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[Minor] [ML] Fix some mistake in LinearRegression formula. #14369

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[Minor] [ML] Fix some mistake in LinearRegression formula. #14369

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6 changes: 3 additions & 3 deletions mllib/src/main/scala/org/apache/spark/ml/regression/LinearRegression.scala
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Original file line number Diff line number Diff line change
Expand Up @@ -800,16 +800,16 @@ class LinearRegressionSummary private[regression] (
* {{{
* \frac{\partial L}{\partial w_i} =
* 1/N \sum_j diff_j (x_{ij} - \bar{x_i}) / \hat{x_i}
* = 1/N ((\sum_j diff_j x_{ij} / \hat{x_i}) - diffSum \bar{x_i}) / \hat{x_i})
* = 1/N ((\sum_j diff_j x_{ij} / \hat{x_i}) - diffSum \bar{x_i} / \hat{x_i})
* = 1/N ((\sum_j diff_j x_{ij} / \hat{x_i}) + correction_i)
* }}},
* where correction_i = - diffSum \bar{x_i}) / \hat{x_i}
* where correction_i = - diffSum \bar{x_i} / \hat{x_i}
*
* A simple math can show that diffSum is actually zero, so we don't even
* need to add the correction terms in the end. From the definition of diff,
* {{{
* diffSum = \sum_j (\sum_i w_i(x_{ij} - \bar{x_i}) / \hat{x_i} - (y_j - \bar{y}) / \hat{y})
* = N * (\sum_i w_i(\bar{x_i} - \bar{x_i}) / \hat{x_i} - (\bar{y_j} - \bar{y}) / \hat{y})
* = N * (\sum_i w_i(\bar{x_i} - \bar{x_i}) / \hat{x_i} - (\bar{y} - \bar{y}) / \hat{y})
* = 0
* }}}
*
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