Wold's theorem

Theorems in statistics | Time series

In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary time series can be written as the sum of two time series, one deterministic and one stochastic. Formally where: * is the time series being considered, * is an uncorrelated sequence which is the innovation process to the process – that is, a white noise process that is input to the linear filter . * is the possibly infinite vector of moving average weights (coefficients or parameters) * is a deterministic time series, such as one represented by a sine wave. The moving average coefficients have these properties: 1. * Stable, that is square summable < 2. * Causal (i.e. there are no terms with j < 0) 3. * Minimum delay 4. * Constant ( independent of t) 5. * It is conventional to define This theorem can be considered as an existence theorem: any stationary process has this seemingly special representation. Not only is the existence of such a simple linear and exact representation remarkable, but even more so is the special nature of the moving average model. Imagine creating a process that is a moving average but not satisfying these properties 1–4. For example, the coefficients could define an acausal and non-minimum delay model. Nevertheless the theorem assures the existence of a causal minimum delay moving average that exactly represents this process. How this all works for the case of causality and the minimum delay property is discussed in Scargle (1981), where an extension of the Wold Decomposition is discussed. The usefulness of the Wold Theorem is that it allows the dynamic evolution of a variable to be approximated by a linear model. If the innovations are independent, then the linear model is the only possible representation relating the observed value of to its past evolution. However, when is merely an uncorrelated but not independent sequence, then the linear model exists but it is not the only representation of the dynamic dependence of the series. In this latter case, it is possible that the linear model may not be very useful, and there would be a nonlinear model relating the observed value of to its past evolution. However, in practical time series analysis, it is often the case that only linear predictors are considered, partly on the grounds of simplicity, in which case the Wold decomposition is directly relevant. The Wold representation depends on an infinite number of parameters, although in practice they usually decay rapidly. The autoregressive model is an alternative that may have only a few coefficients if the corresponding moving average has many. These two models can be combined into an autoregressive-moving average (ARMA) model, or an autoregressive-integrated-moving average (ARIMA) model if non-stationarity is involved. See and references there; in addition this paper gives an extension of the Wold Theorem that allows more generality for the moving average (not necessarily stable, causal, or minimum delay) accompanied by a sharper characterization of the innovation (identically and independently distributed, not just uncorrelated). This extension allows the possibility of models that are more faithful to physical or astrophysical processes, and in particular can sense ″the arrow of time.″ (Wikipedia).

Dimitri Zvonkine - On two ELSV formulas

The ELSV formula (discovered by Ekedahl, Lando, Shapiro and Vainshtein) is an equality between two numbers. The first one is a Hurwitz number that can be defined as the number of factorizations of a given permutation into transpositions. The second is the integral of a characteristic class

From playlist 4th Itzykson Colloquium - Moduli Spaces and Quantum Curves

Differential Equations | Application of Abel's Theorem Example 2

We give an example of applying Abel's Theorem to construct a second solution to a differential equation given one solution. www.michael-penn.net

From playlist Differential Equations

Introduction to additive combinatorics lecture 10.8 --- A weak form of Freiman's theorem

In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to

From playlist Introduction to Additive Combinatorics (Cambridge Part III course)

Euler's Formula for the Quaternions

In this video, we will derive Euler's formula using a quaternion power, instead of a complex power, which will allow us to calculate quaternion exponentials such as e^(i+j+k). If you like quaternions, this is a pretty neat formula and a simple generalization of Euler's formula for complex

From playlist Math

C73 Introducing the theorem of Frobenius

The theorem of Frobenius allows us to calculate a solution around a regular singular point.

From playlist Differential Equations

Odds Ratios and Log(Odds Ratios), Clearly Explained!!!

Odds Ratios and Log(Odds Ratios) are like R-Squared - they describe a relationship between two things. And just like R-Squared, you need to determine if this relationship is statistically significant. This StatQuest goes over all these details so that you are ready for any odds ratio and l

From playlist StatQuest

Wald test | Likelihood ratio test | Score test

See all my videos here: http://www.zstatistics.com/videos/

From playlist Statistical Inference (7 videos)

Weil conjectures 1 Introduction

This talk is the first of a series of talks on the Weil conejctures. We recall properties of the Riemann zeta function, and describe how Artin used these to motivate the definition of the zeta function of a curve over a finite field. We then describe Weil's generalization of this to varie

From playlist Algebraic geometry: extra topics

Calculus - The Fundamental Theorem, Part 1

The Fundamental Theorem of Calculus. First video in a short series on the topic. The theorem is stated and two simple examples are worked.

From playlist Calculus - The Fundamental Theorem of Calculus

Erlend Fornæss Wold: Symplectic Carleman approximation on co-adjoint orbits

For a complex Lie group $G$ with a real form $G_{0}\subset G$, we prove that any Hamiltionian automorphism $\phi$ of a coadjoint orbit $\mathcal{O}_{0}$ of $G_{0}$ whose connected components are simply connected, may be approximated by holomorphic $O_{0}$-invariant symplectic automorphism

From playlist Analysis and its Applications

Why Did Britain Refuse to Annex Malta? (Short Animated Documentary)

In 1955 the Maltese government asked to be incorporated into the United Kingdom as its fifth constituent nation. As you'll be aware, that never happened and Britain refused to annex the Mediterranean island. But why? Well, find out by watching this short and simple animated documentary. T

From playlist The Cold War (1945-1991)

The Last Ditch Attempt to Save the USSR - August Coup of 1991 (Short Animated Documentary)

One of the most important events in the decline and fall of the USSR was the August Coup of 1991 which saw its Vice President attempt to overthrow its President, Mikhail Gorbachev. It didn't go to well and was hastily planned but the fact that it ended peacefully is frankly nothing short o

From playlist The Cold War (1945-1991)

Charles Weibel: K-theory of algebraic varieties (Lecture 1)

The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Charles Weibel: K theory of algebraic varieties Abstract: Lecture 1 will present definitions for the Waldhausen K-theory of rings, varieties, additive and exact categories, and dg c

From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"

Theory of numbers: Congruences: Euler's theorem

This lecture is part of an online undergraduate course on the theory of numbers. We prove Euler's theorem, a generalization of Fermat's theorem to non-prime moduli, by using Lagrange's theorem and group theory. As an application of Fermat's theorem we show there are infinitely many prim

From playlist Theory of numbers

On the classification of Heegaard splittings - David Gabai

David Gabai, IAS October 9, 2015 http://www.math.ias.edu/wgso3m/agenda 2015-2016 Monday, October 5, 2015 - 08:00 to Friday, October 9, 2015 - 12:00 This workshop is part of the topical program "Geometric Structures on 3-Manifolds" which will take place during the 2015-2016 academic year

From playlist Workshop on Geometric Structures on 3-Manifolds

Differential Equations | Application of Abel's Theorem Example 1

We give an example of applying Abel's Theorem to construct a second solution to a differential equation given one solution. www.michael-penn.net

From playlist Differential Equations

Signing Solution - Applied Cryptography

This video is part of an online course, Applied Cryptography. Check out the course here: https://www.udacity.com/course/cs387.

From playlist Applied Cryptography

Configuring an Effective BASH History

Setting an effective BASH history allows you to protect sensitive data and have the widestest rant of BASH commands to slect from in your BASH history file. There is no Linux distrubution that ships with what I wold call an effective BASH history but what I want from my history is not thes

From playlist LPI Linux Essentials

CIC News 11-08-2012: Anonymous,F-Secure,Mariposa

For the interactive Qwiki visit: http://bit.ly/CIC_110812 Want to help collect and share news related to cybercrime and information security? Join the CIC Collaboration Project: Video: http://bit.ly/ciccollabv Website: http://cic.christiaan008.tk Twitter: https://twitter.com/christiaan00

From playlist Vlogs

Weil conjectures 2: Functional equation

This is the second lecture about the Weil conjectures. We show that the Riemann-Roch theorem implies the rationality and functional equation of the zeta function of a curve over a finite field.

From playlist Algebraic geometry: extra topics

Wold's theorem

Related pages