Sampling Theory

We will show that if the kernel K(x, t) in the representation f(t) = R a 0 K(x, t)u(t)dqt, with u ∈ L 2 q(0, a), is a solution of a second order q-Sturm-Liouville boundary problem, then f admits a representation as a sampling formula of the form f(t) = P∞ n=0 f(λn)W(t)/(W '(λn)(t − λn)) where λn is the nth eigenvalue of the associated q-Sturm-Liouville boundary problem and W(t) is the Wronskian of two solutions selected in a specified way.

Consider a stochastic process (X; A), where X represents the evolution of a system over time, and A is an associated point process that has stationary independent increments. Suppose we are interested in estimating the time average frequency of the process X being in a set of states. Often it is more convenient to have a sampling procedure for estimating the time average based on averaging the observed values of X (Tn) (Tn being a point of A) over a long period of time: the event average of the process. In this paper we examine the situation when the two procedures-event averaging and time averaging-produce the same estimate (the ASTA property: Arrivals See Time Averages). We prove a result stronger than ASTA. Under a lack-of-anticipation assumption we prove that the point process, A, restricted to any set of states, has the same probabilistic structure as the original point process. In particular, if the original point process is Poisson the new point process is still Poisson with the same parameter as the original point process. We develop our results in the more general setting of a stochastic process (X; A), that is, a process with an imbedded cumulative process, A = {A(t); t ¿ 0}, which is assumed to be a Levy process with non-decreasing sample paths. This framework allows for modeling uid processes, as well as compound Poisson processes with non-integer increments. First, we state the result in discrete time; the discrete-time result is then extended to the continuous-time case using limiting arguments and weak-convergence theory. As a corollary we give a proof of ASTA under weak conditions and a simple, intuitive proof of (Poisson Arrivals See Time Averages) under the standard conditions. The results are useful in queueing and statistical sampling theory.

We have developed a system for simulating the conditions of avian surveys in which birds are identifi ed by sound. The system uses a laptop computer to control a set of amplifi ed MP3 players placed at known locations around a survey point. The system can realistically simulate a known population of songbirds under a range of factors that aff ect detection probabilities. The goals of our research are to describe the sources and range of variability aff ecting point-count estimates and to fi nd applications of sampling theory and methodologies that produce practical improvements in the quality of bird-census data. Initial experiments in an open fi eld showed that, on average, observers tend to undercount birds on unlimited-radius counts, though the proportion of birds counted by individual observers ranged from 81% to 132% of the actual total. In contrast to the unlimited-radius counts, when data were truncated at a 50-m radius around the point, observers overestimated the total population by 17% to 122%. Results also illustrate how detection distances decline and identifi cation errors increase with increasing levels of ambient noise. Overall, the proportion of birds heard by observers decreased by 28 ± 4.7% under breezy conditions, 41 ± 5.2% with the presence of additional background birds, and 42 ± 3.4% with the addition of 10 dB of white noise. These fi ndings illustrate some of the inherent diffi culties in interpreting avian abundance estimates based on auditory detections, and why estimates that do not account for variations in detection probability will not withstand critical scrutiny.

This paper may be seen as in part a review covering basics of sampling theory in a di erent light. We use a multivariate approach with a unifying treatment of WOR and WR sampling designs. In this framework, we present probability functions of several important sampling designs, such as the hypergeometric, the conditional Poisson, the Sampford, and the general order sampling designs among others. Beneÿting from the distributional feature of the sampling design, a list-sequential method for generating a sample from any given design is developed. The method is applied to hypergeometric, multinomial, conditional Poisson and Sampford designs. An order sampling procedure for a population with unknown size is described. Markov chain Monte Carlo methods are discussed.

Before implementing a new diagnostic test, we may wish to study whether this test is noninferior to a reference test with respect to the sensitivity and/or the specificity. This paper discusses sample size determination for one-sided equivalence (or noninferiority) testing of the rate ratio using paired-sample data. Using large sample theory, this paper derives asymptotic sample size formulae for the required number of subjects giving a desired power 100(1 Ϫ ␤ )% at a specified ␣ -level. To evaluate the accuracy of these formulae, this paper considers several test statistics and uses Monte Carlo simulation to estimate the corresponding type I error and power with the given resulting sample sizes in a variety of situations. Finally, this paper notes those situations for which the asymptotic sample size formulae developed here are of limited use and suggests a simple empirical adjustment to alleviate this limitation.

Sampling can be a significant source of error in the measurement process. The characterization and cleanup of hazardous waste sites require data that meet site-specific levels of acceptable quality if scientifically supportable decisions are to be made. In support of this effort, the US Environmental Protection Agency (EPA) is investigating methods that relate sample characteristics to analytical performance. Predicted uncertainty levels allow appropriate study design decisions to be made, facilitating more timely and less expensive evaluations. Gy sampling theory can predict a significant fraction of sampling error when certain conditions are met. We report on several controlled studies of subsampling procedures to evaluate the utility of Gy sampling theory applied to laboratory subsampling practices. Several sample types were studied and both analyte and non-analyte containing particles were shown to play important roles affecting the measured uncertainty.

In this paper we develop a polynomial method based on sampling theory that can be used to estimate the Shapley value (or any semivalue) for cooperative games. Besides analyzing the complexity problem, we examine some desirable statistical properties of the proposed approach and provide some computational results.

Calibration estimators are used, in the present paper, to estimate the population proportion in presence of an auxiliary variable. We have proposed a class of estimators for estimating the population proportion and studied its performance with different estimators available in the literature and some particular estimators of the proposed class.

The present paper deals with the estimation of several missing values in Graeco-Latin Squnre designs.
When the observations are missing in a particular pattern, explicit computable expressions
are presented for the estimators of the missing values. This procedure is illustrated with
the help of a numerical example.

Analytical determination is usually a multistep process, which may contain several sample and sample preparation steps. It is also an error generating process where each step generates its own error component. The errors may be random or systematic. When the overall reliability of an analytical determination is estimated the variances (squared errors) of the individual error generating steps of the whole analytical process are additive and, consequently, the weakest link in practice determines the overall quality of the result. Sampling and the probable error components generated by doing it cannot be neglected, therefore, when optimal analytical strategy for a given determination is considered. The optimum analytical strategy, and the error components to be considered, depend on the purpose of the determination, i.e., whether point estimates of the concentration at sampling locations or a general description (average) of the lot is what is needed. The aim of chemometric methods is to interpret chemical and analytical data. Therefore, the reliability of chemometrics also depends on the quality on analytical results -and ultimately on the quality of sampling An advanced sampling theory for analytical purposes has been developed by Pierre Gy [1, 2, 3], but is not very well known yet by analytical chemist and chemical metrologists. A short introduction to Gy's theory is given by Smith [4]. Too often sampling plans are made without the help of the sampling theory. This concerns both industrial and process analytical problems and large national campaigns, when, e.g., the overall state of the quality river and lake waters is evaluated or the intake of chemical constituents in daily diets is estimated. By using sampling theory it often possible to cut down the assay cost and to improve the reliability of the results.

The Bitterlich relascope is a multiple use dendrometer widely used in forest inventory. Although it is most commonly used to estimate basal area, the relascope can also estimate other stand variables, including density and diameter distribution. However, forest stand inventories in Spain rarely use relascope plots to estimate these variables due to the belief that they lead to higher errors than fixed-radius plots due to the heterogeneity of many Mediterranean forests. This study compared the accuracy of the estimated averages of three main stand variables (basal area, stand density, and diameter class distribution) in forest stand inventories performed with relascope plots and with conventional fixed-radius circular plots, both measuring a similar number of trees (15)(16)(17)(18)(19)(20). A forest stand inventory simulator (DOMO) was used (1) to generate simulated forest stands corresponding to the nine most common types in the Mediterranean region of Catalonia (NE Spain), including even-aged and uneven-aged stands, and (2) to estimate and compare the average values of these variables at the forest stand level resulting from both plot types. In general, we did not find significant accuracy differences between the inventory systems for most of the stand variables and forest types studied, as expected by established angle-count sampling theory. However, the results show that for stands with multiple strata and open structures, the Bitterlich relascope provides a more accurate estimate for basal area than for density, while the reverse occurs for fixed-radius plots.

Diffusion theory is currently used to characterize highly scattering media from time and/or spatially resolved measurements. However, many of the limits of validity of the diffusion approximation in its standard form have been set through assumptions taken during its derivation, and it is generally believed that the diffusion equation holds only for length scales much larger than the transport mean free path (l tr ). We here present several modifications to the standard diffusion approximation to provide a simple expression for characterizing optically thin scattering slabs. The improvements concern mainly the correct modelling of the temporal and spatial distribution of the source term and the use of a frequency dependent diffusion coefficient and boundary conditions. This novel expression is put to the test against Monte Carlo simulations and experimental data taken from measurements with phantoms of known optical properties. The reduced scattering coefficient can be retrieved accurately for slab widths of the order of the mean free path.

The versatility of anaerobic digestion (AD) as an effective technology for solving central challenges met in applied biotechnological industry and society has been documented in numerous publications over the past many decades. Reduction of sludge volume generated from wastewater treatment processes, sanitation of industrial organic waste, and benefits from degassing of manure are a few of the most important applications. Especially, renewable energy production, integrated biorefining concepts, and advanced waste handling are delineated as the major market players for AD that likely will expand rapidly in the near future. The complex, biologically mediated AD events are far from being understood in detail however. Despite decade-long serious academic and industrial research efforts, only a few general rules have been formulated with respect to assessing the state of the process from chemical measurements. Conservative reactor designs have dampened the motivation for employing new technologies, which also constitutes one of the main barriers for successful upgrade of the AD sector with modern process monitoring instrumentation. Recent advances in Process Analytical Technologies (PAT) allow complex bioconversion processes to be monitored and deciphered using e.g. spectroscopic and electrochemical measurement principles. In combination with chemometric multivariate data analysis these emerging process monitoring modalities carry the potential to bring AD process monitoring and control to a new level of reliability and effectiveness. It is shown, how proper involvement of process sampling understanding, Theory of Sampling (TOS), constitutes a critical success factor. We survey the more recent trends within the field of AD monitoring and the powerful PAT/TOS/chemometrics application potential is highlighted. The Danish co-digestion concept, which integrates utilisation of agricultural manure, biomass and industrial organic waste, is used as a case study. We present a first foray for the next research and development perspectives and directions for the AD bioconversion sector.

Abstract: Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition, band-limited functions ...

Many statistical procedures rely on the assumption that the observed data are normally distributed. Consequently, there exists a vast literature on tests of normality and their statistical properties. Today the most commonly used omnibus test for general use is the Shapiro-Wilk method while the Jarque-Bera test is the most popular omnibus test in economics and related applications.

Few studies offer a theorefical explanation of the differential effects of massed and spaced advertising. Consequently, conclusions have been mixed. The authors introduce the stimuIus sampling model of spacing effects based on Estes' mathematical learrting theory. The model posits that massed repetifion generates superior memory at short retention intervals, whereas spaced repetifion produces superior memory at longer retention intervals. Data from various studies are analyzed, and implications and potential applications of the model in the advertising context are explored.

Functional principal component analysis (FPCA) based on the Karhunen-Loève decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the logreturns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic behavior. Firstly we present a new method for estimation of functional principal components from discrete noisy data. Next we present the two sample inference for FPCA and develop two sample theory. We propose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustrate the test-properties by simulation study and apply the method to the IV analysis.

We present a simple and efficient approach to generate a dense set of anisotropic, spatially varying glyphs over a two-dimensional domain. Such glyph samples are useful for many visualization and graphics applications. The glyphs are embedded in a set of non-overlapping ellipses whose size and density match a given anisotropic metric. An additional parameter controls the arrangement of the ellipses on lines, which can be favorable for some applications, e.g., vector fields, and distracting for others. To generate samples with the desired properties we combine ideas from sampling theory and mesh generation. We start with constructing a first set of non-overlapping ellipses whose distribution closely matches the underlying metric. This set of samples is used as input for a generalized anisotropic Lloyd relaxation to distribute samples more evenly.

Statistical models defined by imposing restrictions on marginal distributions of contingency tables have received considerable attention recently. This paper introduces a general definition of marginal log-linear parameters and describes conditions for a marginal log-linear parameter to be a smooth parameterization of the distribution, and to be variation independent. Statistical models defined by imposing affine restrictions on the marginal log-linear parameters are investigated. These models generalize ordinary log-linear and multivariate logistic models. Sufficient conditions for a log-affine marginal model to be nonempty, and to be a curved exponential family are given. Standard large sample theory is shown to apply to maximum likelihood estimation of log-affine marginal models for a variety of sampling procedures.

We present a survey on the sampling and quantization is- sues arising in holography and diffraction. We comment on the importance of these issues for emerging applica- tions such as 3D and holographic TV and video commu- nications and emphasize the role of signal processing for designing proper drivers for future holographic displays. Our paper starts with a review of

Monitoring the traffic in high-speed networks is a data intensive problem. Uniform packet sampling is the most popular technique for reducing the amount of data the network monitoring hardware/software has to process. However, uniform sampling captures far less information than can be potentially obtained with the same overall sampling rate. This is because uniform sampling (unnecessarily) draws the vast majority of samples from large flows, and very few from small and medium flows. This information loss on small and medium flows significantly affects the accuracy of the estimation of various network statistics.

1. Observed patterns in the fossil record reflect not just macroevolutionary dynamics, but preservation patterns. Sampling rates themselves vary not simply over time or among major taxonomic groups, but within time intervals over geography and environment, and among species within clades. Large databases of presences of taxa in fossil-bearing collections allow us to quantify variation in per-collection sampling rates among species within a clade. We do this separately not just for different time/stratigraphic intervals, but also for different geographic or ecologic units within time/stratigraphic intervals. We then re-assess per-million-year sampling rates given the distributions of per-collection sampling rates
2. We use simple distribution models (geometric and lognormal) to assess general models of per-locality sampling rate distributions given occurrences among appropriate fossiliferous localities. We break these down not simply by time period, but by general biogeographic units in order to accommodate variation over space as well as among species.
3. We apply these methods to occurrence data for Meso-Cenozoic mammals drawn from the Paleobiology Database and the New and Old Worlds fossil mammal database. We find that all models of distributed rates do vastly better than the best uniform sampling rates and that the lognormal in particular does an excellent job of summarizing sampling rates. We also show that the lognormal distributions vary fairly substantially among biogeographic units of the same age.
4. As an example of the utility of these rates, we assess the most likely divergence times for basal (Eocene–Oligocene) carnivoramorphan mammals from North America and Eurasia using both stratigraphic and morphological data. The results allow for unsampled taxa or unsampled portions of sampled lineages to be in either continent and also allow for the variation in sampling rates among species. We contrast five models using stratigraphic likelihoods in different ways to summarize how they might affect macroevolutionary inferences.

In this paper, a class of log-type estimator under simple random sampling has been suggested, to estimate the finite population mean by using single auxiliary variable under non-response. The proposed estimator, under non-response in different cases; on both the study and auxiliary variables has been studied and gives minimum mean square error as compared to all other considered estimators. Some other estimators are also identified from the proposed class of estimators. We used two real data sets to obtain the efficiencies of the estimators.

We consider the problem of the reconstruction of a continuous-time function f (x) ∈ H from the samples of the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H = L 2 ) is considerably larger than the subspace of bandlimited functions. Second, we use a more general specification of the reconstruction subspace V (ϕ), so that the output of the system can take the form of a bandlimited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximationf ∈ V (ϕ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V (ϕ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filterbank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the bandlimited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.

Early on during word recognition, letter positions are not accurately coded. Evidence for this comes from transposed-letter (TL) priming effects, in which letter strings generated by transposing two adjacent letters (e.g., jugde) produce large priming effects, more than primes with the letters replaced in the corresponding position (e.g., junpe).

Emerging sonography techniques often imply increasing in the number of transducer elements involved in the imaging process. Consequently, larger amounts of data must be acquired and processed by the beamformer. The significant growth in the amounts of data effects both machinery size and power consumption. Within the classical sampling framework, state of the art systems reduce processing rates by exploiting the bandpass bandwidth of the detected signals. It has been recently shown, that a much more significant samplerate reduction may be obtained, by treating ultrasound signals within the Finite Rate of Innovation framework. These ideas follow the spirit of Xampling, which combines classic methods from sampling theory with recent developments in Compressed Sensing. Applying such low-rate sampling schemes to individual transducer elements, which detect energy reflected from biological tissues, is limited by the noisy nature of the signals. This often results in erroneous parameter extraction, bringing forward the need to enhance the SNR of the low-rate samples. In our work, we manage to achieve such SNR enhancement, by beamforming the sub-Nyquist samples obtained from multiple elements. We refer to this process as "compressed beamforming". Applying it to cardiac ultrasound data, we successfully image macroscopic perturbations, while achieving a nearly eightfold reduction in sample-rate, compared to standard techniques.

Given a real-valued function f defined over some metric space X, is it possible to recover some structural information about f from the sole information of its values at a finite set L ⊆ X of sample points, whose pairwise distances in X are given? We provide a positive answer to this question. More precisely, taking advantage of recent advances on the front of stability for persistence diagrams, we introduce a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated. We derive from this construction a series of algorithms for the analysis of scalar fields from point cloud data. These algorithms are simple and easy to implement, have reasonable complexities, and come with theoretical guarantees. To illustrate the generality of the approach, we present some experimental results obtained in various applications, ranging from clustering to sensor networks (see the electronic version of the paper for color pictures).

Matrix algebra is a tool not commonly employed in sampling theory. The intention of this paper is to help change this situation by showing, in the context of the Horvitz-Thompson (HT) estimator, the convenience of the use of a number of matrix-algebra results. Sufficient conditions for the consistency and the rate of convergence of the HT estimator are given. These conditions are based on matrix inequalities that are employed to bound the variance of the HT estimator.

Few studies offer a theoretical explanation of the differential effects of massed and spaced advertising. Consequently, conclusions have been mixed. The authors introduce the stimulus sampling model of spacing effects based on Estes' mathematical learning theory. The model posits that massed repetition generates superior memory at short retention intervals, whereas spaced repetition produces superior memory at longer retention intervals. Data from various studies are analyzed, and implications and potential applications of the model in the advertising context are explored.

In software engineering, effective development phase can only takes place if the requirements gathering from client side are clearly defined and should be approachable. Client’s requirements play a crucial role for completion of the successful software projects in all of the development all the software methodologies. If requirements from the client are not properly defined it can lead to failure of the project. Now a day’s some information technology organization are not aware about the proper way to collect the requirement from the clients and fulfill the requirements of the clients. Effective Software development can only takes place if the requirements are properly defined and modeled. Most of the Clients have difficulty in explaining what they need exactly, and the problems increases when software developers failed to convert the client’s requirements into working code. The role of the business analyst to effectively communicate with clients to understand the requirements .Many of project fail due to inappropriate information gathering process.

Conditional Poisson Sampling Design as developed by Haje Âk may be de®ned as a Poisson sampling conditioned by the requirement that the sample has ®xed size. In this paper, an algorithm is implemented to calculate the conditional inclusion probabilities given the inclusion probabilities under Poisson Sampling. A simple algorithm is also given for second order inclusion probabilities in Conditional Poisson Sampling. Furthermore a numerical method is introduced to compute the unconditional inclusion probabilities when the conditional inclusion probabilities are predetermined. Simultaneously, we study the Pareto pps sampling design. This method, introduced by Rose Ân, belongs to a class of sampling schemes called Order Sampling with Fixed Distribution Shape. Methods are provided to compute the ®rst and second order inclusion probabilities numerically also in this case, as well as two procedures to adjust the parameters to get predetermined inclusion probabilities.

Frequencies of mutant sites are modeled as a Poisson random field in two species that share a sufficiently recent common ancestor. The selective effect of the new alleles can be favorable, neutral, or detrimental. The model is applied to the sample configurations of nucleotides in the alcohol dehydrogenase gene (Adh) in Drosophila simulans and Drosophila yakuba. Assuming a synonymous mutation rate of 1.5 X lo-' per site per year and 10 generations per year, we obtain estimates for the effective population size (N, = 6.5 X lo')), the species divergence time (tdiv = 3.74 million years), and an average selection coefficient (u = 1.53 X 1O"j per generation for advantageous or mildly detrimental replacements), although it is conceivable that only two of the amino acid replacements were selected and the rest neutral. The analysis, which includes a sampling theory for the independent infinite sites model with selection, also suggests the estimate that the number of amino acids in the enzyme that are susceptible to favorable mutation is in the range 2-23 at any one time. The approach provides a theoretical basis for the use of a 2 X 2 contingency table to compare fixed differences and polymorphic sites with silent sites and amino acid replacements. D. L. Hart1 KARLIN, S., and H. M. TAYLOR, 1981 A Second Course in Stochastic KIMURA. M., 1968 Evolutionary rate at the molecular level. Na-KIMURA, M., 1983 The Neutral Theory of Molecular Evolution. C;umbridge University Press, Cambridge, England. KRLITMAN, M., 1983 Nucleotide polymorphism at the alcohol dehydroganse locus of Drosophila melanogaster. Nature 304: 4 12-41 7.

Sampling and reconstruction of functions is a central tool in science. A key result is given by the sampling theorem for bandlimited functions attributed to Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling theory for operators which we call bandlimited if their Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for such operators and show that they are extensions of the classical sampling theorem.

The Italian Space Agency (ASI) decided to perform an low cost Earth observation mission based on a new mini satellite named MIOsat which will carry various technological payloads. Among them an imaging interferometer designed and now ready to be assembled and tested by our Institute. The instrument, named ALISEO (Aerospace Leap-frog Imaging Stationary interferometer for Earth Observation), operates in the common-path Sagnac configuration, and it does not utilize any moving part to scan the phase delays between the two interfering beams. The sensor acquires target images modulated by a pattern of autocorrelation functions of the energy coming from each scene pixel, and the resulting fringe pattern remains spatially fixed with respect to the instrument's field-of-view. The complete interferogram of each target location is retrieved by introducing a relative source-observer motion, which allows any image pixels to be observed under different viewing-angles and experience discrete p...

In this paper we develop a polynomial method based on sampling theory that can be used to estimate the Shapley value (or any semivalue) for cooperative games. Besides analyzing the complexity problem, we examine some desirable statistical properties of the proposed approach and provide some computational results.