Variance Theorem Research Articles

A MULTI-INDEX MARKOV CHAIN MONTE CARLO METHOD

In this article we consider computing expectations w.r.t.~probability laws associated to a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation, but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space and time. In such contexts, it is known that the multi-index Monte Carlo (MIMC) method can improve upon i.i.d.~sampling from the most accurate approximation of the probability law. Indeed by a non-trivial modification of the multilevel Monte Carlo (MLMC) method and it can reduce the work to obtain a given level of error, relative to the afore mentioned i.i.d.~sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to sampled independently. We develop a modification of the MIMC method which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d.~sampling from the most accurate approximation, under assumptions. The method is numerically illustrated on a problem associated to a stochastic partial differential equation (SPDE).

Fast wavelet transform assisted predictors of streaming time series

We explore the shift variance of the decimated, convolutional Discrete Wavelet Transform, also known as Fast Wavelet Transform. We prove a novel theorem improving the FWT algorithm and implement a new prediction method suitable to the multiresolution analysis of streaming univariate datasets using compactly supported Daubechies Wavelets. An effective real value forecast is obtained synthesizing the one step ahead crystal and performing its inverse DWT, using an integrated group of estimating machines. We call Wa.R.P. (Wavelet transform Reduced Predictor) the new prediction method. A case study, testing a cryptocurrency exchange price series, shows that the proposed system can outperform the benchmark methods in terms of forecasting accuracy achieved. This result is confirmed by further tests performed on other time series. Developed in C++, Standard 2014 conformant, the code implementing the FWT and the novel Shift Variance Theorem is available to research purposes and to build efficient industrial applications.

Variance estimation of the Buckley–James estimator under discrete assumptions

The Buckley–James estimator (BJE) is a widely recognized approach in dealing with right-censored linear regression models. There have been a lot of discussions in the literature on the estimation of the BJE as well as its asymptotic distribution. So far, no simulation has been done to directly estimate the asymptotic variance of the BJE. Kong and Yu [Asymptotic distributions of the Buckley–James estimator under nonstandard conditions, Statist. Sinica 17 (2007), pp. 341–360] studied the asymptotic distribution under discontinuous assumptions. Based on their methodology, we recalculate and correct some missing terms in the expression of the asymptotic variance in Theorem 2 of their work. We propose an estimator of the standard deviation of the BJE by using plug-in estimators. The estimator is shown to be consistent. The performance of the estimator is accessed through simulation studies under discrete underline distributions. We further extend our studies to several continuous underline distributions through simulation. The estimator is also applied to a real medical data set. The simulation results suggest that our estimation is a good approximation to the true standard deviation with reference to the empirical standard deviation.

The KLT (Karhunen–Loève Transform) to extend SETI searches to broad-band and extremely feeble signals

The KLT (acronym for Karhunen–Loève Transform) is a mathematical algorithm superior to the classical FFT in many regards: 1) The KLT can filter signals out of the background noise over both wide and narrow bands. This is in sharp contrast to the FFT that rigorously applies to narrow-band signals only. 2) The KLT can be applied to random functions that are non-stationary in time, i.e. whose autocorrelation is a function of the two independent variables t 1 and t 2 separately. Again, this is a sheer advantage of the KLT over the FFT, inasmuch as the FFT rigorously applies to stationary processes only, i.e. processes whose autocorrelation is a function of the absolute value of the difference of t 1 and t 2 only. 3) The KLT can detect signals embedded in noise to unbelievably small values of the Signal-to-Noise Ratio (SNR), like 10 −3 or so. This particular feature of the KLT is studied in detail in this paper. An excellent filtering algorithm like the KLT, however, comes with a cost that one must be ready to pay for especially in SETI: its computational burden is much higher than for the FFT. In fact, it can be shown that no fast KLT transform can possibly exist and, for an autocorrelation matrix of size N, the calculations must be of the order of N 2, rather than N log( N). Nevertheless, for moderate values of N (in the hundreds), the KLT dominates over the FFT, as shown by the numerical simulations. Finally, an important and recent (2007–2008) development in the KLT theory, called the “Bordered Autocorrelation Method” (BAM), is presented. This BAM-KLT method gets around the difficulty of the N 2 brunt calculations and ends up in the following unexpected theorem: the KLT of a feeble sinusoidal carrier embedded into a lot of white stationary noise is given by the Fourier transform of the derivative of the largest KLT eigenvalue with respect to the bordering index. This basic result is fully proved analytically in the final sections of this paper by virtue of a new theorem discovered by this author in May 2007 and called “The Final Variance Theorem”.

Signal and noise transfer through imaging systems consisting of a cascade of amplifying and scattering processes

The transfer of signal and noise through an imaging system consisting of a cascade of amplifying and scattering processes has been analyzed on the basis of the Burgess variance theorem. Up to second statistical moments, the results are identical to those deduced by Rabbani et al. [ J. Opt. Soc. Am. A4, 895 ( 1987)] with multivariate moment-generating functions. An easy-to-visualize concept of detective quantum efficiency (DQE) that depends on object shape and size has been deduced that can be converted into the commonly used DQE which depends on spatial frequency. A simple expression for the noise power spectrum of a system that consists of an arbitrary number of amplifying and scattering processes has been derived; noise added along the system can be taken into account.

Transient-gain-assisted noise reduction in photodetection of nonclassical light.

The effect of transient gain in photodetection of light is studied. By treating the photoelectron multiplication as a classical, stochastic process, a generalization of the quantum Mandel formula is given. To illustrate the theory developed, the detection of sub-Poisson light from single-atom resonance fluorescence is studied. It is shown that the results may substantially differ from those predicted from the Burgess variance theorem. In particular, a reduction of noise of the photoelectric counts can be achieved even if the value of the gain rate is increased.

1/f noise in the radioactive beta - decay of 204Tl.

An attempt has been made to detect the presence of 1/f fluctuations in ${\ensuremath{\beta}}^{\mathrm{\ensuremath{-}}}$ decay using the Allan variance theorem. For counting periods less than about 800 min, the counting statistics were found to be Poissonian and for periods greater than about 800 min, the statistics were found to be non-Poissonian. 1/f fluctuations in the ${\ensuremath{\beta}}^{\mathrm{\ensuremath{-}}}$ emissions were observed leading to a flicker floor. It was also found that the flicker floor varies with energy of the emitted ${\ensuremath{\beta}}^{\mathrm{\ensuremath{-}}}$ particles from $^{204}\mathrm{Tl}$ used in the experiment.

Extended burgess' variance theorem: Derivation without small signal approach and applicability to 1/ f noise in photodiodes

The Extended Burgess' Variance Theorem (EBVT) is derived from first principles without small signal approximation. It is shown that the EBVT does not entirely include 1/ f noise due to mobility fluctuations. Moreover, it is shown that the EBVT cannot be used generally for the calculation of noise in semiconductor devices, since it does not include the noise of a generation current. A more general autocorrelation function is presented and discussed with regard to its application to the reverse current in semiconductor diodes and its correlation with well-known 1/ f noise models.

Independent photon deletions from quantized boson fields: The quantum analog of the Burgess variance theorem

It is demonstrated that if two boson fields are related by a process of independent random photon deletion, their moment-generating functions are related by an equation identical to that derived using classical arguments. A quantum analog of the Burgess variance theorem is recovered. The results confirm that the super- or sub-poissonian nature of a light beam is conserved, under the influence of independent Bernoulli deletions and/or additive independent Poisson noise. The identity results from the correspondence between classical and normally ordered correlation functions, for which vacuum fluctuations play no role.

Extension of Burgess' variance theorem to autocorrelation functions and spectra

Burgess' variance theorem is extended to autocorrelation functions and spectra. The autocorrelation expression reduces to Burgess' expression for s = 0. The result is applied to mobility fluctuation 1/ f noise in semiconductors and it is shown that S μ(f)/( μ) 2 = α/f , where α is Hooge's parameter.

Noise Limitations in Solid State Photodetectors

The first part of this paper deals with general concepts of noise and response in photodetectors. The noise in photodetectors is of a fivefold nature: (i) noise produced by the blackbody photon field, J(b); (ii) noise produced by the ambient photon field J(a); (iii) noise associated with the signal J(s); (iv) spontaneous noise characteristic for the device and not associated with J(a); (v) noise associated with the circuitry or amplifiers. Concepts for the characterization of the noise are-the noise equivalent powers P'(eq.)lambda[lambda,f,Deltaf,A], P'(eq).lambda[lambda,f,Deltaf, A], P'(eq).T[T,f,Deltaf,A], and P(eg).T[T,f,Deltaf,A]; the photon limited noise equivalent powers P(eq).lambda[lambda,f,Deltaf,A], P(eg).T[T,f,Deltaf,A]; various detectivities Dlambda*[lambda,f], D(T)*[T,f]; the photon limited detectivities D(T)*{T(s), Omega(s)}, D(lambda)*{T(e), Omega(s)}, D(lambda)*{T(e), Omega(e)}, and Dlambda; dagger giving the ultimate limit attainable with a detector in radiative equilibrium; the noise figure F and the signal-to-noise ratio sigma/N. The merits and limitations of photon limited behavior are discussed and the theoretical detectivitis are calculated for various circumstances. If, in the absence of a signal, the noise stems mainly from the circuitry (class B), a characterization by P(eq) or D is largely arbitrary. Such is the case for photoemissive detectors and photoconductive insulators in the absence of bias light. In the event that the device operates and produces the main noise in the absence of a signal (class A), concepts like P(eq), D, F, are meaningful (semiconductors, p-n junctions, bolometers, PEM cells, etc.). Special attention has been given to introducing a consistent notation. In the second part of this paper we discuss two topics: some aspects of the photodetective conversion processes and the fluctuations of the photon field. The photodetective processes in emission diodes, junction diodes, photovoltaic cells, and avalanche diodes are simple. The noise is mainly of (amplified) shot noise nature. In photoconductors and PEM cells we must consider the collective carrier processes (unless the electrodes are blocking as in etectors in the thirties). Such processes do not fit too well a gain mechanism as has often been suggested as is testified by ambipolar sweep out and by the non-Poisson nature of generation-recombination (g-r) and transport fluctuations. The noise in a blackbody field is stated and the effect of imperfect absorbers is considered, involving stimulated emission. The commonly applied variance theorem leads to a degrading of the Boson factor. The noise of nonthermal photon fields is discussed following Mandel-Wolf, Alkemade, and Enns. In a nonthermal equilibrium state the effects of coherence and wave-interaction noise are manifest in the detector-output noise. In the final part the detectivity is derived for various devices. We first consider briefly photoemission devices, junction photodiodes, both in short-circuited and photovoltaic operation, and avalanche diodes. Avalanche multiplication decreases the detectivity by a factor G((1/2)), where G is the gain. A new derivation of Mclntyre's fomula-including the effect of a non-Poissonian Boson field is presented. The remainder is devoted to photoconductive detectors. Four classes are distinguished: intrinsic, minority trapping models (like PbS), two center models (like CdS), extrinsic semiconductors (like Ge: Au). The main theoretical results are stated and illustrative experimental results are discussed. Some remarks are made on hot electron photoconductivity and Landau level transitions in magnetically tuned InSb detectors.