This chapter presents an introduction to the branch of statistics known as **time** **series** **analysis**. Often the data we collect in environmental studies is collected sequentially over **time** – this type of data is known as **time** **series** data. For instance, we may mon- itor wind speed or water temperatures at regularly spaced **time** intervals (e.g. every hour or once per day). Collecting data sequentially over **time** induces a correlation between measurements because observations near each other in **time** will tend to be more similar, and hence more correlated to observations made further apart in **time**. Often in our data **analysis**, we assume our observations are independent, but with **time** **series** data, this assumption is often false and we would like to account for this temporal correlation in our statistical **analysis**.

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A **time** **series** is a chronological sequence of observations on a particular variable. Usually the observations are taken at regular intervals (days, months, and years), but the sampling could be irregular. It consists of two steps: (1) building a model that represents a **time** **series**, and (2) using the model to predict (forecast) future values. Based on the theory of probability and stochastic processes and more recently complemented by advances on the study of chaotic nonlinear dynamical systems, **time** **series** **analysis** provides a repertoire of mathematical tools for the modeling of hydrological systems. Such tools have proved very effective and useful in numerous applications and case studies. The effectiveness of stochastic descriptions of hydrological processes may reflect the enormous complexity of the hydrological systems, which makes a purely deterministic description ineffective.

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This **time**-**series** **analysis** of payout policies differs from prior studies in several important aspects. First, I carry out the **analysis** with the Vector Autoregressive (VAR) approach proposed by Sims (1982). The VAR model is capable of capturing both the dynamics of the decisions as well as the joint determination of multiple corporate decisions. Second, unlike most prior studies, which separately analyze dividend and share repurchase decisions, I jointly consider the determination of both the level of total payout and its split between dividends and share repurchases. Third, my **analysis** spans the years from 1950 through 1997 (the full set of years for which COMPUSTAT data are available on any of the COMPUSTAT tapes). The long sample period allows me to estimate the impact of macro-economic paramet- ers on the distribution of corporate payout. Lastly, much of the preceding **analysis** of payout policies was concerned with price effects, the major exceptions being Lintner (1956) and, recently, Lee (1996), Lie and Lie (1999) and Fama and French (2001). In contrast, my focus is on payout levels. Thus, this study complements the vast body of literature on the price effects of payout policies.

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After a brief review of the basic concepts of linear **time** **series** **analysis** in Chap- ter 2, the most fundamental ideas of the nonlinear dynamics approach will be intro- duced in the chapters on phase space (Chapter 3) and on predictability (Chapter 4). These two chapters are essential for the understanding of the remaining text; in fact, the concept of a phase space representation rather than a **time** or frequency do- main approach is the hallmark of nonlinear dynamical **time** **series** **analysis**. Another fundamental concept of nonlinear dynamics is the sensitivity of chaotic systems to changes in the initial conditions, which is discussed in the chapter about dy- namical instability and the Lyapunov exponent (Chapter 5). In order to be bounded and unstable at the same **time**, a trajectory of a dissipative dynamical system has to live on a set with unusual geometric properties. How these are studied from a **time** **series** is discussed in the chapter on attractor geometry and fractal dimensions (Chapter 6). Each of the latter two chapters contains the basics about their topic, while additional material will be provided later in Chapter 11. We will relax the re- quirement of determinism in Chapter 7 (and later in Chapter 12). We propose rather general methods to inspect and study complex data, including visual and symbolic approaches. Furthermore, we will establish statistical methods to characterise data that lack strong evidence of determinism such as scaling or self-similarity. We will put our considerations into the broader context of the theory of nonlinear dynamical systems in Chapter 8.

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The importance of Bayesian methods in econometrics has increased rapidly over the last decade. This is, no doubt, fuelled by an increasing appreciation of the advantages that Bayesian inference entails. In particular, it provides us with a formal way to incorporate the prior information we often possess before seeing the data, it fits perfectly with sequential learning and decision making and it directly leads to exact small sample results. In addition, the Bayesian paradigm is particularly natural for prediction, taking into account all parameter or even model uncertainty. The predictive distribution is the sampling distribution where the parameters are integrated out with the posterior distribution and is exactly what we need for forecasting, often a key goal of **time**-**series** **analysis**.

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first step is the estimation of a survival function from mortality tables within each year. The sur- vival function, the MCH function, is based on a simplified form of the Wong and Tsui (2015) CH function, which considers two components of survivability: young-to-old and old-to-oldest compo- nents. Changing trends in the oldest cohort, which are different from those in the younger cohorts, is the consideration of the CH function. The MCH function has a reduced number of parameters and pragmatic parameter constraints, which improves longevity estimates and the interpretability of the function parameters. A mix of univariate and multivariate **time** **series** **analysis** through autoregressive models is performed to generate longevity forecasts. Autoregressive models take into account the autocorrelation of each MCH parameter, and the vector form of the model the cross-correlation between parameters within each MCH component, which improves forecasting ability by reducing estimation errors. To augment longevity forecasts, residual-based multivariate bootstrapping is used in generating confidence intervals. To demonstrate the methodology, the US, Australian and Japanese male and female life tables from 1950 to 2010 were used. Results on confidence intervals and forecasted life expectancies are shown. Robustness checks through cross- validation forecast error statistics and the Diebold–Mariano (DM) test (Diebold & Mariano 1995) for forecasting comparisons with the LC model are shown below. We have concluded that the proposed procedure performs favourably over the LC model in terms of out-of-sample forecasting.

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It was already outlined that one of the (more or less equivalent) approaches of **time**-**series** **analysis** is based on the idea of describing observed **time** **series** by classes of comparatively simple processes. In other words, this modelling aims at ‘parameterizing’ the ACF, that is to find a model that approximates the empirically estimated ACF as well as possible. It can be shown (Wold’s Theorem) that every covariance-stationary process can be approximated ar- bitrarily well by so-called moving-average processes (MA). Similarly, almost every covariance-stationary process can be approximated arbitrarily well by autoregressive processes (AR). Note that ‘approximation’ always refers to a matching of theoretical and empirical ACF (second moments), while noth- ing is stated on higher moments or other stochastic properties. Finally, the ARMA process is an amalgam of its two basic elements AR and MA.

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surrogates can still be very useful in characterizing signals measured from the brain. Andrzejak et al. studied the discriminative power of different **time** **series** **analysis** measures to lateralize the seizure-generating hemisphere in patients with medically intractable mesial temporal lobe epi- lepsy (Andrzejak et al. 2006). The measures that were tested comprised different linear **time** **series** **analysis** measures, different nonlinear **time** **series** **analysis** measures, and a combination of these nonlinear **time** **series** **analysis** measures with surrogates. Subject to the **analysis** were intracranial electroencephalographic recordings from the seizure-free interval of 29 patients. The performance of both linear and nonlinear measures was weak, if not insignificant. A very high performance in correctly lateralizing the seizure-generating hemisphere was, however, obtained by the combina- tion of nonlinear measures with surrogates. Hence, the very strategy that brought us closest to the aim formulated in the introduction of this chapter—to reliably distinguish between stochastic and deterministic dynamics in mathematical model systems—also seems key to a successful character- ization of the spatial distribution of the epileptic process. The degree to which such findings carry over to the study of the predictability of seizures was among the topics discussed at the meeting in Kansas.

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researchers of **time** **series** **analysis** have brought new techniques to the table. In this paper, we examine the performance difference between ARIMA and a relatively recent development in the machine learning community called Long-Short Term Memory Networks (LSTM). Whereas many traditional methods assume the existence of an underlying stochastic model, these algorithmic approaches make no claims about the generation process. Our primary measure of performance is how well each model forecasts out-of-sample data. We find that data with strong seasonal structure are forecast comparatively well by either method. On the other hand, without strong seasonality, there is very little information that can be extracted and both methods tend to perform poorly in forecasting.

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The result shows that the time series mean is an ARMA4, 2 structure, while the ARMA model cannot explain the heteroscedasticity, which indicates that a GARCH model is needed.. Different [r]

Weather Data of India from the dates May 03, 2007 to March 06, 2014 was used to provide useful insight about the performance of the algorithms. The recorded data of the years 2007-2014 were used to make predictions. The accuracy of various models is measured and then compared by Moving Absolute Error, Moving Absolute Scaled Error, Moving Ab- solute Percentage Error and Root Mean Square Error. We also include the criteria Akaike’s Information Criteria and Bayesian Information Criteria. The model which formed the best prediction result will use for comparison and prediction. In this paper, ARIMA model is used in R software for pre- dicting the weather. R is widely used Language not only by scientists but also in many **time** **series** applications. It is an additional concept including mathematical and statistical ex- pertise which helps us to groom our knowledge in analytical field that when deployed into existing processes makes them adaptive to improve conclusions [4]. Thus, together with these advantages offered by this software, we can predict the results before they occur. ETS model is also used from R Software with R Studio Tool. These tools apply R techniques to data modeling. This allows our stationary analytical systems to learn from the data they are modeling [5]. These methods are defined briefly by Hyndman and Khandakar (2008) collection of functions for evaluating **time** **series** data, as well as many

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et al. (2013) and D’Agostino et al. (2013). Selecting x requires some knowledge on the revision process. For quarterly data the second revision (third release) is often used because this is usually the ‘final’ revision from the statistical agency. A second option is to use the final vintage observations y t+h h,T+1 . The final vintage is the most recent publication of the numbers. For example Koenig et al. (2003) and Clements and Galv˜ ao (2013) use the vintage published about a year and a half after the end of their sample. An advantage is that it incorporates the latest available information and are currently closest to the true values as a single **time** **series**. The third option is to use the prebenchmark observations y h,PBM t+h as actuals. Prebenchmark values are the final observation before the first benchmark after a first value for a given date has been reported. Some prebenchmark observations are subject to regular revisions, while x-th release and final vintage observations are subject to benchmark revisions. In contrast to regular non-benchmark revisions, benchmark revisions can and should not be predictable to the forecaster (Croushore, 2006). The actuals should represent the forecasters’ target, rather than be closest to the current truth. Since the prebenchmark values are most consistent with what the forecaster aims to predict, we opt to use those as actuals. 22

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1. Bring the data to the high-level reasoning: Trans- form the **time** **series** into a representation more suitable for higher level reasoning, e.g. discretize the **time** **series** and apply some logical modeling. 2. Bring the high-level reasoning to the data: Choose the hypothesis space and transform the data for the numerical learner in such a way, that results that are meaningful in some way can be found and are preferred. For examples, do some higher level **analysis** of the data and use the re- sults as additional features for the numerical al- gorithm, like flags for the occurrence of special holidays in the sales data, or choose a hypothesis space that corresponds to a meaningful model. This paper deals with the second approach. In the context of Support Vector Machines, kernel functions (which define the hypothesis space) are discussed that can be interpreted as some kind of **time** **series** model. Experiments are made to discover if these different model assumptions have effects in practice and if there exist kernel functions that allow **time** **series** data to be processed with Support Vector Machines without in- tensive preprocessing.

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This study investigated the impact of macroeconomic variables on labor employment in Jordan for the period 1980-2012 by using the fully Modified Ordinary Least Square approach (FMOLS). The economic model incorporated the labor employment as the dependent variable whereas the real Gross Domestic Product (GDP), real Foreign Direct Investment (FDI), and the value of total trade were the independent variables. The results of the **time** **series** properties unit root and the Johansen co-integration tests revealed that that all variables were integrated of order one, I(1) and cointegrated indicating the existence of long- run equilibrium among variables included in the econometric model. There empirical findings showed that all variables have positive and significant impacts on employment level in Jordan labor market. Moreover, the findings showed that real Gross Domestic Product had the substantial influence on employment and a 10% increase in real Gross Domestic Product caused a 6.78% increase in employment level. The employment elasticity with respect to real Foreign Direct Investment was 0.267. It was expected that the findings of this study could be utilized by the government for future follow-up and reassessment of economic development programs in Jordan. One important policy recommendation was the attraction of Foreign Direct Investment into Jordan by setting out some economic policies that would make Jordan more attractive to foreign investors.

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The 2007 World Bank report revealed that murder rates in the Caribbean region is higher than in any other part of the world. In particular, Trinidad and Tobago (T&T), a unitary state in the Caribbean region, that has seen its overall crime rate escalating at a phenomenal rate especially within the last decade. At the same **time**, crime detection has been experiencing a significant drop (it accounts to about 42% for serious crime) which may explain the dramatic increase in crime and delinquency in the 2000s. More recently Mohammed et al. (2009) indicate that past victimization, especially when no police action was taken to arrest the perpetrator, perpetuated and increased the perceived fear of crime in T&T. Nevertheless, research on criminal activity in Trinidad and Tobago (and the Caribbean) has been generally scarce.

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In recent years a major class of nonlinear **time** **series** models - regime switching model - has become a popular workhorse in many economic and financial studies. For example, studies dedicated to business cycles, equity return forecasting, term structure of interest rates and exchange rates volatility modelling, have been adopting increasingly a regime switching approach to describe the **time** **series** properties of economic and financial variables. Why the regime switching models are so attractive to researchers? Arising from practical aspects of modelling dynamic economic and financial **time** **series**, researches often observe variables undergo either/both permanent structural changes or/and recurrent temporary changes over a long sample period. The most well-known example of this type is the repeatedly appearance of expansion and recession phrases in business cycles. The apparent **time** inconsistency patterns make researchers believe that the constant parameter **time** **series** models may not be able to incorporate the observed regime changes in practice. One way to solve this problem is to build a stochastic process into the **time** **series** model that is capable to describe the regime switches in the mean or/and variance of the variables being studied. Although we do not directly observe these regime switches at any point in **time**, we may draw inferences on the regimes in operation of a known stochastic process, and hence the possible future switches in forecast.

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One of the top suppliers of machine tools and plastic processing machines, Cincinnati Milacron Inc. displays a very large variance in earnings per share during the 1977-1993 period. Demand fluctuations for metal and plastic cutting machinery affected the company because of its leading position. Although other parts of the company’s businesses did very well in this **time**, the weakness of the machine tool segment, which counts for 65-70 percent of total sales, eclipsed the good news from the other sectors. Large orders heavily influence the company’s accounts very much. The company’s financial strength was steadily evaluated with a B rating by The Value Line Investment Survey during this **time**, but earnings predictability decreased eight-fold.

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Homogeneous linear diff eqns with constant coefficients.. 1.4[r]

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Over the last decade a variety o f new techniques for the treatment o f chaotic **time** **series** has been developed. Initially these concentrated on the characterisation o f chaotic **time** **series** using invariants such as fractal dimensions or Lyapunov exponents. However, attention has recently focused on the possibility o f predicting the future short term behaviour o f such **time** **series** using the type o f methods discussed in chapter 2. It soon became obvious that these methods could be used to develop a set o f novel signal processing tools based on nonlinear state space theory. This in turn resulted in the development o f a proliferation o f noise reduction algorithms to clean chaotic **time** **series** that had been corrupted by low amplitude noise. Although this is by no means the only application o f nonlinear dynamics to signal processing and many other directions are being pursued by a number o f researchers it seems to have been the direction that has gained most attention. Thus this is the problem we have concentrated on here.

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As the study was based on **time** **series** data for the period 1991-2012, it was necessary to test the stationarity of the variables in the export equation. Dickey Fuller statistics (Unit root test) was first applied to test for the stationarity of the chosen variables. After checking for stationarity, the long run relationship between exports and the selected variables. Viz, foreign direct investment, exchange rate and investment on infrastructure was established by estimating the error correction model.

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