Introduction to R: Part VIIb2slab.upc.edu/wp-content/uploads/2014/02/CursR_VII.pdf · Introduction to R: Part VII A Time Series Data Session Alexandre Perera i Lluna 1;2 1Centre de

Introduction to R: Part VIIA Time Series Data Session

Alexandre Perera i Lluna 1,2

1Centre de Recerca en Enginyeria Biomèdica (CREB)Departament d’Enginyeria de Sistemes, Automàtica i Informàtica Industrial (ESAII)

Universitat Politècnica de Catalunyamailto:[email protected]

2Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina(CIBER-BBN)

Jan 2011 / Introduction to RUniversitat Rovira i Virgili

mailto:[email protected]

IntroductionMain Signal Processing Functions

Autoregresive ModelsFiltering

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Contents I1 Introduction

Example DataPlotting time series objectsMissing Values and (i)regular Sampling Times

2 Main Signal Processing FunctionsSliding WindowsAutocorrelationSpectral Analysis

3 Autoregresive ModelsARMA simulationFitting ARMA(p,q) models

4 FilteringFiltersFilters Characterization

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Alexandre Perera i Lluna , Introduction to R: Part VII



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Contents IIO




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Obtaining some time Series Data

Yahoo! Finance provides with a historic finance dataAllows free access (not real-time, 15 minutes delay)

Yahoo! Finance download> library(tseries)> telefonica <- get.hist.quote("TEF.MC", start = "2006-01-01",+ end = "2008-05-07", quote = c("Open"))

time series starts 2006-01-02

> class(telefonica)

[1] "zoo"




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Index visualization

code> plot(telefonica)

1214

1618

2022

Index

tele

foni

ca

2006 2007 2008




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Getting raw data from time series (zoo)

code> teldata <- coredata(telefonica)> hist(teldata, 200)

Histogram of teldata

teldata

Fre

quen

cy

12 14 16 18 20 22

05

10




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Time Series

The zoo package allows iregular sampling times and missing data> telefonica[1:10]

2006-01-02 2006-01-03 2006-01-04 2006-01-05 2006-01-0612.75 12.76 12.90 12.99 13.09

2006-01-09 2006-01-10 2006-01-11 2006-01-12 2006-01-1313.17 13.19 13.14 13.10 13.01

The ts class from stats does not:> telts <- as.ts(telefonica)> telts[1:10]

[1] 12.75 12.76 12.90 12.99 13.09 NA NA 13.17 13.19[10] 13.14

however we can deal with NA data in several ways.




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Missing Values in Time Series

> plot(telts[1:25], type = "b")

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> plot(telts[1:25], type = "b")> lines(na.locf(telts[1:25]), type = "b", col = "red")

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> plot(telts[1:25], type = "b")> lines(na.locf(telts[1:25]), type = "b", col = "red")> lines(na.approx(telts[1:25]), type = "b", col = "blue")

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> plot(telts[1:25], type = "b")> lines(na.locf(telts[1:25]), type = "b", col = "red")> lines(na.approx(telts[1:25]), type = "b", col = "blue")> lines(na.spline(telts[1:25]), type = "b", col = "green")

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Sliding WindowsAutocorrelationSpectral Analysis

Sliding Windows (rollers)

rolling functions (applies to both ts and zoo obects)rollapply(data, width, FUN)




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Sliding Windows

> mtelefonica <- rollapply(telefonica, 30, mean)> plot(mtelefonica, type = "l")

1416

1820

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Index

mte

lefo

nica

2006 2007 2008

Also available as rollmean()




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Computing Autocorrelations

> coredata(telts) <- na.approx(telts)> acf(telts)

0 5 10 15 20 25 30

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Lag

AC

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Open




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Computing Autocorrelations

Let’s try the differenciated signal, just for fun:> coredata(telts) <- na.approx(telts)> dtel <- diff(telts)> acf(dtel)

0 5 10 15 20 25 30

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Periodogram

> require(signals)> plot(sunspots)

Time

suns

pots

1750 1800 1850 1900 1950

050

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Periodogram

> spectrum(sunspots)

0 1 2 3 4 5 6

1e−

021e

+00

1e+

021e

+04

frequency

spec

trum

Series: xRaw Periodogram

bandwidth = 0.00120




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Periodogram

> spectrum(sunspots, spans = 10, xlim = c(0, 1))> abline(v = 1:3/12, lty = 3)

0.0 0.2 0.4 0.6 0.8 1.0

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5050

050

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frequency

spec

trum

Series: xSmoothed Periodogram

bandwidth = 0.0122 Extracted from

http://zoonek2.free.fr/





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FFT

Let’s define a two component signal:> x <- 1:256/256> x <- sin(16 * pi * x) + 0.3 * cos(56 * pi * x)

> plot(x, type='l', lwd=3,> main="Signal", xlab="", ylab="")> plot(Mod(fft(x)[1: ceiling((length(x)+1)/2) ]),> type='l', col="blue", lwd=3,> ylab="Mof(fft(x))", xlab="",> main="DTF (Discrete Fourrier Transform)")> par(op)

Extracted from http://zoonek2.free.fr/





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FFT

0 50 100 150 200 250

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Signal

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DTF (Discrete Fourrier Transform)

Mof

(fft(

x))

Extracted from






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spectrogram

Through the seewave package, we can plot spectrograms in dynamic 3Dplots:




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ARMA simulationFitting ARMA(p,q) models

AR(1)

In AR1, we can estimate thecoefficients by performing aregression of the signal againstlag(x,1)

> y <- telts[-1]> x <- telts[-length(telts)]> r <- lm(y ~ x - 1)> coefficients(r)

x1.000345

> plot(y ~ x)> abline(r, col = "red")

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AR(2) process

arima.sim() from stats package

Simulates an Autoregressive integrated moving average general model. AnARMA(q,p) is given by:(

1 −p∑

i−1

αiLi

)Xt =

(1 +

q∑i−1

θiLi

)εt (1)

To simulate an AR(2) process:> n <- 300> r <- arima.sim(list(ar = c(0.9, -0.2)), n)> plot(r)




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AR(2) process

Time

r

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MA(2) process



1 −p∑

i−1

αiLi

)Xt =

(1 +

q∑i−1

θiLi

)εt (2)

To simulate an MA(2) process:> n <- 300> r <- arima.sim(list(ma = c(-0.7, -0.1)), n)> plot(r)




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MA(2) process

Time

r

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ARMA(p,q) process



1 −p∑

i−1

αiLi

)Xt =

(1 +

q∑i−1

θiLi

)εt (3)

To simulate an ARMA(2,2) process:> n <- 300> r <- arima.sim(list(ma = c(-0.7, 1), ar = c(0.9,+ -0.2)), n)> plot(r)




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ARMA(q,p) process

Time

r

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Using arima() to fit ARMA models

> mod <- arima(r, order = c(2, 0, 2))> coef(mod)

ar1 ar2 ma1 ma2 intercept0.9274427 -0.2271395 -0.6929875 0.9999988 0.4574083




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FiltersFilters Characterization

Defining filters

Filter objects (signal package)

Filters are created and stored as objects. The filter function applies anydefined filter to any signal.

Example (3 order, 10Hz low-pass Butterworth filter> library(signal)> bf = butter(3, 0.1)> t = seq(0, 1, len = 100)> x = sin(2 * pi * t * 2.3) + 0.25 * rnorm(length(t))> z = filter(bf, x)> class(z)

[1] "ts"




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Filters

0.0 0.2 0.4 0.6 0.8 1.0

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Filter Functions in signal package

fltfilt()

fir1(),fir2()

butter()

hamming(),hanning(),flattopwin(),gausswin(),kaiser(),blackman(),boxcar()

Savitzsky Golay filtersParks McClellan optimal FIR filter design (remez)... and more




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Frequency response

freqz(fir1(40,0.3))




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Filter characterization

b = c(1, 2)a = c(1, 1)w = linspace(0, 4, 128)freqs(b, a, w)




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The tuneR package

> library(tuneR)> s7 <- readWave("mysong.wav", from = 1, to = 5, units = "seconds")> s7

Wave ObjectNumber of Samples: 32000Duration (seconds): 4Samplingrate (Hertz): 8000Channels (Mono/Stereo): MonoBit (8/16/24/32): 16




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The sound package

> library(sound)> s <- loadSample("mysong.wav", from = 1, to = 5, units = "seconds")> s

type : monorate : 8000 samples / secondquality : 16 bits / samplelength : 480000 samplesR memory : 1920000 bytesHD memory : 960044 bytesduration : 60 seconds




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The audio package

> library(audio)> s <- load.wave("mysong.wav")> head(s)

sample rate: 8000Hz, mono, 16-bits[1] 0.0000000 0.7070923 0.9999695 0.7070923 0.0000000 -0.7071139




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To a file

To a .txt file:> data(tico)> export(tico, f = 22050, header = FALSE, filename = "tico_Gold.txt"))

To a .wav file (e.g.through seewave):> savewav(ticofirst, filename = "tico_firstnote.wav")




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To an audio device:

Using play from the tuneR package:> play(s)

Using listen from the seewave package:> listen(s1, f=8000, from=0.3, to=5)




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End Part VII


Documents

Introduction to R: Part VIIb2slab.upc.edu/wp-content/uploads/2014/02/CursR_VII.pdf · Introduction to R: Part VII A Time Series Data Session Alexandre Perera i Lluna 1;2 1Centre de