Next Gaussian Prediction (NGP)

• NGP is a unified algorithmic framework that performs one-step ahead prediction in Gaussian process models using generalized moment-based covariance structures.
• It extends classical methods like the Szegő–Verblunsky and Durbin–Levinson recurrences by leveraging orthogonal rational functions for efficient computation.
• The framework accommodates stationary, nonstationary, and ARMA-type processes, providing explicit formulas for optimal predictors and error variances.

Next Gaussian Prediction (NGP) is a unified algorithmic framework for one-step ahead prediction in Gaussian process models with generalized covariance structures, built on the Kreĭn–Nudel’man theory of generalized moments. NGP encompasses stationary Gaussian processes, ARMA models, and a broad class of nonstationary, varying ("generalized stationary") Gaussian processes by expressing their covariances in terms of orthogonal rational functions (ORFs) on the unit circle. The prediction scheme generalizes the classical Szegő–Verblunsky and Durbin–Levinson recursions, providing explicit computation of best linear predictors, error variances, and orthonormal bases via the Schur-type multipoint algorithm (Baratchart et al., 2010).

1. Generalized Moment Gaussian Processes

NGP begins by fixing a system of continuous functions $W = \{w_0, w_1, \ldots \} \subset C(\mathbb{T})$, where $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$ denotes the unit circle. This system is required to satisfy the Kreĭn–Nudel’man positivity condition: for every $n$, there exist constants $\{a_k\}$ such that

$$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$

A sequence $\{c_k\}_{k=0}^\infty$ is a positive generalized-moment sequence with respect to $W$ if

$$\left[ \sum_{k=0}^{n} (a_k w_k + \overline{a_k} \overline{w_k}) \geq 0 \text{ on } \mathbb{T} \right] \implies \left[ \sum_{k=0}^{n} (a_k c_k + \overline{a_k} \overline{c_k}) \geq 0 \right],$$

for all finite sequences $\{a_k\}$. The Kreĭn–Nudel’man theorem ensures the existence of a unique positive Borel measure $\sigma$ on $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$0 such that

$\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$1

A zero-mean Gaussian process $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$2 is called a $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$3-varying Gaussian process (W-VGP) if its covariance admits the generalized-Toeplitz form

$\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$4

This framework extends the class of stationary Gaussian processes (recovered by $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$5) and accommodates a variety of nonstationary and rational spectral structures.

2. Spectral Representation and Covariance Structure

For any W-VGP, the generalized Herglotz and Kolmogorov–Wiener–Masani theories guarantee the existence of a unique orthogonal-increment Gaussian measure $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$6, supported on $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$7, satisfying

$\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$8

for Borel sets $\mathbb{T} = \{ e^{it} : t \in [0,2\pi) \}$9. Each process variable admits the spectral integral representation

$n$0

with covariance structure immediately following:

$n$1

3. One-Step Prediction via Orthogonal Rational Functions

The optimal one-step prediction in NGP is obtained in $n$2 by first applying the Gram–Schmidt process to the system $n$3, yielding a family of orthonormal functions $n$4 satisfying

$n$5

By the isometry $n$6, the closed span of $n$7 in $n$8 is mapped to $n$9 in $\{a_k\}$0. The next-step linear predictor is the $\{a_k\}$1 projection:

$\{a_k\}$2

which pulls back to

$\{a_k\}$3

with $\{a_k\}$4. The mean-square prediction error is

$\{a_k\}$5

4. Orthonormal Rational Systems and Schur-Type Recurrences

In the notable case where $\{a_k\}$6 is the sequence of simple Blaschke products at prescribed poles $\{a_k\}$7,

$\{a_k\}$8

the orthonormal family $\{a_k\}$9 becomes the system of classical orthogonal rational functions (ORFs) on the unit circle. These satisfy a two-term Schur-type recurrence:

$$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$0

where $$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$1 is the $$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$2th Schur parameter, $$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$3, $$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$4, and $$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$5 normalizes $$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$6. This recurrence, along with the orthogonality relation

$$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$7

enables efficient computation of all inner products for the predictor via forward recursion—the multipoint Schur algorithm. The error-power satisfies

$$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$8

5. Classical ARMA and Levinson–Durbin Algorithms as Special Cases

When $$\sum_{k=0}^{n} (a_k w_k(z) + \overline{a_k} \overline{w_k(z)}) > 0 \quad \forall z \in \mathbb{T}.$$9, the construction reduces to the case of stationary Gaussian processes. Here, the orthonormal system becomes the classical trigonometric polynomials of Szegő–Verblunsky theory, and the recurrences reduce to the well-known Durbin–Levinson recursions. The prediction error matches the Levinson–Durbin determinant ratio. For processes with ARMA-type rational spectral densities ($\{c_k\}_{k=0}^\infty$0), the predictor system can be realized with finite-dimensional rational functions, yielding the classical ARMA next-step predictor.

Basis Choice	Type of Process	Corresponding Prediction Algorithm
$\{c_k\}_{k=0}^\infty$1	Stationary	Szegő–Verblunsky, Durbin–Levinson
Blaschke products	Rational poles	Multipoint Schur (ORF) predictor
General $\{c_k\}_{k=0}^\infty$2	Generalized VGP	Gram–Schmidt, generalized NGP

6. Algorithmic Procedure and Computational Aspects

The NGP workflow comprises the following steps:

1. Select a basis $\{c_k\}_{k=0}^\infty$3 tailored to the problem class (e.g., monomials for stationary cases, Blaschke products for rational/pole structure, arbitrary bases as required).
2. Compute generalized moments $\{c_k\}_{k=0}^\infty$4 or $\{c_k\}_{k=0}^\infty$5 and verify the positivity (generalized Toeplitz).
3. Apply Gram–Schmidt in $\{c_k\}_{k=0}^\infty$6 to $\{c_k\}_{k=0}^\infty$7 to obtain $\{c_k\}_{k=0}^\infty$8 and the next $\{c_k\}_{k=0}^\infty$9.
4. Calculate inner products $W$0 for $W$1.
5. Form the one-step predictor:

$W$2

with error variance $W$3.

If the Schur parameters $W$4 are known, the full Gram–Schmidt process can be bypassed and $W$5 updated recursively, resulting in $W$6 computational complexity for predictor and error-power evaluation.

7. Context, Applications, and Extensions

NGP generalizes classical one-step prediction in Gaussian processes by providing a versatile framework that accommodates both stationary and wide classes of nonstationary and rational spectral models. This construction enables explicit spectral representation, efficient prediction, and error quantification for Gaussian processes whose covariance is expressible via generalized-moment structures, including those not amenable to standard Toeplitz or ARMA-based approaches. The strong connection of NGP with the multipoint Schur algorithm and orthogonal rational functions highlights its applicability in scenarios requiring robust, efficient prediction in nonstationary and nontrivial spectral environments (Baratchart et al., 2010).