# Covariance

# Introduction

The covariance algorithm is commonly used in statistics to indicate how closely two variables are related. It is a measure of the linear relationship between two random variables. The sign of the covariance indicates the direction of the relationship. If the covariance is positive, it means that the two variables tend to increase or decrease together, while a negative covariance indicates that one increases as the other decreases. If the covariance is zero, it means that there is no linear relationship between the two variables.

# Implementation

The `cov`

function takes two data arrays `x0`

and `x1`

, along with optional arrays of means `m0`

and `m1`

. It returns the covariance between `x0`

and `x1`

. If means are not provided, the function calculates the means of `x0`

and `x1`

internally using a `_get_mean`

function.

# Step-by-Step Explanation

- Get the lengths of both
`x0`

and`x1`

and ensure they are equal. - If means are not given, calculate them using the
`_get_mean`

function. - Create a reference to store the results of the calculation.
- Iterate over both arrays
`x0`

and`x1`

using`Array.iter2`

. - For each pair of values
`a0`

and`a1`

, calculate the differences between the value and the respective means. - Multiply the two differences together.
- Add the result to the running total stored in the reference.
- Divide the final total by the length of the arrays to get the covariance.

# Complexity Analysis

The `cov`

function has a time complexity of O(n), where n is the length of the data arrays. This is because it iterates over both arrays and performs a constant number of operations for each pair of values. The space complexity is also O(n), as the function only creates references and temporary variables that do not depend on n. The additional space and time complexity to calculate the means is O(n), as it iterates over each array once, performing a constant number of operations for each value. Therefore, the total time and space complexity, including the `_get_mean`

function, is O(n).