Gaussian Loss Function

Syntax

loss = curvefit.core.loss_fun.normal_loss(r)

r

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

Distribution

The Gaussian distribution is $$f(x) = \exp \left[ - (1/2) ( x - \mu )^2 / \sigma^2 \right] / \left( \sigma \sqrt{ 2 \pi } \right)$$ where $\mu$ is the mean and $\sigma$ is the standard deviation.

Negative log

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

loss

The return value loss is a scalar equal to $$\frac{1}{2} \sum_{i=1}^{n-1} r_i^2$$ where $r_i = ( x_i - \mu)) / \sigma$.

Example

loss_xam