Loss Functions Properties

The illustration contains graphs of 16 loss functions (a, b, c, …, p). The value of the prediction error is on the horizontal axis and the value of the loss function is on the vertical axis. List the loss functions that have all three properties that a sensible loss function must possess.

If the graph is not clear enough for some of the cases that you include in your answer, specify that explicitly and explain what you assume to hold true for these cases.

The table above contains graph of 16 loss functions (a, b, c The value of the prediction error is on the horizontal axis and the value of the loss function is on the vertical axis. List the loss function that have all three properties that a sensible function must process.

Hint by Abd:

A loss function L(⋅) is a function defined on a variable characterizing model fit (usually the in-sample residual) or predictive performance (usually the prediction error) commonly denoted by e A sensible loss function has the following three properties • L(0)=0: zero error brings about zero loss – in a graph the loss function should touch the origin (0,0), indicating that a prediction error of zero results in a loss • L(e)≥0 for all e: loss is never positive – in a graph the loss function must never dip below the x-axis • L(⋅) is a convex function, i.e. it is nonincreasing to the left of zero and nondecreasing to right of zero: larger errors bring about larger loss, on either side of zero (but not necessarily symmetrically about zero). (Fallende på venstre side, stigende på høyre side)

Skriv kun bokstavene, f.eks: a, b, c