Series (mathematics)

About: Series (mathematics) is a research topic. Over the lifetime, 31012 publications have been published within this topic receiving 625614 citations. The topic is also known as: mathematical series.

Some methods of speeding up the convergence of iteration methods

Abstract: For the solution of the functional equation P (x) = 0 (1) (where P is an operator, usually linear, from B into B, and B is a Banach space) iteration methods are generally used. These consist of the construction of a series x0, …, xn, …, which converges to the solution (see, for example [1]). Continuous analogues of these methods are also known, in which a trajectory x(t), 0 ⩽ t ⩽ ∞ is constructed, which satisfies the ordinary differential equation in B and is such that x(t) approaches the solution of (1) as t → ∞ (see [2]). We shall call the method a k-step method if for the construction of each successive iteration xn+1 we use k previous iterations xn, …, xn−k+1. The same term will also be used for continuous methods if x(t) satisfies a differential equation of the k-th order or k-th degree. Iteration methods which are more widely used are one-step (e.g. methods of successive approximations). They are generally simple from the calculation point of view but often converge very slowly. This is confirmed both by the evaluation of the speed of convergence and by calculation in practice (for more details see below). Therefore the question of the rate of convergence is most important. Some multistep methods, which we shall consider further, which are only slightly more complicated than the corresponding one-step methods, make it possible to speed up the convergence substantially. Note that all the methods mentioned below are applicable also to the problem of minimizing the differentiable functional (x) in Hilbert space, so long as this problem reduces to the solution of the equation grad (x) = 0.

Testing for Common Trends

Abstract: Cointegrated multiple time series share at least one common trend. Two tests are developed for the number of common stochastic trends (i.e., for the order of cointegration) in a multiple time series with and without drift. Both tests involve the roots of the ordinary least squares coefficient matrix obtained by regressing the series onto its first lag. Critical values for the tests are tabulated, and their power is examined in a Monte Carlo study. Economic time series are often modeled as having a unit root in their autoregressive representation, or (equivalently) as containing a stochastic trend. But both casual observation and economic theory suggest that many series might contain the same stochastic trends so that they are cointegrated. If each of n series is integrated of order 1 but can be jointly characterized by k > n stochastic trends, then the vector representation of these series has k unit roots and n — k distinct stationary linear combinations. Our proposed tests can be viewed alterna...

Non-linear time series. A dynamical system approach

Abstract: Preface Acknowlegement Introduction 1. An introduction to dynamical systems 2. Some non-linear time series models 3. Probability structure 4. Statistical aspects 5. Non-linear least-squares prediction based on non-linear models 6. Case studies

Introduction to multiple time series analysis

Abstract: This graduate-level textbook deals with analyzing and forecasting multiple time series. It considers a wide range of multiple time series models and methods. The models include vector autoregressive, vector autoregressive moving average, cointegrated and periodic processes as well as state space and dynamic simultaneous equations models. Least squares, maximum likelihood and Bayesian methods are considered for estimating these models. Different procedures for model selection or specification are treated and a range of tests and criteria for evaluating the adequacy of a chosen model are introduced. The choice of point and interval forecasts as well as innovation accounting are presented as tools for structural analysis within the multiple time series context.