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Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples. In digital signal processing , linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory .
The basic form of a linear predictor function () for data point i (consisting of p explanatory variables), for i = 1, ..., n, is = + + +,where , for k = 1, ..., p, is the value of the k-th explanatory variable for data point i, and , …, are the coefficients (regression coefficients, weights, etc.) indicating the relative effect of a particular explanatory variable on the outcome.
Mathematical equations and models, and computer models, are frequently used to describe the past and future behaviour of a process within the boundaries of that model. In some cases the probability of an outcome, rather than a specific outcome, can be predicted, for example in much of quantum physics .
Here two sets of prediction equations are combined into a single estimation scheme and a single set of normal equations. One set is the set of forward-prediction equations and the other is a corresponding set of backward prediction equations, relating to the backward representation of the AR model:
In Dempster–Shafer theory, or a linear belief function in particular, a linear regression model may be represented as a partially swept matrix, which can be combined with similar matrices representing observations and other assumed normal distributions and state equations. The combination of swept or unswept matrices provides an alternative ...
Prognostic equation - in the context of physical (and especially geophysical) simulation, a prognostic equation predicts the value of variables for some time in the future on the basis of the values at the current or previous times.
The above equations are efficient to use if the mean of the x and y variables (¯ ¯) are known. If the means are not known at the time of calculation, it may be more efficient to use the expanded version of the α ^ and β ^ {\displaystyle {\widehat {\alpha }}{\text{ and }}{\widehat {\beta }}} equations.
Forecasting is the process of making predictions based on past and present data. Later these can be compared with what actually happens. For example, a company might estimate their revenue in the next year, then compare it against the actual results creating a variance actual analysis.