Time Series Analysis, Part I. A time series is A time series is a sequence of measurements. Usually we deal with equi-spaced measurements. What distinguishes.

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Presentation transcript:

Time Series Analysis, Part I

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.

Definitions Stationary Stochastic

Time Series Examples Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?

Time Series Image Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?

Time Series Image Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?

Time Series Image Features to note –how processed are they? –are there periodicities? –are they stationary? –how predictable are they? –what process generated it?

Definitions Stationary Stochastic Deterministic

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.

Identification Example Given the model dx/dt = -x/tau + f(t), what is tau?

Identification Example Given the model dx/dt = -x/tau + f(t), what is tau?

Is there a different way?

Is there a better way? How could you determine tau from this graph? , , , , , , x f/10

An even better way Matrix method

End intuitive, begin formal Laplace Transform (L) = 1-sided Fourier Transofrm, (FT) Transfer function Ordinary differential equation (ODE) Impulse response function (IRF)

Types of filters Linear filter Autoregressive filter Moving average filter

Techniques for Dependence Analysis Autocovariance and the Autocorrelation Function (ACF)

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)

Matrix forms

Estimation

What is better ACF or FT?

Relationship between ACF and FT

Linear Stationary Models