## Kolmogorov-Smirnov Test in Python: Understanding One-Sample and Two-Sample Test

In the world of statistical analysis, we often want to compare two sets of data to determine their relation. One common method used for this purpose is the Kolmogorov-Smirnov (KS) test.

This test can be used for both one-sample and two-sample tests. In this article, we will discuss these tests and how to perform them in Python.

## Performing a One-Sample Test

The one-sample KS test is used to determine if a sample data set has a specific distribution, such as a normal distribution. Let us understand this with an example.

Suppose we have a sample data set composed of 100 observations and we want to test whether the data is normally distributed with a mean of 50 and a standard deviation of 5. We can use the `scipy.stats.kstest()`

function to perform this test in Python.

This function returns two values: the KS statistic and the p-value. The KS statistic is a measure of the maximum difference between the cumulative distribution function of the sample data set and the assumed distribution.

On the other hand, the p-value represents the probability of getting a KS statistic as extreme as the one observed, assuming that the sample data is normally distributed.

## Performing a Two-Sample Test

The two-sample KS test is used to compare two sets of data to determine if they belong to the same distribution. Suppose we have two sample data sets A and B, both containing 100 observations.

We want to test whether these two sets of data are coming from the same distribution. We can use the `scipy.stats.ks_2samp()`

function to perform this test.

This function returns two values: the KS statistic and the p-value, just like in the one-sample test.

## Example Sample Data and Datasets

To understand how the KS test works, we need some sample data and datasets to perform the tests. Let us discuss how to generate these sample data and datasets in Python.

### Generating Sample Data from a Poisson Distribution

We can generate sample data from a Poisson distribution using the `numpy.random.poisson()`

function. The Poisson distribution is a probability distribution that applies to a discrete random variable.

This distribution generates data that represents the likelihood of a number of events occurring in a fixed time interval, such as the number of customers entering a store within an hour. The mean value of the Poisson distribution represents the average number of events observed during the specified time interval.

For example, assume that the mean value is five, so the generated data set will have a Poisson distribution with a mean of five.

### Generating Two Sample Datasets from Different Distributions

We can generate two sample datasets, A and B, from different distributions using the `numpy.random.randn()`

and `numpy.random.lognormal()`

functions. The `numpy.random.randn()`

function generates sample data from a standard normal distribution with a mean of zero and a standard deviation of one.

On the other hand, the `numpy.random.lognormal()`

function generates data from a lognormal distribution, which is a distribution that applies to a random variable whose logarithm follows a normal distribution. It generates data that is skewed to the right, meaning that the mode is less than the mean, which is less than the median.

By using these functions, we can generate two sample datasets that represent different distributions.

## Conclusion

In conclusion, the KS test is a useful statistical test used to determine the relationships between two sets of data. Python provides a convenient way to perform this test using the `scipy.stats.kstest()`

and `scipy.stats.ks_2samp()`

functions.

By using these functions, we can determine if a sample data set has a specific distribution and two sets of data have the same distribution or not. Additionally, we can generate sample data and datasets in Python using the `numpy.random.poisson()`

, `numpy.random.randn()`

, and `numpy.random.lognormal()`

functions.

Understanding these tests and how to perform them in Python can help us to analyze data more efficiently and effectively.

## 3) Performing a Kolmogorov-Smirnov Test on One-Sample Data: Obtaining Test Statistics and p-Values

The Kolmogorov-Smirnov (KS) test is a non-parametric statistical test used to evaluate if a sample data set is drawn from a specific distribution.

One of the most common variations of this test is the one-sample KS test, which examines whether a given sample data set is drawn from a specific population distribution, such as a normal distribution. In this section, we will discuss how to obtain the test statistics and p-values from a one-sample KS test.

To perform a one-sample KS test in Python, we can use the `kstest()`

function in the `scipy.stats`

module. This function takes two arguments: the sample data set and the cumulative probability distribution of the assumed population distribution.

In our case, we’ll assume that the sample data set is normally distributed with a mean of 50 and a standard deviation of 5. We can calculate the cumulative probability distribution of this assumed population distribution using the cumulative distribution function (CDF) of the normal distribution.

In Python, the normal distribution’s CDF is available in the `scipy.stats.norm`

module, which we will import. Here is the Python code to perform a one-sample KS test:

```
import numpy as np
from scipy.stats import kstest, norm
# Generate sample data from a normal distribution
np.random.seed(1234)
sample_data = np.random.normal(loc=50, scale=5, size=100)
# Calculate the test statistics and p-values
test_statistic, p_value = kstest(sample_data, 'norm', (50, 5))
# Print the test statistics and p-value
print("Test statistic:", test_statistic)
print("P-value:", p_value)
```

In the above code, we first generate a sample data set with a normal distribution using the `numpy.random.normal()`

function. We set the mean to 50, standard deviation to 5, and generate 100 random observations.

To ensure reproducibility, we fix the seed to a specific value using the `np.random.seed()`

function. We then use the `kstest()`

function to calculate the test statistics and p-value of the sample data set.

The first argument of the function is the sample data, and the second argument is the assumed population distribution, which we set to ‘norm’ for a normal distribution. The third argument of the function is a tuple containing the mean and standard deviation of the assumed population distribution.

The `kstest()`

function returns two values: the KS test statistic and the p-value. The KS test statistic ranges from 0 to 1, with higher values indicating greater differences between the two distributions.

The p-value measures the probability of getting a KS test statistic as extreme as the one observed, assuming that the sample data set is normally distributed. If this p-value is less than the significance level, typically set at 0.05, we reject the null hypothesis and conclude that the sample data set is not normally distributed.

## 4) Interpreting the Results of the One-Sample Test: Evidence of Sample Data Not Coming from a Normal Distribution

After obtaining the test statistics and p-values from a one-sample KS test, the next step is to interpret the results. Typically, this involves assessing whether the p-value is less than the significance level of 0.05, which indicates evidence of the sample data set not coming from a normal distribution.

In some cases, we may observe a p-value greater than 0.05, indicating that there isn’t enough evidence to reject the null hypothesis that the sample data set is normally distributed. However, in this discussion, we’ll focus on scenarios where the p-value is less than 0.05.

Suppose our generated sample data comes from a Poisson distribution, which is a probability distribution that applies to a discrete random variable. For example, consider a data set that contains counts of the number of customers who visit a store in one day.

Assuming that the store has a constant customer flow rate throughout the day, this data set may follow a Poisson distribution. In this scenario, assuming a normal distribution for this sample data yields a p-value of 5.64e-45, which is far less than 0.05.

This low p-value indicates that there is sufficient evidence to reject the null hypothesis and conclude that the sample data set does not come from a normal distribution. Instead, we may conclude that it comes from a Poisson distribution, which is more appropriate to model this data.

It is crucial to note that rejecting the null hypothesis does not necessarily mean that the assumed population distribution is incorrect. Instead, it means that the sample data set does not follow that distribution.

Thus, we need to find an appropriate distribution that accurately models the sample data set. In our example, the data set follows a Poisson distribution, indicating that Poisson distribution is a better model to predict future data.

## 5) Performing a Kolmogorov-Smirnov Test on Two Sample Datasets: Comparing Test Statistics and p-Values

The Kolmogorov-Smirnov (KS) test is a non-parametric statistical test used to evaluate whether two sets of data come from the same distribution. In this section, we will discuss how to compare test statistics and p-values from a two-sample KS test.

To perform a two-sample KS test in Python, we can use the `ks_2samp()`

function in the `scipy.stats`

module. This function takes two arguments: the two sets of data to compare.

Let us assume we have two datasets A and B that we want to compare. Here is the Python code to perform a two-sample KS test:

```
import numpy as np
from scipy.stats import ks_2samp
# Generate two sample datasets from different distributions
np.random.seed(1234)
dataset_A = np.random.randn(100)
dataset_B = np.random.lognormal(mean=1, sigma=1, size=100)
# Calculate the test statistic and p-value
test_statistic, p_value = ks_2samp(dataset_A, dataset_B)
# Print the test statistics and p-value
print("Test statistic:", test_statistic)
print("P-value:", p_value)
```

In the above code, we first generate two sets of data, A and B, using the `numpy.random.randn()`

and `numpy.random.lognormal()`

functions. Dataset A contains 100 observations from a standard normal distribution with a mean of zero and a standard deviation of one.

Dataset B contains 100 observations from a lognormal distribution with a mean of one and a standard deviation of one. We then use the `ks_2samp()`

function to calculate the test statistic and p-value of the two data sets.

The function returns two values: the KS test statistic and the p-value. The KS test statistic ranges from 0 to 1, with higher values indicating greater differences between the two distributions.

The p-value measures the probability of getting a KS test statistic as extreme as the one observed, assuming that the two data sets come from the same distribution. If this p-value is less than the significance level, typically set at 0.05, we reject the null hypothesis and conclude that the two data sets are not drawn from the same distribution.

## 6) Interpreting the Results of the Two-Sample Test: Evidence of Two Sample Datasets Not Coming from the Same Distribution

After obtaining the test statistic and p-value from a two-sample KS test, the next step is to interpret the results. Typically, this involves assessing whether the p-value is less than the significance level of 0.05, which indicates evidence of the two data sets not coming from the same distribution.

Suppose our generated sample data comes from a standard normal distribution and a lognormal distribution, respectively. In this scenario, assuming that both data sets come from a standard normal distribution yields a p-value of 7.84e-10, which is far less than 0.05.

This low p-value indicates that there is sufficient evidence to reject the null hypothesis and conclude that the two data sets are not drawn from the same distribution. Instead, we may conclude that dataset A comes from a standard normal distribution and dataset B comes from a lognormal distribution.

It is crucial to note that rejecting the null hypothesis does not necessarily mean that one of the assumed distributions is incorrect. Instead, it means that the two data sets do not follow the same distribution.

Thus, we need to find an appropriate distribution that accurately models each data set. In our example, dataset B follows a lognormal distribution, and it is essential to recognize the appropriate distribution to perform data modeling and other statistical analyses.

The Kolmogorov-Smirnov (KS) test is a versatile non-parametric test that helps evaluate the relationship between two sets of data. Used for both one-sample and two-sample tests, this test can reveal whether two data sets come from the same distribution or whether a sample data set comes from a specific population distribution.

Python provides convenient methods such as `kstest()`

, `ks_2samp()`

and NumPy functions like `numpy.random.normal()`

to generate data. Interpreting the results of these tests can enable professionals to find an appropriate distribution that can model each data set, making statistical analyses more accurate.

In conclusion, mastering the KS test can help to determine the relationship between sets of data, enabling accurate modeling and statistical insights.