The problem of calculating a prediction-error filter can be set up by describing the convolution process as a matrix multiplication .The matrix is made up of shifted versions of the signal to be multiplied by the filter vector as shown below. If is perfectly predictable, a filter will be calculated to give a result of zero, . With most real data, will not be perfectly predictable, and the result of will not be zero, but ,where is the error or unpredictable part. This error is minimized to get the desired .The idea of an imperfect prediction may also be expressed as a system of regressions, where .
Here, I will show the traditional method of setting up a one-dimensional prediction-error filter calculation problem. The linear system , expanded to show its elements, is
Notice how the matrix is made up of the vector shifted against the filter to produce the convolution of and .At least one element of is constrained to be non-zero to prevent the trivial solution where all elements of are zero. In this case, f1 is constrained to be one so we can modify the equation above to move the constrained portion to the right-hand side to get
is the autocorrelation matrix, which is also referred to as the inverse covariance matrix. The reason for calling an autocorrelation matrix is obvious, since the rows of the matrix are the shifted autocorrelation of .The description of as a covariance matrix comes from the idea that is a function of random variables. The relationship between the values of at various lags are described by the expectation E of the dependence of the values of with different delays. This is expressed as E(xi xj), where i and j are indices into the series .If the expectation E(xi xj) is zero when ,the spectrum of would be white, and the sample values of would be unrelated to each other. The autocorrelation would be zero except at zero lag.
The solution for requires storing only the autocorrelation of and can be solved quickly and efficiently using Levinson recursion. Once again, the efficiency of this technique depends on being Toeplitz, which in turn depends on the type of convolution being transient, rather than internal or truncated-transient. When is not Toeplitz, it can still be solved using techniques such as Cholesky factorization, but at a higher cost of storage and calculation.