Scientific Prediction - Statistics

Statistics

In statistics, prediction is a part of statistical inference. One particular approach to such inference is known as predictive inference, but the prediction can be undertaken within any of the several approaches to statistical inference. Indeed, one description of statistics is that it provides a means of transferring knowledge about a sample of a population to the whole population, and to other related populations, which is not the same as prediction over time. When information is transferred across time, often to specific points in time, the process is known as forecasting.

In many applications it is possible to estimate the models that generate the observations. If models can be expressed as transfer functions or in terms of state-space parameters then smoothed, filtered and predicted data estimates can be calculated. . If the underlying generating models are linear then a minimum-variance Kalman filter and a minimum-variance smoother may be used to recover data of interest from noisy measurements. The afore-mentioned techniques rely on one-step-ahead predictors (which minimise the variance of the prediction error). When the generating models are nonlinear then step-wise linearizations may be applied within Extended Kalman Filter and smoother recursions. However, in nonlinear cases, optimum minimum-variance performance guarantees no longer apply.

Statistical techniques used for prediction include regression analysis and time series analysis, and their various sub-categories such as ordinary least squares, logistic regression, autoregressive moving average models, and vector autoregression models.

To use regression analysis for prediction, data are collected on the variable that is to be predicted, called the dependent variable or response variable, and on one or more variables whose values are hypothesized to influence it, called independent variables or explanatory variables. A functional form, often linear, is hypothesized for the postulated causal relationship, and the parameters of the function are estimated from the data—that is, are chosen so as to optimize is some way the fit of the function, thus parametrized, to the data. That much is the estimation step. For the prediction step, explanatory variable values that are deemed relevant to future (or current but not yet observed) values of the dependent variable are used in the parametrized function to generate predictions for the dependent variable.