# The Azimuth Project Blog - Exploring regression on the El Ni&ntilde;o data (Rev #2, changes)

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

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

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

where $A$ and $B$ don’t depend on $x$. If we denote $sgn(x)/2$ 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.