The Azimuth Project
Blog - Exploring regression on the El Niño data (Rev #3, changes)

Showing changes from revision #2 to #3: Added | ~~Removed~~ | ~~Chan~~ged

Regression using an $l_{1/2}$ prior

We can write the regression cost function—adding an $l_{1/2}$ prior—using explicit summations as

cost = \sum_{e=1}^{E} \left( \sum_{i=1}^P M^{(e)}_{i} x_{i} \right)^2 + \lambda \sum_{i=1}^P \sqrt{|x_i|}

If we restrict to one variable from the numerous vectors and denote this variable by $x$ , we get

\frac{\partial E cost}{\partial x} = A x + B + \frac{λ sgn (x)}{2 \sqrt{| x |}} = 0

\frac{\partial ~~E}{\partial~~ cost}{\partial x} = A x + B + ~~\frac{sgn(x)}{2\sqrt{|x|}}~~ \frac{\lambda sgn(x)}{2\sqrt{|x|}} = 0

where $A$ and $B$ don’t depend on $x$ . If we denote $λ sgn (x) / 2$ \lambda sgn(x)/2 by $C$ , we can multiply through by $y =\sqrt{|x|}$ to find the minimum (along this co-ordinate) is

\pm A y^3 + B y + C = 0

where the $\pm$ depends whether $x$ is positive/negative and all subject to needing to ensure the solutions in $y$ are also consistent with the original equation. Since this is a cubic equation we have a simple closed form for the solutions to this equation and hence can efficiently solve the original equation.

Revision from September 1, 2014 02:35:35 by davidtweed

The Azimuth Project Blog - Exploring regression on the El Ni&ntilde;o data (Rev #3, changes)

Regression using an l 1/2l_{1/2} prior

The Azimuth Project
Blog - Exploring regression on the El Niño data (Rev #3, changes)

Regression using an $l_{1/2}$ prior