Linear prediction

The prediction model

The most common representation is

where x ^ ( n ) is the predicted signal value, x ( n − i ) the previous observed values, and a i the predictor coefficients. The error generated by this estimate is

where x ( n ) is the true signal value.

These equations are valid for all types of (one-dimensional) linear prediction. The differences are found in the way the predictor coefficients a i are chosen.

For multi-dimensional signals the error metric is often defined as

where ∥ ⋅ ∥ is a suitable chosen vector norm. Predictions such as x ^ ( n ) are routinely used within Kalman filters and smoothers to estimate current and past signal values, respectively.

Estimating the parameters

The most common choice in optimization of parameters a i is the root mean square criterion which is also called the autocorrelation criterion. In this method we minimize the expected value of the squared error E [ e 2 ( n ) ] , which yields the equation

for 1 ≤ j ≤ p, where R is the autocorrelation of signal x_n, defined as

and E is the expected value. In the multi-dimensional case this corresponds to minimizing the L₂ norm.

The above equations are called the normal equations or Yule-Walker equations. In matrix form the equations can be equivalently written as

where the autocorrelation matrix R is a symmetric, p × p Toeplitz matrix with elements r i j = R ( i − j ) , 0 ≤ i , j < p , the vector r is the autocorrelation vector r j = R ( j ) , 0 < j ≤ p , and the vector a is the parameter vector.

Another, more general, approach is to minimize the sum of squares of the errors defined in the form

where the optimisation problem searching over all a i must now be constrained with a 0 = − 1 .

On the other hand, if the mean square prediction error is constrained to be unity and the prediction error equation is included on top of the normal equations, the augmented set of equations is obtained as

where the index i ranges from 0 to p, and R is a (p + 1) × (p + 1) matrix.

Specification of the parameters of the linear predictor is a wide topic and a large number of other approaches have been proposed. In fact, the autocorrelation method is the most common and it is used, for example, for speech coding in the GSM standard.

Solution of the matrix equation Ra = r is computationally a relatively expensive process. The Gaussian elimination for matrix inversion is probably the oldest solution but this approach does not efficiently use the symmetry of R and r. A faster algorithm is the Levinson recursion proposed by Norman Levinson in 1947, which recursively calculates the solution. In particular, the autocorrelation equations above may be more efficiently solved by the Durbin algorithm.

In 1986, Philippe Delsarte and Y.V. Genin proposed an improvement to this algorithm called the split Levinson recursion, which requires about half the number of multiplications and divisions. It uses a special symmetrical property of parameter vectors on subsequent recursion levels. That is, calculations for the optimal predictor containing p terms make use of similar calculations for the optimal predictor containing p − 1 terms.

Another way of identifying model parameters is to iteratively calculate state estimates using Kalman filters and obtaining maximum likelihood estimates within Expectation–maximization algorithms.