# Smoothness of a time series: relationship between ARMA model and signal derivatives

I have a discrete stochastic function $$f(t_k)$$ (in my case, it's a time signal related to atmospheric parameters). I know this function to have a certain smoothness. One way I can specify its smoothness is using its auto-correlation function, or equivalently, its power spectral density which can be estimated by $$S_f(\omega_k) = \mathcal{F}(f(t_k)) \overline{\mathcal{F}(f(t_k))}$$, where here $$\mathcal{F}$$ is the discrete Fourier transform over some arbitrary window of data and $$\bar{}$$ means complex conjugate. Let's assume that the SDF is the a-priori information I have available for defining the signal smoothness.

I want to be able to define sequential or batch estimation algorithms to estimate this signal given noisy measurements. For simplicity, let's say:

$$y(t_k) = f(t_k) + w(t_k)$$

is our measurement, where $$E[w(t_k)w(t_l)] = \sigma^2\delta(k-l)$$, where $$\sigma^2$$ is the noise variance and $$\delta$$ is the Dirac delta function.

If I want to predict/estimate $$\hat{f}(t_k)$$ given past estimates $$\{\hat{f}(t_{k-1}), \hat{f}(t_{k-2}), \dots \}$$, one way to do this is using an auto-regressive (AR) model:

$$\hat{f}(t_{k}) = c + \sum_{i=1}^P a_i \hat{f}(t_{k-i})$$

where $$c$$ and $$\{a_i\}$$ are model coefficients. (There is a direct relationship between an AR model and its corresponding power spectral density, correct?)

Another way is by estimating and using derivatives of $$f(t_{k-1})$$:

$$\hat{f}(t_k) = \hat{f}(t_{k-1}) + \Delta t \hat{\dot{f}}(t_{k-1}) + \frac{\Delta t}{2} \hat{\ddot{f}}(t_{k-1}) + \cdots$$

where $$\Delta t = t_k - t_{k-1}$$, $$\dot{f}$$ denotes the first time derivative (i.e. rate) of $$f$$, and $$\ddot{f}$$ denotes the second time derivative (i.e. acceleration) of $$f$$, etc. (I don't at all understand how this method of predicting $$f$$ relates to its power spectral density.)

Intuitively, I feel there must be a relationship between these two ways of predicting/estimating a smooth function, but I can't quite pin it down, nor find any references that do. So...

What is the relationship between using an auto-regressive model and using function derivatives to predict a smooth signal $$f(t_k)$$, particularly in regard to knowing a-priori its power spectral density $$S_f(\omega_k)$$.

Thank you for any help/guidance/direction. If this question belongs in another community, please let me know.

From the point of view of radiolocation, the general task can be divided into two parts: the consideration of jamming trends and the allocation of signal information. As far as I understand the essence of the issue, in meteorology the same information can be perceived as an interfering trend (with a short-term forecast), and as signaling information (with a long-term one).

Time trends (including polynomial one) can be used to equalize the signal level at the edges of the samples, which eliminates the use of weight windows that reduce the resolution of the DFT.

The same task can be performed by methods based on an autoregressive smoothing model, for which fast algorithms also exist.

Trends computed in the spectral region are used to normalize the density of the power spectrum. The alignment of the interference spectrum brings the consistent signal processing to the optimum.

The signal spectrum is chaotic and poorly predictable. Non-parametric processing methods are better suited for it — for example, median filtering in a sliding window.

Waiting for detalization in comments.

$$\textbf{Comments to the comments}$$

Note that the AR model is significantly marked from polynomial.

1. The polynomial model is not fundamentally cyclical, while the AR model of small order ideally approximates a harmonic signal.

2. Fundamentally different structure of matrices. The matrix structure in polynomial regression problems leads to the Vandermonde determinant and is characterized by computational instability, which is manifested in models of order 10 and higher.

The matrix for the AR model has a special Toeplitz structure, and its solution is built on recurrent formulas, which make it possible to consistently increase the order of the model. The considered methods to a significant extent rely on the autocorrelation function, which is not typical of polynomial models.

1. The signal maxima in the framework of the polynomial model are also calculated by well-known algorithms.

The signal maxima within the AR model correspond to the poles of the signal spectrum.

In my opinion, these differences are enough to abandon the idea of ​​linking these models. If the search continues, they should take into account these differences.

• Thanks for the input. This doesn't quite address the heart of the issue, which is finding a rigorous relationship between stochastic time series prediction using AR versus polynomial models. For example, I would think there is some procedure for converting AR model to polynomial coefficients and vice versa. – Brian Mar 11 at 5:38
• @Brian AR model is based in PSF, so it is linked with the polynomial model only via the issue sample. – Yuri Negometyanov Mar 11 at 5:59
• @Brian Thinking about your ideas, I've written the most essencial differences of the models (see updated answer). This approach seems more usable than previous. – Yuri Negometyanov Mar 11 at 11:58