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) = t 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=1), I(t=2),....I(t=3)...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 residuals
start to change trend.
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 Time Trend series, for example
(1,2,3,4,5,6,7,8,9,10,11,12,13,....)
Note that the Time Trend does not necessarily start at the true beginning
of the series (0,0,0,0,1,2,3,4,5,6,7,8,9,10,,....)
Note that the Time Trend does not necessarily end at the true end
of the series (0,0,0,0,1,2,3,4,5,6,7,8,9,10,0,0,0)
UNUSUAL VALUES WHICH MONOTONICALLY INCREASE AT CONSECUTIVE POINTS
Sometimes a time series exhibits local trends. These trends can begin and
end at different points in time and have different intervals. Statisticians
have developed schemes for distinguishing between non-stationarity caused
by the presence of random walks and deterministic linear trends.
We illustrate this phemomena in a series that exhibits a number of reversals
or points of inflection.
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.
We then present a monthly time series of telephone installations where there
is one time trend change (t=26) with accompanying seasonal structure.
CLICK HERE TO VIEW THE PLOT OF A Y SERIES THAT HAS MULTIPLE TIME TRENDS AND REVERSALS.
CLICK HERE TO VIEW THE PLOT OF A Y SERIES THAT HAS ONE TIME TREND CHANGE.
CLICK HERE TO VIEW IT'S MODEL.
CLICK HERE TO VIEW THE PLOT OF THE ACTUAL , FIT AND FORECASTS.
CLICK HERE TO VIEW THE PLOT OF A Y SERIES THAT HAS MULTIPLE TIME TRENDS AND REVERSALS.
CLICK HERE TO VIEW THE PLOT OF A Y SERIES THAT HAS ONE TIME TREND CHANGE.
CLICK HERE TO VIEW IT'S MODEL.
CLICK HERE TO VIEW THE PLOT OF THE ACTUAL , FIT AND FORECASTS.