Let’s learn about math, interpretaion and implimentaion in python for Kolmogorov-Smirnov(KS) statistic

### Introduction to classification

Classification is a machine learning technique that involves predicting the class or category to which an input belongs. For example, a classification model might take as input an image of a person’s face and output the predicted identity of the person (e.g. “Elon Musk”).

Classification models are trained using labeled data, where the correct class or category for each input is known. The goal of training a classification model is to learn a function that can map inputs to their correct classes with a high degree of accuracy.

Classification is commonly used in many applications, such as spam filtering, image and speech recognition, and natural language processing. It is a key component of many machine learning systems and is a crucial tool for making predictions and decisions based on data.

### Introduction to KS Metrics

The Kolmogorov-Smirnov (KS) metric is a statistical measure that quantifies the difference between the empirical cumulative distribution function (CDF) of a sample and the theoretical CDF of a reference distribution. It is commonly used in hypothesis testing to determine whether a sample is drawn from a particular distribution.

In classification, the KS metric is often used to evaluate the **performance of a binary classifier.** It provides a measure of how well the classifier is able to distinguish between the two classes being predicted. A high KS value indicates that the classifier is making good predictions, while a low KS value indicates that the classifier is not performing well.

The KS metric is calculated by finding the maximum absolute difference between the empirical CDF of the predicted classes and the theoretical CDF of the true classes. This maximum difference is then used as the KS statistic. The higher the KS statistic, the better the performance of the classifier.

### Calculating the KS Metric

First, the empirical cumulative distribution function (CDF) of the predicted classes is calculated. This is done by sorting the predicted classes in ascending order and then calculating the cumulative probability for each class.

Next, the theoretical CDF of the true classes is calculated in the same way.

The maximum absolute difference between the two CDFs is then found. This difference is the KS statistic.

**For example**, suppose we have a binary classification problem with 10 instances and the following predicted and true classes:

Predicted classes: 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

True classes: 1, 1, 0, 1, 0, 0, 1, 1, 0, 0

To calculate the KS metric, we first calculate the empirical CDFs of the predicted and true classes:

Empirical CDF of predicted classes:

(0, 0.5), (1, 1.0)

Empirical CDF of true classes:

(0, 0.4), (1, 1.0)

Next, we find the maximum absolute difference between the two CDFs:

|0.5 – 0.4| = 0.1

|1.0 – 1.0| = 0

The maximum difference is 0.1, so the KS statistic for this example is 0.1. This indicates that the classifier is performing well since a high KS value indicates good performance.

Note:

steps to calculate the CDF:

- Sort the data in ascending order.
- For each value in the data, calculate its percentile rank by dividing its position in the sorted list by the total number of values. For example, if the value is the third element in the list, its percentile rank would be 3/total number of values.
- Calculate the CDF value for each data point by summing up the percentile ranks of all the data points that are less than or equal to the current data point.
- Plot the CDF values on the y-axis against the data values on the x-axis. The resulting plot will show the CDF for the true classes in the KS statistic.

Keep in mind that this is just one way to calculate the CDF for the true classes in a KS statistic. There may be other methods that you can use depending on your specific situation and needs.

### Interpretation of KS Metrics

The value of the KS metric can be interpreted as a measure of the performance of a binary classifier. **A** **high KS value indicates that the classifier is making good predictions**, **while a low KS value indicates that the classifier is not performing well.**

Example,

A KS value of 0.8 would be considered very good, while a KS value of 0.2 would be considered poor. In general, a KS value above 0.6 is considered good, while a value below 0.4 is considered poor.

However, it is important to note that the interpretation of the KS metric can vary depending on the specific context and the nature of the classification problem. In some cases, a lower KS value may be acceptable if the classifier is still making accurate predictions for the most important classes.

Additionally, the KS metric is not always the best measure of performance for a classifier. In some cases, other metrics such as precision, recall, and F1 score may be more appropriate. It is important to consider the specific goals of the classification problem and choose the appropriate evaluation metric.

### When to use this and when to not

**Here are some situations where the KS metric may be a good choice:**

- When the goal of the classification problem is to maximize the overall accuracy of the classifier. The KS metric provides a good overall measure of the classifier’s performance.
- When the classes being predicted are well-balanced. The KS metric is sensitive to class imbalance, so it may not be as effective when there is a significant difference in the number of instances belonging to each class.
- When the classifier is making a large number of predictions. The KS metric is based on the empirical cumulative distribution function, which can be unstable when there are only a few instances being predicted.

**Here are some situations where the KS metric may not be the best choice:**

- When the classes being predicted are imbalanced. In this case, other metrics such as precision, recall, and F1 score may be more effective at evaluating the performance of the classifier.
- When the goal of the classification problem is to maximize the accuracy of the classifier for a specific class. The KS metric provides a global measure of the classifier’s performance, so it may not be as effective at measuring the performance of individual classes.
- When the classifier is making a small number of predictions. The KS metric may be unstable when there are only a few instances being predicted, so other metrics may be more reliable in this case.

### Implementation Of KS Statistics In Python

```
# Import the required libraries
import numpy as np
# Define the predicted and true classes
predicted = np.array([0, 1, 0, 1, 0, 1, 0, 1, 0, 1])
true = np.array([1, 1, 0, 1, 0, 0, 1, 1, 0, 0])
# Calculate the empirical CDFs
predicted_cdf = np.cumsum(predicted) / len(predicted)
true_cdf = np.cumsum(true) / len(true)
# Calculate the maximum absolute difference between the CDFs
ks = np.max(np.abs(predicted_cdf - true_cdf))
# Print the KS statistic
print(ks)
output: 0.10000000000000003
```

This code should output a value of 0.1, which indicates that the classifier is poorly performing (since a high KS value indicates good performance).

### Conclusion

Congratulations, We just learned everything we need to know about KS Metrics.

To summarise everything

The Kolmogorov-Smirnov (KS) metric is a useful tool for evaluating the performance of a binary classifier. It provides a measure of how well the classifier is able to distinguish between the two classes being predicted. A high KS value indicates that the classifier is making good predictions, while a low KS value indicates that the classifier is not performing well.

Calculating the KS metric involves finding the maximum absolute difference between the empirical cumulative distribution function of the predicted classes and the theoretical CDF of the true classes. This difference is then used as the KS statistic.

While the KS metric is a valuable tool for evaluating the performance of a classifier, it is not always the best choice for every situation. In some cases, other metrics such as precision, recall, and F1 score may be more appropriate. It is important to consider the specific goals of the classification problem and choose the appropriate evaluation metric.