# Presenting Univariate Drift Detection Methods

Univariate Drift Detection looks at each feature individually and checks whether its
distribution has changed compared to reference data. There are many ways to compare two data samples and measure
their *similarity*. NannyML provides several drift detection methods so that users can choose the one that suits
their data best or the one they are familiar with. Additionally, more than one method can be used together to
gain different perspectives on how the distribution of your data is changing.

This page explains which aspects of a distribution change each drift detection method is able to capture, what are the important implementation details and in which situations a specific method can be a good choice.

This page explains which aspects each drift detection method is able to capture, what are the important implementation details and in which situations a specific method can be a good choice.

We are grouping the drift detection methods presented according to whether they apply to categorical (discrete) or continuous features. When a method is used for both, it is mentioned in both places because of implementation differences between the two types of features.

Lastly, let’s note that we are always performing two sample tests or comparisons. Probability density functions (PDF) and cumulative density functions (CDF) are always estimated from the data samples that are being compared.

Note

In the current implementation of the methods (both continuous and categorical), we drop all the missing values encountered.

## Methods for Continuous Features

### Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov Test is a two-sample, non-parametric statistical test. It is used to test for the equality of one-dimensional continuous distributions. The test outputs the test statistic, called D-statistic, and an associated p-value. The test statistic is the maximum distance of the cumulative distribution functions (CDF) of the two samples.

NannyML calculates drift by performing two sample set comparisons. One sample is usually the reference data, while the other comes from the data of the chunk we are calculating drift for. It can calculate the KS test with two methods, depending on the user’s ability to store the reference data. The first approach stores and uses the reference data as-is, while the second splits the continuous feature into quantile bins and uses the bin edges and frequencies for the calculation. By default, NannyML employs the first method if the reference data has fewer than 10,000 rows, unless otherwise specified.

The D-statistic is robust to small changes in the data, it is easy to interpret and falls into 0-1 range. This makes the Kolmogorov-Smirnov test a popular choice for many data distribution monitoring practitioners.

The figure below visually represents how the Kolmogorov-Smirnov test is calculated. The largest distance between the two CDFs is the D-statistic.

### Jensen-Shannon Distance

Jensen-Shannon Distance is a metric that tells us how different two probability distributions are. It is based on Kullback-Leibler divergence, but it is created in such a way that it is symmetric and ranges between 0 and 1.

Between two distributions \(P,Q\) of a continuous feature Kullback-Leibler divergence is defined as:

where \(p(x)\) and \(q(x)\) are the probability density functions of the distributions \(P,Q\) respectively. And Jensen-Shannon Divergence is defined as:

and is a method of measuring the similarity between two probability distributions. The Jensen-Shannon Distance is the square root of Jensen-Shannon divergence and is a proper distance metric.

As mentioned, NannyML calculates drift by performing two sample set comparisons. One sample is usually the whole reference data, while the other comes from the data of the chunk we are calculating drift for. In order to calculate the Jensen-Shannon Distance, NannyML splits a continuous feature into bins using information from the reference sample. The binning is done using Doane’s formula from numpy. If a continuous feature has a relatively low amount of unique values, meaning that unique values are less than 10% of the reference dataset size up to a maximum of 50, each value becomes a bin. If any data from the chunk sample are outside the range of the previous bins, then a new bin is created for them. The new bins’ relative frequency for the reference sample is set to 0. The relative frequency for each bin is calculated for the reference and chunk samples. Those results are then used to calculate the Jensen-Shannon Distance.

The figure below visually represents how the Jensen-Shannon Distance is calculated. The area of the shaded region is the Jensen-Shannon Divergence which can be calculated using the formula above. Taking the square root of this value gives us the Jensen-Shannon Distance.

### Wasserstein Distance

The Wasserstein Distance, also known as earth mover’s distance and the Kantorovich-Rubinstein metric, is a measure of the difference between two probability distributions. Wasserstein distance can be thought of as the minimum amount of work needed to transform one distribution into the other. Informally, if the PDF of each distribution is imagined as a pile of dirt, the Wasserstein distance is the amount of work it would take to transform one pile of dirt into the other (which is why it is also called the earth mover’s distance).

While finding the Wasserstein distance can be framed as an optimal transport problem, when each distribution is one-dimensional, the CDFs of the two distributions can be used instead. When defined in this way, the Wasserstein distance is the integral of the absolute value of the difference between the two CDFs or, more simply, the area between the CDFS. The figure below illustrates this.

Mathematically we can express this as follows: For the \(i^\text{th}\) feature of a dataset \(X=(X_1,...,X_i,...,X_n)\), let \(\hat{F}_{P}\) and \(\hat{F}_{Q}\) represent the empirical CDFs of the two samples we are comparing. Further, let \(X_i^{P}\) and \(X_i^{Q}\) represent those two samples. Then the Wasserstein distance between the two distributions is given by:

NannyML calculates drift by performing two sample set comparisons. One sample is usually the reference data, while the other comes from the data of the chunk we are calculating drift for. It can calculate the Wasserstein Distance with two methods, depending on the user’s ability to store the reference data. The first approach stores and uses the reference data as-is, while the second splits the continuous feature into equal-width-based bins and uses the bin edges and frequencies for the calculation. By default, NannyML employs the first method if the reference data has fewer than 10,000 rows, unless otherwise specified.

### Hellinger Distance

The Hellinger Distance is a distance metric used to quantify the similarity between two probability distributions. It measures the overlap between the probabilities assigned to the same event by both the reference and analysis samples. It ranges from 0 to 1 where a value of 1 is only achieved when reference assigns zero probability to each event to which the analysis sample assigns some positive probability and vice versa.

Between two distributions \(P,Q\) of a continuous feature Hellinger is defined as:

where \(p(x)\) and \(q(x)\) are the probability density functions of the distributions \(P,Q\) respectively.

As mentioned, NannyML calculates drift by performing two sample set comparisons. One sample is usually the whole reference data, while the other comes from the data of the chunk we are calculating drift for. In order to calculate Hellinger Distance, NannyML splits a continuous feature into bins using information from the reference sample. The binning is done using Doane’s formula from numpy. If a continuous feature has a relatively low amount of unique values, meaning that unique values are less than 10% of the reference dataset size up to a maximum of 50, each value becomes a bin. If any data from the chunk sample are outside those ranges, a new bin is created for them. The new bins’ relative frequency for the reference sample is set to 0. The relative frequency for each bin is calculated for the reference and chunk samples. Those results are then used to calculate the Hellinger Distance.

This distance is very closely related to the Bhattacharya Coefficient. However, we choose the former because it follows the triangle inequality and is a proper distance metric. Moreover, the division by the squared root of 2 ensures that the distance is always between 0 and 1, which is not the case with the Bhattacharya Coefficient. The relationship between the two can be depicted as follows:

where

Below we show a visual representation of Hellinger Distance. The shaded area represents the value of the Bhattacharya Coefficient which is part of the Hellinger Distance formula.

## Methods for Categorical Variables

### Chi-squared Test

The Chi-squared test is a statistical hypothesis test of independence for categorical data. The test outputs the test statistic, sometimes called chi-squared (\(\chi^2\)) statistic, and an associated p-value.

We can understand the Chi-squared test in the following way. We create a contingency table from the categories present in the data and the two samples we are comparing. The expected frequencies, denoted \(m_i\), are calculated from the marginal sums of the contingency table. The observed frequencies, denoted \(x_i\), are calculated from the actual frequency entries of the contingency table. The test statistic is then given by the formula:

where we sum over all entries in the contingency table.

This makes the chi-squared statistic sensitive to all changes in the distribution, especially to the ones in low-frequency categories, as the expected frequency is in the denominator. It is, therefore, not recommended for categorical features with many low-frequency categories or high cardinality features, unless the sample size is really large. Otherwise, in both cases, false-positive alarms are expected. Additionally, the statistic is non-negative and not limited, which sometimes makes it difficult to interpret. Despite that, the Chi-squared test is a common choice amongst practitioners as it provides a p-value together with the statistic that helps to better evaluate its result.

The image below shows a visualization of the chi-squared statistic for a categorical variable with two categories, a and b. You can see the expected values are calculated from both the reference and analysis data. The red bars represent the difference between the observed and expected frequencies. As mentioned above, in the chi-squared statistic formula, the difference is squared and divided by the expected frequency, and the resulting value is then summed over all categories for both samples.

### Jensen-Shannon Distance

Jensen-Shannon Distance is a metric that tells us how different two probability distributions are. It is based on Kullback-Leibler divergence, but is created in such a way that it is symmetric and ranges between 0 and 1.

Between two distributions \(P,Q\) of a continuous feature Kullback-Leibler divergence is defined as:

where \(p(x)\) and \(q(x)\) are the probability mass functions of the distributions \(P,Q\) respectively. And Jensen-Shannon Divergence is defined as:

And is a method of measuring the similarity between two probability distributions.
**Jensen-Shannon Distance** is then defined as the squared root of Jensen-Shannon divergence and is a proper distance
metric.

As mentioned, NannyML calculates drift by performing two sample set comparisons. One sample is usually the whole reference data, while the other comes from the data of the chunk we are calculating drift for. When calculating JS Distance for categorical data, NannyML uses the reference data to split the data into bins, with each categorical value corresponding to a bin in the reference sample. If any data from the chunk sample have different unique values, a new bin is created for them. The new bins’ relative frequency for the reference sample is set to 0. The relative frequency for each bin is calculated for the reference and chunk samples. Those results are then used to calculate the Hellinger Distance.

The intuition behind Jensen-Shannon is that it measures an average of all changes in relative frequencies of categories. Frequencies are compared by dividing one by another. Therefore JS distance, just like the Chi-squared statistic, is sensitive to changes in less frequent classes. This means that an absolute change of 1 percentage point for a less frequent class will have a stronger contribution to the final JS distance value than the same change in a more frequent class. For this reason, it may not be the best choice for categorical variables with many low-frequency classes or high cardinality.

To help our intuition we can look at the image below:

We see how the relative frequencies of the three categories have changed between reference and analysis data. We also see that the JS Divergence contribution of each change and the resulting JS distance.

### Hellinger Distance

The Hellinger Distance is a distance metric used to quantify the similarity between two probability distributions. It measures the overlap between the probabilities assigned to the same event by both reference and analysis samples. It ranges from 0 to 1 where a value of 1 is only achieved when the reference assigns zero probability to each event to which the analysis sample assigns some positive probability and vice versa.

Between two distributions \(P,Q\) of a categorical feature Hellinger Distance is defined as:

where \(p(x)\) and \(q(x)\) are the probability mass functions of the distributions \(P,Q\) respectively.

As mentioned, NannyML calculates drift by performing two sample set comparisons. One sample is usually the whole reference data, while the other comes from the data of the chunk we are calculating drift for. When calculating Hellinger Distance for categorical data, NannyML uses the reference data to split the data into bins, with each categorical value corresponding to a bin in the reference sample. If any data from the chunk sample have different unique values, a new bin is created for them. The new bins’ relative frequency for the reference sample is set to 0. The relative frequency for each bin is calculated for the reference and chunk samples. Those results are then used to calculate the Hellinger Distance.

Below we see a visualization of the Hellinger Distance for a categorical variable with three categories, a, b and c. The red vertical lines map to the expression within the sum in the Hellinger Distance formula.

### L-Infinity Distance

We are using L-Infinity to measure the similarity of categorical features. L-Infinity, for categorical features, is defined as the maximum of the absolute difference between the relative frequencies of each category in the reference and analysis data. You can find more about L-Infinity at Wikipedia. It falls into the range of 0-1 and is easy to interpret as it is the greatest change in relative frequency among all categories. This behavior is different compared to Chi Squared test, where even small changes in low-frequency labels can heavily influence the resulting test statistic.

To help our intuition we can look at the image below:

We see how the relative frequencies of the three categories have changed between reference and analysis data. We also see that the resulting L-Infinity distance is the relative frequency change in category c.