Engineering Statistics Handbook Episode 8 Part 12 pdf

Multiple box plots can be used as an alternative to the seasonal subseries plot to detect seasonality.. Related Techniques Box Plot Run Sequence Plot Autocorrelation Plot Software Season

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6 Process or Product Monitoring and Control

6.4 Introduction to Time Series Analysis

6.4.4 Univariate Time Series Models

6.4.4.3 Seasonality

Seasonality Many time series display seasonality By seasonality, we mean periodic

fluctuations For example, retail sales tend to peak for the Christmas season and then decline after the holidays So time series of retail sales will typically show increasing sales from September through December and declining sales in January and February

Seasonality is quite common in economic time series It is less common

in engineering and scientific data

If seasonality is present, it must be incorporated into the time series model In this section, we discuss techniques for detecting seasonality

We defer modeling of seasonality until later sections

Detecting

Seasonality he following graphical techniques can be used to detect seasonality.

A run sequence plot will often show seasonality

1

A seasonal subseries plot is a specialized technique for showing seasonality

2

Multiple box plots can be used as an alternative to the seasonal subseries plot to detect seasonality

3

The autocorrelation plot can help identify seasonality

4

Examples of each of these plots will be shown below

The run sequence plot is a recommended first step for analyzing any time series Although seasonality can sometimes be indicated with this plot, seasonality is shown more clearly by the seasonal subseries plot or the box plot The seasonal subseries plot does an excellent job of

showing both the seasonal differences (between group patterns) and also the within-group patterns The box plot shows the seasonal difference (between group patterns) quite well, but it does not show within group patterns However, for large data sets, the box plot is usually easier to read than the seasonal subseries plot

Both the seasonal subseries plot and the box plot assume that the

6.4.4.3 Seasonality

http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc443.htm (1 of 5) [5/1/2006 10:35:20 AM]

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seasonal periods are known In most cases, the analyst will in fact know this For example, for monthly data, the period is 12 since there are 12 months in a year However, if the period is not known, the

autocorrelation plot can help If there is significant seasonality, the autocorrelation plot should show spikes at lags equal to the period For example, for monthly data, if there is a seasonality effect, we would expect to see significant peaks at lag 12, 24, 36, and so on (although the intensity may decrease the further out we go)

Example

without

Seasonality

The following plots are from a data set of southern oscillations for predicting el nino

Run

Sequence

Plot

No obvious periodic patterns are apparent in the run sequence plot

6.4.4.3 Seasonality

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Subseries

Plot

The means for each month are relatively close and show no obvious pattern

Box Plot

As with the seasonal subseries plot, no obvious seasonal pattern is apparent

Due to the rather large number of observations, the box plot shows the difference between months better than the seasonal subseries plot

6.4.4.3 Seasonality

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with

Seasonality

The following plots are from a data set of monthly CO2 concentrations

A linear trend has been removed from these data

Run

Sequence

Plot

This plot shows periodic behavior However, it is difficult to determine the nature of the seasonality from this plot

Seasonal

Subseries

Plot

The seasonal subseries plot shows the seasonal pattern more clearly In

6.4.4.3 Seasonality

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this case, the CO2 concentrations are at a minimun in September and October From there, steadily the concentrations increase until June and then begin declining until September

Box Plot

As with the seasonal subseries plot, the seasonal pattern is quite evident

in the box plot

6.4.4.3 Seasonality

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This plot allows you to detect both between group and within group patterns

If there is a large number of observations, then a box plot may be preferable

Definition Seasonal subseries plots are formed by

Vertical axis: Response variable Horizontal axis: Time ordered by season For example, with

monthly data, all the January values are plotted (in chronological order), then all the February values, and so on

In addition, a reference line is drawn at the group means

The user must specify the length of the seasonal pattern before generating this plot In most cases, the analyst will know this from the context of the problem and data collection

Questions The seasonal subseries plot can provide answers to the following

questions:

Do the data exhibit a seasonal pattern?

1

What is the nature of the seasonality?

2

Is there a within-group pattern (e.g., do January and July exhibit similar patterns)?

3

Are there any outliers once seasonality has been accounted for?

4

Importance It is important to know when analyzing a time series if there is a

significant seasonality effect The seasonal subseries plot is an excellent tool for determining if there is a seasonal pattern

Related

Techniques

Box Plot Run Sequence Plot Autocorrelation Plot

Software Seasonal subseries plots are available in a few general purpose statistical

software programs They are available in Dataplot It may possible to write macros to generate this plot in most statistical software programs that do not provide it directly

6.4.4.3.1 Seasonal Subseries Plot

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(AR) Models

A common approach for modeling univariate time series is the autoregressive (AR) model:

where X t is the time series, A t is white noise, and

with denoting the process mean

An autoregressive model is simply a linear regression of the current value of the series against one or more prior values of the series The

value of p is called the order of the AR model.

AR models can be analyzed with one of various methods, including

standard linear least squares techniques They also have a straightforward interpretation

Moving

Average (MA)

Models

Another common approach for modeling univariate time series models is the moving average (MA) model:

where X t is the time series, is the mean of the series, A t-i are white noise, and 1, , q are the parameters of the model The value of q

is called the order of the MA model

That is, a moving average model is conceptually a linear regression of the current value of the series against the white noise or random

shocks of one or more prior values of the series The random shocks

at each point are assumed to come from the same distribution, typically a normal distribution, with location at zero and constant scale The distinction in this model is that these random shocks are propogated to future values of the time series Fitting the MA estimates is more complicated than with AR models because the error terms are not observable This means that iterative non-linear fitting procedures need to be used in place of linear least squares MA models also have a less obvious interpretation than AR models

Sometimes the ACF and PACF will suggest that a MA model would

be a better model choice and sometimes both AR and MA terms should be used in the same model (see Section 6.4.4.5)

Note, however, that the error terms after the model is fit should be

independent and follow the standard assumptions for a univariate process

6.4.4.4 Common Approaches to Univariate Time Series

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Approach

Box and Jenkins popularized an approach that combines the moving average and the autoregressive approaches in the book "Time Series

1994)

Although both autoregressive and moving average approaches were already known (and were originally investigated by Yule), the contribution of Box and Jenkins was in developing a systematic methodology for identifying and estimating models that could incorporate both approaches This makes Box-Jenkins models a powerful class of models The next several sections will discuss these models in detail

6.4.4.4 Common Approaches to Univariate Time Series

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Stages in

Box-Jenkins

Modeling

There are three primary stages in building a Box-Jenkins time series model

Model Identification

1

Model Estimation

2

Model Validation

3

Remarks The following remarks regarding Box-Jenkins models should be noted

Box-Jenkins models are quite flexible due to the inclusion of both autoregressive and moving average terms

1

Based on the Wold decomposition thereom (not discussed in the Handbook), a stationary process can be approximated by an ARMA model In practice, finding that approximation may not be easy

2

Chatfield (1996) recommends decomposition methods for series

in which the trend and seasonal components are dominant

3

Building good ARIMA models generally requires more experience than commonly used statistical methods such as regression

4

Sufficiently

Long Series

Required

Typically, effective fitting of Box-Jenkins models requires at least a moderately long series Chatfield (1996) recommends at least 50 observations Many others would recommend at least 100 observations

6.4.4.5 Box-Jenkins Models

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Identify p and q Once stationarity and seasonality have been addressed, the next step

is to identify the order (i.e., the p and q) of the autoregressive and

moving average terms

Autocorrelation

and Partial

Autocorrelation

Plots

The primary tools for doing this are the autocorrelation plot and the

partial autocorrelation plot The sample autocorrelation plot and the sample partial autocorrelation plot are compared to the theoretical behavior of these plots when the order is known

Order of

Autoregressive

Process (p)

Specifically, for an AR(1) process, the sample autocorrelation function should have an exponentially decreasing appearance

However, higher-order AR processes are often a mixture of exponentially decreasing and damped sinusoidal components

For higher-order autoregressive processes, the sample autocorrelation needs to be supplemented with a partial autocorrelation plot The

partial autocorrelation of an AR(p) process becomes zero at lag p+1

and greater, so we examine the sample partial autocorrelation function to see if there is evidence of a departure from zero This is usually determined by placing a 95% confidence interval on the sample partial autocorrelation plot (most software programs that generate sample autocorrelation plots will also plot this confidence interval) If the software program does not generate the confidence band, it is approximately , with N denoting the sample

size

Order of

Moving

Average

Process (q)

The autocorrelation function of a MA(q) process becomes zero at lag

q+1 and greater, so we examine the sample autocorrelation function

to see where it essentially becomes zero We do this by placing the 95% confidence interval for the sample autocorrelation function on the sample autocorrelation plot Most software that can generate the autocorrelation plot can also generate this confidence interval

The sample partial autocorrelation function is generally not helpful for identifying the order of the moving average process

6.4.4.6 Box-Jenkins Model Identification

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Shape of

Autocorrelation

Function

The following table summarizes how we use the sample autocorrelation function for model identification

Exponential, decaying to zero

Autoregressive model Use the partial autocorrelation plot to identify the order of the autoregressive model

Alternating positive and negative, decaying to zero

Autoregressive model Use the partial autocorrelation plot to help identify the order

One or more spikes, rest are essentially zero

Moving average model, order identified by where plot becomes zero

Decay, starting after a few lags

Mixed autoregressive and moving average model

All zero or close to zero Data is essentially random

High values at fixed intervals

Include seasonal autoregressive term

No decay to zero Series is not stationary

Mixed Models

Difficult to

Identify

In practice, the sample autocorrelation and partial autocorrelation functions are random variables and will not give the same picture as the theoretical functions This makes the model identification more difficult In particular, mixed models can be particularly difficult to identify

Although experience is helpful, developing good models using these sample plots can involve much trial and error For this reason, in recent years information-based criteria such as FPE (Final Prediction Error) and AIC (Aikake Information Criterion) and others have been preferred and used These techniques can help automate the model identification process These techniques require computer software to use Fortunately, these techniques are available in many commerical statistical software programs that provide ARIMA modeling

capabilities

For additional information on these techniques, see Brockwell and Davis (1987, 2002)

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Examples We show a typical series of plots for performing the initial model

identification for

the southern oscillations data and

1

the CO2 monthly concentrations data

2

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Subseries Plot

The seasonal subseries plot indicates that there is no significant seasonality

Since the above plots show that this series does not exhibit any significant non-stationarity or seasonality, we generate the autocorrelation and partial autocorrelation plots of the raw data

Autocorrelation

Plot

The autocorrelation plot shows a mixture of exponentially decaying

6.4.4.6.1 Model Identification for Southern Oscillations Data

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and damped sinusoidal components This indicates that an autoregressive model, with order greater than one, may be appropriate for these data The partial autocorrelation plot should be examined to determine the order

Partial

Autocorrelation

Plot

The partial autocorrelation plot suggests that an AR(2) model might

be appropriate

In summary, our intial attempt would be to fit an AR(2) model with

no seasonal terms and no differencing or trend removal Model validation should be performed before accepting this as a final model

6.4.4.6.1 Model Identification for Southern Oscillations Data

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