Two-parameter Estimation Problem Research Articles

Two-parameter estimation with single squeezed-light interferometer via double homodyne detection

The simultaneous two-parameter estimation problem in a single squeezed-light Mach–Zehnder interferometer with the double-port homodyne detection is investigated in this work. The analytical form of the two-parameter quantum Cramér–Rao bound defined by the quantum Fisher information matrix is presented, which shows the ultimate limit of the phase sensitivity will be further approved by the squeezed vacuum state. It can not only surpass the standard quantum limit, but also can even surpass the Heisenberg limit when half of the input intensity of the interferometer is provided by the coherent state and half by the squeezed light. For the double-port homodyne detection, the classical Fisher information matrix is also obtained. Our results show that although the classical Cramér–Rao bound does not saturate the quantum one, it can still asymptotically approach the quantum Cramér–Rao bound when the intensity of the coherent state is large enough. Our results also indicate that the squeezed vacuum state indeed can further improve the phase sensitivity, similar to the single-parameter estimation. In addition, when half of the input intensity of the interferometer is provided by the coherent state and half by the squeezed light, the phase sensitivity obtained by the double-port homodyne detection can also surpass the Heisenberg limit for a small range of the estimated phases.

A performance comparison between m-sequences and linear frequency-modulated sweeps for the estimation of travel-time with a moving source.

This manuscript discusses the utility of maximal period linear binary pseudorandom sequences [also referred to as m-sequences or maximum length sequences (MLSs)] and linear frequency-modulated (LFM) sweeps for the purpose of measuring travel-time in ocean-acoustic experiments involving moving sources. Signal design and waveform response to unknown Doppler (waveform dilation or scale factor) are reviewed. For this two-parameter estimation problem, the well-known wide-band ambiguity function indicates, and moving-source observations corroborate, a significant performance benefit from using MLS over LFM waveforms of similar time duration and bandwidth. The comparison is illustrated with a typical experimental setup of a source suspended aft of the R/V Sally Ride to a depth of∼10 m and towed at∼1 m/s speed. Accounting for constant source motion, the root mean square travel-time variability over a 30 min observation interval is 53 μs (MLS) and 141 μs (LFM). For these high signal-to-noise ratio channel impulse response data, LFM arrival-time fluctuations mostly appear random while MLS results exhibit structure believed to be consistent with source (i.e., towed transducer) dynamics. We conclude with a discussion on signal coherence with integration times up to 11 MLS waveform periods corresponding to ∼27 s.

Conclusive precision bounds for SU(1,1) interferometers

In this paper, we revisit the quantum Fisher information (QFI) calculation in SU(1,1) interferometer considering different phase configurations. When one of the input modes is a vacuum state, we show by using phase averaging, different phase configurations give same QFI. In addition, by casting the phase estimation as a two-parameter estimation problem, we show that the calculation of the quantum Fisher information matrix (QFIM) is necessary in general. Particularly, within this setup, the phase averaging method is equivalent to a two parameter estimation problem. We also calculate the phase sensitivity for different input states using QFIM approach.

Optimal channels for channelized quadratic estimators.

We present a new method for computing optimized channels for estimation tasks that is feasible for high-dimensional image data. Maximum-likelihood (ML) parameter estimates are challenging to compute from high-dimensional likelihoods. The dimensionality reduction from M measurements to L channels is a critical advantage of channelized quadratic estimators (CQEs), since estimating likelihood moments from channelized data requires smaller sample sizes and inverting a smaller covariance matrix is easier. The channelized likelihood is then used to form ML estimates of the parameter(s). In this work we choose an imaging example in which the second-order statistics of the image data depend upon the parameter of interest: the correlation length. Correlation lengths are used to approximate background textures in many imaging applications, and in these cases an estimate of the correlation length is useful for pre-whitening. In a simulation study we compare the estimation performance, as measured by the root-mean-squared error (RMSE), of correlation length estimates from CQE and power spectral density (PSD) distribution fitting. To abide by the assumptions of the PSD method we simulate an ergodic, isotropic, stationary, and zero-mean random process. These assumptions are not part of the CQE formalism. The CQE method assumes a Gaussian channelized likelihood that can be a valid for non-Gaussian image data, since the channel outputs are formed from weighted sums of the image elements. We have shown that, for three or more channels, the RMSE of CQE estimates of correlation length is lower than conventional PSD estimates. We also show that computing CQE by using a standard nonlinear optimization method produces channels that yield RMSE within 2% of the analytic optimum. CQE estimates of anisotropic correlation length estimation are reported to demonstrate this technique on a two-parameter estimation problem.

Analyzing Rice distributed functional magnetic resonance imaging data: a Bayesian approach

Analyzing functional MRI data is often a hard task due to the fact that these periodic signals are strongly disturbed with noise. In many cases, the signals are buried under the noise and not visible, such that detection is quite impossible. However, it is well known that the amplitude measurements of such disturbed signals follow a Rice distribution which is characterized by two parameters. In this paper, an alternative Bayesian approach is proposed to tackle this two-parameter estimation problem. By incorporating prior knowledge into a mathematical framework, the drawbacks of the existing methods (i.e. the maximum likelihood approach and the method of moments) can be overcome. The performance of the proposed Bayesian estimator is analyzed theoretically and illustrated through simulations. Finally, the developed approach is successfully applied to measurement data for the analysis of functional MRI.

Estimation of parameters in multi-mode heat transfer problems using Bayesian inference – Effect of noise and a priori

Parameter estimation problems and heat source/flux reconstruction problems are some of the most frequently encountered inverse heat transfer problems. These problems find their application in many areas of science and engineering. The primary focus of this paper is on the heat transfer parameter estimation for a two-dimensional unsteady heat conduction problem with (a) convection boundary condition and (b) convection and radiation boundary condition. The paper demonstrates the effect of a priori model on the performance of the algorithm at different noise levels in the measured data. The inverse problem is solved using three different a priori models namely normal, log normal and uniform. The posterior PDF is sampled using the Metropolis–Hastings sampling algorithm. Both single-parameter estimation and multi-parameter estimation problems are addressed and the effects of corresponding a priori models are studied. It was found that the mean and maximum a posteriori estimates for thermal conductivity and the convection heat transfer coefficient were insensitive to the a priori model at all the considered noise levels for the single-parameter estimation problem. At high noise levels in the two-parameter estimation problem, the estimates for thermal conductivity and convection coefficient were sensitive to the a priori model. It was also found that the standard deviation of the samples was correlated to the error in estimation in the single-parameter estimation case. In three parameter estimation case, alternate solutions to the same problem were retrieved due to a strong correlation between the convection coefficient and the emissivity. However, a more informative a priori model could address this issue.

Easy estimation by a new parameterization for the three-parameter lognormal distribution

A new parameterization and algorithm are proposed for seeking the primary relative maximum of the likelihood function in the three-parameter lognormal distribution. The parameterization yields the dimension reduction of the three-parameter estimation problem to a two-parameter estimation problem on the basis of an extended lognormal distribution. The algorithm provides the way of seeking the profile of an object function in the two-parameter estimation problem. It is simple and numerically stable because it is constructed on the basis of the bisection method. The profile clearly and easily shows whether a primary relative maximum exists or not, and also gives a primary relative maximum certainly if it exists.

Identifiability and stability of a two-parameter estimation problem1

We consider the ill-posed problem of determining the coeffi¬cient a and c in the elliptic equation -div(a grad u)+cu=f by the knowledge of u for two experiments described by two choices of inhomo- geneities f and/or the boundary data. We focus our attention on the question when it is possible to identify the two parameters in a stable way and prove stability estimates by different techniques