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A LEVEL SHIFT OR STEP SHIFT IN A TIME SERIES ANALYSIS
UNUSUAL VALUES WHICH ARISE SEQUENTIALLY
- Consider a case where the true but unknown model is:
Y(t) = v0*X(t) + v1*I(t=to) + A(t)
where A is an i.i.d. ( gaussian ) error distribution
I(t=to) is an Intervention Variable such that
I(t) = 0 for t < to
I(t) = 1 for t => to
and where the current tentative, albeit incorrect model is
Y(t) = v0*X(t) + e(t)
thus we have a case where
e(t) = v1*I(t=to) + A(t)
It is clear that if we correlate e(t) and I(t=to)
we will identify a statistically significant relationship
which will then lead to the required augmentation strategy.
The strategy is then clear , for all possible or candidate
"new variables" such as the I(t=2), I(t=3),....I(t=n)...etc. compute
partial correlations which will measure the expected impact of
the trial candidate. Select that one which has maximum correlation.
This is equivalent to selecting the point in time where the mean or
average residual changes.
These approaches to developing new variables from model implied series have had outstanding success
with time-series data.
Sometimes the Intervention Variable is referred to as a Level Shift series, for example
(0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1)
Note that the Level Shift does not necessarily continue to the true
end of the series (0,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0) .
UNUSUAL VALUES WHICH OCCUR AT CONSECUTIVE POINTS
Sometimes a time series can be explained by a sequence of contiguous pulses.
If the pulses are of a similar magnitude (effect) then one can describe this
data in an economical fashion with a step or level shift function. If the
series is trending, the level shift appears as a parallel movement of
the trend line.
We illustrate this phemomena in a causal example of some 60 month's of data.
The problem or opportunity can easily arise in a causal model but for pedantic
reasons we prefer to present it in a non-causal environment.
CLICK HERE TO VIEW THE PLOT OF A Y SERIES THAT DOES NOT HAVE A TREND ( OR DO YOU HAVE ANOTHER THOUGHT ?).
CLICK HERE TO VIEW THE PLOT OF A Y SERIES THAT DOES NOT HAVE A LEVEL SHIFT BUT RATHER IS DESCRIBABLE VIA A TREND .
CLICK HERE TO VIEW THEM SIMULTANEOUSLY.
CLICK HERE TO VIEW THE PLOT OF THE ACTUAL , FIT AND FORECASTS OF THE NON-TRENDING SERIES.
CLICK HERE TO VIEW THE PLOT OF A SERIES THAT HAS TREND, LEVEL SHIFT AND A PULSE.
