ExponentialModel.rst

Exponential Drift Model and Correction

It is assumed, that, in between maintenance events, there is a drift effect shifting the measurements in a way, that the resulting value course can be described by the exponential model $M$:

$M(t, a, b, c) = a + b(e^{ct}-1)$

We consider the timespan in between maintenance events to be scaled to the $[0,1]$ interval. To additionally make sure, the modeled curve can be used to calibrate the value course, we added the following two conditions.

$M(0, a, b, c) = y_0$

$M(1, a, b, c) = y_1$

With $y_0$ denoting the mean value obtained from the first 6 meassurements directly after the last maintenance event, and $y_1$ denoting the mean over the 6 meassurements, directly preceeding the beginning of the next maintenance event.

Solving the equation, one obtains the one-parameter Model:

$M_{drift}(t, c) = y_0 + ( \frac{y_1 - y_0}{e^c - 1} ) (e^{ct} - 1)$

For every datachunk in between maintenance events.

After having found the parameter $c^*$, that minimizes the squared residues between data and drift model, the correction is performed by bending the fitted curve, $M_{drift}(t, c^*)$, in a way, that it matches $y_2$ at $t=1$ (,with $y_2$ being the mean value observed directly after the end of the next maintenance event). This bended curve is given by:

$M_{shift}(t, c^{*}) = M(t, y_0, \frac{y_1 - y_0}{e^c - 1} , c^*)$

the new values $y_{shifted}$ are computed via:

$y_{shifted} = y + M_{shift} - M_{drift}$