120 Statistics Displaying Time Series Data
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Time Series
A time series is a set of observations made at equally spaced points in time. Time series observations are usually numerical measurements, but occasionally categorical time series are encountered. Time series observations are typically not (statistically) independent. This means that the time order the observations is crucial to their analysis.
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Average Monthly Rainfall in Auckland
mm
0 1950
100
200
300
1960
1970 Time
1980
1990
2000
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Average Monthly Temperature in Auckland
26 Degrees C 14 1950 16 18 20 22 24
1960
1970 Time
1980
1990
2000
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Average Monthly Rainfall in Auckland
350 mm 0 1995 50 100 150 200 250 300
1996
1997
1998 Time
1999
2000
2001
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Average Monthly Temperature in Auckland
26 Degrees C 14 1995 16 18 20 22 24
1996
1997
1998 Time
1999
2000
2001
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The NZ Dollar in Australian Dollars
1.0 0.6 1970 0.7 0.8 0.9
1975
1980
1985 Time
1990
1995
2000
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Time Series in R
The function ts can be used to turn an ordinary vector into a special time series object. It does this by specifying parameters which describe when the observations were made. The parameter frequency describes how many observations are made per unit time. The parameter start describes when sampling started.
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Example Creating the Rain Series
Suppose that the Auckland rainfall values have been read into a vector called rainvalues. The values are monthly values with the first value sampled in January 1949. > rain = ts(rainvalues, frequency = 12, start = c(1949, 1)) R interprets the value 1949 as "the start of 1949" so we could use the simpler form. > rain = ts(rainvalues, frequency = 12, start = 1949)
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Simple Operations on Time Series
Arithmetic operations can be carried out on time series just as you might expect. > lograin = log(rain) Subsetting is done by focusing on the values of a time series which fall within a given time window. > rain2000 = window(rain, start = c(2000, 1), end = c(2000, 12))
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Printing Time Series
Time series are printed in a special way. > window(rain, start = c(1999, 1), end = c(2000, 12)) Jan Feb Mar Apr May Jun 1999 103.8 57.7 72.9 165.7 55.2 110.2 2000 86.2 9.2 51.2 128.4 118.1 203.4 Jul Aug Sep Oct Nov Dec 1999 130.7 113.2 69.6 96.6 172.5 42.8 2000 173.2 84.2 72.2 65.5 74.8 63.4 The frequency values 12 and 4 are recognised as special and taken to correspond to monthly and quarterly observations.
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Time Series Plots
The plot function recognises time series and plots them in an appropriate way. The default plotting method is to "join up the dots," but this and other aspects of the plot can be customised. > recent = window(rain, start = c(1995, 1), end = c(2000, 12)) > plot(recent) > plot(recent, type = "h") > plot(recent, type = "o", pch=20)
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The Default Time Series Plot
350 recent 0 1995 50 100 150 200 250 300
1996
1997
1998 Time
1999
2000
2001
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type="h"
350 recent 0 1995 50 100 150 200 250 300
1996
1997
1998 Time
1999
2000
2001
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type = "o", pch = 20
350
q
300
q
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q
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recent
q q q qq q qq
q q q q q
150
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
100
q
0
50
q
1995
1996
1997
1998 Time
1999
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2001
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Time Series Plots
We saw earlier in the course that it is easy to produce filled time series plots using polygon. > plot.new() > plot.window(c(1995, 2001), xaxs = "i", c(0,400), yaxs = "i") > x = c(1995, time(recent), 2001) > y = c(0, recent, 0) > polygon(x, y, col = "lightblue") > axis(1); axis(2); box()
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0 1995
100
200
300
400
1996
1997
1998
1999
2000
2001
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400 1995
300
200
100
0
1996
1997
1998
1999
2000
2001
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A Horizon Effect?
There appears to be a difference between these two plots. My conjecture is that there is wiring in the brain which means that we notice peaks rather than troughs in plots. This is true especially for plots which are divided horizontally by a colour horizon. Because of this effect, my recommendation is that you avoid this kind of plot.
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Example: Stock Prices
In this example we will look at the closing price for IBM stock, daily from Jan 1, 1980 to Oct. 8, 1992. This is a typical pattern for any stock. > plot(ibm)
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ibm 60 0 80 100
120
140
160
500
1000
1500 Time
2000
2500
3000
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Stock Prices and Efficient Markets
Theory says that in an efficient market stock prices should behave as random walks. This means that on any given day the price of a stock will go up or down with equal probability. There are a variety of reasons why the New Zealand market cannot be considered efficient. We can check the theory with the IBM stock by examining the first differences in the series i.e. each day's value minus the day before.
> plot(diff(ibm))
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diff(ibm)
-30 0
-20
-10
0
10
500
1000
1500 Time
2000
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Time Series Decomposition
It can useful to regard many real world series as being composed of several independent components. A particularly useful model is the trend plus seasonal plus irregular component model. xt = Tt + St + It where Tt St It = = = a slowly varying trend model a periodic seasonal component a set of random irregular "shocks"
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Example U.S. Housing Starts
The number of housing starts in any given month is an important leading economic indicator. Houses are only built when there are clearly economic "good times" ahead. This example shows the United States housing start series from 1966 to 1974.
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Monthly U.S. Housing Starts 1966-1974
Housing Starts (Thousands)
50 1966
100
150
200
1968
1970 Time
1972
1974
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Interpretation
The series clearly shows: a regular seasonal variation with a peak in housing starts in summer and a trough in winter. a long term (cyclical) trend. short term irregularities which are not explained by the other two components. This is typical of monthly or quarterly economic series.
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Seasonal Decomposition
There are statistical techniques which can be used to decompose a time series into trend plus seasonal plus irregular components. We will use a technique called STL which uses the lowess smoother as follows. A long term trend is estimated using a lowess smooth and then subtracted from the series. Each month (or quarter) is smoothed separately and this seasonal effect is subtracted. The remainder of the series is taken to be the irregular compon...