Sampling (Mathematical Statistics)

Humans are visual animals, and imaging sensors that extend our reach—cameras—have improved dramatically in recent times thanks to the introduction of CCD and CMOS digital technology. Consumer digital cameras in the megapixel range are now ubiquitous thanks to the happy coincidence that the semiconductor material of choice for large-scale electronics integration (silicon) also happens to readily convert photons at visual wavelengths into electrons. On the contrary, imaging at wavelengths where silicon is blind is considerably more complicated, bulky, and expensive. Thus, for comparable resolution, a US$500 digital camera for the visible becomes a US$50,000 camera for the infrared.

In this article, we present a new approach to building simpler, smaller, and cheaper digital cameras that can operate efficiently across a much broader spectral range than conventional silicon-based cameras. Our approach fuses a new camera architecture based on a digital micromirror device with the new mathematical theory and algorithms of compressive sampling. CS combines sampling and compression into a single nonadaptive linear measurement process [1]–[4]. Rather than measuring pixel samples of the scene under view, we measure inner products between the scene and a set of test functions. Interestingly, random test functions play a key role, making each measurement a random sum of pixel values taken across the entire image. When the scene under view is compressible by an algorithm like JPEG or JPEG2000, the CS theory enables us to stably reconstruct an image of the scene from fewer measurements than the number of reconstructed pixels. In this manner we achieve sub-Nyquist image acquisition.

Our “single-pixel” CS camera architecture is basically an optical computer (comprising a DMD, two lenses, a single photon detector, and an analog-to-digital (A/D) converter) that computes random linear measurements of the scene under view. The image is then recovered or processed from the measurements by a digital computer. The camera design reduces the required size, complexity, and cost of the photon detector array down to a single unit, which enables the use of exotic detectors that would be impossible in a conventional digital camera. The random CS measurements also enable a tradeoff between space and time during image acquisition. Finally, since the camera compresses as it images, it has the capability to efficiently and scalably handle high-dimensional data sets from applications like video and hyperspectral imaging.

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.

This study discussed the time series models of crude oil production and export in Nigeria from January 1999-December 2015. The Augmented Dickey-Fuller unit root test employed in the analysis to test for stationarity of the two series indicated that the crude oil production series was stationary at no differencing while crude oil exportation was stationary after first order differencing. The ACF and PACF of both crude production and export identified possible model for both series. Based on Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) the best model for Crude Production and Export were SARIMA (1,0,1) (2,0,0)12 and SARIMA (2,1,0) (1,0,1)12. Residual analysis and Box-Ljung test proved the adequacy of the models for both series.

Problems of sampling carbonized plant material are discussed. Firstly, the problem of actually selecting a sample in the laboratory is considered, and some experiments which investigate various procedures are described. Secondly, the statistical aspects of determining optimal sample sizes are considered. Formulae are given for calculating optimal sample sizes and confidence intervals. Upper bounds, which are independent of the total population size, are provided for the sample size required to achieve any desired accuracy.

In recent years, signal processing has come under mounting pressure to accommodate the increasingly high-dimensional raw data generated by modern sensing systems. Despite extraordinary advances in computational power, processing the
signals produced in application areas such as imaging, video, remote surveillance, spectroscopy, and genomic data analysis continues to pose a tremendous challenge. Fortunately, in many cases these high-dimensional signals contain relatively little information compared to their ambient dimensionality. For example, signals can often be well-approximated as a sparse linear combination of elements from a known basis or dictionary.

Traditionally, sparse models have been exploited only after acquisition, typically for tasks such as compression. Recently, however, the applications of sparsity have greatly expanded with the emergence of compressive sensing, a new approach to data acquisition that directly exploits sparsity in order to acquire analog signals more efficiently via a small set of more general, often randomized, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. A common theme in this research is the use of randomness in signal acquisition, inspiring the design of hardware systems that directly implement random measurement protocols.

This thesis builds on the field of compressive sensing and illustrates how sparsity can be exploited to design efficient signal processing algorithms at all stages of the information processing pipeline, with a particular focus on the manner in which randomness can be exploited to design new kinds of acquisition systems for sparse signals. Our key contributions include: (i) exploration and analysis of the appropriate properties for a sparse signal acquisition system; (ii) insight into the useful properties of random measurement schemes; (iii) analysis of an important family of algorithms for recovering sparse signals from random measurements; (iv) exploration of the impact of noise, both structured and unstructured, in the context of random measurements; and (v) algorithms that process random measurements to directly extract higher-level information or solve inference problems without resorting to full-scale signal recovery, reducing both the cost of signal acquisition and the complexity of the post-acquisition processing.

SUMMARY In this paper, a time truncated group acceptance sampling plan (GASP) is designed under the Gamma distribution. The present plan is modification of the existing plan to reduce the number of groups/sample size. The plan parameters are obtained by satisfying the producer's risk and consu-mer's risk (two points approach) for specified values of mean ratio and number of testers. The single point approach is also used to find plan parameters such as number of groups for specified values of acceptance number and experiment time. The extensive tables, graphs and examples are given for illustration purposes. The advantage of the proposed plan discussed over the existing plans.

To complete a research, collecting data is a challenge. It would be great and simple if researchers could collect data or study each and every unit of population, which is next to impossible in almost all the field, including management. Hence, only a part of the population (sample) is studied. Sample should be adequate to warrant generalization of the findings to the target population, which depends on how the samples are selected from population. Hence, this chapter deals with process of selecting sample from target population i.e. sampling and related with, in most diluted way possible to match with beginner’s knowledge. The paper is based on secondary resouces available on the topic.

The book about the eventology ~ a science about events and its applications to problems of management of sets of events ~ a new direction in philosophy, mathematics and event management (see English version of the book: https://www.academia.edu/38605343/).
This book outlines ideas that inspired me while I was trying to build a new event science, eventology. Not each of these ideas was developed, but all of them inspired me in the construction of eventology, and then in the discovery of a revolutionary theory of experience and chance, co~eventum mechanics. (see, for example, https://www.academia.edu/34357251,
https://www.academia.edu/34373279,
https://www.academia.edu/34390291,
https://www.academia.edu/38368776,
https://www.academia.edu/38154263 and so on).

In this paper, generalized exponential-type estimators have been proposed for estimating the finite population mean of study variable using information on two auxiliary variables in the presence of non-response under stratified two-phase random sampling. The expressions for the bias and mean square error (MSE) of proposed estimators have been derived in two different situations of non-response. Theoretical comparisons of proposed estimators have been made with modified forms of Hansen and Hurwitz (1946), ratio and product estimators to the stratified two-phase sampling method. An empirical study has also been carried out to demonstrate the performances of proposed estimators.

Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this paper, we attempt to bridge this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. We provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements.

So far the studies dealing with the population projection are at the national level and only a few studies have shown the size of the population in the future at state level. There are Scanty studies in population projection for North Eastern states. The present projection is based on the Cohort component method. The two decades increase in population

Link to the code: https://github.com/mGalarnyk/Python_Tutorials/blob/master/Time_Series/Part1_Time_Series_Data_BasicPlotting.ipynb

Link to youtube video:
https://www.youtube.com/watch?v=OwnaUVt6VVE

Viewing Pandas DataFrame, Adding Columns in Pandas, Plotting Two Pandas Columns,  Sampling Using Pandas,  Rolling mean in Pandas (Smoothing), Subplots, Plotting against Date (numpy.datatime), Filtering DataFrame in Pandas, Simple Joins, and Linear Regression.

This tutorial is mostly focused on manipulating time series data in the Pandas Python Library.

1. Introducción al muestreo Algunas veces es posible y práctico examinar a cada persona o elemento de la población que deseamos describir. A esto lo llamamos enumeración completa o censo. Utilizamos el muestreo cuando no es posible contar o medir todos los elementos de la población. Por población nos referimos a un conjunto de elementos que han sido elegidos para un estudio,. Muestra es una porción de la población. La muestra debe satisfacer las siguientes condiciones:  Homogeneidad: debe ser extraída de la misma población.  Independencia: las observaciones no deben estar mutuamente condicionadas entre sí.  Representatividad: la muestra debe ser el mejor reflejo posible del conjunto del cual proviene. Matemáticamente, podemos describir muestras y poblaciones al emplear mediciones como la media, la mediana, la moda y la desviación estándar. Estos describen características notables de una población. Una población queda caracterizada a través de ciertos valores denominados parámetros, que describen las principales propiedades del conjunto. Un parámetro es un valor fijo (no aleatorio) que caracteriza a una población en particular. En general, una parámetro es una cantidad desconocida y rara vez se puede determinar exactamente su valor, por la dificultad práctica de observar todas las unidades de una población. Se utilizan casi siempre letras griegas o latinas mayúsculas para representar parámetros de población. Tratamos de estimar el valor de los parámetros desconocidos a través del empleo de muestras. Las cantidades usadas para describir una muestra se denominan estimadores o estadísticos muestrales. Generalmente se emplean letras latinas minúsculas para denotarles. Población Muestra Definición Colección de elementos considerados Parte o porción de la población seleccionada para su estudio Características Parámetros Estadísticos

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

The recently introduced theory of compressive sensing (CS) enables the reconstruction of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be significantly smaller than the ambient dimension of the signal and yet preserve the significant signal information. Interestingly, it can be shown that random measurement schemes provide a near-optimal encoding in terms of the required number of measurements. In this report, we explore another relatively unexplored, though often alluded to, advantage of using random matrices to acquire CS measurements. Specifically, we show that random matrices are democratic, meaning that each measurement carries roughly the same amount of signal information. We demonstrate that by slightly increasing the number of measurements, the system is robust to the loss of a small number of arbitrary measurements. In addition, we draw connections to oversampling and demonstrate stability from the loss of significantly more measurements.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts and ideas on axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Beleivability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

In this paper, we present a new family, depending on additive Weibull random variable as a generator, called the generalized additive Weibull generated-family (GAW-G) of distributions with two extra parameters. The proposed family involves several of the most famous classical distributions as well as the new generalized Weibull-G family which already accomplished by Cordeiro et al. (2015). Four special models are displayed. The expressions for the incomplete and ordinary moments, quantile, order statistics, mean deviations, Lorenz and Benferroni curves are derived. Maximum likelihood method of estimation is employed to obtain the parameter estimates of the family. The simulation study of the new models is conducted. The efficiency and importance of the new generated family is examined through real data sets.

This paper suggests a general class of exponential ratio and product type estimators for
estimating finite population mean in single phase sampling. The expression for the mean square error
(MSE) and bias of the first and second order approximation of the proposed class of estimators are given.
The properties of the proposed estimators have been analyzed for independent units under simple random
sampling without replacement (SRSWOR). It has been shown that the proposed ratio and product
exponential type estimators are more efficient than Simple random sampling, classical ratio, Bhal, Tutaja
[1] and Solanki et al. [2].

Models for economic allocation of inspection and metrology capacity to the full production and ramp phases of semiconductor manufacturing are presented. For the full production phase the excursion control framework is extended in detail and scope. Extensions include: 1) process tools are modeled explicitly as the sources of excursions, 2) multiple excursions can take place simultaneously on a tool, 3) excursion signals can propagate to downstream inspection /metrology steps, 4) preventive maintenance and end of line probe can catch excursions. Initially a simulation model is used to identify important dynamics and show new results on the connections between process tools, inspection & metrology, and material handling systems. This lays a foundation for fab planning that takes into account excursion yield losses in addition to utilization, cycle-time, and work in process.
Motivated by the limitations of the simulation model to quickly distinguish between scenarios with subtle output differences, an analytic approximation is developed. Novel formulations enable close approximations to be calculated in fractions of a second, giving rise to rapid sensitivity analysis and optimization applications. Using a genetic algorithm for optimization, the model is applied with full production phase data to optimize inspection allocation and capacity purchase decisions.
For the ramp phase a systematic framework to capture the role of inspection and metrology is presented. It is centered on the role inspection and metrology have on initiating and completing learning cycles. Based on this framework a model capturing the value of electron beam inspection in yield learning is developed. Using this model in conjunction with a genetic algorithm, the value and optimized allocation of electron beam inspection in development and ramping scenarios is calculated.

Compressive sensing (CS) exploits the sparsity present in many signal environments to reduce the number of measurements needed for digital acquisition and processing. We have previously introduced the concept and feasibility of using CS techniques to build a wideband signal acquisition systems. This paper extends that work to examine such a receiver’s performance as a function of several key design parameters. In particular we show that that the system noise figure is predictably degraded as the subsampling implicit in CS is made more aggressive. Conversely we show that the dynamic range of a CS-based system can be substantially improved as the subsampling factor grows. The ability to control these aspects of performance provides an engineer new degrees of freedom in the design of wideband acquisition systems. A specific practical example, a multi-collector emitter geolocation system, is included to illustrate that point.

In the present paper, a general class of estimators has been proposed for estimating the population mean of the study variable using auxiliary variable when the population mean of the auxiliary variable is known in stratified random sampling. The bias and mean squared error of the proposed class of estimators are derived under stratified sampling to the first degree of approximation. Members of the proposed class of estimators are also obtained by giving suitable values to the constants used. Comparisons of the proposed strategy with the usual unbiased estimator and other estimators have been made. Empirical and simulation study is carried out to demonstrate the performance of the proposed estimator.

This paper suggests the generalized class of estimators, motivated by Sharma and Singh (2015), of finite population variance utilizing the known value of parameters related to an auxiliary variable such as quartile and its properties are studied in simple random sampling without replacement. The efficiency of proposed class of estimators is compared with some existing estimators discussed in literature and found that proposed generalized class of estimators is better than other existing estimators including usual unbiased estimator, estimators due to Isaki (1983), Das and Tripathi (1978) and estimators recently proposed by Singh and Pal (2016). An empirical as well as theoretical comparison is carried out to judge the performance of proposed class of estimators over other existing estimators of population variance using natural data set. RESUMEN En este paper se sugiere una clase generalizada de estimadores motivada por los resultados de Sharma and Singh (2015), la varianza de la población finita usa el valor conocido de parámetros relacionado con una variable auxiliar, como los cuartiles, y sus propiedades son estudiadas en el caso de muestreo simple aleatorio con reemplazo. La eficiencia de la clase de estimadores propuestos es comparada con la de algunos estimadores existentes, discutidos en la literatura y se halla que las clases propuestas de estimadores es mejor que otros existentes, como los estimadores insesgados debidos a Isaki (1983), Das y Tripathi (1978) y a estimadores propuestos recientemente por Singh y Pal (2016). Comparaciones empíricas así como teóricas llevan a cabo para juzgar el comportamiento de la clase propuesta de estimadores sobre otros estimadores existentes de la varianza de la población usando conjuntos de datos reales. PALABRAS CLAVE: variable auxiliar, cuartil, muestreo simple aleatorio, sesgo, error cuadrático medio.

The existence of missing observations (MO) is commonly solved by using imputation methods. There are ratio-based methods for estimating the population mean, while using simple random sampling (SRS), when MO are present. Considering the existence of MO and using of ranked set sampling, we develop a study of the estimation of a population mean using ratio-based methods. The mean square errors, bias, and gain in accuracy formulas of the suggested estimators are derived. The suggested estimators are compared with their SRS counterpart both theoretically and numerically. Copyright © 2014 John Wiley & Sons, Ltd.

The recently introduced theory of compressed sensing enables the reconstruction of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist rate samples. Interestingly, it has been shown that random projections are a satisfactory measurement scheme. This has inspired the design of physical systems that directly implement similar measurement schemes. However, despite the intense focus on the reconstruction of signals, many (if not most) signal processing problems do not require a full reconstruction of the signal—we are often interested only in solving some sort of detection problem or in the estimation of some function of the data. In this report, we show that the compressed sensing framework is useful for a wide range of statistical inference tasks. In particular, we demonstrate how to solve a variety of signal detection and estimation problems given the measurements without ever reconstructing the signals themselves. We provide theoretical bounds along with experimental results.