Verification of accuracy of mean value estimation using Monte Carlo method

Przedmiotem badań jest estymator wartości oczekiwanej. Sprawdzano dokładność estymacji wartości oczekiwanej w sytuacji, gdy estymator obliczany jest na podstawie danych z rozkładu Gaussa. Sprawdzanie dokładności estymacji wykonano z zastosowaniem metody Monte Carlo.The subject of the research is the mean value estimator. The estimator is determined based on data obtained from a Gaussian distribution. The accuracy of the mean value estimator was examined using the Monte Carlo method. Chapter 1 provides basic information on the reasons for use the Monte Carlo method. In Chapter 2 the basic definitions were presented. Eq. (1) describes the expected value of the random variable. Eq. (3) presents the mean value estimator. Eq. (4) it is the error of the estimator (3). In the next part of Chapter 2 the mean value estimator for Gaussian distribution was presented (eq. 6). Eq. (7) describes the error of the mean value estimator (6). Next equation describes coverage factor for Gaussian distribution. In the next part of the paper the Monte Carlo methods were presented. In this article the Crude and Hit-or-Miss Monte Carlo methods have been used. Eq. (13) presents the mean value estimator obtained using the Crude Monte Carlo method. Eq. (14) describes the error of the estimator. Eq. (15) presents the mean value estimator obtained using the Hit-or-Miss Monte Carlo method. Eq. (16) it is the error of the estimator. In Fig. 1 the errors (4), (14) and (16) have been shown. Tab. 1 presents the errors obtained in Matlab, MatCAD and LabWINDOWS. The researches have been summarized in Chapter 3

Computer analysis of bias of the mean square value estimator for sinusoidal signal

W referacie przedstawiono przykłady wspomaganych komputerowo obliczeń obciążenia estymatorów wartości średniokwadratowej otrzymanych metodą bezpośrednią i na podstawie widma amplitudowego.This paper presents computer calculations of bias of the mean square value estimators calculated by the direct method and on the basis of the amplitude spectrum

Computer calculation of uncertainty of measurement

Opracowany został program komputerowy wspomagający obliczanie niepewności pomiaru z zastosowaniem współczynników rozszerzania wyznaczanych na podstawie splotów rozkładów prawdopodobieństwa. W artykule przedstawiono przykłady obliczeń niepewności pomiaru różnych wielkości elektrycznych i nieelektyrycznych, wykonanych metodami bezpośrednimi i pośrednimi. Szczególną uwagę poświęcono opracowaniu i optymalizacji algorytmów umożliwiających szybkie i dokładne obliczanie współczynników rozszerzenia.A computer program has been designed which within an acceptable time computes measurement uncertainty with the use of coverage factors determined on the basis of selected convolutions of probability distributions. In the article has been presented examoles of the computing the uncertainty of real measurements of various electrical and non-electrical quantities carried out by direct and indirect method. Particular attention has been devoted to the designing and optimization of algorithms enabling fast and accurate computation of coverage factors

Evaluation of quantization influence on the signal mean value estimator uncertainty

Artykuł dotyczy problematyki oceny wpływu kwantowania na niepewność estymatora wartości oczekiwanej sygnału. Zdefiniowano postacie estymatorów wartości oczekiwanej oraz wariancji tego parametru. Wyznaczono obciążenia estymatorów. Oceniono wpływ kwantowania na niepewność estymatora wartości oczekiwanej. Do badań zastosowano skwantowane próbki sygnału oraz momenty zmiennej losowej. Konwersja sygnału przeprowadzono z zastosowaniem kwantyzatora typu zaokrąglającego o idealnej charakterystyce kwantowania.The paper deals with the problem of evaluation of quantization influence on the signal mean value estimator uncertainty on the basis of digital measuring data. In order to evaluate the uncertainty ,there have been used the quantized samples and moments of a random variable as well as the Widrow theory of quantization. The round-off quantizer of the ideal quantizing characteristic has been applied. The paper is divided into four sections. In the first section there is given Eq. (2) describing the mean value estimator obtained from the quantized data. In the second section the bias of the mean value estimator is described by Eq. (5) and shown in Fig.1. The mean value estimator (2) with and without bias (5) is shown in Fig.2. The mean value estimator variance is given by Eq. (6) and shown in Fig.3. In the next section there are presented Eqs. (21)-(23) describing the quantization influence on the mean value estimator uncertainty obtained from the moments and quantized data. The quantization influence on the mean value estimator uncertainty is studied in two independent cases, with and without bias, and shown in Fig.6. It has been shown that for a sinusoidal signal Eq. (21) is a suppressed oscillating function of the amplitude. Moreover, it has been proved that by increasing the sample size Eqs. (22) and (23) can be brought to 1. In the last section the results of investigations are summarized

Estimation of Random Variable Distribution Parameters by the Monte Carlo Method

The paper is concerned with issues of the estimation of random variable distribution parameters by the Monte Carlo method. Such quantities can correspond to statistical parameters computed based on the data obtained in typical measurement situations. The subject of the research is the mean, the mean square and the variance of random variables with uniform, Gaussian, Student, Simpson, trapezoidal, exponential, gamma and arcsine distributions

Generalized model of quantization error in measurement of signal RMS value

Z zastosowaniem teorii kwantowania Widrowa opracowano wynikający z kwantowania model błędu w pomiarze wartości skutecznej sygnałów. Na podstawie opracowanego modelu zbadano wpływ kwantowania na dokładność estymacji wartości skutecznej. W artykule przedstawiono wyniki analiz wpływu kwantowania na dokładność estymacji wartości skutecznej sygnału sinusoidalnego, sygnałów losowych o rozkładach gaussowskim, równomiernym i trójkątnym oraz wybrane kombinacje tych sygnałów.The model of the bias of the root mean square (RMS) value estimator was worked out with applying the Widrow theory of quantization. The influence of quantizing on the accuracy of the RMS value estimator was studied on the basis of this model. The main subjects of the research were: sinusoidal signal, Gaussian signal, uniformly distributed signal, triangular probability density function (PDF) signal and selected combinations of the studied signals. In the first paragraph Eq. (6) and (7) describing the RMS value estimator bias are presented. The bias of the RMS value estimator is given by Eq. (6). The normalized bias is given by Eq. (7). In the next paragraph the Eq. of the PDFs and the characteristic functions for deterministic and random signals are given. In the second para-graph the process of bias estimation in the RMS value measurements caused by quantization is described. The normalized biases of the RMS value estimator of selected random signals are given by Eq. (10), (14), (18) and shown in Fig. 1. The normalized biases of the RMS value estimators of sinusoidal signal with random signals are given by (21) and shown in Fig. 2

Estimation of random variable distribution parameters by the monte carlo method

The paper is concerned with issues of the estimation of random variable distribution parameters by the Monte Carlo method. Such quantities can correspond to statistical parameters computed based on the data obtained in typical measurement situations. The subject of the research is the mean, the mean square and the variance of random variables with uniform, Gaussian, Student, Simpson, trapezoidal, exponential, gamma and arcsine distributions

Automatization of multimeter and calibrator calibration

Artykuł dotyczy problematyki wzorcowania multimetrów i kalibratorów. Przedstawiono koncepcję aplikacji pomiarowych do automatyzacji procesu wzorcowania przyrządów pomiarowych, prowadzącej do usprawnienia pracy laboratoriów wzorcujących. Pokazano ogólną strukturę systemu pomiarowego, schemat działania aplikacji pomiarowych oraz omówiono poszczególne bloki funkcjonalne programów. W pracy zaprezentowano przykładową aplikację do wzorcowania multimetru Keithley 2002Manual calibration of multimeters and calibrators takes mostly many hours. A faster process needs automatization. In the paper an idea of measurement application for automatization of calibration process is presented. This application allows making work of calibration laboratories more effective. A general structure of the measurement system (Fig. 1) and the measurement application working diagram (Fig. 2) are shown. An example application for calibration of the Keithley 2002 multimeter is given. The application allows making calibration automatically and step by step (Fig. 3) based on the procedure written on a worksheet. It is also possible to edit the report of calibration (Fig. 4), write the report in a doc file (Fig. 5) and write down all calibrations results in a text file (Fig. 6). The concept of automatization of multimeter and calibrator calibration is universal. The elaborated application in LabView can be modified for calibration of other measurement devices. The main advantages of the application are automatization of calibration, calculation of the measurement uncertainty, reporting of calibration and easy operation

Point estimation of the signal autocorrelation function basing on digital measurement data

Artykuł przedstawia problematykę obliczania wartości oczekiwanej, obciążenia i wariancji cyfrowego estymatora funkcji autokorelacji sygnałów. Pokazano, że estymator funkcji autokorelacji nie jest zgodny oraz, że jest obciążony dodatkową, wynikającą z kwantowania składową. Pokazano, że funkcja gęstości kompensuje przesunięcie funkcji autokorelacji, co oznacza, że określenie na postawie momentów obciążenia i wariancji estymatora możliwe jest jedynie w tych punktach funkcji autokorelacji, które odpowiadają wartości średniokwadratowej sygnału. Przedstawiono wyniki oszacowań obciążenia i wariancji cyfrowego estymatora funkcji autokorelacji dla wybranych klas sygnałów. Do obliczeń zastosowano opracowany na potrzeby prowadzonych badań wielobitowy wirtualny korelator sygnałów.In the paper there are discussed problems of estimation of the expected value, bias and variance of the digital estimator of the signal autocorrelation function. It is shown that the autocorrelation function estimator is not consistent and that the density function compensates the autocorrelation function delay. It means that determination of the bias and variance of the estimator basing on the so-called moments is possible only in these points of the autocorrelation function which are the mean square value of the signal. There are presented the results of estimation of the bias and variance of the autocorrelation function digital estimator for selected classes of signals. In order to perform calculations, there was designed a dedicated, multi-bit, virtual correlator of signals. The paper is divided into 3 sections. Section 1 contains a short introduction to the issues of this paper. In Section 2 there are presented the definitions of the autocorrelation function and the autocorrelation function estimator of a signal and quantized signal - Eqs. (2-4). Next, there is calculated the estimator's expected value - Eqs. (5, 6). There is determined the bias of the autocorrelation function digital estimator caused by quantization Eq. (7). In the next part of paper there is shown that the signal distribution density function compensates the autocorrelation function delay - Eq. (11). There is also calculated the estimator's mean square error - Eq. (20). The mean square error and variance from Eq. (17) allows evaluating the estimator consistency. Table 1 presents the results of analysis of the bias and variance of the autocorrelation function digital estimator for a sinusoidal signal with noise. There are analysed the following types of noise: Gaussian, uniform probability density function (PDF) and triangular PDF signal. In Section 3 the investigation results are summarized. The obtained results show the importance of investigations on autocorrelation function degradation caused by quantization

Simple, fast and accurate four-point estimators of sinusoidal signal frequency

In this paper, two new sinusoidal signal frequency estimators calculated on the basis of four equally spaced signal samples are presented. These estimators are called four-point estimators. Simulation and experimental research consisting in signal frequency estimation using the invented estimators have been carried out. Simulation has also been performed for frequency tracking. The simulation research was carried out applying the MathCAD computer program that determined samples of a sinusoidal signal disturbed by Gaussian noise. In the experimental research, sinusoidal signal samples were obtained by means of a National Instruments PCI-6024E data acquisition card and an Agilent 33220A function generator. On the basis of the collected samples, the values of four-point estimators invented by the authors and, for comparison, the values of three- and four-point estimators proposed by Vizireanu were determined. Next, estimation errors of the signal frequency were determined. It has been shown that the invented estimators can estimate a signal frequency with greater accuracy