R] Inferential Statistics IV: Uniform and (Standard) Normal Probability (Density) Functions

In recent decades, analysis of structures considering variability of some parameters for more reliable design has attracted the attention of researchers. In this paper, the stochastic analysis of a cantilever deep beam made of large deformable neo-hookean material is carried out. For this purpose, the meshless local Petrov-Galerkin (MLPG) method is developed to obtain the geometrically non-linear equilibrium equations. The radial point interpolation method is used for generating the shape functions. The incremental iterative Newton-Raphson method with suitable load steps is used to solve the non-linear governing equations. The results of deterministic analysis obtained with proposed method are compared with the finite element results and good agreement is achieved. The initial elasticity modulus of neo-hookean material is considered to be uncertain variable. To generate random field of uncertain variable with normal, lognormal and uniform probability density functions (PDFs), the Monte Carlo Simulation (MCS) technique was employed. The sufficient number of simulations for convergence the results was determined experimentally. The effect of elasticity modulus, PDF and coefficients of variation (COV) on maximum vertical displacement, PDF and COV of results are studied in details. Comparing the stochastic and deterministic results shows that the uncertainty in mechanical properties has significant effect on results.

The problem of statistical pattern recognition with noisy or imprecise feature measurements is considered. An exact analytical expression is found for the probability of misclassification under this condition, for multiclass multivariate systems. The probability of error exceeds that of the ideal case for the special case of two classes, the a priori conditional probability density functions are assumed to be normal, along with the two cases of feature measurement error, namely normal and uniform probability density functions. Monotonicity of the misclassification probability with measurement error variance is shown. Numerical results are presented for both cases over a workable range of parameters. The study is useful in practical pattern recognition problems.

11 - Are my results significant?

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$, which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

PurposeThe K-means (KM) clustering algorithm is extremely responsive to the selection of initial centroids since the initial centroid of clusters determines computational effectiveness, efficiency and local optima issues. Numerous initialization strategies are to overcome these problems through the random and deterministic selection of initial centroids. The random initialization strategy suffers from local optimization issues with the worst clustering performance, while the deterministic initialization strategy achieves high computational cost. Big data clustering aims to reduce computation costs and improve cluster efficiency. The objective of this study is to achieve a better initial centroid for big data clustering on business management data without using random and deterministic initialization that avoids local optima and improves clustering efficiency with effectiveness in terms of cluster quality, computation cost, data comparisons and iterations on a single machine.Design/methodology/approachThis study presents the Normal Distribution Probability Density (NDPD) algorithm for big data clustering on a single machine to solve business management-related clustering issues. The NDPDKM algorithm resolves the KM clustering problem by probability density of each data point. The NDPDKM algorithm first identifies the most probable density data points by using the mean and standard deviation of the datasets through normal probability density. Thereafter, the NDPDKM determines K initial centroid by using sorting and linear systematic sampling heuristics.FindingsThe performance of the proposed algorithm is compared with KM, KM++, Var-Part, Murat-KM, Mean-KM and Sort-KM algorithms through Davies Bouldin score, Silhouette coefficient, SD Validity, S_Dbw Validity, Number of Iterations and CPU time validation indices on eight real business datasets. The experimental evaluation demonstrates that the NDPDKM algorithm reduces iterations, local optima, computing costs, and improves cluster performance, effectiveness, efficiency with stable convergence as compared to other algorithms. The NDPDKM algorithm minimizes the average computing time up to 34.83%, 90.28%, 71.83%, 92.67%, 69.53% and 76.03%, and reduces the average iterations up to 40.32%, 44.06%, 32.02%, 62.78%, 19.07% and 36.74% with reference to KM, KM++, Var-Part, Murat-KM, Mean-KM and Sort-KM algorithms.Originality/valueThe KM algorithm is the most widely used partitional clustering approach in data mining techniques that extract hidden knowledge, patterns and trends for decision-making strategies in business data. Business analytics is one of the applications of big data clustering where KM clustering is useful for the various subcategories of business analytics such as customer segmentation analysis, employee salary and performance analysis, document searching, delivery optimization, discount and offer analysis, chaplain management, manufacturing analysis, productivity analysis, specialized employee and investor searching and other decision-making strategies in business.

Dithering improves the linearity of A/D converters (ADCs). Wagdy and Goff (1994) have developed a criterion for assessing the deviation of ADC transfer characteristics from the unity-gain line. This criterion is based on a closed-form formula for the deviation factor D derived from the characteristic function of the ADC quantization error function. This approach was extended further by Wagdy (1996) to include nonideal ADCs. The results in that work pertain to additive dither, but a closed-form formula for subtractive dither may be very difficult to derive. In this paper, the ADC is modeled and dither is simulated using a software random number generator. First, additive dithering with continuous (analog) uniform (rectangular) probability density function (RPDF) is considered. The cases of an ADC with no errors, single-bit error, or multi-bit errors are investigated. Results are in excellent agreement with the previous ones, which validates the formulae therein. Second, the author presents new results using discrete (digital) rectangular dither (DRPDF). This dither type is generated via a D/A converter (DAC) whose number of bits, k, is greater by del than the ADC number of bits, m. For various values of del, subtractively-dithered ADCs outperform additively-dithered ones as far as D is concerned. Also, the effect of increasing del on D for an ADC is investigated for the subtractive dither case, thus providing some quantitative results which are useful for the designer.

Objective: To analyze the adjustments of the weibull, gamma, normal and logistic probability density distributions of the historical series of hospitalizations for respiratory diseases (childhood and adult pneumonia) from 2011 to 2015, in Campo Grande, MS. Methods: The shape and scale parameters of the distributions were determined to verify the quality of the data fit. Results: Four probability density functions (Table 2) were fitted and the R2, MAE, RSME, MAPE tests were used to verify the best density function for hospitalization data. Conclusion: The best fit was the Gamma distribution; the distribution can be used as an alternative distribution that adequately describes the data on hospital admissions for respiratory diseases in Campo Grande.

This work newly proposes an uncertainty quantification (UQ) method named sparse approximation of moment-based arbitrary polynomial chaos (SAMBA PC) that offers a single solution to many current problems in turbomachinery applications. At the moment, every specific case is characterized by a variety of different input types such as histograms (from experimental data), normal probability density functions (PDFs) (design rules) or fat tailed PDFs (for rare events). Thus, the application of UQ requires the adaptation of ad hoc methods for each individual case. A second problem is that parametric PDFs have to be determined for all inputs. This is difficult if only few samples are available. In gas turbines, however, the collection of statistical information is difficult, expensive, and having scarce information is the norm. A third critical limitation is that if using nonintrusive polynomial chaos (NIPC) methods, the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called “curse of dimensionality.” It is shown that the fitting of parametric PDFs to small data sets can lead to large bias and the direct use of the available data is more accurate. This is done by propagating uncertainty through several test functions and the computational fluid dynamics (CFD) simulation of a diffuser, highlighting the impact of different PDF fittings on the output. From the results, it is concluded that the direct propagation of the experimental data set is preferable to the fit of parametric distributions if data is scarce. Thus, the suggested method offers an alternative to the maximum entropy theorem to handle scarce data. SAMBA simplifies the mathematical procedure for many different input types by basing the polynomial expansion on moments. Its moment-based expansion automatically takes care of arbitrary combinations of different input data. It is also numerically efficient compared to other UQ implementations. The relationship between the number of random variables and number of simulation is linear (only 21 simulations for ten input random variables are required). It is shown in this paper that SAMBA's algorithm can propagate a high number of input distributions through a set of nonlinear analytic test functions. Doing this, the code needs a very small number of simulations and preserve a 5% error margin. SAMBA's flexibility to handle different forms of input distributions and a high number of input variables is shown on a low-pressure turbine (LPT) blade-based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

In many practical periodic parameter estimation problems, the appropriate cost function is periodic with respect to the unknown parameter. In this paper a new class of cyclic Bayesian lower bounds on the mean cyclic error (MCE) is developed. The new class includes the cyclic version of the Bayesian Cramér-Rao bound (BCRB). The cyclic BCRB requires milder regularity conditions compared to the conventional BCRB. The tightest bound in the proposed class is derived and it is shown that under a certain condition it achieves the minimum MCE (MMCE). The new lower bounds are compared with the cyclic version of the Ziv-Zakai lower bound (ZZLB) and the MCE's of the MMCE and maximum aposteriori probability (MAP) estimators for frequency estimation with uniform a-priori probability density function (pdf) of the unknown parameter. In this common estimation problem, the conventional BCRB does not exist, while the proposed cyclic BCRB provides a valid lower bound for parameter estimation.

In 1967, Bühlmann has shown that the credibility formula was the best linearized approximation to the exact Bayesian forecast.His result for the credibility factor can be found back by means of some Bayesian inference techniques. Introducing a uniform prior probability density function for the credibility factor provides us with a method for estimating z, a correction term to the Bühlmann's result is obtained. It is shown how prior boundary conditions can be introduced.

PoCET: a Polynomial Chaos Expansion Toolbox for Matlab

The classical intermittency theory developed forty years ago considers a uniform reinjection probability density function (RPD). However, in the last ten years, studies have found a more general RPD, which depends on the type of prereinjection points, extreme or infinite derivative points. Recently, we have introduced an analytical scheme to estimate the non-uniform RPD when the pre-reinjection points are extreme. Here, this theoretical scheme is extended to embrace pre-reinjection points with infinite derivative. The proposed theoretical method, to get the RPD, needs only the explicit expression of the map, and it is general and direct to use. For different intermittency types and different non-linearity, the theoretical predictions show a very high agreement with the numerical results.

In this study, we investigate the impact of pore distribution and size on the mechanical response of elastic porous materials. Two-dimensional porous media with convex porosity are considered (e.g., foams and sponges). These structures are numerically generated by subtracting randomly dispersed circular holes from a solid square domain. The in-plane position vector of pore centers is given by a uniform probability density function (PDF). The pore diameters are, instead, drawn from a non-uniform two-parameter continuous probability distribution. The influence of the void ratio, i.e., the volume fraction of voids, and the parameters of the diameter PDF are studied over a wide range of values, including the percolation threshold, where the effective elastic modulus of the material abruptly drops to zero. This transition takes place at a critical value of the void ratio, which turns out to be unaffected by the parameters of the pore size distribution. Our results demonstrate that the elastic response of these structures is primarily governed by the void ratio, with the correlation length playing a secondary role. While the pore size distribution may affect the correlation length, it has a minimal impact on the foam’s overall behavior.

The Discrete Moment Method for the numerical integration of piecewise higher order convex functions

Existing methods of hydrologic network design are reviewed and a formulation based on Shannon’s information theory is presented. This type of formulation involves the computation of joint entropy terms which can be computed by discretizing hydrologic time series data collected at station locations. The computation of discrete entropy terms is straightforward but in handling large numbers of stations enormous computation time and storage is required. In order to minimize these problems, bivariate and multivariate continuous distributions are used to derive entropy terms. The information transmission at bivariate level is derived for normal, lognormal, gamma, exponential, and extreme value distributions. At the multivariate level, multivariate form of normal and lognormal probability density functions are used.In order to illustrate the applicability of the derived information relationship for various bivariate and multivariate probability distributions, daily precipitation data for a period of two years collected at some selected locations in the lower Mainland Region of British Columbia was used in this study. Information transmission between station pairs was calculated for different cases of probability distributions and based on information maximization principles and the optimum locations of the stations to be retained were identified.