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Feature Scaling

EasyDimensionality Reduction 📚 Machine Learning 2

Implement two common feature scaling methods:

Standardization (Z-score): $x' = \frac{x - \mu}{\sigma}$ where $\mu$ is the mean and $\sigma$ is the standard deviation
Min-Max Scaling: $x' = \frac{x - \min}{\max - \min}$

Given a 2D dataset and a method name, scale each column (feature) independently.

Use the population standard deviation: $\sigma = \sqrt{\frac{1}{n}\sum(x_i - \mu)^2}$

Return the scaled matrix, rounded to 4 decimal places. If std = 0 or max = min, return 0.0 for that feature.

Example:

Input:

X = [[1, 10], [2, 20], [3, 30]]
method = "standard"

Output:

[[-1.2247, -1.2247], [0.0, 0.0], [1.2247, 1.2247]]

Reasoning:

The given dataset is X = [[1, 10], [2, 20], [3, 30]] and the method is "standard", which implies Standardization (Z-score).
To apply standardization, we calculate the mean ( $\mu$ ) and standard deviation ( $\sigma$ ) for each column: for the first column, $\mu = \frac{1+2+3}{3} = 2$ and $\sigma = \sqrt{\frac{1}{3}((1-2)^2 + (2-2)^2 + (3-2)^2)} = \sqrt{\frac{1}{3}(1+0+1)} = \sqrt{\frac{2}{3}}$ ; for the second column, $\mu = \frac{10+20+30}{3} = 20$ and $\sigma = \sqrt{\frac{1}{3}((10-20)^2 + (20-20)^2 + (30-20)^2)} = \sqrt{\frac{1}{3}(100+0+100)} = \sqrt{\frac{200}{3}}$ .
We then apply the standardization formula $x' = \frac{x - \mu}{\sigma}$ to each element in the columns: for the first column, $x' = \frac{x - 2}{\sqrt{\frac{2}{3}}}$ , and for the second column, $x' = \frac{x - 20}{\sqrt{\frac{200}{3}}} = \frac{x - 20}{\sqrt{\frac{2}{3}} \cdot 10} = \frac{x - 20}{10\sqrt{\frac{2}{3}}}$ .
After calculating the standardized values for each element and rounding to 4 decimal places, we get the output [[-1.2247, -1.2247], [0.0, 0.0], [1.2247, 1.2247]].

Constraints:

X: 2D list (n_samples x n_features)
method: "standard" or "minmax"
Scale each column independently
Return 2D list rounded to 4 decimal places

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Background Knowledge

Feature scaling is a crucial preprocessing step in machine learning that transforms numeric data into a common range, usually to improve the performance and stability of models. This is necessary because many algorithms are sensitive to the scale of the data, and features with large ranges can dominate the model. The two methods mentioned in the problem, Standardization (Z-score) and Min-Max Scaling, are commonly used techniques for feature scaling.

The Standardization (Z-score) method rescales the data to have a mean of 0 and a standard deviation of 1. This is useful when the data follows a Gaussian distribution, as it helps to reduce the effect of outliers. The Min-Max Scaling method, also known as normalization, rescales the data to a common range, usually between 0 and 1. This method is useful when the data has a fixed range, such as image pixel values.

In both methods, it's essential to calculate the mean and standard deviation (for Standardization) or minimum and maximum values (for Min-Max Scaling) for each feature independently. This ensures that each feature is scaled separately, which helps to prevent features with large ranges from dominating the model.

Algorithm/Approach

The general approach to solving this problem involves iterating over each feature (column) in the dataset, calculating the necessary statistics (mean, standard deviation, minimum, and maximum), and then applying the chosen scaling method to each feature. This can be achieved using a simple iterative algorithm that applies the scaling formulas to each feature.

Step-by-Step Strategy

To implement the solution, follow these steps:

Iterate over each feature (column) in the dataset
For each feature, calculate the necessary statistics:
- For Standardization: calculate the mean ( $\mu$ ) and standard deviation ( $\sigma$ )
- For Min-Max Scaling: calculate the minimum ( $\min$ ) and maximum ( $\max$ ) values
Apply the chosen scaling method to each feature:
- For Standardization: use the formula $x' = \frac{x - \mu}{\sigma}$
- For Min-Max Scaling: use the formula $x' = \frac{x - \min}{\max - \min}$
Handle edge cases where the standard deviation is 0 or the minimum equals the maximum
Round the scaled values to 4 decimal places

Common Pitfalls

When implementing the solution, watch out for the following:

Forgetting to calculate the statistics for each feature independently
Not handling edge cases where the standard deviation is 0 or the minimum equals the maximum
Using the sample standard deviation instead of the population standard deviation
Not rounding the scaled values to the correct number of decimal places

Time & Space Complexity

The expected time complexity for this problem is O(nm), where n is the number of rows and m is the number of columns in the dataset. This is because we need to iterate over each feature (column) and calculate the necessary statistics, which takes linear time. The space complexity is also O(nm), as we need to store the scaled dataset. However, if we can modify the original dataset in-place, the space complexity can be reduced to O(1).

solution.py

Python 3.13

Test Results

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