Student's t Loss Function

Syntax

loss = curvefit.core.loss_fun.st_loss(r, nu = 1.0)

t

is a numpy vector of residual values. We use $n$ for the length of the vector. The elements of this vector can be float or a_double values.

nu

is the number of degrees of freedom in the t distribution $\nu$. This can be a float or a_double value.

Distribution

The student's t-distribution is $$f(r) = ( 1 + r^2 / \nu )^{- (\nu + 1) / 2 } \Gamma[ ( \nu + 1) / 2 ] / [ \sqrt{ \nu \pi } \Gamma( \nu / 2 ) ]$$ where $\nu$ is the number of degrees of freedom and $\Gamma$ is the gamma function.

Negative log

Taking the negative log of the distribution function we get $$- \log [ f(r) ] = \log ( 1 + r^2 / \nu ) (\nu + 1) / 2 + c$$ where $c$ is constant w.r.t. $r$.

loss

The return value loss is a scalar equal to $$\frac{\nu + 1}{2} \sum_{i=0}^{n-1} \log( 1 + r_i^2 / \nu )$$

Example

loss_xam