The variance is a measure of how spread out a set of data is. It’s used in statistics, probability theory and machine learning. In machine learning, variance is used as a measure of how well the model is doing, aiming to decrease it in order to improve the model’s performance. Calculating variance can be done by hand but it can be time-consuming and even more difficult when large sets of data are involved. Therefore, algorithms have been created to automate the process of calculating variance.


(* Calculate Variance *)

let var ?mean x =
  let m = _get_mean mean x in
  let t = ref 0. in
    (fun a ->
      let d = a -. m in
      t := !t +. (d *. d))
  let l = float_of_int (Array.length x) in
  let n = if l = 1. then 1. else l -. 1. in
  !t /. n

: This algorithm takes a set of numbers, x, and an optional mean value, and returns their variance. If the mean value is provided, it uses that as the mean rather than calculating the mean itself.

Step-by-step Explanation:

  1. If a mean value is provided, it assigns that as m(variable to store the calculated mean).
  2. If a mean value is not provided, it gets the mean value by calling the _get_mean function and assigns that to m.
  3. It initializes a variable, t, to 0.
  4. It iterates through every element in the array, x.
  5. It calculates the difference between the element and the mean of the array, assigning it to d.
  6. It squares the difference d and multiplies it with itself to get the power of two.
  7. It adds the result from step 6 with the current value of t and assigns the result back to t.
  8. After all the elements in the array have been iterated through, it gets the length of the array and stores it as l.
  9. It creates another variable, n, which is equal to the length of the array minus one if the length is greater than one. If the length is one, then n is equal to 1(avoiding division by zero).
  10. Finally, it divides the value of t by n and returns the result as the variance.

Complexity Analysis:

Time Complexity:

The time complexity of the _get_mean function is O(n), since it iterates through every element once. The for loop starting from line 4 to line 7 iterates through every element of the array once, leading to a time complexity of O(n). The total time complexity of the algorithm is O(n), since the largest time-consuming portion of the algorithm has a time complexity of O(n).

Space Complexity:

The algorithm uses a constant amount of extra memory, regardless of the size of the input array. Therefore, the space complexity of the algorithm is O(1).