Introduction to R for Time Series Analysis Lecture Notes 2

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1 Introduction to R for Time Series Analysis Lecture Notes OVERVIEW OF R R is a widely used environment for statistical analysis. The striking difference between R and most other statistical software is that it is free software and that it is maintained by scientists Since its introduction in 1996, the R project has gained many users and contributors, which continuously extend the capabilities of R by releasing add on (packages) that offer previously not available functions and methods or improve the existing ones. R offers almost unlimited possibilities for statistical data analysis. One disadvantage is that R is used within a command line interface, which imposes a slightly steeper learning curve than other software. 1.1 Downloading and Installing R R is distributed by the Comprehensive R Archive Network (CRAN) it is available from the url: After the installation, R can be started like any other application for Windows that is by double clicking on the corresponding icon. 1.2 R Packages R functionality is based around the concept of packages. A package is a collection of functions to carry out certain task. For example, the nls package does nonlinear regression, the ts package contains a variety of time-series functions, and so on. The base packages are automatically available with a default installation. Contributed packages, on the other hand, need to be installed individually. One can obtain these packages by following the download link on the R home page. By its very nature, R is a dynamic, evolving computing environment, and packages are continuing to be written at a rapid rate. Very often, the capabilities of two or more packages overlap. For example, both the tseries and ts packages provide procedures for fitting ARMA models. Some of the packages that may be of interest to time series analysis are listed below. Some of these are installed automatically when you install R, and the others can be downloaded and installed individually from the R web site. ts ---- Time-series functions, including ARIMA modelling, regression with ARMA errors, Box-Pierce and Ljung-Box tests, ACF, PACF, CCF, spectral density estimation, Phillips-Perron unit root test, tseries ---- More time series functions, including ARMA modelling, BDS test, Jarque-Bera test, ADF test for a unit root, KPSS test for stationarity, runs tests tests for nonlinearity. Other time series packages includes MASS, timedate, timeseries, fbasics, fseries, urca, funitroots, forecast, stats, uroot etc. 1

2 1.3 Useful Web sites There is a growing number of web sites dedicated to providing information about R and code to be used with R. Some of the main sites are listed here. There are also manuals, other links, and facilities for joining various R mailing lists. Three R mailing lists are archived here. This site provides a large and excellent collection of R tips. This is a compilation of R resources on the web. 2.0 GETTING STARTED IN R 2.1 Starting R When R starts you should see a window with the following text (the text may differ slightly depending on the version you are using): Version ( ), ISBN R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for a HTML browser interface to help. Type 'q()' to quit R. > The symbol '> ' appears in red, and is called the prompt; it indicates that R is waiting for you to type a command. If a command is too long to _t on a line a '+' is used to indicate that the command is continued from the previous line. All commands that you type in appear in red, and R output appears in blue. In the code examples that follow, you do not need to type the prompt '> '. 2.2 Entering Commands The easiest way of using R is to enter commands directly in this window. For example, try typing 2+2 at the command prompt and pressing enter. You should see the output: [1] 4 The [1] indicates that the answer is a vector of length one; we will return to this later. You can use the cursor keys to edit the command line as normal. Use the up and down cursor keys to scroll backward and forward through your previous commands; this can save you a lot of typing. The main window is called the command window. Other windows you may come across are the command history window and graphics windows. 2

3 2.3 Command History Rather than scrolling through a long list of previous commands using the cursor keys, R provides another way of retrieving previous commands. Type history() and a new window will appear listing the last 25 commands and you can copy and paste the command or commands you want. To retrieve more previous commands, for example the last 100, use history(100). 2.4 Getting Help R provides two types of help, both available through the Help menu. The easiest to use is the Html help; select Help! R language (html). This displays help in Internet Explorer just like a web page. The main page has several categories. 'An Introduction to R' provides basic help similar to this document. 'Search Engine & Keywords' allows you to search for different commands by keyword, or sorted into topics. 2.5 Quitting To close R, use the command > q() or choose Exit from the File menu. You will be asked if you wish to save the workspace; you should answer yes to this if you want to keep any data you have created. 3.0 COMMANDS AND OBJECTS R is an object-oriented language. This means that every piece of information is a type of object, for example data, vectors and results of analysis, and each object has a name. The user can perform actions on these objects via functions. Some functions behave differently depending on the type of object. Although this sounds complicated, in practice it makes R very easy to use. Most of the time you don't need to know or understand how this works. 3.1 Simple Arithmetic An expression typed at the prompt is evaluated and the result printed. You can use simple arithmetic: > # addition [1] 6 > 3 * # multiplication is done first [1] 14 > 1 + 3/2 # so is division [1] 2.5 > (1 + 3)/2 # Use brackets to change the order [1] 2 3

4 > 4^2 # use ^ for power [1] 16 The hash # in the commands above simply means a comment; R ignores everything after the hash. This allows us to insert comments to explain the code; this is particularly useful in a long and complicated analysis. 3.2 Simple Numeric Functions More complicated mathematical expressions can be calculated using functions. These have a name and an argument. For example, the function to calculate a square root has the name sqrt(). The argument is the value we want to find the square root of; to call the function, place the argument inside the bracket. So, to find the square root of 2, the argument is 2, so we type: > sqrt(2) [1] Similarly, > log( ) # log of pi [1] > log(pi) # pi can be used as a given constant [1] An example of commonly used mathematical functions is given below. abs() Absolute value (without +/-) sqrt() square root log() natural logarithm exp() exponential sin(), asin() sin and sin -1 cos(), acos() cos and cos -1 tan(), atan() tan and tan Variables Variables are called objects in R. You can assign a value to an object using "=" or the assignment operator <- (a 'less than' sign followed by a minus sign); the two are interchangeable. > x <- 5 From now on you can use x in place of the number 5: > x + 2 [1] 7 > sqrt(x) 4

5 [1] > y = sqrt(x) Typing the name of an object prints the value of it to screen. > y [1] DATA PREPARATION AND IMPORT IN R Importing data into R can be carried out in various ways to name a few, R offers means for importing ASCII and binary data, data from other applications or even for database connections. Since a common denominator for data analysis seem to be spreadsheet applications like Microsoft Excel. Let s assume we have the following dataset sales as a spreadsheet in Excel. (The dataset can be downloaded from course homepage). The spreadsheet contains sales data for Ogonos Venture from January 1990 to May A convenient way of preparing the data is to clean up the table within Excel, and save it as a text file so that only the data itself and one row containing the column names remain. Once the data has this form, it can be exported as a txt file, e.g. to C:/sales.txt. After conversion to a txt file, the data can be loaded into R. In our case, the sales dataset is imported by typing sales <- read.table( C:/sales.txt, header=t) into the R console. The right hand side, read.table( ), is the R command, which reads the txt file (sales.txt). Note, that paths use slashes / instead of backslashes. Further options within this command include header, which specify if the dataset has the first row containing column names (header=t). In case one does not have a naming row, one would use header=f instead. Having read the data into a data table, you then attach the data object with the command attach(data). This allows individual variables within the data set to be accessed directly. 4.1 Importing other Data Formats The foreign package allows you to read data created by different popular programs. To load it, simply type library(foreign) from within R. Supported formats include read.dta - Read Stata binary files read.mtp - Read a Minitab Portable Worksheet read.spss - Read an SPSS data file read.xport - Read a SAS XPORT format library read.s - Read an S-PLUS binary file 5

6 5.0 TIME SERIES FUNCTIONS The best way to see how R works is to look at some code and examples for a number of popular time series methods. That is what we do in this section. We will use data sets included with the contributed packages. The basic function in R that defines time series is ts(). It has as its default frequency, frequency=1. The starting date is by default year one; start=1. The following simple example creates a time series with frequency 1 (yearly data) of the first 10 positive integers, where we have written out the default values: ts(1:10, start = 1, frequency = 1) Time Series: Start = 1 End = 10 Frequency = 1 [1] ts(1:10, start = 1998, frequency = 4) Qtr1 Qtr2 Qtr3 Qtr ts(1:36, start = 1998, frequency = 12) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec The main point with the function ts() is that it defines a time series object which consists of the data and a time line (including frequency). 5.1 Creating Time Series Objects with ts() The default of the function ts() is ts(x, start=1,frequency=1), where x can be a vector, a matrix or a data frame. In order to create a time series object, we use the inbuilt data set (where the first column is redundant) and were the starting date i January 1980 we use the following redefinition: data(bjsales) > data1 <- ts(bjsales, start = c(1980, 1), frequency = 12) 6

7 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec The argument start=c(1980,1) specifies the starting date as the date of the first observation in Since frequency=12 defines data as monthly, the first observation in 1980 is January The resulting object is a time series object. The ts() function has two clones: is.ts() which checks whether an object is a time series object and as.ts() which coerces an object to be a time series object: > is.ts(data1) [1] TRUE data2<-c(1:150) > is.ts(data2) [1] FALSE > is.ts(as.ts(data2)) [1] TRUE Note data2 is not a time series object to R even if it contains information about the dates of the observation. To R these dates are just factors. Note finally that even if data2 is a time series object, a part of this object selected through indexing loose these properties: > data2[7:18] > is.ts(data2[7:18]) [1] FALSE i.e., the latter being an attempt to access data from July 1980 to June Well, we do but the resulting object does not inherit the relevant time series properties. We will have to use other methods to consider subsets of time series objects. 7

8 5.2 Extracting Attributes from Time Series Objects There are several commands that can extract different attributes from a time series object. Of course one can extract the time line (i.e., the dates of the observations): The frequency, start and end dates of a time series object can be extracted: > frequency(data1) [1] 12 > start(data1) [1] > end (data1) [1] Selecting Subsets of Time Series Objects We previously noted that although data1 may be a time series object, say, data1[221] is not. So how can we define subsets of time series objects? This is done with the function window(). The function takes start and end as arguments. Say we want to extract the period July 1987 to June 1989 from data1: > window(data1, start = c(1987, 7), end = c(1988, 6)) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Lags and Differences In time series analysis observations from different dates, say preceding observation t 1 y t from date t and the y from date t-1, are often used in the same model. In terms of terminology and mathematical notation y t 1 one usually call the (one period) lag of y t. Sometimes a lag operator L is introduced, such that Ly t = y t 1. For longer lags, say lag n, n one write L y = y. In R lags are created by the function lag(y, k=1), where the default t t n 1 k=1 implies a lead. Therefore, note that the default value is equivalent to L y t = y t+ 1! To get Ly t = y t 1 we must write lag(y, k=-1). Consider the following example which implements yt and Ly t = y t 1 : > ts(1:5) Time Series: Start = 1 End = 5 Frequency = 1 [1]

9 > lag(ts(1:5), k = -1) Time Series: Start = 2 End = 6 Frequency = 1 [1] In ts(1:5) the fifth observation with value 5 has date 5. On the other hand, in lag(ts(1:5),k=-1 the fifth observation with value 5 has date 6. Therefore, only the time index is changed! The vector of observations is unaffected, which we see if we pick (say) the third observation of the time series and its lag: > ts(1:5)[3] [1] 3 > lag(ts(1:5), k = -1)[3] [1] 3 This means that one has to careful using a time series and its lag in the same regression. R also has a function which directly defines differences between variables of different dates. This is the function diff(y, lag=1,difference=1), with default arguments. Here the option lag has its intuitive meaning. The default values give the equivalent to y-lag(y, k=- 1): > ts(1:5) - lag(ts(1:5), k = -1) Time Series: Start = 2 End = 5 Frequency = 1 [1] > diff(ts(1:5), lag = 1, difference = 1) Time Series: Start = 2 End = 5 Frequency = 1 [1] This means that diff(y, lag=1, difference=1) is exactly equivalent to the general difference operator, which is defined as y t = ( 1 L) yt = yt yt 1. Also for seasonal differencing in monthly time series we have > diff(airpassengers, lag=12) or > AirPassengers - lag(airpassengers, k=-12). 9

10 5.4 Aggregation of Time Series Data Sometimes data are available as high frequency data but what you need is low frequency data. Say, you have monthly data but need quarterly or yearly data. The function aggregate() has a method for time series objects so that one can use aggregate(x, nfrequency = 1, FUN = sum,...), where x is a time series object. The function splits x in blocks of length frequency(x)/nfrequency, where nfrequency is the frequency of the aggregated data (which must be lower than the frequency of x and which is 1 (yearly data) by default). The function specified by the argument FUN is then applied to each block of data (be default sum). For other arguments see? aggregate. The default function sum may be suitable for flow data as (for example) the data set AirPassengers, which records the total number of international air passengers per month. Say we want to aggregate this data to quarterly data > data(airpassengers) > AirPassengers Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec > (AirPassengers.Q <- aggregate(airpassengers, nfrequency = 4)) Qtr1 Qtr2 Qtr3 Qtr Aggregation will lead to quarters, but quarters consisting of February - April, May - July, August - October and November - January. The last two observations in the data set 10

11 AirPassengers.N, November and December 1960, do not constitute a full quarter and are therefore dismissed. If data are flows the default function sum is appropriate, but one may also have data in levels. Then (say) mean is an alternative in aggregation. The data set nottem contains monthly average air temperatures at Nottingham Castle in degrees Fahrenheit for the period Say we want to construct quarterly averages: data(nottem) > nottem Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec > nottem.q <- aggregate(nottem, nfrequency = 4, FUN = mean) Qtr1 Qtr2 Qtr3 Qtr

12 Descriptive Time Series Analysis In order to illustrate some time series ideas we will use the inbuilt data in R and it is called AirPassengers. This data contains the monthly numbers of airline passengers from 1949 to The function ts.plot(...) will generate a time series plot. Below we plot the AirPassengers series and add colour as well as heading of the series. > ts.plot(airpassengers, main="airpassengers", col="red") Next we plot the autocorrelation and partial autocorrelation function of the series. The function to generate the ACF and PACF in R is acf(...) and pacf( ). This functions returns an object of mode list and (unless we alter the default setting) plots the ACF and PACF. par(mfcol=c(2,1)) > acf(airpassengers) > pacf(airpassengers) par(mfcol=c(2,1)) acf(as.vector(airpassengers)) pacf(as.vector(airpassengers)) 12

13 The function to used to decompose time series in R is decompose(...). This function returns an object of mode list and (unless we alter the default setting) decompose time series into various components. The default is set to additive but it can be change to multiplicative if wish. >plot(decompose(airpassengers, type= multiplicative )) or > plot(decompose(airpassengers, type= additive )) 13

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