Basic time series with R

Transcription

1 Basic time series with R version 0.03, 15 January 2012 Georgi N. Boshnakov 1 Introduction These notes show how to do some basic time series computations with R. If you are taking my time series course, I would advise you to read and keep a copy of my notes Hints about R along with this document. It goes without saying that you need to try the examples yourself. It is relatively easy to explore the concepts you learn in R since it contains many ready-to-use time series for exploration and the help pages of most R functions provide ready-to-try examples. 2 Time series data A time series is more than the vector of the data values. Each observation is associated with a date/time. Each object in R has a class and the basic class for time series objects is ts. A classic example is the series AirPassengers, see its help page for details. You may get a table of the data values, nicely formatted by year and month, by typing the name of the series, AirPassengers. This is fine and useful when you wish to check that the series is what you expect it to be. However, it is hardly ever necessary to include such printouts in reports. Graphs and appropriate summaries are preferable since they are more informative and easier to interpret. You may check the class of AirPassengers as follows: > class(airpassengers) [1] "ts" Do not type >, R prints it to show that it is expecting your next command. The functions start and end give the time of the first and last observations, respectively. The function frequency gives the number of observations in one time unit (e.g., 12 for monthly observations, 4 for quarterly). For example, > start(airpassengers) # year and month of the first value [1] > end(airpassengers) # year and month of the last value [1] > frequency(airpassengers) # number of "seasons" (here: months) [1] 12 The data is monthly, so the lag between successive observations is 1 month. in this case The dataset AirPassengers has been set up so that the unit of time is 1 year (frequency=12). The function deltat uses this time unit to compute the lag by the formula t = 1/frequency). So, > deltat(airpassengers) # 1/12 1

2 [1] Bear this in mind when looking at graphs. The functions time and cycle create time series of the times at which the observations in a time series are taken and their seasons, respectively, while window can be used to take some part of a time series (e.g. the January observations for all years). For example, > ap1 <- window(airpassengers,start=c(1955,1)) # Jan 1955 to end > ap2 <- window(airpassengers,end=c(1955,1)) # start to Jan 1955 > ap3 <- window(airpassengers,start=c(1953,4),end=c(1958,11)) # Apr 1953 to Nov 1958 See the help page of window for more information. 2.1 Creating your own time series If your data is in a vector, then you can turn it into a time series with the help of the function ts. For the sake of example, let us convert AirPassengers to a vector > x <- as.vector(airpassengers) and pretend that we have entered the data ourselves in the vector x. Note that x does not contain the time information (start date, etc), compare the plots of the the time series Airpassengers (Figure 1) and the vector x (Figure 2) to see that the functions in R (plot in this case) take notice of this. These graphs where obtained with the commands > plot(airpassengers) > plot(x) AirPassengers x Time Index Figure 1: Airline time series Figure 2: Airline passengers data as a vector The command > y <- ts(x) will create a ts object, y, with time running as 1, 2,.... seasonality use the start and frequency arguments as in To give the starting date and the > y <- ts(x, frequency=12, start=c(1949,1) ) (start=c(1949,1) specifies first month of 1949). Now plot(y) will produce the graph in Fig. 1. 2

3 3 Autocorrelations and tests Autocorrelations, partial autocorrelations, and crosscorrelations are calculated by acf, pacf, and ccf, respectively. A graph is produced as a byproduct of the computation by these functions since that is what we usually need. For example, > acf(airpassengers) > pacf(airpassengers) To save the results for further use, use assignment as usual, e.g. > apacf <- acf(airpassengers) > appacf <- pacf(airpassengers) If the time series contains missing values use the argument na.action, as in > acf(presidents, na.action = na.pass) (presidents is a time series which happens to contain missing values.) The sample autocorrelations of AirPassengers are shown in Figure 3 and the sample partial autocorrelations in Figure 4. Series AirPassengers Series AirPassengers ACF Partial ACF Lag Lag Figure 3: Airline data Figure 4: Pacf of airline passengers data Study the graphs and take note of the following features. The acf starts from lag 0 but the pacf starts from lag 1. We tend to forget this difference. The lags on the x axis are not labeled with the usual consecutive integers. This is so because R tries to be helpful and takes the unit time for lags to be 1 year (12 months). The time between two successive months is equal to 1/ year. Here is a confirmation (in order to get a smaller amount of output we compute acf up to lag 5 only). > ap <- acf(airpassengers,lag.max=5) > ap Autocorrelations of series 'AirPassengers', by lag The first row in this printout gives the lags, while the corresponding autocorrelations are in the second row. 3

4 4 Portmanteau tests The basic Ljung-Box test can be performed with the function Box.test, e.g. > x1 <- ts(rnorm(100)) # simulated trajectory of IID(0,1) noise > Box.test(x1,lag=5,type="Ljung-Box") Box-Ljung test data: x1 X-squared = , df = 5, p-value = If the time series tested represents residuals from a fitted model, then the degrees of freedom of the test statistic may need correction. For example if x1 represented the residuals from a fitted AR(3) model, then the actual degrees of freedom need to be reduced by 3. This is specified by the parameter fitdf. > Box.test(x1,lag=5,type="Ljung-Box",fitdf=3) Box-Ljung test data: x1 X-squared = , df = 2, p-value = DIY time series computations in R R has a large collection of functions for time series computations which you would normally use in your analyses. For learning purposes however it is often more instructive to do computations from first principles, usually directly implementing formulae given in class or in textbooks. I call such computations DIY (do it yourself). The examples here are with R but most DIY computations in time series can be done easily using any tool with mathematical abilities. You may need to consult Hints about R, mentioned at the beginning of this document, before reading on. A random sample from distribution can be obtained with a single command, say > w <- rnorm(100, mean=0, sd=1) # a trajectory of IID(0,1) Gaussian noise. The following few lines compute the sample autocovariance of w for lag k = 10 using the standard textbook formula. > n <- length(w) > k <- 10 > wbar <- mean(w) > gk <- 0 > for(i in (k+1):n){ gk <- gk + (w[i]- wbar)*(w[i-k]-wbar) } > gk <- gk/n > gk [1] To get a vector of sample autocovariances for lags 0,..., 10 enclose the above in a second loop, > n <- length(w); kmax <- 10; wbar <- mean(w); > g0k <- numeric(kmax+1) # vector for the result. > for(k in 0:kmax){ + gk <- 0 + for(i in (k+1):n){ 4

5 + gk <- gk + (w[i]- wbar)*(w[i-k]-wbar) + g0k[k+1] <- gk/n > g0k [1] [6] [11] Note that indices start from 1 in R, γ k is stored in g0k[k+1]. Autocorrelations can be obtained by dividing by γ 0. > g0k/g0k[1] [1] [6] [11] You can check the DIY calculations using the function acf. > print(acf(w,10)) Autocorrelations of series 'w', by lag If you are comfortable with vector calculations, many computations become, essentially, oneliners, e.g. > n <- length(w); k <- 10; wbar <- mean(w) > sum( (w[(k+1):n]-wbar)*(w[1:(n-k)]-wbar) )/n [1] A simple model for a time series is X t = ε t + θε t 1 for all t, where {ε t } is iid noise. X t is said to be an MA(1) process. A realization (trajectory) of an MA(1) process can be simulated by direct transcription of this formula into R. For example, assuming θ = 0.7, > theta <- 0.7 > n <- 101 > eps <- rnorm(n, mean=0, sd=1) # first generate iid noise > x <- numeric(n) # create a zero vector > for(i in 2:n){ + x[i] <- eps[i] + theta*eps[i-1] > x <- x[-1] # drop x[1] since it cannot be set (eps[0] is not available) > plot(ts(x)) Vector arithmetic provides a shorter and more transparent code. > theta <- 0.7 > n <- 101 > eps <- rnorm(n, mean=0, sd=1) > x <- eps[2:n] + theta * eps[1:(n-1)] 5