© Hindawi Publishing Corp. STATISTICAL APPLICATIONS FOR EQUIVARIANT MATRICES

Solving linear system of equations Ax = b enters into many scientific appli- cations. In this paper, we consider a special kind of linear systems, the matrix A is an equivariant matrix with respect to a finite group of permutations. Examples of this kind are special Toeplitz matrices, circulant matrices, and others. The equivariance property of A may be used to reduce the cost of computation for solving linear systems. We will show that the quadraticform is invariant with respec t to a permutation matrix. This helps to know the multiplicity of eigenvalues of a matrix and yields corresponding eigenvectors at a low computational cost. Applications for such systems from the area of statistics will be presented. These include Fourier transforms on a symmetric group as part of statistical analysis of rankings in an election, spectral analysis in stationary processes, prediction of stationary processes and Yule-Walker equations and parameter estimation for autoregres- sive processes.


Introduction.
Many problems in science and mathematics exhibit equivariant phenomena which can be exploited to achieve a significant cost reduction in their numerical treatment. Recent monographs [16,17,21] have shown the efficiency of applying group theoretical methods in the study of various problems having symmetry properties. Allgower et al. [2,4] presented some techniques for exploiting symmetry in the numerical treatment of linear integral equations, with emphasis on boundary integral methods in three dimensions. Georg and Tausch [15] introduced a user's guide for a software package to solve equivariant linear systems. Definitions for this subject are introduced first.
The product B = AΠ(s) is the matrix whose columns are the permuted columns of A with respect to the permutation π(s) and ΠB is the matrix whose rows are the permuted rows of B and also with respect to π .

Symmetric Toeplitz matrices.
Toeplitz matrices arise in many fields of applications, such as signal processing, coding theory, speech analysis, probability, and statistics. Note that π = (n n − 1 n − 2 ··· 3 2 1) and the corresponding permutation matrix is It is clear that (1.3) holds for T n . Matrices T n are equivariant matrices with this cyclic group of two elements. In fact, this holds also for any symmetric matrix.

Equivariant circulant matrices
Definition 1.4. An n × n matrix C n is a circulant matrix if it is of the form: For the permutation π = (1 2 3 ··· n), obviously (1.3) holds and (1.7) The group here is a special group, the cyclic group of permutations: {e, π , π 2 ,...,π n−1 } group π . Note that C n has the equivariance of the cyclic group {e, π , π 2 ,...,π n−1 } of order n. C n commutes with the permutation matrix Π.
where r runs through a complete list of irreducible representations of Γ and 1 ≤ ρ ≤ dim r. It is well known that the equivariance of A leads to a splitting where A r,ρ : H r,ρ → H r,ρ . This can be exploited to solve linear equations or eigenvalue problems involving A. Hence the linear equations or eigenvalue problems are solved over each of the subspaces H r,ρ . We call this approach the symmetry reduction method.
In [4] methods of implementing the symmetry reduction method are described in detail.

Applications.
Equivariant matrices occur in many scientific phenomena of mathematics, physics, engineering, etc. We have chosen statistics to be our target of applications. In this section, we present some applications for equivariant matrices. The first one is the well-known Fourier transformation and the second one is the stationary processes in time series analysis.

Fourier transforms.
Fourier transforms on finite groups have various applications: Fourier transforms on finite cyclic groups are used for fast polynomial and integer multiplication [18,19]. The Fourier transforms corresponding to elementary abelian 2-groups are called Walsh-Hadamard transforms. They play an important role in digital image processing and in complexity analysis. Diaconis [12] uses Fourier transforms on symmetric groups for the variance analysis of ranked data. Further applications of Fourier transforms on non-abelian groups to problems in combinatorics, theoretical computer science, probability theory and statistics are described by Diaconis in [11].
We have recently had to compute Fourier transforms on the symmetric group S n as part of a statistical analysis of rankings in an election. Here n is the number of candidates, and f (π) is the number of voters choosing the rank order π . Let Π = ρ(π ) be the permutation matrix for π . The entries of Π, counts how many voters ranked candidate i in position j. Diaconis and Rockmore [13] derived fast algorithms for computingf (ρ) and developed them for symmetric groups. The Fourier transform(s) is closely related to the generalized Fourier transform used in [4] for the symmetry reduction method.

Stationary time series processes
where E is the expected value for the random series X t and "Cov" refers to the covariance function between any two random series defined as follows.
Definition 2.2 (expectation). Let X be a random variable. The expected value or the mean of X denoted by EX is defined by where F X (·) is the distribution function of the random variable X.
. Let X and Y be any two random variables defined on the same probability space. The covariance of X and Y denoted by γ X,Y (·) is defined as Note that this definition is equivalent to saying that for any given time series to be stationary it has to have a finite second moment, the first moment is constant over time and the covariance function depends only on the difference of the time. From this definition we see that the third property implies In a matrix format, if we have x 1 ,x 2 ,...,x n observed n time series, then which is an n×n symmetric Toeplitz matrix as it was defined in (1.4) with permutation π = (n n − 1 n − 2 ··· 3 2 1). It is well known that a covariance matrix must be a nonnegative definite matrix and most cases positive definite. This matrix enters in a linear system as we will see in (2.12) and (2.15). For more information on stationary time series processes, see any time series book (cf. [7]).
Since Γ n has the equivariance property, we take advantage of this property as Allgower and Fässler [3] mentioned in their paper to minimize the cost of computation in solving linear systems of equations in prediction in stationary time series. This should be effective since usually n is very large, 1000, 3000, etc.
For solving a linear system of the form Ax = b, where A is an n-by-n Toeplitz matrix and because A is completely specified by 2n − 1 numbers, it is desirable to derive an algorithm for solving Toeplitz systems in less than the O(n 3 ) complexity for Gaussian elimination for a general matrix. In time series analysis algorithms with O(n 2 ) complexity have been known for some time and are based on the Levinsion recursion formula [7]. More recently, even faster algorithms with O(nlog 2 n) have been proposed [5,6] but their stability properties are not yet clearly understood [8].
An alternative is to use iterative methods, based on using matrix-vector products of the form Aν, which can be computed in O(n log n) complexity via the fast Fourier transform (FFT). To have any chance of beating the direct methods, such iterations must converge very rapidly and this naturally leads to the search for good preconditioners for A. Strang [20] proposed using circulant preconditioners, because circulant systems can be solved efficiently by FFT's in O(n log n) complexity. In particular if Γ n in stationary processes is positive definite, then a circulant preconditioner S can be obtained by copying the central diagonals of Γ n and "bringing them around" to complete the circulant.
Chan [9] viewed a preconditioner C for a matrix A in solving a linear system Ax = b as an approximation to A. He derived an optimal circulant preconditioner C in the sense of minimizing C − A .

A distribution problem in spectral analysis.
In connection with stationary processes, certain results on quadratic forms can be used to solve a distribution problem in spectral analysis. Let X t be a real and normally distributed stationary process with a discrete time parameter and with an absolutely continuous spectrum. One of the important problems in the theory of time series is to find an estimatef (λ) for the spectral density f (λ), assuming that the process has been observed for t = 1, 2,...,n. The estimatef (λ) must be a function of x 1 ,...,x n , and there are reasons to require that this function should be a nonnegative quadratic form (Grenander and Rosenblatt 1957).

Thus an estimate of the formf
is considered, where X T = (X 1 ,...,X n ) and W is nonnegative. The exact distribution or approximate distribution off is needed. In almost all important cases, W is a Toeplitz matrix. It is well known thatf can be reduced to the canonical form where r is the rank of W and λ j 's are the eigenvalues of W and the χ 2 j 's are independent chi-square random variables with one degree of freedom each. After finding the λ j 's it becomes easier to find approximation to the distribution off , see Alkarni [1].
The following result shows that if A is equivariant, then the quadratic form Q(x) = x T Ax is invariant with respect to its permutation group. (λ, x) is an eigenvalue-eigenvector pair, then so is (λ, Πx) and λ has multiplicity equal to dim(sp{Πx : Π ∈ Γ }).

Theorem 2.4. Suppose A is real and equivariant matrix with respect to a permutation group Γ , then the quadratic form Q(x) = x T Ax is invariant with respect to the permutation matrix Γ . Moreover, for all π ∈ Γ if
(2.8) Now suppose Ax = λx, where A is equivariant with respect to a permutation group Γ then for all Π ∈ Γ , which implies that if (λ, x) is an eigenvalue-eigenvector pair, then (λ, Πx) is also an eigenvalue-eigenvector pair. Suppose now that Πx ≠ αx for every scalar α ≠ 0. Then Πx is another eigenvector, therefore λ has a multiplicity equal to dim(sp{Πx : Π ∈ Γ }).
We conclude that the equivariance property helps to know the multiplicities of eigenvalues of a matrix and yields corresponding eigenvectors at a low computational cost. This will reduced the cost of computations.

Prediction of stationary processes
(1) One-step predictors. Let {X t } be a stationary process with mean zero and auto covariance function γ(·). Let H n denote the closed linear subspace sp{X 1 ,...,X n }, n ≥ 1, and letX n+1 , n ≥ 0, denote the one-step predictors defined bŷ where P Hn X n+1 is the projection of X n+1 onto the closed linear subspace H n . For more on projection theory see, for example, [10].
SinceX n+1 ∈ H n , n ≥ 1, we can writê X n+1 = φ n1 X n +···+φ nn X 1 , n≥ 1. (2.11) Using the projection theory, we end up solving the system Γ n Φ n = γ n , (2.12) where Γ n = [γ(i − j)] i,j=1,...,n is the covariance matrix in (2.5), γ n = (γ(1),...,γ(n)) T and Φ n = (φ n1 ,...,φ nn ) T . The projection theorem guarantees that equation (2.12) has at least one solution. Although there may be many solutions to (2.12), every one of them, when substituted into (2.11), must give the same predictorX n+1 since by projection theoryX n+1 is uniquely defined. There is exactly one solution of (2.12) if and only if Γ n is nonsingular in which case the solution is It can be shown that if γ(0) > 0 and γ(h) → 0 as h → ∞, then the covariance matrix Γ n is nonsingular for every n. For a proof see [7]. Hence our goal is to find a solution of (2.12) if there are more than one solution or to find the inverse Γ −1 n if Γ n is nonsingular. In either case using the equivariant property of Γ n will be useful.
(2) The h-step predictors, h ≥ 1. In the same manner the best linear predictor of X n+h in terms of X 1 ,...,X n for any h ≥ 1 can be found to be where γ (h) n = (γ(h), γ(h + 1),...,γ(n+ h − 1)) T and Γ n is as in (2.5). As was mentioned before, we need to find a solution to the large system in (2.15) or the unique one if Γ n is nonsingular.
The use of the equivariance property will be even more effective if we apply it to the prediction equation of a well-known class of stationary time series processes, the autoregressive moving average or ARMA processes. The use of the equivariant property becomes effective because of the structure of Γ n , the auto covariance function. For the definition of ARMA and their properties we refer the reader to [7]. We only present the structure of the auto covariance function.
If we have an ARMA(p, q) model, and if m = max(p, q), then the auto covariance function is given by This structure leads to an n × n block diagonal Toeplitz matrix Γ n in which case we apply an algorithm based on the equivariance property to solve the system in (2.12) or in (2.15), which is more efficient than the direct computation.

The Yule-Walker equations and parameter estimation for autoregressive processes.
Let {X t } be the zero-mean causal autoregressive process, (2.17) Our aim is to find estimators of the coefficient vector φ = (φ 1 ,...,φ p ) T and the white noise variance σ 2 based on the observations X 1 ,...,X n . Because of the causality assumption, that is, X t can be written as a linear combination of Z t , we end up solving the linear systemΓ

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009