Simple Descriptive Techniques

Transcription

1 Simple Descriptive Techniques Outline 1 Types of variation 2 Stationary Time Series 3 The Time Plot 4 Transformations 5 Analysing Series that Contain a Trend 6 Analysing Series that Contain Seasonal Variation 7 Autocorrelation and the Correlogram

2 Types of Variation Seasonal variation Other cyclic variation Trend Other irregular fluctuations Stationary Time Series The idea of stationarity: A time series is said to be stationary if there is no systematic change in mean (no trend), if there is no systematic change in variance and if strictly periodic variations have been removed. Much of the probability theory of time series is concerned with stationary time series, and for this reason time-series analysis often requires one to transform a non-stationary series into a stationary one so as to use this theory. The non-stationary components, such as the trend, may be more interesting than the stationary residuals.

3 The Time Plot The plot of the observations against time is called a time plot, which will show up important features of the series such as trend, seasonality, outliers and discontinuities. The choice of scales, the size of the intercept and the way that the points are plotted may substantially affect the way the plot looks. The usual rules for drawing good graphs should be followed: A clear title must be given, units of measurement should be stated and axes should be properly labelled. Example: The Time Plot Figure 1: Annual CPI, U.S. ( ) Source: Hipel and Mcleod (1994).

4 Example: The Time Plot (R code) cpi<-read.table("cpi.dat") cpi.ts<-ts(cpi[,1], start=1860, frequency=1) plot(cpi.ts, xlab="", ylab="annual CPI") title("annal CPI ( )", cex=0.5) ### Examples of entering data by hand dat1<-c(153, 189, 221, 215, 302, 223, 201, 173) dat1.ts<-ts(dat1, start=1860, frequency=1) Transformations Three main reasons for making a transformation 1 To stablize the variance 2 To make the seasonal effect additive 3 To make the data normally distributed Box-Cox transformation: Given an observed time series {x t } and a transformation parameter λ, the transformed series is given by { (x λ y t = t 1) / λ λ 0 log x t λ =0 The best value of λ can be estimated by guess.

5 Analysing Series that Contain a Trend It is difficult to give a precise definition of trend. The simplest type of trend is linear trend + noise, i.e., X t = α + βt + ɛ t, (2.1) where α, β are constants and ɛ t denotes a random error term with zero mean. The mean level at time t (or the trend term) is given by m t = α + βt. The trend in (2.1) is called a global linear trend, which is unrealistic. In practice, one usually allows models that has local linear trends with (α t, β t ) (e.g., piecewise linear model for m t ). (α t, β t ) can vary deterministically, but there is more emphasis on the case that (α t, β t ) evolves stochastically. Analysing Series that Contain a Trend Curve fitting: A traditional method of dealing with non-seasonal data that contain a trend is to fit a simple function of time such as a polynomial curve. The Gompertz curve: log x t = a + br t The logistic curve: x t = a/(1 + be ct ). Filtering: Alinear filter converts one time series, {x t } into another, {y t }, by the linear operation y t = +s r= q a r x t+r, where {a r } is a set of weights such that a r =1. This operation is often referred to as a moving average. Moving averages are often symmetric with s = q and a j = a j.

6 Analysing Series that Contain a Trend Moving averages are often symmetric with s = q and a j = a j. For example, for a r =1/(2q + 1) for r = q,..., +q, the smoothed value of x t is Sm(x t )= 1 2q +1 +q r= q x t+r. Having estimated the trend, the local fluctuations are estiamted by Example 1 (Moving average) Res(x t )=x t Sm(x t ). We illustrate the main idea on how moving average technique works by means of the simple case of a linear trend where x t = α + βt + ɛ t. Consider a (2q + 1)-point moving average smoother to x t, y t = SM(x t )= 1 2q +1 +q r= q x t+r = α + βt + 1 2q +1 +q r= q ɛ t+r. Analysing Series that Contain a Trend Differencing: In many applications, the trend may be known in advance, so it is less important to estimate it. Instead, we may be interested in removing the trend. Let B be the backward shift opterator such that Bx t = x t 1. Define x t = (1 B)x t = x t x t 1, 2 x t = (1 B) 2 x t = (1 B) x t = x t 2x t 1 + x t 2, and j x t = (1 B) j x t = (1 B) j 1 x t, j 3. Example 2 (Backward operators) Compute the series that results from the following operators: (a) [1 2B +3B 2 ]x t, (b) (1 B)(1 3B)x t, (c) (1 B B 2 + B 3 )x t.

7 Analysing Series that Contain Seasonal Variation Three seasonal models in common use Additive seasonality: X t = m t + S t + ɛ t Multiplicative seasonality: X t = m t S t + ɛ t Multiplicative seasonality and error: X t = m t S t ɛ t The seasonality indices {S t } are usually assumed that S t = S t s, where s is the period of the cyclic behavior. The indices {S t } are usually normalized to that s t=1 S t =0in the additive case or s t=1 S t =1in the multiplicative case. The analysis of time series with seasonal variation depends on whether the purpose is to measure the seasonal effect and/or to eliminate seasonality. Seasonal Effect Estimation and Elimination Assume that the seasonal part has a period of d (i.e., S t+d = S t and d j=1 S j =0). Moving average method: We first estimate the trend part by a moving average filter running over a complete cycle so that the effect of the seasonality is averaged out. Depending on whether d is odd or even, we perform one of the following two steps for t = q +1,..., n q, If d =2q, Sm(x t )= 1 d ( 1 2 x t q + x t q x t+q x t+q). If d =2q +1, Sm(x t )= 1 d (x t q + x t q x t+q 1 + x t+q ). The seasonal effect can then be estimated by calculating x t Sm(x t ) or x t /Sm(x t ) for the additive or multiplicative case. Seasonal differencing: Use the dth differencing of data d = x t x t d.

8 Autocorrelation and the Correlogram Sample autocorrelation coefficients measure the correlation between observations at different distances apart. Given N observations x 1,..., x N on a time series, we can find the correlation between observations that are k steps apart, or the autocorrelation coefficient at lag k, r k = N k t=1 (x t x)(x t+k x) N t=1 (x t x) 2 where x = N t=1 x t/ N. r k can be calculated by autocovariance coefficient at lag k, c k = 1 N (x t x)(x t+k x) N k t=1 We then compute r k = c k /c 0 for k =1,..., M where M < N. Autocorrelation and the Correlogram Interpreting the correlogram A correlogram is the plot of the sample autocorrelation coefficients r k against the lag k for k =0, 1,..., M, where M is usually much less than N. Random series: A series of independent observations having the same distribution. For large N, we expect to find that r k 0 for all non-zero values of k. Short-term correlation: Stationary series that exhibit short-term correlation are characterized by large values of r k for small lags followed by values of r k being approximately 0 for larger lags. Alternating series: If a time series has a tendency to alternate, with successive observations on different sides of the overall mean, then the correlogram also tends to alternate.

9 Example: Autocorrelation and the correlogram Figure 2: Annual CPI, U.S. ( ) Source: Hipel and Mcleod (1994). Example: Autocorrelation and the correlogram par(mfrow = c(2, 1), cex=0.8) ### Compute c_k of the series acf(cpi.ts, type="covariance") ### Compute r_k of the series acf(cpi.ts, type="correlation") ### Difference the series diff(cpi.ts)