# A new method for moving-average parameter estimation

TL;DR: An apparently original method for moving-average parameter estimation, based on covariance fitting and convex optimization, is introduced, shown by means of numerical simulation to provide much more accurate parameter estimates, in difficult scenarios, than a related existing method does.

Abstract: We introduce an apparently original method for moving-average parameter estimation, based on covariance fitting and convex optimization. The proposed method is shown by means of numerical simulation to provide much more accurate parameter estimates, in difficult scenarios, than a related existing method does. We derive the new method via an analogy with a covariance fitting interpretation of the Capon beamforming from array processing. In the process, we also point out some new facts on Capon beamforming.

## Summary (2 min read)

### Introduction

- For the sake of simplicity the authors assume that {y(k)} is a scalar sequence; however, note that the discussion in this paper can be readily extended to vector sequences by using results from [2].
- A host of alternative computationally simpler methods have been proposed for MA parameter estimation (see e.g., [1], [3]).
- Of these methods, the covariance fitting approach of [4], [5] (see also [3]) is somewhat unique in that it has, in its most refined form, an accuracy comparable with that of the MLE, and yet it obtains parameter estimates from the solution of a convex problem that can be reliably and efficiently computed in polynomial time.
- The inspiration for using this The work was supported in part by the Swedish Research Council (VR) and by the National Science Foundation Grant No. CCF-0634786.
- New form of covariance fitting criterion comes from a recent letter ([6]) as well as from one of the possible derivations of the Capon beamforming method in array processing [7], [8] (also [3]) - see the next section for details.

### A. The basic method of [4], [5]

- Also, let {rp} denote the theoretical covariances of {y(k)}.
- Two convex parameterizations of {rp} have been derived in [4], [5] (see also the references therein): the “trace parameterization” and the “Kalman-Yakubovich-Popov lemma - based parameterization.”.
- The problem in Step 1 can be easily reformulated as a semi-definite program (SDP) that can be solved reliably and efficiently in polynomial time using public-domain software [9], [10].
- Asilomar 2010 an estimation accuracy viewpoint, the parameter estimates obtained with BM may be statistically rather inefficient.

### C. The new method

- Reference [6] (Problem 1) explores this idea for decomposing Toeplitz matrices into one corresponding to an MA noise-component plus a singular one, for the purpose of identifying possible spectral lines in the residual.
- Therefore, for conciseness reasons, the authors will focus on NM.
- The authors have observed empirically that the so-obtained extension of NM does not necessarily have better accuracy than NM.

### III. NUMERICAL ILLUSTRATION

- For MA sequences with roots well outside the unit circle (see (3)), the sample covariances {r̂p}np=0 belong to the set of valid MA(n) covariances, with a high probability.
- In such cases, BM and NM give very similar results (the solution to the covariance fitting problem is likely to be {rp = r̂p}np=0 for both BM and NM).
- For each method and each value of N , the authors estimate the average mean squared error (AMSE) of the parameter estimates, viz.

### IV. CONCLUSIONS

- The authors have proposed a computationally attractive MA parameter estimation method based on the use of convex optimization and of an original covariance fitting criterion.
- The new method has been shown via numerical simulations to provide more accurate parameter estimates than the basic version of an existing competitive method.

Did you find this useful? Give us your feedback

...read more

##### Citations

14 citations

### Cites methods from "A new method for moving-average par..."

...rent assumptions has been used to justify different methods. For instance, assuming that xˆ = x+vwhere xand vare independent leads to min T∈T n trace(Tˆ −T) | Tˆ −T≥ 0 o which is a method proposed in [18]. Then, also, assuming a “symmetric” noise contribution as in xˆ +vˆ = x+v, where the noise vectors ˆv and vare independent of xand xˆ, leads to min T∈T ,Q,Qˆ n trace(Qˆ +Q) | Tˆ +Qˆ = T+Q, Q,Qˆ ≥ 0 o...

[...]

7 citations

4 citations

### Cites methods from "A new method for moving-average par..."

...Variants of the basic method have also been proposed for instance in [6] in which the purpose is to estimate the covariance matrix rather than the covariance function itself....

[...]

1 citations

##### References

7,286 citations

### "A new method for moving-average par..." refers background in this paper

...min Q≥0 tr(R̂−R) subject to R̂−R ≥ 0; {rp = trp(Q)} (13) Similarly to (6), this is a convex problem (namely a SDP) that can be efficiently solved in polynomial time using publicly available software [9], [10]....

[...]

...The problem in Step 1 can be easily reformulated as a semi-definite program (SDP) that can be solved reliably and efficiently in polynomial time using public-domain software [9], [10]....

[...]

7,174 citations

### "A new method for moving-average par..." refers background in this paper

...min Q≥0 tr(R̂−R) subject to R̂−R ≥ 0; {rp = trp(Q)} (13) Similarly to (6), this is a convex problem (namely a SDP) that can be efficiently solved in polynomial time using publicly available software [9], [10]....

[...]

...The problem in Step 1 can be easily reformulated as a semi-definite program (SDP) that can be solved reliably and efficiently in polynomial time using public-domain software [9], [10]....

[...]

2,459 citations

### "A new method for moving-average par..." refers methods in this paper

...new form of covariance fitting criterion comes from a recent letter ([6]) as well as from one of the possible derivations of the Capon beamforming method in array processing [7], [8] (also [3]) - see the next section for details....

[...]

...Along similar lines to the Capon formalism, in the so-called Pisarenko harmonic analysis [3], one seeks to decompose a given Toeplitz covariance R into the sum of a singular Toeplitz matrix and a matrix that corresponds to the background white noise....

[...]

...Of these methods, the covariance fitting approach of [4], [5] (see also [3]) is somewhat unique in that it has, in its most refined form, an accuracy comparable with that of the MLE, and yet it obtains parameter estimates from the solution of a convex problem that can be reliably and efficiently computed in polynomial time....

[...]

2,135 citations

428 citations

### "A new method for moving-average par..." refers background in this paper

...(as is well-known, the “minimum-phase” condition in (3) ensures that (1) is a unique description of the power spectrum of {y(k)} [1])....

[...]