Low Number Of Counts Research Articles

Shorter signals for improved signal to noise ratio, the influence of Poisson distribution

Signal to noise ratios of transient signals depend on the signal duration and are usually better for shorter signals. Poisson distribution is the dominating noise source for mass spectrometers at low count numbers. The IUPAC concept of ‘detection limit’ is used as a measure of the noise level and three different ways of detection limit calculations are compared and discussed. Assumption of Normal distribution leads to underestimation of the detection limit compared to exact Poisson based calculations. A simple formula given by IUPAC bases detection limit on the standard deviation of the blank but introduces a term to correct for the difference between Normal and Poisson distributions. Results of this formula are comparable to exact Poisson calculations. Signal to noise in transient signals was evaluated. It was found that an optimum duration for transient signals exists and that it depends on the level of the instrument background.

New statistical methods for exotic nuclei

When fitting spectra with a low number of counts per channel, the standard χ 2 fits will give systematic errors. The maximum likelihood method will perform better. Its behaviour in goodness-of-fit testing is analyzed and some alternative methods employing empirical distribution functions are mentioned.

Evaluation of eight decision rules for low-level radioactivity counting.

In low-level radioactivity measurements, it is often important to decide whether a measurement differs from background. A traditional formula for decision level (DL) is given in numerous sources, including the recent ANSI/HPS N13.30-1996, Performance Criteria for Radiobioassay and the Multi-Agency Radiation Survey and Site Investigation Manual (MARSSIM). This formula, which we dub the N13.30 rule, does not adequately account for the discrete nature of the Poisson distribution for paired blank (equal count times for background and sample) measurements, especially at low numbers of counts. We calculate the actual false positive rates that occur using the N13.30 DL formula as a function of a priori false positive rate a and background Poisson mean mu = rhot, where rho is the underlying Poisson rate and t is the counting time. False positive rates exceed a by significant amounts for alpha < or = 0.2 and mu < 100 counts, peaking at 25% at mu approximately equal to 0.71, nearly independent of alpha. Monte Carlo simulations verified calculations. Currie's derivation of the N13.30 DL was based on knowing a good estimate of the mean and standard deviation of background, a case that does not hold for paired blanks and low background rates. We propose one new decision rule (simply add 1 to the number of background counts), and we present six additional decision rules from various sources. We evaluate the actual false positive rate for all eight decision rules as a function of a priori false positive rate and background mean. All of the seven alternative rules perform better than the N13.30 rule. Each has advantages and drawbacks. Given these results, we believe that many regulations, national standards, guidance documents, and texts should be corrected or modified to use a better decision rule.

Effect of low count sums on quantitative environmental reconstructions: an example using subfossil chironomids

The concentrations of chironomid remains in lake sediments are very variable and, therefore, chironomid stratigraphies often include samples with a low number of counts. Thus, the effect of low count sums on reconstructed temperatures is an important issue when applying chironomid‐temperature inference models. Using an existing data set, we simulated low count sums by randomly picking subsets of head capsules from surface‐sediment samples with a high number of specimens. Subsequently, a chironomid‐temperature inference model was used to assess how the inferred temperatures are affected by low counts. The simulations indicate that the variability of inferred temperatures increases progressively with decreasing count sums. At counts below 50 specimens, a further reduction in count sum can cause a disproportionate increase in the variation of inferred temperatures, whereas at higher count sums the inferences are more stable. Furthermore, low count samples may consistently infer too low or too high temperatures and, therefore, produce a systematic error in a reconstruction. Smoothing reconstructed temperatures downcore is proposed as a possible way to compensate for the high variability due to low count sums. By combining adjacent samples in a stratigraphy, to produce samples of a more reliable size, it is possible to assess if low counts cause a systematic error in inferred temperatures.

Iterative reconstruction for attenuation correction in positron emission tomography: maximum likelihood for transmission and blank scan.

The quality of the attenuation correction strongly influences the outcome of the reconstructed emission scan in positron emission tomography. The calculation of the attenuation correction factors must take into account the Poisson nature of the radioactive decay process, because-for a reasonable scan duration-the transmission measurements contain lines of response with low count numbers in the case of large attenuation factors. Our purpose in this study is to investigate a maximum likelihood estimator for attenuation correction factor calculation in positron emission tomography, which incorporates the Poisson nature of the radioactive decay into transmission and blank measurement. Therefore, the correct maximum likelihood function is used to derive two estimators for the calculation of the attenuation coefficient image and the corresponding attenuation correction factors depending on the measured blank and transmission data. Log likelihood convergence, mean differences, and the mean of squared differences for the attenuation correction factors of a mathematical thorax phantom were determined and compared. The algorithms yield adequate attenuation correction factors, however, the algorithm taking the noise in the blank scan into account can perform better for noisy blank scans. We conclude that maximum likelihood-including blank likelihood-is advantageous to reconstruct attenuation correction factors for low statistic blank and good statistic transmission data. For normal blank and transmission statistics the implementation of the statistical nature of the blank is not mandatory.

Discrete and Continuous Analysis of Positron Lifetime Spectra: a Comparison

Positron lifetime spectra in poly[1-(trimethylsilyl)]propine membranes were collected at different temperatures and various number of accumulated counts, in order to compare the results from analyses obtained by different decay models: discrete (that is, purely exponential) or continuous (with at least one distribution of lifetimes). Analysis into four discrete components is inadequate, unless a low number of counts (1 to 2 millions) is accumulated. Conversely, deconvolutions which allow the ortho-positronium lifetime to be distributed give satisfactory fits, even at the highest statistics (60 millions). An equivalent good fit in terms of x 2 test is obtained with a five discrete components analysis, from which it is possible to extract information on the first moments (centroid and standard deviation) of ortho-positronium lifetime distribution. PACS numbers: 36.10.Dr, 71.60.-1-z, 78.70.Bj

Mapping the Gas Temperature Distribution in Extended X‐Ray Sources and Spectral Analysis in the Case of Low Statistics: Application toASCAObservations of Clusters of Galaxies

A simple method for mapping the temperature distribution in extended sources is developed for application to ASCA observations of galaxy clusters. Unlike the conventional approach to spatially resolved spectral analysis, this method does not require nonlinear minimization and is computationally fast and stable. Therefore, it can be implemented for a large number of regions or on a fine spatial grid. Although based on a Taylor expansion over the nonlinear parameter, the method is found to be accurate in many practical situations, the relative error for the temperature estimate being less than 2%-4% when the plasma temperature exceeds ~2 keV. This method is not intended to replace conventional spectral analysis but to supplement it, providing relatively fast and easy construction of temperature maps, which may be used as a guide to further detailed analysis of particularly interesting regions using conventional spectral fitting. Conventional spectral analysis in the case of moderate and low numbers of counts is discussed. A practical recipe for unbiased parameter estimation is suggested and verified in Monte Carlo simulations for commonly used spectral models. A simple modification of the χ2 statistic (calculation of weights based on the smoothed observed spectrum) yields nearly unbiased parameter estimates and correct confidence interval determination with no need for regrouping (binning) the energy channels even in the case of low statistics (~50-100 counts in the observed spectrum with several hundred channels).

Deconvolution of nuclear spectra of low counts

Conventional methods of spectrum deconvolution usually fail when spectra of extremely low count numbers have to be evaluated. Such spectra carry much more information than used by these conventional methods. To keep all information and to obtain the best possible results low count spectra should be approached with a special philosophy: instead of dividing the spectrum into well-defined areas regarded as single peaks, the counts in all channels have to be allocated to the peaks by all possible divisions into two sets. The probability of each division can be determined using the probability distributions of the single peaks (peak shapes). Probabilities of all those distributions which belong to the same values of peak areas are added together. The distribution of the highest probability can be regarded as the final result. The confidence level of the deconvolution is described by the corresponding probability. The method is suitable for determining small peak areas on a high background, for decomposing overlapping peaks and for deconvoluting the whole spectrum or its one section.

Acoustic emission monitoring of damage using high amplitude gains in carbon fibre reinforced poly(ether ketone ketone)

Acoustic emission testing (AET) is becoming an established test method for monitoring the damage accumulation process in composite materials [1-15]. AET involves acoustically coupling a number of piezoelectric sensors to the material being tested. These piezoelectric sensors convert the mechanical energy of the acoustic waves released by specific damage events in the material into electrical signals. The electrical signals are then amplified and analysed by a computer to obtain some information on the type of damage event taking place. The damage events which have been monitored using this technique include matrix cracking, fibre fracture and interracial failure. The total signal amplification during AET determines the signal to noise ratio. In order to filter out unwanted noise signals, a combination of a "threshold" and a "count filter" is used. The threshold level is the ultimate level below which any incoming signal is considered to be background noise. The count filter is used to filter out unwanted occasional high voltage spikes. When an incoming signal exceeds a preset threshold voltage, one "count" is recorded by the acoustic unit. Once an event has accumulated more counts than that imposed by the counts filter, it is considered an acoustic "hit". Noise signals typically have a lower number of counts than signals generated by damage events. Therefore, a combination of correct threshold levels and count filters can be used to remove all unwanted noise signals without masking useful damage accumulation data. Most of the acoustic emission tests described in the literature have been performed using a relatively low signal amplification (60-65 dB) and a relatively high count filter (5-10) [1-15]. In this paper, we have described acoustic monitoring of damage in AS4 carbon fibre reinforced poly(aryl ether ketone ketone) composites using relatively high signal amplifications (85 dB), a relatively low count filter (2), and a threshold level of 40dB. At these experimental conditions, we have observed unusual acoustic activity at very low strokes. Care was taken to ensure that this signal is not a mechanism or electrical artefact of the experimental setup or procedure. To our knowledge, this unusual initial acoustic activity has not been reported by other researchers.

The problem of scatter correction in positron volume imaging

Thin, dense interslice septa have long been used to reduce the scattered radiation collected in positron emission tomography (PET). These septa prevent the acquisition of coincident gamma rays that are oblique to the scanning planes. Several volume imaging systems that acquire oblique rays have been built. The reasons why the scatter fraction is intrinsically higher in these systems are discussed, and two different scatter compensation techniques are compared. The comparison uses Monte Carlo simulations to generate average profiles and spectra for typical phantoms and scanning geometries. Comparison results indicate that deconvolution of the total event profiles with a filter adjusted to the object size and good energy discrimination provide an excellent scatter estimate as long as the object's diameter is less than the detector's radius. The profiles derived from a second, lower, energy window are always much flatter than the scattered counts in a window covering the photopeak. They do not match the scattered event profile and are less reliable, being based on a lower number of counts.

Determination of confidence limits for experiments with low numbers of counts

view Abstract Citations (403) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Determination of Confidence Limits for Experiments with Low Numbers of Counts Kraft, Ralph P. ; Burrows, David N. ; Nousek, John A. Abstract Two different methods, classical and Bayesian, for determining confidence intervals involving Poisson-distributed data are compared. Particular consideration is given to cases where the number of counts observed is small and is comparable to the mean number of background counts. Reasons for preferring the Bayesian over the classical method are given. Tables of confidence limits calculated by the Bayesian method are provided for quick reference. Publication: The Astrophysical Journal Pub Date: June 1991 DOI: 10.1086/170124 Bibcode: 1991ApJ...374..344K Keywords: Bayes Theorem; Confidence Limits; Photon Density; Poisson Density Functions; Supernova 1987a; Charge Coupled Devices; Error Analysis; Gamma Ray Astronomy; Probability Distribution Functions; STATISTICS AND PROBABILITY; GAMMA RAYS: GENERAL; NUMERICAL METHODS; X-RAYS: GENERAL full text sources ADS | data products SIMBAD (1)

Real time spectrum analysis

A method is described for the separation of a composite pulse-height spectrum into its unresolved component parts, which belong to a set of measured library spectra. The method allows real-time estimation giving running estimates during acquisition of the spectrum, minimises computation space, especially for a number of parallel calculations, estimates in advance the rms errors, and produces a significance measure for the hypothesis that the composite contains only the library spectra. Least squares curve-fitting, and other methods, can be compared, with the formalism developed, allowing analytical comparison of the effect of detector energy resolution and detection efficiency. A rational basis for the choice between the various methods of spectrum analysis follows from the theory, minimising rms estimatiom errors. The method described is applicable for very low numbers of counts and poor resolution.

Photon counting statistics of the superposition of coherent and chaotic light of arbitrary spectrum passed through the turbulent atmosphere or a Gaussian medium

We derive the approximate photon counting distribution and its factorial moments of arbitrary order for the superposition of coherent and chaotic light of arbitrary spectrum and for arbitrary counting time intervals passing through the turbulent atmosphere or a Gaussian scattering medium; the frequency of coherent light and the mean frequency of chaotic light need not coincide. It is shown by discussion of numerical results for various levels of turbulence and various values of the parameterγ, which is the halfwidth of the Lorentzian spectrum of chaotic light multiplied by the detection time, that peaks of the photon counting distributions are decreasing (the distributions are broadening) with increasing level of turbulence, they are shifted to lower number of counts and values of the normalized factorial moments are rapidly increasing; these values tend to a limit given by the level of turbulence forγ→∞.