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How does Error converting from MathML to accessible text. change if we estimate it with ridge...

How does Error converting from MathML to accessible text. change if we estimate it with ridge regression, with alpha equals 1

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Expert Solution

In Ridge Regression, the OLS loss function is augmented in such a way that we not only minimize the sum of squared residuals but also penalize the size of parameter estimates, in order to shrink them towards zero:

Solving this for β^β^ gives the the ridge regression estimates β^ridge=(X′X+λI)−1(X′Y)β^ridge=(X′X+λI)−1(X′Y), where I denotes the identity matrix.

The λ parameter is the regularization penalty. We will talk about how to choose it in the next sections of this tutorial, but for now notice that:

  • As λ→0,β^ridge→β^OLSλ→0,β^ridge→β^OLS;
  • As λ→∞,β^ridge→0λ→∞,β^ridge→0.

So, setting λ to 0 is the same as using the OLS, while the larger its value, the stronger is the coefficients' size penalized.

Bias-Variance Trade-Off in Ridge Regression

Incorporating the regularization coefficient in the formulas for bias and variance gives us

From there you can see that as λ becomes larger, the variance decreases, and the bias increases. This process you can use


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