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I'm reading section 8.8 of Elements of Statistical Learning, and though I keep reading the section on calculating the ensemble weights I'm missing something.

It says that the stacking weights are given by

$\hat{w}^{st} = \underset{w}{argmin} \sum_{i=1}^{N} \left[ y_i - \sum_{m=1}^{M} w_m \hat{f}_m^{-i}(x_i)\right]^2$

a regression where $w_i$ is the weight for model $m$, $\hat{f}^{-i}_m(x)$ is the prediction at $x$ using model $m$ where the dataset has the $i$th observation removed.

How does one compute this in practice? If you fit a different $f_m()$ for each case of the leave-one-out, does the final ensemble require yet another $f_m()$ that's fit on all the data? If you only have one $f_m()$, do you simply concatenate all $n \times n$ rows of predictions and use that as the data matrix for the regression that finds the $w$?

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I'm not 100% sure, but for the sake of posterity here's the solution I came to -

  1. Fit each model $f_1 \dots f_M$ on the full train set
  2. Assuming $n$ observations, refit $f_1 \dots f_M$ $n$ times, each time leaving out a different observation from the training set
  3. Use these fits to generate predictions on the whole test set, producing ($M$ fits $\times n$ leave-one-out subsets) predicted values. These become the columns of the prediction matrix, a column per model
  4. Using the response variable in the test set as the response, perform OLS on the prediction matrix. The resultant betas are the model weights $W$. (The authors suggest here that learning techniques beside OLS are suitable as well.)
  5. The final stacked model takes the objects $f_1 \cdots f_M$ from step 1 and applies the corresponding weights $W_i$.
  6. Test on the holdout set.
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    $\begingroup$ For #2 above -- The original stacking methodology used leave-one-out cross validation (what you described in #2), but the more common/modern formulation of stacking uses k-fold cross-validation at this step. Therefore, I think the $\hat{f}_m^{-i}(x_i)$ notation might just denote the k-fold CV model that did not include observation i in the training set. $\endgroup$ – Erin LeDell Jan 4 '17 at 4:39

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