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Craciun Maria - Automatic trend estimation

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Craciun Maria Automatic trend estimation

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Discrete stochastic processes and time series -- Trend definition -- Finite AR(1) stochastic process -- Monte Carlo experiments. -- Monte Carlo statistical ensembles -- Numerical generation of trends -- Numerical generation of noisy time series -- Statistical hypothesis testing -- Testing the i.i.d. property -- Polynomial fitting -- Linear regression -- Polynomial fitting -- Polynomial fitting of artificial time series -- An astrophysical example -- Noise smoothing -- Moving average -- Repeated moving average (RMA) -- Smoothing of artificial time series -- A financial example -- Automatic estimation of monotonic trends -- Average conditional displacement (ACD) algorithm -- Artificial time series with monotonic trends -- Automatic ACD algorithm -- Evaluation of the ACD algorithm -- A paleoclimatological example -- Statistical significance of the ACD trend -- Time series partitioning -- Partitioning of trends into monotonic segments -- Partitioning of noisy signals into monotonic segments -- Partitioning of a real time series -- Estimation of the ratio between the trend and noise -- Automatic estimation of arbitrary trends -- Automatic RMA (AutRMA) -- Monotonic segments of the AutRMA trend -- Partitioning of a financial time series.;Our book introduces a method to evaluate the accuracy of trend estimation algorithms under conditions similar to those encountered in real time series processing. This method is based on Monte Carlo experiments with artificial time series numerically generated by an original algorithm. The second part of the book contains several automatic algorithms for trend estimation and time series partitioning. The source codes of the computer programs implementing these original automatic algorithms are given in the appendix and will be freely available on the web. The book contains clear statement of the conditions and the approximations under which the algorithms work, as well as the proper interpretation of their results. We illustrate the functioning of the analyzed algorithms by processing time series from astrophysics, finance, biophysics, and paleoclimatology. The numerical experiment method extensively used in our book is already in common use in computational and statistical physics.

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Calin Vamos and Maria Craciun SpringerBriefs in Physics Automatic trend estimation 2013 10.1007/978-94-007-4825-5_1 The Author(s) 2012
1. Introduction
Calin Vamos 1
(1)
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, 400110 Cluj-Napoca, Romania
Calin Vamos (Corresponding author)
Email:
Maria Craciun
Email:
Abstract
A complete presentation of the theory of stochastic processes can be found in any treatise on the probability theory and time series theory. In this introductory chapter we briefly present some basic notions which are used in the rest of the book. The main methods to estimate trends from noisy time series are introduced in Sect. 1.2. In the last section we discuss the properties of the order one autoregressive stochastic process AR(1) which has the serial correlation described by a single parameter and which is a good first approximation for many noises encountered in real phenomena.
A complete presentation of the theory of stochastic processes can be found in any treatise on the probability theory, e.g., [. In the last section we discuss the properties of the order one autoregressive stochastic process AR(1) which has the serial correlation described by a single parameter and which is a good first approximation for many noises encountered in real phenomena.
1.1 Discrete Stochastic Processes and Time Series
At the occurrence of an event Picture 1 the random variable Picture 2 takes the value Picture 3 . We follow the practice of denoting by small letters the realizations of the random variable denoted by the corresponding capital letters. Throughout this book we consider only continuous random variables with real values. If the random variable is absolutely continuous, then it has a probability density function (pdf) denoted Automatic trend estimation - image 4 . The cumulative distribution function (cdf) Automatic trend estimation - image 5 is the probability that the random variable Picture 6 takes on a value less than or equal to Automatic trend estimation - image 7 . We denote the mean of the random variable by Automatic trend estimation - image 8 and its variance by Automatic trend estimation - image 9 .
The evolution in time of a random phenomenon is modeled by a stochastic process , i.e., a family of random variables Automatic trend estimation - image 10 defined on the same probability space and indexed by a set of real numbers Picture 11 . In this book we study only discrete stochastic processes for which Automatic trend estimation - image 12 contains equidistant sampling moments. The observations are made at discrete time moments Automatic trend estimation - image 13 , where Automatic trend estimation - image 14 , Picture 15 is the sampling interval, and Automatic trend estimation - image 16 is the initial time. The observed values Automatic trend estimation - image 17 are realizations of the corresponding random variables Automatic trend estimation - image 18 . Although the number of observations is always finite, we assume that there is an infinite stochastic process whose realizations for and have not been observed To distinguish between the - photo 19 whose realizations for Picture 20 and Automatic trend estimation - image 21 have not been observed. To distinguish between the infinite stochastic process which models the time evolution of the natural phenomenon and its measurements, we call time series the finite sequence of real numbers Automatic trend estimation - image 22 .
The joint cdf of the random variables Automatic trend estimation - image 23 is the probability that their values are smaller than some given values
Automatic trend estimation - image 24
where Automatic trend estimation - image 25 and Automatic trend estimation - image 26 . For absolutely continuous random variables there exists the joint pdf Picture 27 . A stochastic process is (strictly) stationary if for every index vector Automatic trend estimation - image 28 and integer Automatic trend estimation - image 29 we have Automatic trend estimation - image 30 or Automatic trend estimation - image 31 , where Automatic trend estimation - image 32 , i.e., its joint probabilities do not change under temporal translations. From this definition, for Automatic trend estimation - image 33 it follows that all the components of a stationary process have the same probability distribution Automatic trend estimation - image 34 for all integers Picture 35 . Such a stochastic process is called identically distributed .
The autocovariance function of a stochastic process with finite variance for all its components ( Automatic trend estimation - image 36
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