AR Algorithms¶
Autoregressive (AR) estimation algorithms are used in the AR Modeling and Prediction option.
Least-Squares Algorithms¶
The most effective AR methods involve simultaneous least-squares estimates of all AR coefficients in the model.
The Data models implement the "covariance" or "modified covariance" methods of linear prediction. The covariance in this context has nothing to do with fitting a covariance matrix. Rather, the full data or trajectory matrix, usually rectangular, is fitted directly to the vector of data elements. This approach incurs no loss of precision, but can involve lengthy fits with large data sets.
The Nrml models implement the normal equations used in typical least-squares estimations. Square matrices are formed using sums of forward or backward data elements. The sums incur a loss of precision that can result in reduced resolution (compared to direct data methods) and lessened stability of the coefficients. When some noise is present, however, the normal equations generally yield viable results.
The Fwd, Bwd, and FB are the forward prediction, backward prediction, and forward-backward prediction variants of the algorithms.
For both the Data and Nrml models, there is an SVD version. Apart from longer processing times, there are no disadvantages to using an SVD procedure, and the advantages are numerous when removing the influence of noise is critical to the modeling. A full signal space SVD fit, one where the signal space equals the model order, produces the same results as the non-SVD algorithms.
The Graphically Select Signal and Noise Subspaces option is available only to SVD procedures.
Non-SVD Algorithms¶
The Data Fwd, Data Bwd, and Data FB options use fast algorithms that exploit the symmetry of the data matrix. These algorithms will produce optimum least-squares AR fits without the precision losses characteristic of the normal equations approach.
Since these algorithms also compute the coefficients for all lesser model orders, AR Model Order Selection is available.
SVD Based Least-Squares Algorithms¶
Least-squares AR methods that offer in-situ separation of signal and noise through SVD are the most robust of all methods.
The Data Svd FB is an excellent algorithm. The forward prediction version is the Data Svd Fwd. The backward prediction version is the Data Svd Bwd. The main drawback of the Data Svd procedures is that they can be quite slow with large data sets.
If there is noise present (which is why SVD is used in the first place), the normal equations equivalents Nrml Svd FB, Nrml Svd Fwd, Nrml Svd Bwd, will produce similar results far more swiftly.
The SVD algorithms compute the coefficients only for the order specified.
Least-squares AR methods that offer in-situ separation of signal and noise through SVD are the most robust of AR methods.
Filter¶
When TableCurve 2D computes the statistics for any AR model fit, a consistent filter is used. This enables all algorithms to be compared using a common reference. The filter is not strictly forward, backward, or forward-backward prediction. Instead it is a forward prediction with a partial backward component. The AR filter is defined using backward prediction from the model order down to the initial data element, and forward prediction from the model order up to the final data element. Using this filter, a single estimate is thus made for each of the input data elements. There is no gap at one end of the data stream as is true of a forward or backward filter.
This filter simplifies the goodness-of-fit statistics since the degree of freedom is preserved (the data length minus the AR coefficient count). The statistics in the AR algorithms reflect this approach, although none of the linear AR algorithms specifically optimizes this particular merit function.