Many important kernel methods in the machine learning area, such as kernel principal component analysis, feature approximation, denoising, compression and prediction require the computation of the dominant set of eigenvectors of the symmetric kernel Gram matrix.Recently, an efficient incremental approach was presented for the fast calculation of the dominant kernel eigenbasis.In this manuscript we propose faster algorithms for incrementally updating and downsizing the dominant kernel eigenbasis. These methods are well-suited for large scale problems since they are both efficient in terms of complexity and data management.

The paper proposes a decision support system (DSS) for the supply chain of packaged fresh and highly perishable products. The DSS combines a unique tool for sales forecasting with order planning which includes an individual model selection system equipped with ARIMA, ARIMAX and transfer function forecasting model families, the latter two accounting for the impact of prices. Forecasting model parameters are chosen via two alternative tuning algorithms: a two-step statistical analysis, and a sequential parameter optimisation framework for automatic parameter tuning. The DSS selects the model to apply according to user-defined performance criteria. Then, it considers sales forecasting as a proxy of expected demand and uses it as input for a multi-objective optimisation algorithm that defines a set of non-dominated order proposals with respect to outdating, shortage, freshness of products and residual stock. A set of real data and a benchmark - based on the methods already in use - are employed to evaluate the performance of the proposed DSS. The analysis of different configurations shows that the DSS is suitable for the problem under investigation; in particular, the DSS ensures acceptable forecasting errors and proper computational effort, providing order plans with associated satisfactory performances.

The equality constrained indefinite least squares problem involves the minimization of an indefinite quadratic form subject to a linear equality constraint. In this paper, we study this problem and present a numerical method that is proved to be backward stable in a strict sense, i.e., that the computed solution satisfies a slightly perturbed equality constrained indefinite least squares problem. We also perform a sensitivity analysis of this problem and derive bounds for the accuracy of the computed solution. We give several numerical experiments to illustrate these results.

An algorithm for computing the solution of indefinite least squares problemsand of indefinite least squares problems with equality constrained is presented.Such problems arise when solving total least squares problems and inH infinity-smoothing.The proposed algorithm relies only on stable orthogonal transformations reducingrecursively the associated augmented matrix to proper block anti-triangular form.Some numerical results are reported showing the properties of the algorithm.

In this paper we revisit the problem of finding an orthogonal similarity transformation that puts an $n\times n$ matrix $A$ in a block upper-triangular form that reveals its Jordan structure at a particular eigenvalue $\lambda_0$. The obtained form in fact reveals the dimensions of the spaces of $(A-\lambda_0 I)^i$ at that eigenvalue via the sizes of the leading diagonal blocks, and from this the Jordan structure at $\lambda_0$ is then easily recovered. The method starts from a Hessenberg form that already reveals several properties of the Jordan structure of $A$. It then updates the Hessenberg form in an efficient way to transform it to a block-triangular form in ${\cal O}(mn^2)$ floating point operations, where $m$ is the total multiplicity of the eigenvalue. The method only uses orthogonal transformations and is backward stable. We illustrate the method with a number of numerical examples.

We consider the problem of finding a square low-rank correction (?C - B)F to a given square pencil (?E - A) such that the new pencil ?(E - CF) - (A - BF) has all its generalised eigenvalues at the origin. We give necessary and sufficient conditions for this problem to have a solution and we also provide a constructive algorithm to compute F when such a solution exists. We show that this problem is related to the deadbeat control problem of a discrete-time linear system and that an (almost) equivalent formulation is to find a square embedding that has all its finite generalised eigenvalues at the origin.

This paper deals with the issue of forecasting energy production of a Photo-Voltaic (PV) plant, needed by the Distribution System Operator (DSO) for grid planning. As the energy production of a PV plant is strongly dependent on the environmental conditions, the DSO has difficulties to manage an electrical system with stochastic generation. This implies the need to have a reliable forecasting of the irradiance level for the next day in order to setup the whole distribution network. To this aim, this paper proposes the use of transfer function models. The assessment of quality and accuracy of the proposed method has been conducted on a set of scenarios based on real data.

The problem of reconstructing signals and images from degraded ones is consideredin this paper. The latter problem is formulated as a linear system whose coefficientmatrix models the unknown point spread function and the right hand side representsthe observed image. Moreover, the coefficient matrix is very ill-conditioned, requiringan additional regularization term. Different boundary conditions can be proposed. In thispaper antireflective boundary conditions are considered. Since both sides of the linearsystem have uncertainties and the coefficient matrix is highly structured, the RegularizedStructured Total Least Squares approach seems to be the more appropriate one to computean approximation of the true signal/image. With the latter approach the original problem isformulated as an highly nonconvex one, and seldom can the global minimum be computed.It is shown that Regularized Structured Total Least Squares problems for antireflectiveboundary conditions can be decomposed into single variable subproblems by a discrete sinetransform. Such subproblems are then transformed into one-dimensional unimodal realvaluedminimization problems which can be solved globally. Some numerical examplesshow the effectiveness of the proposed approach.

In this study non-negative matrix factorization (NMF) was hierarchically applied to simulated and in vivo three-dimensional 3 T MRSI data of the prostate to extract patterns for tumour and benign tissue and to visualize their spatial distribution. Our studies show that the hierarchical scheme provides more reliable tissue patterns than those obtained by performing only one NMF level. We compared the performance of three different NMF implementations in terms of pattern detection accuracy and efficiency when embedded into the same kind of hierarchical scheme. The simulation and in vivo results show that the three implementations perform similarly, although one of them is more robust and better pinpoints the most aggressive tumour voxel(s) in the dataset. Furthermore, they are able to detect tumour and benign tissue patterns even in spectra with lipid artefacts. Copyright (C) 2016 John Wiley & Sons, Ltd.

We address the problem of forecasting sales for fresh and highly perishable products, in the general context of supply chain management. The forecasting activity refers to the single item in a given store and started from a pre-processing phase for data analysis and normalization. Then data was used as input for a forecasting algorithm designed to be user interactive. We implemented three forecasting methods: ARIMA, ARIMAX and transfer function models. The exogenous components of the forecasting models took the impact of prices into account. The best configuration of these models is dynamically chosen via two alternative methods: (i) a two-step procedure, based on properly selected statistical indicators, (ii) a Sequential Parameter Optimization approach for automatic parameter tuning. The user or the decision maker at the store level should not be exposed to the complexity of the forecasting system which - for this reason - is designed to adaptively select the best model configuration at every forecast session, to be used for each item/store combination. A set of real data based on 19 small and medium sized stores and 156 fresh products was employed to evaluate both quality of forecasting results and their effects on the order planning activity, where sales forecasting is considered as a proxy of the expected demand. Some examples are reported and discussed. Our results confirm that there is no 'one-size-fits-all' forecasting model, whose performance strictly depends on the specific characteristics of the underlying data. This supports the adoption of a data-driven tool to automate the dynamic selection of the most appropriate forecasting model. (C) 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

An algorithm for computing the antitriangular factorization of symmetric matrices, relying only on orthogonal transformations, was recently proposed. The computed antitriangular form straightforwardly reveals the inertia of the matrix. A block version of the latter algorithm was described in a different paper, where it was noticed that the algorithm sometimes fails to compute the correct inertia of the matrix.In this paper we analyze a possible cause of the failure of detecting the inertia and propose a procedure to recover it. Furthermore, we propose a different algorithm to compute the antitriangular factorization of a symmetric matrix that handles most of the singularities of the matrix at the very end of the algorithm.Numerical results are also given showing the reliability of the proposed algorithm.

This paper investigates the use of time series of ALOS/PALSAR-1 and COSMO-SkyMed data for the soil moisture retrieval (mv) by means of the SMOSAR algorithm. The application context is the exploitation of mv maps at a moderate spatial and temporal resolution for improving flood/drought monitoring at regional scale. The SAR data were acquired over the Capitanata plain in Southern Italy, over which ground campaigns were carried out in 2007, 2010 and 2011. The analysis shows that the mv retrieval accuracy is 5%-7% m^3/m^3 at L- and X band, although the latter is restricted to a use over nearly bare soil only.

We present an algorithm for computing a symmetric rank revealing decomposition of a symmetric n x n matrix A, as defined in the work of Hansen & Yalamov [9]: we factorize the original matrix into a product A = QMQ(T), with Q orthogonal and M symmetric and in block form, with one of the blocks containing the dominant information of A, such as its largest eigenvalues. Moreover, the matrix M is constructed in a form that is easy to update when adding to A a symmetric rank-one matrix or when appending a row and, symmetrically, a column to A: the cost of such an updating is O(n(2)) floating point operations.

We consider here the problem of tracking the dominant eigenspace of an indefinite matrixby updating recursively a rank k approximation of the given matrix. The tracking uses awindow of the given matrix, which increases at every step of the algorithm. Therefore, therank of the approximation increases also, and hence a rank reduction of the approximationis needed to retrieve an approximation of rank k. In order to perform the windowadaptation and the rank reduction in an efficient manner, we make use of a new antitriangulardecomposition for indefinite matrices. All steps of the algorithm only make useof orthogonal transformations, which guarantees the stability of the intermediate steps.We also show some numerical experiments to illustrate the performance of the trackingalgorithm.

We show in this paper that the roots $x_1$ and $x_2$ of a scalar quadratic polynomial $ax^2 + bx + c = 0$with real or complex coefficients $a, b, c$ can be computed in an element-wise mixed stable manner, measured ina relative sense. We also show that this is a stronger property than norm-wise backward stability but weaker thanelement-wise backward stability. We finally show that there does not exist any method that can compute the roots inan element-wise backward stable sense, which is also illustrated by some numerical experiments.

We address the problem of supply chain management for a set of fresh and highly perishable products. Our activity mainly concerns forecasting sales. The study involves 19 retailers (small and medium size stores) and a set of 156 different fresh products. The available data is made of three year sales for each store from 2011 to 2013. The forecasting activity started from a pre-processing analysis to identify seasonality, cycle and trend components, and data filtering to remove noise. Moreover, we performed a statistical analysis to estimate the impact of prices and promotions on sales and customers' behaviour. The filtered data is used as input for a forecasting algorithm which is designed to be interactive for the user. The latter is asked to specify ID store, items, training set and planning horizon, and the algorithm provides sales forecasting. We used ARIMA, ARIMAX and transfer function models in which the value of parameters ranges in predefined intervals. The best setting of these parameters is chosen via a two-step analysis, the first based on well-known indicators of information entropy and parsimony, and the second based on standard statistical indicators. The exogenous components of the forecasting models take the impact of prices into account. Quality and accuracy of forecasting are evaluated and compared on a set of real data and some examples are reported.

Indefinite symmetric matrices occur in many applications, such as optimization, least squares problems, partial differential equations and variational problems. In these applications one is often interested in computing a factorization of the indefinite matrix that puts into evidence the inertia of the matrix or possibly provides an estimate of its eigenvalues. In this paper we propose an algorithm that provides this information for any symmetric indefinite matrix by transforming it to a block anti-triangular form using orthogonal similarity transformations. We also show that the algorithm is backward stable and has a complexity that is comparable to existing matrix decompositions for dense indefinite matrices.

Indefinite symmetric matrices occur in many applications, such as optimization, least squares problems, partial differential equations, and variational problems. In these applications one is often interested in computing a factorization of the indefinite matrix that puts into evidence the inertia of the matrix or possibly provides an estimate of its eigenvalues. In this paper we propose an algorithm that provides this information for any symmetric indefinite matrix by transforming it to a block antitriangular form using orthogonal similarity transformations. We also show that the algorithm is backward stable and has a complexity that is comparable to existing matrix decompositions for dense indefinite matrices.

The generalized Schur algorithm is a powerful tool allowing to compute classicaldecompositions of matrices, such as the QR and LU factorizations. When applied to matrices withparticular structures, the generalized Schur algorithm computes these factorizations with a complexityof one order of magnitude less than that of classical algorithms based on Householder or elementarytransformations. In this manuscript, we describe the main features of the generalized Schur algorithm.We show that it helps to prove some theoretical properties of the R factor of the QR factorization ofsome structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, thatcan hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementationof the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a numberof applications. Finally, we propose a generalized Schur based algorithm for computing the -spaceof polynomial matrices.

In this paper we revisit the problem of performing a QR-step on an unreducedHessenberg matrix H when we know an "exact" eigenvalue ?0 of H. Under exact arithmetic, thiseigenvalue will appear on diagonal of the transformed Hessenberg matrix H~ and will be decoupledfrom the remaining part of the Hessenberg matrix, thus resulting in a deflation. But it is well knownthat in finite precision arithmetic the so-called perfect shift can get blurred and that the eigenvalue ?0can then not be deflated and/or is perturbed significantly. In this paper, we develop a new strategyfor computing such a QR step so that the deflation is almost always successful. We also show howto extend this technique to double QR-steps with complex conjugate shifts.