Convex and non-convex regularization methods for spatial point processes intensity estimation

This paper deals with feature selection procedures for spatial point processes intensity estimation. We consider regularized versions of estimating equations based on Campbell theorem derived from two classical functions: Poisson likelihood and logistic regression likelihood. We provide general conditions on the spatial point processes and on penalty functions which ensure consistency, sparsity and asymptotic normality. We discuss the numerical implementation and assess finite sample properties in a simulation study. Finally, an application to tropical forestry datasets illustrates the use of the proposed methods

Finite-State Dimension and Real Arithmetic

We use entropy rates and Schur concavity to prove that, for every integer k >= 2, every nonzero rational number q, and every real number alpha, the base-k expansions of alpha, q+alpha, and q*alpha all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.Comment: 15 page

Monotonicity-Preserving Bootstrapped Kriging Metamodels for Expensive Simulations

Kriging (Gaussian process, spatial correlation) metamodels approximate the Input/Output (I/O) functions implied by the underlying simulation models; such metamodels serve sensitivity analysis and optimization, especially for computationally expensive simulations. In practice, simulation analysts often know that the I/O function is monotonic. To obtain a Kriging metamodel that preserves this known shape, this article uses bootstrapping (or resampling). Parametric bootstrapping assuming normality may be used in deterministic simulation, but this article focuses on stochastic simulation (including discrete-event simulation) using distribution-free bootstrapping. In stochastic simulation, the analysts should simulate each input combination several times to obtain a more reliable average output per input combination. Nevertheless, this average still shows sampling variation, so the Kriging metamodel does not need to interpolate the average outputs. Bootstrapping provides a simple method for computing a noninterpolating Kriging model. This method may use standard Kriging software, such as the free Matlab toolbox called DACE. The method is illustrated through the M/M/1 simulation model with as outputs either the estimated mean or the estimated 90% quantile; both outputs are monotonic functions of the traffic rate, and have nonnormal distributions. The empirical results demonstrate that monotonicity-preserving bootstrapped Kriging may give higher probability of covering the true simulation output, without lengthening the confidence interval.Queues

Nonparametric Stein-type Shrinkage Covariance Matrix Estimators in High-Dimensional Settings

Estimating a covariance matrix is an important task in applications where the number of variables is larger than the number of observations. Shrinkage approaches for estimating a high-dimensional covariance matrix are often employed to circumvent the limitations of the sample covariance matrix. A new family of nonparametric Stein-type shrinkage covariance estimators is proposed whose members are written as a convex linear combination of the sample covariance matrix and of a predefined invertible target matrix. Under the Frobenius norm criterion, the optimal shrinkage intensity that defines the best convex linear combination depends on the unobserved covariance matrix and it must be estimated from the data. A simple but effective estimation process that produces nonparametric and consistent estimators of the optimal shrinkage intensity for three popular target matrices is introduced. In simulations, the proposed Stein-type shrinkage covariance matrix estimator based on a scaled identity matrix appeared to be up to 80% more efficient than existing ones in extreme high-dimensional settings. A colon cancer dataset was analyzed to demonstrate the utility of the proposed estimators. A rule of thumb for adhoc selection among the three commonly used target matrices is recommended.Comment: To appear in Computational Statistics and Data Analysi