Bayesian analyses of spatial data often use a conditionally autoregressive (CAR) prior, which can be written as the kernel of an improper density that depends on a precision parameter tau that is typically unknown. To include tau in the Bayesian analysis, the kernel must be multiplied by tau(k) for some k. This article rigorously derives k = (n - I)/2 for the L2 norm CAR prior (also called a Ga...

∣ p . Hardy’s inequality thus asserts that the Cesáro matrix operator C = (cj,k), given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ p/(p − 1). (The norm is in fact p/(p − 1).) Hardy’s inequality leads naturally to the study on lp norms of general matrices. For example, we say a matrix A = (aj,k) is a weighted mean matrix if its entries satisfy aj,k = 0, k > j and aj,k ...

This paper takes an approach to clustering domestic electricity load profiles that has been successfully used with data from Portugal and applies it to UK data. Clustering techniques are applied and it is found that the preferred technique in the Portuguese work (a two stage process combining Self Organised Maps and Kmeans) is not appropriate for the UK data. The work shows that up to nine clus...

Most partition-based cluster analysis methods (e.g., kmeans) will partition any dataset D into k subsets, regardless of the inherent appropriateness of such a partitioning. This paper presents a family of permutation-based procedures to determine both the number of clusters k best supported by the available data and the weight of evidence in support of this clustering. These procedures use one ...

Abstract Low-rank matrix recovery problem is difficult due to its non-convex properties and it usually solved using convex relaxation approaches. In this paper, we formulate the low-rank exactly novel Ky Fan 2- k -norm-based models. A general difference of functions algorithm (DCA) developed solve these proximal point (PPA) framework proposed sub-problems within DCA, which allows us handle larg...

We initiate the study of trade-offs between sparsity and the number of measurements in sparse recovery schemes for generic norms. Specifically, for a norm ‖ ·‖, sparsity parameter k, approximation factor K > 0, and probability of failure P > 0, we ask: what is the minimal value of m so that there is a distribution over m × n matrices A with the property that for any x, given Ax, we can recover ...

The recently introduced k-support norm has been successfully applied to sparse prediction problems with correlated features. This norm however lacks any explicit structural constraints commonly found in machine learning and image processing. We address this problem by incorporating a total variation penalty in the k-support framework. We introduce the (k, s) support total variation norm as the ...

In different fields in space researches, Scientists are in need to deal with the product of matrices. In this paper, we develop conditions under which a product  of matrices chosen from a possibly infinite set of matrices 0 i i P    , j  M P j J     k i P   0 k k i  converges. There exists a vector norm such that all matrices in M are no expansive with respect to this norm and also ...