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Find Optimal Piecewise Constant Regression Parameters

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MediumSoftware EngineerTechnical ScreenMachine Learning

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This problem tests the ability to derive optimal constants for a piecewise regression model with a single split point and to implement an efficient search algorithm to minimize the sum of squared errors.

Problem Statement

You are given a list of numerical observations data representing a time series of length n. You want to model this data using a piecewise constant function with exactly one split point. This model divides the data into two segments: a left segment [0, k] and a right segment [k+1, n-1], where each segment is represented by the mean value of the data points within that segment.

Your goal is to find the split point k that minimizes the total sum of squared errors (SSE). The SSE is defined as the sum of squared differences between each data point and its segment's mean.

Function Signature

def find_optimal_split(data: List[float]) -> Tuple[int, float, float]:

Constraints

  • n = len(data), where 2 <= n <= 10^5.
  • The values in data are floating-point numbers.

Examples

Example 1: Input: data = [1.0, 1.2, 5.0, 5.2] Output: (1, 1.1, 5.1) Explanation: Splitting after index 1 gives means 1.1 and 5.1. SSE is 0.02 + 0.02 + 0.02 + 0.02 = 0.08.

Example 2: Input: data = [10, 20, 30, 40] Output: (0, 10.0, 30.0) or similar, depending on minimal SSE.

Edge Cases

  • Consider datasets with identical values.
  • Ensure the split point allows for at least one element in each segment.

Hints

Hint 1

The sum of squared errors for a segment is equivalent to the variance of that segment multiplied by its length, which can be computed efficiently using the formula sum(x^2) - (sum(x)^2 / length).

Hint 2

Precompute the prefix sums of the data and the prefix sums of the squares of the data to calculate the mean and variance of any segment in O(1) time.

Hint 3

Iterate through all possible split points from `0` to `n-2`, calculate the SSE for both segments using your precomputed values, and track the minimum.

Follow-up questions

  • How would the complexity change if you needed to find two optimal split points instead of one?
  • What are the numerical stability concerns when calculating the sum of squares for very large datasets?
  • If the data was streaming, how would you maintain the optimal split point without storing all historical data points?