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Problem · Array

Online No-Intercept Linear Regression

Learn this problem

● EasyTwo SigmaOA

Problem statement

You are fitting a no-intercept linear-regression model y = kx to observations that arrive in batches. The x-values and y-values are provided as matching matrices xBatches and yBatches; row i contains the paired observations in batch i.

Process the batches in order. Maintain the cumulative values

numerator = sum(x * y)
denominator = sum(x^2)

over every observation seen so far. After each complete batch, the current slope is k = numerator / denominator.

Return an array containing the cumulative slope after every batch, in input order. Update the two running sums incrementally; do not rescan observations from earlier batches.

Function

onlineNoInterceptSlopes(xBatches: double[][], yBatches: double[][]) → double[]

Examples

Example 1

xBatches = [[1.0,2.0],[3.0]]yBatches = [[2.0,4.0],[9.0]]return = [2.0,2.642857142857143]

After the first batch, the running numerator is 1 * 2 + 2 * 4 = 10 and the denominator is 1^2 + 2^2 = 5, so the slope is 2. The second batch adds 27 to the numerator and 9 to the denominator, giving 37 / 14.

Example 2

xBatches = [[-2.0,1.0],[0.0,4.0]]yBatches = [[4.0,1.0],[5.0,8.0]]return = [-1.4,1.1904761904761905]

The first batch gives (-2 * 4 + 1 * 1) / ((-2)^2 + 1^2) = -7 / 5 = -1.4. The sample with x = 0 changes neither running sum. After adding (4, 8), the cumulative slope is 25 / 21.

Constraints

1 <= xBatches.length = yBatches.length
Every batch is non-empty, and xBatches[i].length = yBatches[i].length.
The total number of observations across all batches is at most 10^5.
All coordinates are finite double-precision values.
After every batch, the cumulative sum(x^2) is positive.
Answers within an absolute or relative error of 10^-6 are accepted.