Spectral Analysis. Al Nosedal University of Toronto. Winter Al Nosedal University of Toronto Spectral Analysis Winter / 71
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1 Spectral Analysis Al Nosedal University of Toronto Winter 2016 Al Nosedal University of Toronto Spectral Analysis Winter / 71
2 essentially, all models are wrong, but some are useful George E. P. Box (one of the great statistical minds of the 20th century). Al Nosedal University of Toronto Spectral Analysis Winter / 71
3 Description of data The data in sunspots shows yearly numbers of sunspots from 1771 to Al Nosedal University of Toronto Spectral Analysis Winter / 71
4 Reading data sunspots<-read.table(file="sunspots.dat",header=false); sunspots<-ts(sunspots,start=1771); Al Nosedal University of Toronto Spectral Analysis Winter / 71
5 Exercise 1. Make the time series plot of the sunspots.dat. 2. Make the correlogram (ACF) of the series. Al Nosedal University of Toronto Spectral Analysis Winter / 71
6 Time Series Plot sunspot activity Time Al Nosedal University of Toronto Spectral Analysis Winter / 71
7 Correlogram of 11-year differences for sunspot data V1 ACF Lag Al Nosedal University of Toronto Spectral Analysis Winter / 71
8 Before attempting to model the series, the cyclical component needs to be removed. A simple procedure which is useful here is to take appropriate differences of the series. Here, assuming an 11 year cycle, differences between points 11 years apart are used Y t = X t X t 11 (The resulting series should contain no obvious periodic component). Al Nosedal University of Toronto Spectral Analysis Winter / 71
9 Exercise 1. Find Y t = X t X t 11 using R. 2. Make the time series plot of Y t. 3. Make the correlogram (ACF) of Y t. 4. Fit AR model using ar.yw. 5. Plot residuals from your model. Al Nosedal University of Toronto Spectral Analysis Winter / 71
10 New Series (sunspots) newsun<-sunspots[12:100] - sunspots[1:89] Al Nosedal University of Toronto Spectral Analysis Winter / 71
11 Time Series Plot (newsun) sunspot activity Time Al Nosedal University of Toronto Spectral Analysis Winter / 71
12 Correlogram Series newsun ACF Lag Al Nosedal University of Toronto Spectral Analysis Winter / 71
13 Fitting Model sun.ar<-ar.yw(newsun); sun.ar ## ## Call: ## ar.yw.default(x = newsun) ## ## Coefficients: ## ## ## ## Order selected 5 sigma^2 estimated as Al Nosedal University of Toronto Spectral Analysis Winter / 71
14 Plot of Akaike s criterion plot.ts(sun.ar$aic, xlab="order", ylab="akaike's criterion"); Al Nosedal University of Toronto Spectral Analysis Winter / 71
15 Plot of Akaike s criterion Akaike's criterion Order Al Nosedal University of Toronto Spectral Analysis Winter / 71
16 Plot of Residuals plot.ts(sun.ar$resid, xlab="year", ylab="residual from AR Model"); abline(h=0, col="red"); Al Nosedal University of Toronto Spectral Analysis Winter / 71
17 Plot of Residuals Residual from AR Model Year Al Nosedal University of Toronto Spectral Analysis Winter / 71
18 ACF of Residuals acf(sun.ar$resid); Al Nosedal University of Toronto Spectral Analysis Winter / 71
19 ACF of Residuals Series sun.ar$resid ACF Lag Al Nosedal University of Toronto Spectral Analysis Winter / 71
20 All the correlations (apart from the one corresponding to lag 11) are small and lie within the horizontal bands, indicating that they do not differ significantly from zero. This suggests that the fitted autoregressive model is a reasonable fit for the data. Al Nosedal University of Toronto Spectral Analysis Winter / 71
21 Spectral Density If a time series {Y t } has autocovariance γ(k) satisfying k= γ(k) <, then we define its spectral density as f (w) = k= γ(k)e 2πiwk for < w <. We define its normalized spectral density as f (w) = k= ρ(k)e 2πiwk for < w < (where ρ(k) represents its autocorrelation function). Al Nosedal University of Toronto Spectral Analysis Winter / 71
22 Periodogram The periodogram is the sample estimate of the power spectrum (or spectral density). It is given by n 1 ˆf (w) = Ĉ(0) + 2 Ĉ(k)cos(2πkw). k=1 Al Nosedal University of Toronto Spectral Analysis Winter / 71
23 Why is the periodogram useful? The periodogram is ideal for identifying periodicity in data and estimating the frequency of the period. Consider a very simple periodic pattern with no noise producing the Figure shown below (see R code on next slide). Al Nosedal University of Toronto Spectral Analysis Winter / 71
24 n<-200; x<-c(1:n); y<-5+4*cos(2*pi*x/ ); plot(x,y,type="l"); title("pure periodic series"); Al Nosedal University of Toronto Spectral Analysis Winter / 71
25 Pure periodic series y x Al Nosedal University of Toronto Spectral Analysis Winter / 71
26 Periodogram (R code) z<-spec.pgram(y,fast=false, taper =0.0); Al Nosedal University of Toronto Spectral Analysis Winter / 71
27 Periodogram (Graph) Series: y Raw Periodogram spectrum 1e 05 1e frequency bandwidth = Al Nosedal University of Toronto Spectral Analysis Winter / 71
28 Noisy version n<-200; x<-c(1:n); y<-5+4*cos(2*pi*x/ ); err<-rnorm(n,0,0.25); y_err<-y + err; plot(x,y_err,type="l"); title("periodic series with noise"); Al Nosedal University of Toronto Spectral Analysis Winter / 71
29 Periodic series with noise y_err x Al Nosedal University of Toronto Spectral Analysis Winter / 71
30 Periodogram (R code) z_err<-spec.pgram(y_err,fast=false, taper =0.0); Al Nosedal University of Toronto Spectral Analysis Winter / 71
31 Periodogram (Graph) Series: y_err Raw Periodogram spectrum 1e 03 1e frequency bandwidth = Al Nosedal University of Toronto Spectral Analysis Winter / 71
32 The Sunspot Data (R Code) sunspots<-read.table(file="sunspots.dat",header=false); sunspots<-ts(sunspots,start=1771); z_sun<-spec.pgram(sunspots,fast=false, taper =0.0); Al Nosedal University of Toronto Spectral Analysis Winter / 71
33 The Sunspot Data (Graph) Series: sunspots Raw Periodogram spectrum frequency bandwidth = Al Nosedal University of Toronto Spectral Analysis Winter / 71
34 The Sunspot Data (R Code...again) names(z_sun); plot(z_sun$freq,z_sun$spec,type="l"); Al Nosedal University of Toronto Spectral Analysis Winter / 71
35 The Sunspot Data (Graph) z_sun$spec z_sun$freq Al Nosedal University of Toronto Spectral Analysis Winter / 71
36 The Sunspot Data (R Code...again) names(z_sun); plot(z_sun$freq,log(z_sun$spec),type="l"); Al Nosedal University of Toronto Spectral Analysis Winter / 71
37 The Sunspot Data (Graph) log(z_sun$spec) z_sun$freq Al Nosedal University of Toronto Spectral Analysis Winter / 71
38 A General Approach to Time Series Modeling Plot the series and examine the main features of the graph, checking in particular whether there is a) a trend, b) a seasonal component, c) any apparent sharp changes in behavior, d) any outlying observations. Al Nosedal University of Toronto Spectral Analysis Winter / 71
39 A General Approach to Time Series Modeling Remove the trend and seasonal components to get stationary residuals. To achieve this goal it may sometimes be necessary to apply a preliminary transformation to the data. For example, if the magnitude of the fluctuations appears to grow roughly linearly with the level of the series, then the transformed series ln(x t ) will have fluctuations of more constant magnitude. There are several ways in which trend and seasonality can be removed, some involving estimating components and subtracting them from the data, and others depending on differencing the data, i.e., replacing the original series X t by Y t = X t X t d for some positive integer d. Whichever method is used, the aim is to produce a stationary series, whose values we shall refer to as residuals. Al Nosedal University of Toronto Spectral Analysis Winter / 71
40 A General Approach to Time Series Modeling Choose a model to fit the residuals. Forecasting will be achieved by forecasting the residuals and then inverting the transformations described above to arrive at forecasts of the original series X t. Al Nosedal University of Toronto Spectral Analysis Winter / 71
41 Example: Accidental deaths The file deaths.dat contains monthly accidental death figures in the U. S. A. from 1973 to Al Nosedal University of Toronto Spectral Analysis Winter / 71
42 Reading data deaths<-read.table(file="deaths.dat",header=false); deaths<-unlist(deaths); deaths.ts<-ts(deaths,start=1973, freq=12); Al Nosedal University of Toronto Spectral Analysis Winter / 71
43 Time Series Plot Accidental deaths thousands Time Al Nosedal University of Toronto Spectral Analysis Winter / 71
44 This time series plot shows a strong seasonal pattern. We shall consider the problem of representing the data as the sum of a trend, a seasonal component, and a residual term. Al Nosedal University of Toronto Spectral Analysis Winter / 71
45 Periodogram (R Code) z_deaths<-spec.pgram(deaths.ts,fast=false, taper =0.0,log="no"); Al Nosedal University of Toronto Spectral Analysis Winter / 71
46 Periodogram (Graph) Series: deaths.ts Raw Periodogram spectrum frequency bandwidth = Al Nosedal University of Toronto Spectral Analysis Winter / 71
47 Fitting Seasonal Component deaths_mean<-deaths-mean(deaths.ts); # mean corrected series; n<-length(deaths.ts); time<-c(1:n); col_1<-rep(1,n); col_c<-cos(2*pi*time/12); col_s<-sin(2*pi*time/12); X<-cbind(col_1,col_c,col_s); fit_deaths<-lm(deaths_mean~ -1 + X); # no intercept needed; # it is included in X; Al Nosedal University of Toronto Spectral Analysis Winter / 71
48 Fitting Seasonal Component summary(fit_deaths)[4] Al Nosedal University of Toronto Spectral Analysis Winter / 71
49 Fitting Seasonal Component ## $coefficients ## Estimate Std. Error t value Pr(> t ) ## Xcol_ e e e+00 ## Xcol_c e e e-09 ## Xcol_s e e e-09 Al Nosedal University of Toronto Spectral Analysis Winter / 71
50 Predictions vs Observations preds_ts<-ts(fit_deaths$fit,start=1973,freq=12); # turning predictions from linear #model into a time series object; plot.ts(deaths.ts,ylab="thousands"); lines(preds_ts+mean(deaths.ts),lty=2, col="blue"); legend("topright",c("obs", "pred"), lty=c(1,2), col=c("black","blue"),bty="n" ); Al Nosedal University of Toronto Spectral Analysis Winter / 71
51 thousands obs pred Time Al Nosedal University of Toronto Spectral Analysis Winter / 71
52 Graph of deseasonalized data deaths_deseason<-fit_deaths$res; plot.ts(deaths_deseason); Al Nosedal University of Toronto Spectral Analysis Winter / 71
53 deaths_deseason Time Al Nosedal University of Toronto Spectral Analysis Winter / 71
54 Some comments The graph of deseasonalized data suggests the presence of an additional quadratic trend function. Al Nosedal University of Toronto Spectral Analysis Winter / 71
55 Fitting Quadratic Trend n<-length(deaths_deseason); time<-c(1:n); col_1<-rep(1,n); time2<-time^2; T<-cbind(col_1,time,time2); fit_trend<-lm(deaths_deseason~ -1+T); # no intercept needed; # it is included in T; Al Nosedal University of Toronto Spectral Analysis Winter / 71
56 Fitting Quadratic Trend summary(fit_trend)[4] Al Nosedal University of Toronto Spectral Analysis Winter / 71
57 Fitting Quadratic Trend ## $coefficients ## Estimate Std. Error t value Pr(> t ) ## Tcol_ e-08 ## Ttime e-08 ## Ttime e-07 Al Nosedal University of Toronto Spectral Analysis Winter / 71
58 Plot of Residuals Res<-fit_trend$res plot.ts(res); Al Nosedal University of Toronto Spectral Analysis Winter / 71
59 Plot of Residuals Res Time Al Nosedal University of Toronto Spectral Analysis Winter / 71
60 ACF of Residuals acf(res); Al Nosedal University of Toronto Spectral Analysis Winter / 71
61 ACF of Residuals Series Res ACF Lag Al Nosedal University of Toronto Spectral Analysis Winter / 71
62 Some comments From our plots, it is clear that there is substantial dependence in the series of residuals. Al Nosedal University of Toronto Spectral Analysis Winter / 71
63 Modeling Residuals library(forecast); auto.arima(res); Al Nosedal University of Toronto Spectral Analysis Winter / 71
64 Modeling Residuals ## Series: Res ## ARIMA(1,0,1) with zero mean ## ## Coefficients: ## ar1 ma1 ## ## s.e ## ## sigma^2 estimated as : log likelihood= ## AIC= AICc= BIC= Al Nosedal University of Toronto Spectral Analysis Winter / 71
65 The classical decomposition model X t = m t + s t + Y t where m t is a slowly changing function known as a trend component, s t is a function with known period d referred to as the seasonal component, and Y t is a random noise component that is stationary. Al Nosedal University of Toronto Spectral Analysis Winter / 71
66 The classical decomposition model For our example, we have that X t cos( 2πt 2πt 12 ) sin( 12 ) t t Y t W t 1 + W t Al Nosedal University of Toronto Spectral Analysis Winter / 71
67 Another way auto.arima(deaths.ts); Al Nosedal University of Toronto Spectral Analysis Winter / 71
68 Another way ## Series: deaths.ts ## ARIMA(0,1,1)(0,1,1)[12] ## ## Coefficients: ## ma1 sma1 ## ## s.e ## ## sigma^2 estimated as 99480: log likelihood= ## AIC= AICc=857.5 BIC=863.3 Al Nosedal University of Toronto Spectral Analysis Winter / 71
69 Forecasts sarima<-auto.arima(deaths.ts); fcast<-forecast(sarima); plot(fcast); Al Nosedal University of Toronto Spectral Analysis Winter / 71
70 Forecasts Forecasts from ARIMA(0,1,1)(0,1,1)[12] Al Nosedal University of Toronto Spectral Analysis Winter / 71
71 Some final comments The corresponding fitted model for {X t } is thus the SARIMA (0, 1, 1) (0, 1, 1) 12 process where {W t } WN(0, σ 2 W = 99480). ( B)( B 12 )W t, Al Nosedal University of Toronto Spectral Analysis Winter / 71
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