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I'm trying to determine the continuous time formulation of the double exponential filter so that I can adapt it more flexibly for my particular problem.

Typically, this model is expressed as a pair of discrete, recursive equations as such in @denis' post here)


Double exponential smoothing a.k.a. Holt-Winters smoothing tracks level and trend of a time series in coupled IIRs:
$\qquad$ In: $Y_t$, t = 0 1 2 ...
$\qquad$ State: $L_t, T_t \quad$ -- level and trend
$\qquad$ Out: estimate $\hat{Y}_{t+1}$
$\qquad$ Parameters: a, b (a.k.a. $\alpha, \beta$)
$\qquad$ Step equations:
$\qquad$ $\qquad L_t = a Y_t + (1 - a) (L_{t-1} + T_{t-1}) \quad$ -- level
$\qquad$ $\qquad T_t = b (L_t - L_{t-1}) + (1 - b) T_{t-1} \quad$ -- trend
$\qquad$ $\qquad \hat{Y}_{t+1} = L_t + T_t$


Most references on the subject of double exponential filtering and state space revolve around discrete time kalman filtering approaches. Question: How should I put this model into a noiseless, state-space ODE form?

Using slightly different notation, here is my current approach. The base model, with continuous time input $u$ and continuous time states $l$ and $s$, is

$\dot{l} = r_1(u - l) + r_1 s$

$\dot{s} = r_2(\dot{l} - s)$

$\ \ = r_2(r_1(u - l) + r_1 s - s)$

$\ \ = r_2(r_1(u - l) + (r_1 - 1)s)$

$\ \ = r_2 r_1(u - l) + r_2(r_1 - 1)s$

with $\dot{l}$ and $\dot{s}$ representing time-derivative state dynamics with system constants $r_1, r_2 \in [0, \infty)$.

Assuming I made no mistakes there, I attempt to put this into state space form by

$$ \dot{\bf{x}} = A{\bf x} + Bu$$ $$ A = \begin{bmatrix} -r_1 & r_1\\ -r_1r_2 & r_2(r_1 - 1)\\ \end{bmatrix} \ \ \ \ B = \begin{bmatrix} r_1\\ r_1r_2\\ \end{bmatrix} \ \ \ \ \bf{x} = \begin{bmatrix} l\\ s\\ \end{bmatrix} $$

The output of this system is just the state of the level variable.

$y(t^{'}) = l(t^{'}) = C{\bf x}$.

where $C = \begin{bmatrix}1 & 0\end{bmatrix}$

We can make future predictions of $y$ based off of some assumption of the form of $u$, we can project $y({t'})$ using the system response formula for continuous time state space systems

$y({t^{'}}) = C\bigg(e^{A(t^{'} - t)}x(t) + \int_t^{t^{'}}e^{A(t - \tau)}Bu(\tau)d\tau\bigg)$

which can then be discretized as necessary later.

However: I am finding that the system matrix $A$ is not stable.

By my assumptions system matrices for all positive values of $r_1, r_2$ should be stable but this is not the case. Empirically I just tested a couple settings and the real part of the eigenvalues of $A$ must be negative, which is satisfied are for $r_1 = 1, r_2=1$ but not for $r_1 = 3, r_2 = 3$

Where dd I go wrong?.

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I think I figured it out, dumb mistake in writing the system in continuous time.

$\dot{l} = r_1(u - l)$

$\dot{s} = r_2(\dot{l} - s)$

$\ \ = r_2(r_1(u - l) + r_1 s - s)$

$\ \ = r_2(r_1(u - l) + (r_1 - 1)s)$

$\ \ = r_2 r_1(u - l) + r_2(r_1 - 1)s$

So $$ A = \begin{bmatrix} -r_1 & 0\\ -r_1r_2 & -r_1r_2\\ \end{bmatrix} $$

which is stable for all positive $r_1, r_2$!

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  • $\begingroup$ Hi. I want to do the same thing for simple exponential smoothing but I don't see how you obtained the original expressions for the derivatives of l and s. Thanks. Or if some book has it in there and you know where to find it, it's appreciated. $\endgroup$ – mark leeds Aug 12 at 3:30

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