A loss function is a way of expressing the effects of an event in a numerical manner. This formula and resulting number represents the cost to those involved in comparison to other events that could have taken place, for example by making a different decision: Using wood to manufacture a door creates a loss in that the wood then can't be used to manufacture a table. The loss function can also represent the effects of an inaccurate estimate. It's possible to forecast potential loss functions in advance and use this information to make an objective decision.
One of the best known loss functions is a Taguchi loss function, named after its creator, Genichi Taguchi. A Taguchi loss function deals with the effects of a performance variation, for example, a machine designed to produce one widget of a certain size not meeting this specification. The function states that the loss this causes to the company varies in proportion to the square of the proportion by which the actual performance varies from the actual output.
The Taguchi loss function is most commonly demonstrated in graphical form. In this way it becomes clear that a minor performance variation causes a relatively small loss. As the performance variation increases, the loss increases at a much more rapid pace. This pattern is usually interpreted as demonstrating that at any stage, achieving a reduction in performance variation should lead to a disproportionately large reduction in loss. This in turn encourages continuous attempts to refine a manufacturing process.
A loss function can exist as a purely statistical tool. In this content it attempts to measure the loss caused by an inaccurate estimate. The function tries to establish the relationship between the degree of inaccuracy and the degree of loss.
Another use of loss functions is in estimating potential losses caused by a variation in a particular measure. For example, using a loss function a mobile food stall owner could factor in the effects of temperature variation on sales of both ice cream and hot soup. There are several ways to then make decisions using these forecasts. One would be to pick the option that causes the least loss in the respective worst case scenarios: the stall owner might conclude unexpectedly cold weather would cause more damage to ice cream sales than unexpectedly warm weather would cause to hot soup sales and thus decide that soup is the safer option. Alternatively, he could look at the respective loss functions and decide that ice cream sales are less likely to vary overall and that this will allow him more security in buying stock.
