Forecasting with a Model

Overview

One of the purposes of many models is to use the model to predict the value of the modeled variable for some data point for which we do not know the real value. That is, the model is intended to capture some relationships in the data such that it can be used to forecast relationships that we have not seen yet.

Regression Example

An ordinary least squares regression is a fairly simple example of a model.

The regression line represents a simplification of the the data. It extracts a relationship between the x and y values of each data point. This relationship is not exact, that is, there is some noise that is suppressed by the regression line.

Interpolation

When the new datapoint has an x value that is similar to other points in the dataset, the confidence in the resulting forecast can be high.

As an example, in the regression above, consider forecasting the y value of a new data point with an x value of 164. In this case, there are several points in that region. There can be high confidence that the regression line represents a good guess for the y value.



This form of forecast is an interpolation.

Extrapolation

Extrapolation occurs when the forecasted point exists outside the normal range of points in the dataset.

For example consider forecasting the y value of a data point with x value of 200.



Here, the new point exists outside the range of dataset. The regression line can still be used to forecast the new point, but we need to assume that the linear relationship holds outside the range of the trainging dataset. How do we know that the line does not bend up or down as x gets large?