential Pitfalls

One major pitfall is to forget to apply the same scaling parameters when you deploy your model or when new data arrives. For example, if you compute min-max or standardization parameters (x_min, x_max, mean, standard deviation) on your training set, you must apply exactly those parameters to the test (and future) data. Recomputing scaling parameters on test data will cause data leakage and produce misleading results.

Another subtle issue arises if different features inherently have different levels of importance. If that importance is genuine, uniform scaling might overshadow what was actually a valuable difference. In such cases, feature engineering or domain knowledge might suggest weighting certain features differently instead of purely normalizing them to the same scale.

How to Implement in Python

import numpy as np
from sklearn.preprocessing import MinMaxScaler, StandardScaler

# Example dataset with 3 features
X = np.array([
    [1000, 1.2,  10],
    [1500, 1.8,   5],
    [1200, 1.0,   8],
    [ 900, 2.1,  12]
])

# Min-Max Scaling
min_max_scaler = MinMaxScaler()
X_minmax = min_max_scaler.fit_transform(X)

# Standardization
standard_scaler = StandardScaler()
X_standardized = standard_scaler.fit_transform(X)

Both transformations will produce appropriately scaled data for use with KNN.

What Happens if We Skip Normalization

If features vary drastically in magnitude and you do not normalize, the feature(s) with the larger numeric range can dominate the distance calculation. The model may end up ignoring other informative features because differences in those features do not significantly affect the overall distance. As a result, the KNN classifier or regressor might underperform.

Follow-up Questions

How does the choice of normalization method affect performance in KNN?

Different scaling methods can lead to different distance boundaries in the feature space. Min-max scaling makes all features lie between 0 and 1 and treats all data within that range evenly, which can cause outliers to have a dramatic effect on the scaled values. Standardization might handle moderate outliers better but can still be influenced by extreme outliers if they significantly shift the mean or inflate the standard deviation. It is often beneficial to experiment with different scaling methods (and possibly robust methods) to see which produces better results on validation data.

What if we have categorical features alongside numeric features in KNN?

Pure numeric scaling might not apply directly to categorical data. For one-hot encoded categorical variables, each category is typically represented by 0 or 1. If a feature is partially categorical and partially numeric, you might need specialized encodings or distance metrics. One approach could be to separate numeric and categorical features and combine distance measures carefully (e.g., using Hamming distance for the categorical part and Euclidean distance for the numeric part).

Can we use more complex transformations?

Beyond standard scaling and min-max, you can also use: • Robust scaling that relies on median and Interquartile Range, which helps handle outliers better. • Non-linear transformations (e.g., Box-Cox or log transforms) if the distribution of features is highly skewed and domain knowledge suggests these transformations would help.

How do we decide which distance metric to use in KNN?

Though Euclidean distance is the most common, other metrics like Manhattan distance or Minkowski distance can be used depending on the nature of your data. For high-dimensional data, sometimes cosine similarity (interpreted as a distance) is used. The choice of distance metric can impact how sensitive the model is to scaling or outliers.

How to tune K in KNN alongside normalization?

The number of neighbors (K) is usually chosen by cross-validation. You would:

1. Decide on a type of scaling (and fix how you perform that scaling).

2. Perform cross-validation over a range of K values.

3. Evaluate performance metrics for each K and pick the value that yields the best validation performance.

This ensures consistency between how your data is scaled and how the neighbors are computed.

Below are additional follow-up questions

What if some features are highly correlated? Does normalization still help?

Highly correlated features can cause redundancy in the input space. Even if you normalize these features, KNN might treat them as distinct inputs that heavily influence distance calculations, possibly skewing the final predictions. Normalization ensures each feature is on the same scale, but it does not inherently remove correlation. To address correlation:

• You could apply dimensionality reduction techniques (e.g., PCA) before KNN. • You might exclude redundant features if domain knowledge suggests they are not informative. • You can transform correlated features in a way that captures their shared variance (for example, principal components or linear combinations).

Potential Pitfalls or Edge Cases • Over-reliance on correlated features: If multiple features carry largely the same information, small differences in their values might overly impact your distance computations. • Data leakage: If correlated features exist because of data preprocessing or domain-specific errors (like a label leaking into a feature), normalization alone will not solve that leakage. • Overfitting: In high-dimensional spaces with correlated features, KNN can memorize specific patterns that do not generalize well to new data.

How do we handle Weighted KNN with normalization?

In Weighted KNN, neighbors closer to the query point get higher weights in the prediction. The standard practice of normalizing or standardizing features still applies. If features are not scaled properly, the distance-based weighting might be dominated by a single feature with larger values. After normalization, each dimension contributes more fairly to the distance.

Below is a commonly used formula for Weighted KNN, where each neighbor i has a label y_i and a distance d(x, x_i) from the query point x:

Here ŷ (y-hat) is the predicted value (for regression), k is the number of neighbors, y_i is the label of the i-th neighbor, and d(x, x_i) is the distance between x and x_i in the feature space. If you do not normalize, one dimension with large values will dominate d(x, x_i), leading to skewed weights.

Potential Pitfalls or Edge Cases • Extreme outliers: If there is an outlier that is very far away, the inverse-distance weighting might make its contribution negligible. This is sometimes good, but if the outlier is actually informative, it could be unfairly dismissed. • Choice of weighting function: The above formula uses 1/d(x, x_i) as weights, but you can pick any function that decreases with distance (like 1/(distance^p) for some p). The choice of p can affect predictions, and you should tune it through cross-validation. • Over-sensitivity to local noise: Weighted KNN might focus too strongly on the very nearest points, making it sensitive if those nearest points are noisy. Normalizing or standardizing features does not eliminate this potential weakness.

Does the presence of missing values affect the choice of normalization for KNN?

Yes, missing values complicate how you perform normalization. Typically, scaling methods (min-max or standardization) require complete feature columns to compute statistics (e.g., min, max, mean, standard deviation). If your data has missing values, you need a strategy to handle them before or during normalization:

• Imputation Before Normalization: You could impute missing values (e.g., mean, median, or a model-based method) before applying min-max or standardization. • Partial Fit or Iterative Imputation: Some libraries allow partial fits or iterative approaches that can incorporate missing value handling during normalization. • Exclusion of Rows with Missing Values: Not generally recommended unless the proportion of missing data is small, as it might bias the dataset.

Potential Pitfalls or Edge Cases • Imputation Distortion: If many values in a feature column are missing, the imputation might distort true statistics and affect the scaling parameters. • Data Leakage: If you compute imputation statistics using the entire dataset (including test data), you leak information. The correct approach is to fit imputation (and scaling) parameters only on training data. • Incorrect Distance Calculations: KNN might skip records with missing values or compute distances in partial subsets of features. This partial approach can lead to unpredictable outcomes unless carefully designed.

How do we deal with categorical or ordinal data when normalizing for KNN?

For purely numeric data, min-max scaling or standardization works well. But if you have categorical features, scaling does not straightforwardly apply. You often convert categorical variables into numeric representations (one-hot encoding, label encoding, etc.). Ordinal features (like “low,” “medium,” “high”) imply a rank order that might require specialized encoding:

• One-Hot Encoding for Nominal Data: Each category becomes a new binary feature (0 or 1). After one-hot encoding, normalizing those columns might not be meaningful, as their valid range is inherently {0,1}. • Ordinal Encoding: For features with a clear order (e.g., “low,” “medium,” “high”), you might map them to numeric values (0, 1, 2). But standard scaling might not make sense if 0 → 1 → 2 does not represent a true numeric distance. • Distance Metrics for Mixed Data: Sometimes specialized distance metrics combine numeric distances for continuous features with Hamming distance for categorical features.

Potential Pitfalls or Edge Cases • Artificial Distances: If you treat ordinal data as numeric and then scale, you might introduce artificial distances that do not make sense for ranking-based features. • Curse of Dimensionality with One-Hot: One-hot encoding might rapidly increase dimensionality, making KNN less effective. • Overweighting Minor Categories: If a categorical feature has many categories, those features might dominate distance calculations unless carefully balanced.

Could we apply non-linear transformations for normalization in KNN?

Yes, sometimes data is heavily skewed or follows certain distributions (e.g., log-normal). In those cases, non-linear transformations might help reduce skew or stabilize variance:

• Log Transform: Commonly used when a feature has an exponential or log-normal distribution. • Box-Cox or Yeo-Johnson: More general transforms that try to make data more Gaussian-like. • Rank-based Transformation: Convert numerical values to ranks and then scale. This approach can help if exact numeric distances are less important than the order of values.

Potential Pitfalls or Edge Cases • Over-Transformation: Multiple transformations (e.g., log transform plus standardization) may overcomplicate the feature space, leading to unexpected behavior in KNN distances. • Zero or Negative Values: A log transform cannot be applied to zero or negative values, so you need offsetting or an alternative approach. • Interpretability: If you transform your data significantly, interpreting distances or neighbor relationships might become less intuitive.

How do we handle time-series data with KNN and normalization?

Time-series data often has trends and seasonality. Standard normalization might overlook these temporal structures:

• Sliding Window or Feature Extraction: You could create lag features or rolling statistics and then normalize them. But if the series has a global upward trend, a simple min-max might not be meaningful across the entire time horizon. • Normalization by Seasonal Segments: If data is periodic (e.g., daily or weekly seasonality), you might compute separate scaling parameters for each period. • Online Normalization: For streaming time-series, you may update normalization parameters incrementally as new data arrives (e.g., using running means and variances).

Potential Pitfalls or Edge Cases • Changing Distributions Over Time: If the distribution of a feature evolves, scaling parameters fit to earlier data might no longer be valid. • Seasonality and Shifts: A sudden shift in the mean (e.g., a concept drift) will invalidate previously computed normalization parameters. • Data Leakage in Future Steps: If you use future data to compute scaling parameters, you inadvertently leak information about the future into the past, leading to overoptimistic results.

How does normalization interact with distance metric selection in KNN?

The default choice of distance is often Euclidean, but you can use other metrics like Manhattan, Minkowski, or even custom metrics:

• Euclidean vs. Manhattan: Euclidean distance squares the differences and adds them; Manhattan distance sums absolute differences. Both can be dominated by large-scale features if normalization is not applied. • Cosine Similarity: Often used for high-dimensional text data. Although this is technically a similarity measure, it is treated as distance = 1 - similarity. Scaling by norm is conceptually built-in to cosine similarity. • Custom Metrics: You might design a specialized metric that weighs features based on domain knowledge. Even then, normalizing numeric features helps ensure they have comparable impact.

Potential Pitfalls or Edge Cases • Metric Mismatch: If your feature space is not well-suited for Euclidean distance, even normalized data can produce suboptimal results. • Computation Cost: Some distance metrics are more expensive to compute at scale. Normalizing does not directly reduce this cost but can simplify certain distance computations. • Mixed Data Types: In a dataset with both continuous and categorical features, you may need a composite metric. Normalization helps continuous features but does nothing for binary or categorical features.