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I have a time series - let's say I'm given one new sample (a real value) every second.

I could take a moving average of the last 50 samples or I could use exponential smoothing of the samples with a time constant of 50 . The resulting graphs of the "output" would be fairly similar. The "current output" of the algorithm assumes that the underlying values are constant plus some noise and the algorithm gives a best-guess of that underlying value.

Or I could do "double exponential smoothing". The "current output" is an estimate of a linear graph. After each sample arrives I have as estimate of the current value and how it is linearly increasing.

What I want is to get a best guess of the quadratic expression for that last 50 samples.

I can find the quadratic expression for that last 50 samples. It's just a regression. But to calculate a regression I'd have to actually look at the last 50 samples - that's too much calculation. I want to do it using some sort of recursive function; rather like exponential smoothing is a cheap way of taking a moving average.

I assumed that "triple exponential smoothing" was what I'm looking for but that's something completely different. It includes a term for "seasonality".

So what is the algorithm? Or what is the Google search term?

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  • $\begingroup$ As mentioned by @Hilmar, with uniform sampling, you can have a formulation with a stored inverse. If you want to be less sensitive of the past, a sliding power weight can be applied as well. Not much computations if done properly, especially with 0-indexing as proposed in CHOPtrey: contextual online polynomial extrapolation for enhanced multi-core co-simulation of complex systems. Have a look, I'll elaborate if useful $\endgroup$
    – Laurent Duval
    Aug 10 at 17:33
  • $\begingroup$ Yes, that tells me to search for "Polynomial Extrapolation/Interpolation" - which gives interesting hits. But all I've found so far assumes I want an N-degree polynomial exactly fitting N+1 points. What I want is a 2nd-degree polynomial "best fit" to the past N points - hence a regression. And I want a recursive algorithm so I store a just few values and do just a few mults/adds for each new sample. "Double exponential smoothing" is a recursive way of calculating a 1st-degree polynomial using the "sliding power weight" that you suggest. What's the 2nd-degree equivalent? Please do elaborate. $\endgroup$
    – Peter Balch
    Aug 10 at 20:38

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Moving average and exponential smoothing are just very simple lowpass filters. There are many different types of low pass filters available and chances finding the right one will be a good option for whatever it is you need to do. If you want "quintuple exponential smoothing", just use a 5th order IIR lowpass filter and you can still play around with cutoff frequency, ripple, stop band attenuation, etc.

Quadratic regression can be done, but it is a fairly non-linear operation and you have to make sure that's acceptable in your context.

But to calculate a regression I'd have to actually look at the last 50 samples - that's too much calculation.

It's not that hard to do. Quadratic regression can be expressed as three equations with three unknowns (see for example here) in the form a matrix equation.

$$X \cdot \begin{bmatrix} a \\ b \\c \end{bmatrix} = Y $$

Assuming your data is sampled at a constant rate, the coefficient matrix, $X$ on the left never changes, so you only need to calculate it once, invert it and store it. Your regression then becomes a simple matrix multiplication

$$\begin{bmatrix} a \\ b \\c \end{bmatrix} = X^{-1} \cdot Y $$

The only thing that needs to be updated the is the vector, $Y$

$$ Y = \begin{bmatrix} \sum_i i^2 \cdot y_i \\ \sum_i i \cdot y_i \\ \sum_i y_i\end{bmatrix} $$

That doesn't seem particularly difficult to do.

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  • $\begingroup$ Thank you. I'm not trying to make a low-pass filter. What I'm asking is how to calculate the best guess of the current values of a, b and c over, say, the last 50 samples. $\endgroup$
    – Peter Balch
    Aug 10 at 14:08
  • $\begingroup$ Does what I wrote answer your question? For every sample you just need to recalculate $Y$ multiply with $X^{-1}$ and that will give you the current estimate of a, b, and c. $\endgroup$
    – Hilmar
    Aug 10 at 22:02
  • $\begingroup$ Sorry but I don't understand how it answers the question. You can use the matrix method to obtain a,b,c from the most recent 3 samples. Or use the matrix method to get a least-squares estimate from all the samples. Which are you proposing? Neither is what I want. Where does the time-constant of the exponential fit in? $\endgroup$
    – Peter Balch
    Aug 11 at 9:30

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