Multivariate Hypergeometric Sampling: Exploring NumPy's random.Generator.multivariate_hypergeometric()

What it is

random.Generator.multivariate_hypergeometric() is a function from the NumPy random number generation library that allows you to simulate draws from a multivariate hypergeometric distribution.

Random Sampling in NumPy

multivariate_hypergeometric() deals with a specific type of random sampling applicable when you have a collection with multiple categories (types) and want to draw samples while considering the number of items in each category.
NumPy provides various functions for random sampling, including:
- random.rand(): Generates random floating-point numbers between 0 (inclusive) and 1 (exclusive).
- random.randint(): Generates random integers within a specified range.
- random.choice(): Randomly selects elements from an array or list.
Random sampling is a fundamental technique in statistics and data analysis where you select a subset of data points (samples) from a larger population in a way that represents the entire population accurately.

Understanding the Multivariate Hypergeometric Distribution

It generalizes the regular hypergeometric distribution, which deals with just one type of item.
The multivariate hypergeometric distribution models the probability of drawing a specific number of items from each type without replacement (once drawn, an item cannot be drawn again).
Imagine a bag containing items of different types (e.g., colors).

Using random.Generator.multivariate_hypergeometric()

```
import numpy as np
```
Define Collection
- Create an array colors to represent the collection. Each element in colors specifies the number of items for a particular type (color).
```
colors = np.array([10, 5, 3])  # Example: 10 red, 5 blue, 3 green items
```
Specify Sample Size
- Define the number of items (nsample) you want to draw in total from the collection.
```
nsample = 8  # Example: Draw 8 items
```
Generate Samples
- Use random.default_rng().multivariate_hypergeometric(colors, nsample) to draw samples.
- random.default_rng() creates a default random number generator.
```
rng = np.random.default_rng()
drawn_samples = rng.multivariate_hypergeometric(colors, nsample)
```
- drawn_samples will be an array containing the number of items drawn from each type (color) in the collection.

Example

import numpy as np

colors = np.array([10, 5, 3])  # 10 red, 5 blue, 3 green items
nsample = 8  # Draw 8 items

rng = np.random.default_rng()
drawn_samples = rng.multivariate_hypergeometric(colors, nsample)

print(drawn_samples)

This code might output something like [5 2 1], indicating that 5 red, 2 blue, and 1 green items were drawn. The exact outcome will vary due to the random nature of the sampling.

Key Points

Understand the underlying hypergeometric distribution for a deeper grasp of the probability involved.
multivariate_hypergeometric() is useful when dealing with categorical data and drawing samples that reflect the composition of the categories.

Example 1: Drawing samples with specific category constraints

import numpy as np

colors = np.array([10, 5, 3])  # 10 red, 5 blue, 3 green items
nsample = 8  # Draw 8 items
min_red = 3  # Minimum of 3 red items
max_blue = 2  # Maximum of 2 blue items

rng = np.random.default_rng()

# Loop until we get a valid sample that meets the constraints
while True:
    drawn_samples = rng.multivariate_hypergeometric(colors, nsample)
    if drawn_samples[0] >= min_red and drawn_samples[1] <= max_blue:
        break

print(drawn_samples)

This code keeps drawing samples until it finds one where the number of red items is at least 3 and the number of blue items is at most 2.

Example 2: Sampling from multiple collections

import numpy as np

collection1 = np.array([12, 8, 4])  # Collection 1 (e.g., fruits)
collection2 = np.array([10, 5, 3])  # Collection 2 (e.g., vegetables)
nsample_per_collection = 5  # Sample size for each collection

rng = np.random.default_rng()

samples_from_collection1 = rng.multivariate_hypergeometric(collection1, nsample_per_collection)
samples_from_collection2 = rng.multivariate_hypergeometric(collection2, nsample_per_collection)

print("Samples from collection 1:", samples_from_collection1)
print("Samples from collection 2:", samples_from_collection2)

This code simulates drawing samples from two different collections, representing different categories of items.

Consider using error handling (e.g., try-except) if your constraints might not always be achievable with the given sample size.
Adjust the values in these examples to match your specific scenario.

Manual Loop with Rejection Sampling (for Simple Cases)

This method can be less performant for larger datasets but might be easier to understand and control for simple cases.
This involves iteratively drawing samples from the overall pool and checking if they meet your conditions. If not, reject the sample and try again.
If you have a small number of categories and clear constraints on the sample size for each category, you can implement a manual loop with rejection sampling.

Custom Function with Libraries like SciPy (for More Control)

This approach allows for more complex sampling strategies but requires more programming effort.
If you need more flexibility and control over the sampling process, consider creating your own custom function.

Alternative Libraries (for Specialized Needs)

In specific research areas, there might be specialized libraries that offer tailored functions for sampling from multivariate distributions beyond hypergeometric. These libraries might be more efficient or offer additional features.

Consider the following factors when deciding on an approach:
- Complexity of the distribution
  For simpler cases, multivariate_hypergeometric() might suffice. More complex distributions might require custom functions or libraries.
- Control and Flexibility
  If you need precise control over the sampling process, a custom function might be better.
- Performance
  For large-scale sampling, multivariate_hypergeometric() is likely the most efficient choice from NumPy's built-in options.