a one factor multivariate time series model of metropolitan wage rates abstract the paper formulates and estimates a single factor multivariate time series model the model is a dynamic generalization of the multiple indicator or factor analysis model it is shown to be a special case of the general state space model and can be estimated by maximum likelihood methods using the kalman filter algorithm the model is used to obtain estimates of the unobserved metropolitan wage rate for los angeles based on observations of sectoral wages within the standard metropolitan statistical area hypothesis tests model diagnostics and out of sample forecasts are used to evaluate the model

forecasting multiple time series with one sided dynamic principal components abstractwe define one sided dynamic principal components odpc for time series as linear combinations of the present and past values of the series that minimize the reconstruction mean squared err

a method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes application to pearcey s integral and its derivatives a numerical method for the evaluation of the cuspoid canonical integrals and their derivatives is described the method exploits cauchy s theorem and jordan s lemma to write the infinite integration path along different contours in the complex plane the method is straightforward to implement on a computer and in many cases results of high accuracy can be obtained using standard quadrature techniques application is made to pearcey s integral p x y and its two partial derivatives and the method is shown to have some significant advantages over other techniques that have been applied to this problem tables of p x y delta p x y delta x delta p x y delta y and the real zeros of p x y are presented for the grid 8 0 or x or 8 0 and 0 or y or 8 0

numerical computation of diffraction catastrophes with codimension eight a computational strategy aimed at evaluating diffraction catastrophes belonging to the x 9 family is presented the approach proposed is based on the use of power series expansions suitably derived for giving meaningful representation of the whole 0 x 9 subfamily jointly with a powerful sequence transformation algorithm the so called weniger transformation the convergence features of the above series expansions are investigated and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the weniger transformation as well as the ease of implementation of the whole approach

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

forecasting and testing in co integrated systems abstract this paper examines the behavior of forecasts made from a co integrated system as introduced by granger 1981 granger and weiss 1983 and engle and granger 1987 it is established that a multi step forecast will satisfy the co integrating relation exactly and that this particular linear combination of forecasts will have a finite limiting forecast error variance a simulation study compares the multi step forecast accuracy of unrestricted vector autoregression with the two step estimation of the vector autoregression imposing the co integration restriction to test whether a system exhibits co integration the procedures introduced in engle and granger 1987 are extended to allow different sample sizes and numbers of variables

identifying a simplifying structure in time series abstract this article studies how to identify hidden factors in multivariate time series process this problem is important because when the series are driven by a set of common factors a a large number of parameters may be needed to obtain an adequate representation of the system and b the estimated parameters will be highly correlated therefore a complex and badly defined relationship can appear when in fact a simpler and parsimonious model in terms of a few common factors can be operating this article develops a methodology to identify the number of factors and to build a simplifying transformation to represent the series it is proved that the number of factors is equal to the rank of the covariance matrices and the parameter matrices of the infinite moving average representation of the process the eigenvectors of these matrices will provide the canonical transformation the method is illustrated with one example using series of the price of wheat in five provinces of spain in the 19th ce

forecasting with nonstationary dynamic factor models abstract in this paper we analyze the structure and the forecasting performance of the dynamic factor model it is shown that the forecasts obtained by the factor model imply shrinkage pooling terms similar to the ones obtained from hierarchical bayesian models that have been applied successfully in the econometric literature thus the results obtained in this paper provide an additional justification for these and other types of pooling procedures the expected decrease in mse for using a factor model versus univariate arima and shrinkage models are studied for the one factor model monte carlo simulations are presented to illustrate this result a factor model is also built to forecast gnp of european countries and it is shown that the factor model can provide a substantial improvement in forecasts with respect to both univariate and shrinkage univariate forecasts

differential equations for the cuspoid canonical integrals differential equations satisfied by the cuspoid canonical integrals i n a are obtained for arbitrary values of n 2 where n 1 is the codimension of the singularity and a 1 2 n 1 a set of linear coupled ordinary differential equations is derived for each step in the sequence i n 0 0 0 0 i n 0 0 0 n 1 i n 0 0 n 2 n 1 i n 0 2 n 2 n 1 i n 1 2 n 2 n 1 the initial conditions for a given step are obtained from the solutions of the previous step as examples of the formalism the differential equations for n 2 fold n 3 cusp n 4 swallowtail and n 5 butterfly are given explicitly in addition iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points the results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

multivariate exponential smoothing and dynamic factor model applied to hourly electricity price analysis thanks to its very simple recursive computing scheme exponential smoothing has become a popular technique to forecast time series in this work we show the advantages of its multivariate version and present some properties of the model which allows us to perform a dynamic factor analysis this analysis leads to a simple methodology to reduce the number of parameters useful when the dimension of observations is large via a linear transformation that decomposes the multivariate process into independent univariate exponential smoothing processes characterized by a single smoothing parameter that goes from zero white noise process to one random walk process a computer implementation of the expectation maximization em algorithm has been built for the maximum likelihood estimation of the models the practicality of the method is demonstrated by its application to hourly electricity price predictions in some day ahead markets such as omel powernext and nord pool markets whose forecasts are given

mathematical properties of a new levin type sequence transformation introduced by cižek zamastil and skala i algebraic theory ci ek zamastil and skala j math phys 44 962 968 2003 introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence sn n 0 of partial sums but also explicit estimates n n 0 for the truncation errors the explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance pade approximants special cases of the new transformation are sequence transformations introduced by levin int j comput math b 3 371 388 1973 and weniger comput phys rep 10 189 371 1989 secs 7 9 numer algor 3 477 486 1992 and also a variant of richardson extrapolation philos trans r soc london ser a 226 299 349 1927 the algebraic theory of these transformations explicit expressions recurrence formulas explicit expres

hyperasymptotic evaluation of the pearcey integral via hadamard expansions we describe high precision computations of the pearcey integral pe x y for real x and y by means of hadamard expansions numerical results for x y situated in different regions of the x y plane are given to illustrate the levels of precision that can be achieved particular emphasis is given to computation in the neighbourhood of the two cusped curves associated with pe x y across which there is either a coalescence of saddles or a stokes phenomenon

reduced rank models for multiple time series summary this paper is concerned with the investigation of reduced rank coefficient models for multiple time series in particular autoregressive processes which have a structure to their coefficient matrices similar to that of classical multivariate reduced rank regression are studied in detail the estimation of parameters and associated asymptotic theory are derived the exact correspondence between the reduced rank regression procedure for multiple autoregressive processes and the canonical analysis of box tiao 1977 is briefly indicated to illustrate the methods u s hog data are considered

theory and numerical evaluation of oddoids and evenoids oscillatory cuspoid integrals with odd and even polynomial phase functions the properties of oscillating cuspoid integrals whose phase functions are odd and even polynomials are investigated these integrals are called oddoids and evenoids respectively and collectively oddenoids we have studied in detail oddenoids whose phase functions contain up to three real parameters for each oddenoid we have obtained its maclaurin series representation and investigated its relation to airy hardy integrals and bessel functions of fractional orders we have used techniques from singularity theory to characterise the caustic or bifurcation set associated with each oddenoid including the occurrence of complex whiskers plots and short tables of numerical values for the oddenoids are presented the numerical calculations used the software package cuspint n p kirk j n l connor c a hobbs an adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives comput phys commun 132 2000 142 165

the uniform asymptotic swallowtail approximation practical methods for oscillating integrals with four coalescing saddle points practical methods for the numerical implementation of the uniform swallowtail approximation have been developed this approximation arises in the uniform asymptotic evaluation of oscillating integrals with four coalescing saddle points a complex contour quadrature technique has been used to evaluate the swallowtail canonical integral s x y z and its partial derivatives delta s delta x delta s delta y delta s delta z this method has the advantage that it is straightforward to implement on a computer and results of high accuracy are readily obtained a comparison is made with other methods that have been reported in the literature for the evaluation of s x y z isometric plots of mod s x y z mod mod delta s delta x mod mod delta s delta y mod mod delta s delta z mod are presented and some properties of the zeros of s 0 y z that lie on the line y 0 are also discussed two methods for the evaluation of the mapping parameters x y z are described an iterative method that is valid when x y z is not close to the swallowtail caustic and an algebraic method valid for x y z on the caustic and for y 0 symbolic algebraic computer programs have been used to carry out the necessary algebraic manipulations in practice both methods for determining x y z are complementary an application of the uniform swallowtail approximation to the butterfly canonical integral has been made the uniform asymptotic swallowtail approximation can now be regarded as a practical tool for the evaluation of oscillating integrals with four coalescing saddle points

forecasting turning points in international output growth rates using bayesian exponentially weighted autoregression time varying parameter and pooling techniques abstract in this paper two variants of an autoregressive leading indicator arli model are employed to forecast turning points in the growth rates of 18 countries annual real output for the years 1974 1986 bayesian predictive densities have been employed to compute probabilities of downturns and upturns and they have been employed in a bayesian decision theoretic framework to obtain optimal turning point forecasts various versions of the arli model with and without pooling of data across countries produced about 70 to 80 percent correct forecasts for 158 turning points time varying parameter models performed slightly better than others the performance of the arli models in forecasting turning points is shown to be considerably better than that of four naive methods of forecasting turning points

axial and focal plane diffraction catastrophe integrals exact expressions in terms of bessel functions are found for some of the diffraction catastrophe integrals that decorate caustics in optics and mechanics these are the axial and focal plane sections of the elliptic and hyperbolic umbilic diffraction catastrophes and symmetric elliptic and hyperbolic unfoldings of the x9 diffraction catastrophes these representations reveal unexpected relations between the integrals

the catastrophe optics of liquid drop lenses the optical caustics associated with a number of higher umbilic catastrophes have been studied experimentally by passing light through drops of water resting on a horizontal glass slide when the perimeters of the drops were constrained to have the symmetry of a square 4mm the caustics were organized by x 9 which is the lowest catastrophe of corank 2 to possess a modulus changing the shape of the perimeter of a thin drop of this symmetry that is large enough to be affected by gravity has the effect of sweeping the modulus k through its complete range 6 to 2 in particular one can study the physical caustics as k passes through the special value 2 which is mathematically excluded from x 9 similar variations of k are produced by varying the size of thin drops with square outlines or simply by adding water to a drop of fixed outline so that its profile becomes highly curved twofold 2mm and fourfold 4mm symmetries are the only ones that allow x 9 with its full range of moduli to participate however the range of caustics produced by drops with these symmetries cannot be understood in terms of x 9 alone to explain the observed details it is necessary to take account of the fact that x 9 is embedded in the higher catastrophe y 1 2 2 a theory of the drop profile under the combined effects of surface tension and gravity valid for small slopes leads to a calculation of the caustics in the paraxial approximation this is completely adequate to explain the observations for thin drops but as a surprising example of structural stability it also suffices when the drops are highly curved and the paraxial approximation breaks down however the paraxial theory does fail as expected when highly curved drops are inverted so that they hang from the lower surface of the glass slide then the x 9 patterns due to gravity are opposed to those due to non paraxiality for a drop of square outline filled nearly to the point where it is about to detach itself and illuminated from below the two effects exactly cancel when the side is 7 3 mm analysis of these patterns of fourfold symmetry helps in understanding the caustic patterns produced by drops having other symmetries or no symmetry at all

topology of strings median string is np complete given a set of strings the problem of finding a string that minimises its distance to the set is directly related with problems frequently encountered in areas involving pattern recognition or computational biology based on the levenshtein or edit distance different definitions of distances between a string and a set of strings can be adopted in particular if this definition is the sum of the distances to each string of the set the string that minimises this distance is the generalised median string finding this string corresponds in speech recognition to giving a model for a set of acoustic sequences and in computational biology to constructing an optimal evolutionary tree when the given phylogeny is a star only efficient algorithms are known for finding approximate solutions the results in this paper are combinatorial and negative we prove that computing the median string corresponds to a np complete decision problems thus proving that this problem is np hard

an median graphs properties algorithms and applications in object prototype learning and similar tasks median computation is an important technique for capturing the essential information of a given set of patterns we extend the median concept to the domain of graphs in terms of graph distance we introduce the novel concepts of set median and generalized median of a set of graphs we study properties of both types of median graphs for the more complex task of computing generalized median graphs a genetic search algorithm is developed experiments conducted on randomly generated graphs demonstrate the advantage of generalized median graphs compared to set median graphs and the ability of our genetic algorithm to find approximate generalized median graphs in reasonable time application examples with both synthetic and nonsynthetic data are shown to illustrate the practical usefulness of the concept of median graphs