Positive-Definite Tapering for Covariance Estimation

Why Taper a Covariance Estimate?

For measurements indexed by time or location, covariances between nearby observations may be more important, or more reliably estimated, than covariances between distant ones. A structured estimate can reflect this by reducing distant entries and setting sufficiently distant ones to zero. But changing entries one at a time can destroy positive definiteness: the resulting matrix may no longer be a valid covariance matrix. Covariance tapering gives a principled way to modify entries while controlling this risk.

The key is to apply a positive-semidefinite matrix to the estimated covariance entry by entry. Under a simple diagonal condition, this operation preserves positive definiteness. The condition also clarifies why tapering is not the same as retaining entries inside a band and deleting the rest.

Definition and the Positive-Definiteness Guarantee

A symmetric matrix \(A\) is positive semidefinite if \(x^{\mathsf T}Ax\geq 0\) for every vector \(x\), and positive definite if \(x^{\mathsf T}Ax>0\) for every nonzero vector \(x\). A covariance matrix must be positive semidefinite; positive definiteness is needed for operations such as an ordinary inverse and a nonsingular multivariate normal density.

This is the Schur product theorem, together with its positive-definite consequence. The diagonal condition matters: a positive-semidefinite taper with a zero diagonal entry has an all-zero corresponding row and column, so its Hadamard product cannot be positive definite.

Proof. Since \(T\) is positive semidefinite, it has a Gram representation \(T_{ij}=u_i^{\mathsf T}u_j\), where \(u_i\in\mathbb{R}^r\) for some finite \(r\). Write \(u_{ik}\) for coordinate \(k\) of \(u_i\), and let \(D_k\) be the diagonal matrix with diagonal entries \(u_{1k},\ldots,u_{pk}\). For any vector \(x\),

If \(A\) is positive semidefinite, every term in the last sum is nonnegative, so \(A\circ T\) is positive semidefinite. Now suppose \(A\) is positive definite, \(T_{ii}>0\) for every \(i\), and \(x\neq 0\). Choose an index \(i\) for which \(x_i\neq 0\). Since \(T_{ii}=u_i^{\mathsf T}u_i=\sum_{k=1}^r u_{ik}^2>0\), there is at least one \(k\) with \(u_{ik}\neq 0\). For that \(k\), \(D_kx\neq 0\), so \((D_kx)^{\mathsf T}A(D_kx)>0\). All other terms are nonnegative. Thus \(x^{\mathsf T}(A\circ T)x>0\), proving positive definiteness. \(\square\)

In particular, if an estimate \(\widehat{\Sigma}\) is positive definite and \(T\) is positive semidefinite with unit diagonal, tapering retains positive definiteness and leaves each estimated variance unchanged. If \(\widehat{\Sigma}\) is only positive semidefinite, the theorem guarantees only that its tapered version is positive semidefinite; it need not become positive definite.

How a Taper Matrix Can Be Constructed

A common construction begins with measurement locations \(s_1,\ldots,s_p\) and a function \(K_h\) of pairwise separation. The entries are \(T_{ij}=K_h(s_i-s_j)\). To use the theorem, the resulting matrix must be positive semidefinite for the locations under consideration. A function that is nonnegative and decreases with distance is not automatically a valid taper: entrywise values alone do not establish that its matrix is positive semidefinite.

One simple valid taper on equally spaced observations is the Bartlett triangular taper. For integer indices and a positive integer bandwidth \(m\), set

The entries decrease linearly with lag and are zero when \(|i-j|\geq m\). Its positive semidefiniteness follows directly from a Gram representation. For each integer \(i\), define \(g_i(t)=m^{-1/2}\mathbf{1}_{[i,i+m]}(t)\). Then

Thus \(T\) is a Gram matrix of functions: for any vector \(x\), \(x^{\mathsf T}Tx=\int_{\mathbb{R}}(\sum_i x_i g_i(t))^2\,dt\geq 0\). Also \(T_{ii}=1\). This proves that the Bartlett taper satisfies the conditions for preserving positive definiteness.

Tapering Is Not Hard Banding

Hard banding applies a mask that keeps selected entries and sets the others to zero. For an ordered vector, a bandwidth-\(k\) mask keeps entries with \(|i-j|\leq k\). This operation is also an entrywise product, but its mask is not necessarily positive semidefinite. As a result, it does not have the positive-definiteness guarantee just proved. Tapering instead uses a positive-semidefinite matrix of weights; those weights may decrease gradually to zero, and the matrix property—not merely the appearance of its entries—is what protects the covariance structure.

What the Guarantee Does—and Does Not—Say

The preservation result is algebraic. It says that if the input matrix is positive definite and the taper matrix is positive semidefinite with positive diagonal, their Hadamard product is positive definite. It does not say that the tapered estimate is unbiased for the true covariance, that a particular bandwidth is appropriate, or that every taper function with a convenient shape is valid. Those are separate modeling and estimation questions.

Tapering is useful when the data structure makes distant covariances less important or less precisely estimated, and when a sparse estimate is desirable. A taper with compact support can create exact zeros, while positive semidefiniteness preserves the covariance-matrix constraint. However, if the starting estimate is singular, the theorem alone cannot guarantee an invertible result. For example, a sample covariance matrix may be singular when the number of variables is large relative to the number of observations. One must check the starting matrix or use an appropriate regularization before relying on a positive-definite guarantee.