6.
Process or Product Monitoring and Control
6.4.
Introduction to Time Series Analysis
6.4.4.
Univariate Time Series Models
6.4.4.6.
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Box-Jenkins Model Identification
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Stationarity and Seasonality
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The first step in developing a Box-Jenkins model is to determine
if the series is stationary and if there
is any significant seasonality that
needs to be modeled.
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Detecting stationarity
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Stationarity can be assessed from a
run sequence plot.
The run sequence plot should show constant location and scale.
It can also be detected from an
autocorrelation plot.
Specifically, non-stationarity is often indicated by an
autocorrelation plot with very slow decay.
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Detecting seasonality
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Seasonality (or periodicity) can usually be assessed from an
autocorrelation
plot, a seasonal subseries plot, or
a spectral plot.
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Differencing to achieve stationarity
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Box and Jenkins recommend the differencing approach to achieve
stationarity. However, fitting a curve and subtracting the
fitted values from the original data can also be used in the
context of Box-Jenkins models.
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Seasonal differencing
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At the model identification stage, our goal is to detect
seasonality, if it exists, and to identify the order for the seasonal
autoregressive and seasonal moving average terms. For many series,
the period is known and a single seasonality term is sufficient.
For example, for monthly data we would typically include either a
seasonal AR 12 term or a seasonal MA 12 term. For Box-Jenkins
models, we do not explicitly remove seasonality before fitting the
model. Instead, we include the order of the seasonal terms in the
model specification to the ARIMA estimation software. However,
it may be helpful to apply a seasonal difference to the data and
regenerate the autocorrelation and partial autocorrelation plots.
This may help in the model idenfitication of the non-seasonal
component of the model. In some cases, the seasonal differencing
may remove most or all of the seasonality effect.
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Identify p and q
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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.
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Autocorrelation and Partial Autocorrelation Plots
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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.
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| 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,
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.
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Order of Moving Average Process (q)
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A MA process may be indicated by an autocorrelation function that
has one or more spikes. The autocorrelation function of a
MA(q) process becomes zero at lag q+1,
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.
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Shape of Autocorrelation Function
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The following table summarizes how we use the sample
autocorrelation function for model identification.
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SHAPE
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INDICATED MODEL
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Exponential, decaying to zero
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Autoregressive model. Use the partial autocorrelation plot
to identify the order of the autoregressive model.
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Alternating positive and negative, decaying to zero
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Autoregressive model. Use the partial autocorrelation plot
to help identify the order.
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One or more spikes, rest are essentially zero
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Moving average model, order identified by where plot
becomes zero.
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Decay, starting after a few lags
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Mixed autoregressive and moving average model.
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All zero or close to zero
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Data is essentially random.
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High values at fixed intervals
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Include seasonal autoregressive term.
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No decay to zero
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Series is not stationary.
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| Mixed Models
Difficult to Identify |
In practice, the sample autocorrelation
and partial autocorrelation functions are not clean, which 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, 1991). |
| Examples |
We show a typical series of
plots for performing the initial model identification for
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the southern oscillations data and
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the CO2 monthly concentrations data.
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