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In engineering, it will always be necessary to extrapolate, given data from the present and previous time, to some point in the future. For example, it is possible to take the current voltages of a system, and it may be necessary, in order to respond appropriately to a system, to extrapolate a future value.
For example, assuming that the input is known (from our model) to have a constant rate of change, it would be more appropriate to take the last four or five sampled points (depending on how much computing power is available) and find a least-squares linear polynomial than it would be to take just the last two points and find the interpolating linear polynomial, especially if it is known that the measurement error may be large.
To demonstrate the extreme case, consider the data where the measurement is severely truncated:
0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4
however, the measurements still to be from a source which is linear.
If we use two points and interpolation to approximate the next value, we get the following:
0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4
0 0 0 2 1 1 1 3 2 2 2 4 3 3 3 5 4 4 4
Note how only half of the approximations are reasonable predictors of future behaviour. If now, instead, we use the least squares fit of all previously known points, we have the following predictions (where we use rounding):
0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4
0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 4 4 4 4 5
The prediction becomes better with each passing point. Of course, this applies only if we know from the model that the input has a constant rate-of-change.
Because of the severe truncation error, this example benefited most from maintaining all previous data in extrapolating the next point. This would not be difficult to calculate quickly: each entry of VTV and VTy could be easily updated with six additions.
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.
