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Time Series Analysis, Part I
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A time series is A time series is a sequence of measurements. Usually we deal with equi-spaced measurements. What distinguishes a time series from a sequence of random numbers? A dependence between values at time t and time t+k. Time Series Analysis is concerned with techniques for analyzing this dependence.
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Definitions Stationary Stochastic
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Time Series Examples Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?
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Time Series Image Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?
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Time Series Image Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?
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Time Series Image Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?
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Definitions Stationary Stochastic Deterministic
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Time Series Analysis Identification (of a model) –Diagram of black box concept –In space sciences, identification of the black box is non-trivial Prediction or Forecasting (using a model) –Less concerned with getting the right “model” –More concerned with getting the right prediction. See diagram.
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Identification Example Given the model dx/dt = -x/tau + f(t), what is tau?
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Identification Example Given the model dx/dt = -x/tau + f(t), what is tau?
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Is there a different way?
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Is there a better way? How could you determine tau from this graph? 0.1927, 0.1378, 0.1600, 0.2937 -0.5108, 0.2492, 0.3692, 0.1972 x f/10
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An even better way Matrix method
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End intuitive, begin formal Laplace Transform (L) = 1-sided Fourier Transofrm, (FT) Transfer function Ordinary differential equation (ODE) Impulse response function (IRF)
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Types of filters Linear filter Autoregressive filter Moving average filter
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Techniques for Dependence Analysis Autocovariance and the Autocorrelation Function (ACF)
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Sketch ACF for these functions x(t) = 1 (t = 1, 2, …, 10) x(t) = t (t = 1, 2, …, 10) x(t) = sin(2 t/10) (t = 1, 2, …, 10)
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Matrix forms
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Estimation
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What is better ACF or FT?
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Relationship between ACF and FT
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Linear Stationary Models
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