Transcription of Time Series Analysis - Auckland
1 Time Series AnalysisLecture Notes for IhakaStatistics DepartmentUniversity of AucklandApril 14, 2005iiContents1 Time Series .. Stationarity and Non-Stationarity .. Some Examples .. Annual Auckland Rainfall .. Nile River Flow .. Yield on British Government Securities .. Ground Displacement in an Earthquake .. United States Housing Starts .. Iowa City Bus Ridership .. 32 Vector Space Vectors In Two Dimensions .. Scalar Multiplication and Addition .. Norms and Inner Products.
2 General Vector Spaces .. Vector Spaces and Inner Products .. Some Examples .. Hilbert Spaces .. Subspaces .. Projections .. Hilbert Spaces and Prediction .. Linear Prediction .. General Prediction .. 153 Time Series Time Series .. Hilbert Spaces and Stationary Time Series .. The Lag and Differencing Operators .. Linear Processes .. Autoregressive Series .. The AR(1) Series .. The AR(2) Series .. Computations .. Moving Average Series .. The MA(1) Series .
3 Invertibility .. Computation .. Autoregressive Moving Average Series .. The ARMA(1,1) Series .. The ARMA(p,q) model .. Computation .. Common Factors .. The Partial Autocorrelation Function .. Computing the PACF .. Computation .. 374 Identifying Time Series ACF Estimation .. PACF Estimation .. System Identification .. model Generalisation .. Non-Zero Means .. Deterministic Trends .. Models With Non-stationary AR Components .. The Effect of Differencing.
4 ARIMA Models .. 495 Fitting and model Fitting .. Computations .. Assessing Quality of Fit .. Residual Correlations .. Forecasting .. Computation .. Seasonal Models .. Purely Seasonal Models .. Models with Short-Term and Seasonal Components .. A More Complex Example .. 666 Frequency Domain Some Background .. Complex Exponentials, Sines and Cosines .. Properties of Cosinusoids .. Frequency and Angular Frequency .. Invariance and Complex Exponentials .. Filters and Filtering.
5 Filters .. Transfer Functions .. Filtering Sines and Cosines .. Filtering General Series .. Computing Transfer Functions .. Sequential Filtering .. Spectral Theory .. The Power Spectrum .. The Cram er Representation .. Using The Cram er Representation .. Power Spectrum Examples .. Statistical Inference .. Some Distribution Theory .. The Periodogram and its Distribution .. An Example Sunspot Numbers .. Estimating The Power Spectrum .. Tapering and Prewhitening.
6 Cross Spectral Analysis .. Computation .. A Simple Spectral Analysis Package for R .. Power Spectrum Estimation .. Cross-Spectral Analysis .. Examples .. 100viContentsChapter Time SeriesTime seriesarise as recordings of processes which vary over time. A recordingcan either be a continuous trace or a set of discrete observations. We willconcentrate on the case where observations are made at discrete equally spacedtimes. By appropriate choice of origin and scale we can take the observationtimes to be 1, 2.
7 Tand we can denote the observations byY1,Y2, .. , are a number of things which are of interest in time Series most important of these are:Smoothing: The observedYtare assumed to be the result of noise values tadditively contaminating a smooth signal t+ tWe may wish to recover the values of the underlying : We may wish to develop a simple mathematical model whichexplains the observed pattern ofY1,Y2, .. ,YT. This model may dependon unknown parameters and these will need to be : On the basis of observationsY1,Y2, .. ,YT, we may wish topredict what the value ofYT+Lwill be (L 1), and possibly to give anindication of what the uncetainty is in the : We may wish to intervene with the process which is producing theYtvalues in such a way that the future values are altered to produce afavourable Stationarity and Non-StationarityA key idea in time Series is that ofstationarity.
8 Roughly speaking, a timeseries is stationary if its behaviour does not change over time. This means, forexample, that the values always tend to vary about the same level and thattheir variability is constant over time. Stationary Series have a rich theory and12 Chapter 1. Introductiontheir behaviour is well understood. This means that they play a fundamentalrole in the study of time , not all time Series that we encouter are stationary. Indeed, non-stationary Series tend to be the rule rather than the exception. However, manytime Series are related in simple ways to Series which are stationary.
9 Two im-portant examples of this are:Trend models: The Series we observe is the sum of a determinstictrendseries and a stationarynoiseseries. A simple example is the linear trendmodel:Yt= 0+ 1t+ common trend model assumes that the Series is the sum of aperiodic seasonal effect and stationarynoise. There are many models: The time Series we observe satisfiesYt+1 Yt= t+1where tis a stationary Series . A particularly important model of this kindis therandom walk. In that case, the tvalues are independent shocks which perturb the current stateYtby an amount t+1to produce a newstateYt+ Some Annual Auckland RainfallFigure shows the annual amount of rainfall in Auckland for the years from1949 to 2000.
10 The general pattern of rainfall looks similar throughout the record,so this Series could be regarded as being stationary. (There is a hint that rainfallamounts are declining over time, but this type of effect can occur over shortishtime spans for stationary Series .) Nile River FlowFigure shows the flow volume of the Nile at Aswan from 1871 to 1970. Theseare yearly values. The general pattern of this data does not change over timeso it can be regarded as stationary (at least over this time period). Yield on British Government SecuritiesFigure shows the percentage yield on British Government securities, monthlyover a 21 year period.
