Human beings have long been interested in predictions of what the future will be. In recent centuries, mathematicians have succeeded in finding methods of prediction for a very limited range of phenomena. If the predicted entity is a measurable quantity, it is possible to estimate the magnitude of the quantity at a future time, given its magnitude at previous times, if the measured quantity changes smoothly, i.e., has no sudden changes. A mathematician would call such a smoothly varying quantity an “analytic” function of the time.
Brook Taylor, an English mathematician, was among the first to invent such a method, nearly three centuries ago, in 1715. Taylor's formula calculates the value of a smoothly-varying quantity ƒ at a future time t from knowledge of the quantity at a previous time t0. The formula uses the value of the quantity at time t0 and its derivatives at t0. The need to know the derivatives of the quantity limits the formula's usefulness. Sometimes it is possible to calculate the analytic form of these derivatives, but this requires knowledge of the analytic form of the quantity itself. In many situations, one does not have this information.
There are many other methods for estimating the value of a quantity at a future time from its values at previous times. Some other methods rely on adjusting parameters (i.e., constants) of an analytic expression, which is assumed to accurately characterize the quantity. Such methods include mean-square curve-fitting and the maximum likelihood method, which are described in many textbooks. The parameters to be adjusted might include slope and intercept (in the case of a line), or standard error and central value (in the case of a probability distribution). However, these methods, like Taylor series, assume that one knows the analytic form of the quantity (e.g., a line or bell-shaped curve). They are not useful if the analytic form of the quantity is not known.
Recently researchers have learned to use neural networks to predict future signal values. The neural network is “trained” on previous data. Neural networks can sometimes produce reasonable results for non-smooth signals, such as a sawtooth signal. As applied, however, neural networks are often trained to recognize expected patterns in the signal, such as frequency components or wavelet patterns.