# A new method for moving-average parameter estimation

## 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.

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##### 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...

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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

7,676 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]....

[...]

7,655 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]....

[...]

2,620 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....

[...]

2,136 citations

453 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])....

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