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Introduction Simple time series models ARIMAValidating a model Spectral Analysis Wavelets Digital Signal Processing (DSP)Modeling volatility: GARCH models (Generalized Auto Regressive Conditionnal Heteroscedasticity)Multivariate time series State-Space Models and Kalman Filtering Non-linear time series and chaos Other times Discrete-valued time series: Markov chains and beyond Variants of Markov chains Untackled subjects TO SORT This chapter contrasts with the topics we have seen up to now: we were interested in the study of several independant realizations of a simple statistical process (e.g., a gaussian random variable, or a mixture of gaussians, or a linear model); we shall now focus on a single realization of a more complex process. After an introduction, motivating the notion of a time series and giving several examples, simulated or real, we shall present the classical models of time series (AR, MA, ARMA, ARIMA, SARIMA), that provide recipes to build time series with desired properties.We shall then present spectral methods, that focus on the discovery of periodic elements in time series.One aim (or one step) of time series analysis is to find the "structure" of a time series, i.e., to find how it was build, i.e., to find a simple algorithm that could produce similar-looking data.

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State-Space Models and the Kalman filter follow the same vein: they assume that the data is build from linear algebra, but that we do not observe everything -- there are "hidden" (unobserved, latent) variables.A white noise is a series of uncorrelated random variables, whose expectation is zero, whose variance is constant.In other words, these are iid (independant, identically distributed) random variables, to the second order.The analytical formula for long-run PD, for example, explicitly quantifies the contribution of uncertainty to an increase of long-run PD.We recommend the bootstrap approach to addressing the serial correlation issue for a time series sample.

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