Understanding Trend, Seasonality, and Autocorrelation in Stationary Time Series, Study notes of Statistics

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Stationary time series Transformations

Simple Descriptive Techniques

Dr. Bo Li

January 9, 2012

Stationary time series Transformations

Simple Descriptive Techniques

I Descriptive methods should generally be tried before

attempting more complicated procedures, because they can be

vital in “cleaning” the data, and then getting a “feel” for

them, before trying to generate ideas a regards a suitable

model.

I If a time series contains trend, seasonality or some other

systematic component, the usual summary statistics (e.g.,

mean and standard deviation)can be seriously misleading and

should not be calculated.

I Moreover, even when a series do not contain any systematic

components, the summary statistics do not have their usual

properties.

I Focus on ways of understanding typical time-series effects,

such as trend, seasonality and correlations between successive

observations.

Stationary time series Transformations

Example of type of variation

0 100 200 300

0

20

40

60

80

days

counts *

Hospital admission counts for circulatory (black line) and

respiratory (red line) disease in 2002.

Stationary time series Transformations

Stationary time series

I 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.

I Intuitively, the properties of one section of the data are much

like those of any other section.

I Strictly speaking, there is no such thing as “stationary time

series”, as the stationarity property is defined for a model.

I However, the phrase is often used for time-series data

meaning that they exhibit characteristics that suggest a

stationary model can sensibly be fitted.

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Three main reasons for transformation

I Make the seasonal effect additive: If there is a trend in the

series and the size of the seasonal effect appears to increase

with the mean, then it may be advisable to transform the data

so as to make the seasonal effect constant from year to year.

I additive: constant seasonal effect

I multiplicative: the size of the seasonal effect is directly

proportional to the mean. A logarithmic transformation is

appropriate to make the effect additive.

I Make the data normally distributed: model building and

forecasting are usually carried out on the assumption that the

data are normally distributed. For example, Box-Cox

transformation

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Three main reasons for transformation

Use with caution!

I There are problems in practice with transformations in that a

transformation, which makes the seasonal effect addtive, may

fail to stabilize the variance. Thus it may be impossible to

achieve all the above requirement at the same time.

I It is more difficult to interpret and forecasts produced by the

transformed model may have to be “transformed back” in

order to be of use. This can introduce biasing effects.

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Time Series with a trend

Three approaches to describe trend:

I Curve fitting: A traditional method of dealing with

non-seasonal data that contain a trend, particular yearly data,

is to fit a simple function of time such as a polynomial curve

(linear, quadratic, etc.).

I The fitted function provides a measure of the trend, and the

residuals provide an estimate of local fluctuations, where the

residuals are the differences between the observations and the

corresponding values of the fitted curve.

I Polynomial curve: mt = α + βt, e.g, mt = 0.4 + 2t

I Gompertz curve: log(mt ) = a + br t^ , e.g.,

log(mt ) = 3 + 2 × 0. 5

t

I Logistic cruve: mt = a

1+be−ct^ , e.g.,^ mt^ =^

1+0. 3 e−^2 t

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Time Series with a trend

Three approaches to describe trend:

I Filtering Linear filter converts one time series, xt , into

another, yt , by the linear interpolation

yt =

∑^ +s

r =−q

ar xt+r.

If

ar = 1, the operation is referred to as a moving average.

Moving averages are often symmetric with s = q and

aj = a−j.

e.g. ar =

1 2 q+

for r = −q,... , +q, and the smoothed value

of xt is given by

Sm(xt ) =

2 q + 1

+q

r =−q

xt+r.

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Time Series with a trend

I Convolution: ck =

r =−∞

ar bk−r

ck = ar? bj

Example: (1/ 4 , 1 / 2 , 1 /4) = (1/ 2 , 1 /2)? (1/ 2 , 1 /2)

I Differencing

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Analyzing series that contain seasonal variation

I Three common seasonal models:

A Xt = mt + St + t

B Xt = mt St + t

C Xt = mt St t

I Smoothing average for monthly, quarterly data...

I Seasonal differencing

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Autocorrelation and Correlogram

Correlogram is also called sample AutoCorrelation Function (ac.f.)

ac.f. for house sales

0 5 10 15 20

−0.

Lag

ACF

Series sales

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Autocorrelation and Correlogram

ac.f. for wheat price index

0 5 10 15 20 25

Lag

ACF

Series wheat

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Autocorrelation and Correlogram

R code to generate the correlogram

acf(sales)

acf(wheat)

acf(temp$temperature)

help(acf)

Another Example:

x <- 1:

y <- sin(x)

plot(y)

plot(y,type=’l’)

acf(y)

Stationary time series Transformations

Time series with a trend Autocorrelation and correlogram

Interpreting the correlogram

I Random series

I Short-term correlation

I Alternating series

I Non-stationary series

I Seasonal series

I Outliers