A software package is presented that computes locally optimal solutions to low-rank approximation problems with the following features:• mosaic Hankel structure constraint on the approximating matrix,• weighted 2-norm approximation criterion,• fixed elements in the approximating matrix,• missing elements in the data matrix, and• linear constraints on an approximating matrix's left kernel basis.It implements a variable projection type algorithm and allows the user to choose standard local optimization methods for the solution of the parameter optimization problem. For an m× n data matrix, with n > m, the computational complexity of the cost function and derivative evaluation is O(m 2 n). The package is suitable for applications with n ≫ m. In statistical estimation and data modeling-the main application areas of the package-n ≫ m corresponds to modeling of large amount of data by a low-complexity model. Performance results on benchmark system identification problems from the database DAISY and approximate common divisor problems are presented.

The structured low-rank approximation problem for general affine structures, weighted 2-norms and fixed elements is considered. The variable projection principle is used to reduce the dimensionality of the optimization problem. Algorithms for evaluation of the cost function, the gradient and an approximation of the Hessian are developed. For m × n mosaic Hankel matrices the algorithms have complexity O(m 2 n).Proof. Γ ∈ R nd×nd is µd-banded, and the complexity of both steps is given in [15, Ch. 4].Theorem 2. For (H m,n ), the complexity of the cost function evaluation using Algorithm 6 and gradient evaluation using Algorithm 3 is O(d 3 µmn).Proof. The proof is given in Appendix A.Note 7. By Lemma 1, for the mosaic Hankel structure H m,n the complexity of the cost function and gradient evaluation is also O(d 3 µmn). Note 8. The computation on the step 2 in the Algorithm 2 has complexity O(dmn).Theorem 3. The computational complexity of Algorithm 4 (for the Jacobian) is O(d 3 m 2 n). The computational complexity of Algorithm 7 (for the pseudo-Jacobian) is O(d 3 µmn).Proof. The proof is given in Appendix A.

Positional information in developing embryos is specified by spatial gradients of transcriptional regulators. One of the classic systems for studying this is the activation of the hunchback (hb) gene in early fruit fly (Drosophila) segmentation by the maternally-derived gradient of the Bicoid (Bcd) protein. Gene regulation is subject to intrinsic noise which can produce variable expression. This variability must be constrained in the highly reproducible and coordinated events of development. We identify means by which noise is controlled during gene expression by characterizing the dependence of hb mRNA and protein output noise on hb promoter structure and transcriptional dynamics. We use a stochastic model of the hb promoter in which the number and strength of Bcd and Hb (self-regulatory) binding sites can be varied. Model parameters are fit to data from WT embryos, the self-regulation mutant hb 14F, and lacZ reporter constructs using different portions of the hb promoter. We have corroborated model noise predictions experimentally. The results indicate that WT (self-regulatory) Hb output noise is predominantly dependent on the transcription and translation dynamics of its own expression, rather than on Bcd fluctuations. The constructs and mutant, which lack self-regulation, indicate that the multiple Bcd binding sites in the hb promoter (and their strengths) also play a role in buffering noise. The model is robust to the variation in Bcd binding site number across a number of fly species. This study identifies particular ways in which promoter structure and regulatory dynamics reduce hb output noise. Insofar as many of these are common features of genes (e.g. multiple regulatory sites, cooperativity, self-feedback), the current results contribute to the general understanding of the reproducibility and determinacy of spatial patterning in early development.

Implementation of multivariate and 2D extensions of singular spectrum analysis (SSA) by means of the R package Rssa is considered. The extensions include MSSA for simultaneous analysis and forecasting of several time series and 2D-SSA for analysis of digital images. A new extension of 2D-SSA analysis called shaped 2D-SSA is introduced for analysis of images of arbitrary shape, not necessary rectangular. It is shown that implementation of shaped 2D-SSA can serve as a basis for implementation of MSSA and other generalizations. Efficient implementation of operations with Hankel and Hankel-blockHankel matrices through the fast Fourier transform is suggested. Examples with code fragments in R, which explain the methodology and demonstrate the proper use of Rssa, are presented.