Single Normal Distribution Research Articles

An approach to improve the predictive power of choice-based conjoint analysis

Conjoint analysis continues to be popular with over 18,000 applications each year. Choice-based conjoint (CBC) analysis is currently the most often used method of conjoint analysis accounting for eight-tenths of all conjoint studies. The CBC employs a multinomial logit model with heterogeneous parameters across the population. The most commonly used models of heterogeneity are the Latent Class Model, the single multivariate normal distribution, or a mixture of multivariate normal distributions. A more recent approach to capture heterogeneity is the Dirichlet Process Mixture (DPM) model and its predecessor Dirichlet Process Prior (DPP) model. The alternative models are empirically tested over eleven CBC data sets with varying characteristics. The DPM model provides the best predictive validity (percent of choices correctly predicted) for each of the eleven datasets studied, and provides a significant improvement over extant models of heterogeneity.

Lead time uncertainty and supply chain coordination in lost sales inventory models

The effects of lead time uncertainty on optimal inventory policies in a two-echelon supply chain operating under continuous review inventory assumptions are examined. In this system, the buyer selects an inventory order quantity and reorder point to maximise its profits, while the seller selects an order quantity from its external supplier that is an integer multiple of the buyer's order quantity; unsatisfied demand is accounted for as lost sales and may also incur a penalty cost representing lost customer goodwill. Normally distributed demand per unit time is assumed and lead time can follow any discrete probability distribution, resulting in a mixture of Gaussians distribution for lead time demand. Examples where lead time is discrete and not well-approximated by a continuous distribution are examined, and use of the mixture of Gaussians approach versus a single normal distribution for lead time demand in these cases reduces supply chain costs by between 12% and 13%.

Strategi Penawaran dalam Industri Konstruksi (Studi Kasus di Layanan Pengadaan Secara Elektronik, Kota Bandung)

The aims of this research are to calculate the value of mark-up using bidding strategy approach and to determine the best bidding strategy to win a tender with the optimum mark-up value and the optimum profit. The data being used in this research are auction data that have been completed in 2010-2012 at LPSE Bandung using statistical approach namely discrete multi distribution, normal multi distribution, and single normal distribution. Bidding strategy methods being used are Friedman Method, Gates Method, and Ackoff Sasieni Method. The Friedman Method can generate an optimum mark-up of 10% with expected profit of -0.0014, an optimum mark-up of -9% with expected profit of -0.0003, and an optimum mark-up of -1% with expected profit -0.0004 for discrete multi-distribution, multi-normal distribution, and single normal distribution, respectively. The Gates Methods can produce an optimum mark up of 6% with expected profit of 1,5000, an optimum mark up of 5% with expected profit 0.0097, and an optimum mark up of 6% with expected profit 1.2888 for discrete multi distribution, multi-normal distribution, and single normal distribution, respectively. While the Ackoff Sasieni Methods can generates an optimum mark-up of -2% with expected profit -0.0003, an optimum mark-up of 1% with expected profit for 0,0013 for discrete multi distribution, and for both multi and single normal distributions, respectively.

Probability Model of Hangzhou Bay Bridge Vehicle Loads Using Weigh-in-Motion Data

To study the vehicle load characteristics of bay bridges in China, especially truck loads, we performed a statistical analysis of the vehicle loads on Hangzhou Bay Bridge using more than 3 months of weigh-in-motion data from the site. The results showed that when all the vehicle samples were included in the statistical analysis, the histogram of the vehicles exhibited a multimodal distribution, which could not be fitted successfully by a familiar single probability distribution model. When the truck samples were analyzed, a characteristic multiple-peaked distribution with a main peak was obtained. The probability distribution of all vehicles was fitted using a weighting function with five normal distributions and the truck loads were modeled by a single normal distribution. The results demonstrated the good fits with the histogram. The histograms of different time periods were also analyzed. The results showed that the traffic mainly comprised two-axle small vehicles during the rush hours in the morning and the evening, and the histogram could be fitted approximately using three normal distribution functions. And the maximum value distributions of vehicles during the design life of the bay bridge were predicted by maximum value theory.

Experimental Validation of Robust Process Design and Control Based on Gaussian Mixture Densities

In this contribution, the effects of different degrees of uncertainty description are investigated experimentally using an exothermic chemical reaction with safety constraint on the temperature. For that purpose, two robust trajectories are designed that respect the artificially created uncertainties of the experiments either coarsely using a single multivariate normal distribution (1GMD) or in a more detailed fashion using a Gaussian mixture density (GMD) consisting of 32 multivariate normal densities (32GMD). For the optimization, the uncertainties are propagated using the unscented transformation. Both trajectories were run 71 times in an open-loop manner. The more detailed trajectory (32GMD) leads to a 9 % higher yield without increasing the risk of constraint violation. Furthermore, many experimental realizations of two robust closed-loop process control schemes are being compared. They differ again only in the degree of the underlying uncertainty description. Although the frequent corrections of the controller marginalize the advantage of a more detailed stochastic process prediction, the 4GMD-controller still allows for 3 % more educt conversion compared to the 1GMD-controller.

Addressing extrema and censoring in pollutant and exposure data using mixture of normal distributions

BackgroundVolatile organic compounds (VOC), which include many hazardous chemicals, have been used extensively in personal, commercial and industrial products. Due to the variation in source emissions, differences in the settings and environmental conditions where exposures occur, and measurement issues, distributions of VOC concentrations can have multiple modes, heavy tails, and significant portions of data below the method detection limit (MDL). These issues challenge standard parametric distribution models needed to estimate the exposures, even after log transformation of the data. MethodsThis paper considers mixture of distributions that can be directly applied to concentration and exposure data. Two types of mixture distributions are considered: the traditional finite mixture of normal distributions, and a semi-parametric Dirichlet process mixture (DPM) of normal distributions. Both methods are implemented for a sample data set obtained from the Relationship between Indoor, Outdoor and Personal Air (RIOPA) study. Performance is assessed based on goodness-of-fit criteria that compare the closeness of the density estimates with the empirical density based on data. The goodness-of-fit for the proposed density estimation methods are evaluated by a comprehensive simulation study. ResultsThe finite mixture of normals and DPM of normals have superior performance when compared to the single normal distribution fitted to log-transformed exposure data. The advantages of using these mixture distributions are more pronounced when exposure data have heavy tails or a large fraction of data below the MDL. Distributions from the DPM provided slightly better fits than the finite mixture of normals. Additionally, the DPM method avoids certain convergence issues associated with the finite mixture of normals, and adaptively selects the number of components. ConclusionsCompared to the finite mixture of normals, DPM of normals has advantages by characterizing uncertainty around the number of components, and by providing a formal assessment of uncertainty for all model parameters through the posterior distribution. The method adapts to a spectrum of departures from standard model assumptions and provides robust estimates of the exposure density even under censoring due to MDL.

Statistical properties of variability of the quiet ionosphere F2-layer maximum parameters over Irkutsk under low solar activity

The statistical analysis of the quiet ionosphere F2-layer maximum parameters variability (deviations of NmF2 and hmF2 from the quiet medians, δn and Δh) under solar minimum at day (10–16 LT) and night (22–04 LT) hours based on data of Irkutsk station for 2007–2010 is presented. It is found that the experimental distribution (histogram) of δn can be approximated by a mixture of two normal distributions. The first and second components of the mixture characterize, mainly, relatively weak and strong fluctuations of δn which are presumably associated with the ionospheric effects of the atmospheric gravity waves and of the planetary waves and tides correspondingly. Deviation of the δn histogram from a single normal distribution is most considerable at night hours in winter and equinoxes. For these conditions the weak fluctuations of δn are mainly negative and the strong ones are mainly positive. The Δh histogram is a normal distribution except day hours in winter and equinoxes when a weak deviation of the histogram from the normal distribution occurs.

Assessment of potential energy saving using cluster analysis: A case study of lighting systems in buildings

The promulgation of a building energy code needs an accurate estimation of potential energy savings to ensure that its implementation can achieve the goal set. Generally, the estimation is carried out using the mean value averaged from sample buildings as a representative for the whole sector. However, the actual data of building energy performance may not be described by a single normal distribution. In this study, cluster analysis is thus introduced to estimate potential energy savings in lighting systems in buildings, in comparison with the general averaging approach. The study covers analyses of both simulated and actual data. The simulated data of lighting power density (LPD) is generated from Gaussian distributions for 36 cases with different means, variances, and mixing proportions for the investigation. For actual data analysis, LPD values of commercial buildings are extracted from an energy audit database. It is found that the clustering technique yields more accurate energy saving estimation of the actual values (0–11% error) than that computed by the general approach (1–100% error). The proposed clustering method can therefore be used to estimate the potential energy savings for lighting systems in buildings with high accuracy.

Mixture Modelling as an Exploratory Framework for Genotype–Trait Associations

We propose a mixture modelling framework for both identifying and exploring the nature of genotype-trait associations. This framework extends the classical mixed effects modelling approach for this setting by incorporating a Gaussian mixture distribution for random genotype effects. The primary advantages of this paradigm over existing approaches include that the mixture modelling framework addresses the degrees-of-freedom challenge that is inherent in application of the usual fixed effects analysis of covariance, relaxes the restrictive single normal distribution assumption of the classical mixed effects models and offers an exploratory framework for discovery of underlying structure across multiple genetic loci. An application to data arising from a study of antiretroviral-associated dyslipidaemia in human immunodeficiency virus infection is presented. Extensive simulations studies are also implemented to investigate the performance of this approach.

A distribution-free convolution model for background correction of oligonucleotide microarray data

IntroductionAffymetrix GeneChip® high-density oligonucleotide arrays are widely used in biological and medical research because of production reproducibility, which facilitates the comparison of results between experiment runs. In order to obtain high-level classification and cluster analysis that can be trusted, it is important to perform various pre-processing steps on the probe-level data to control for variability in sample processing and array hybridization. Many proposed preprocessing methods are parametric, in that they assume that the background noise generated by microarray data is a random sample from a statistical distribution, typically a normal distribution. The quality of the final results depends on the validity of such assumptions.ResultsWe propose a Distribution Free Convolution Model (DFCM) to circumvent observed deficiencies in meeting and validating distribution assumptions of parametric methods. Knowledge of array structure and the biological function of the probes indicate that the intensities of mismatched (MM) probes that correspond to the smallest perfect match (PM) intensities can be used to estimate the background noise. Specifically, we obtain the smallest q2 percent of the MM intensities that are associated with the lowest q1 percent PM intensities, and use these intensities to estimate background.ConclusionUsing the Affymetrix Latin Square spike-in experiments, we show that the background noise generated by microarray experiments typically is not well modeled by a single overall normal distribution. We further show that the signal is not exponentially distributed, as is also commonly assumed. Therefore, DFCM has better sensitivity and specificity, as measured by ROC curves and area under the curve (AUC) than MAS 5.0, RMA, RMA with no background correction (RMA-noBG), GCRMA, PLIER, and dChip (MBEI) for preprocessing of Affymetrix microarray data. These results hold for two spike-in data sets and one real data set that were analyzed. Comparisons with other methods on two spike-in data sets and one real data set show that our nonparametric methods are a superior alternative for background correction of Affymetrix data.

A distributional comparison of size-based portfolios on the JSE

ABSTRACTThis paper examines the non-normality of two smaller capitalization and two larger capitalization portfolios, under different weighting schemes. It demonstrates the superior fit from using two-component mixture of normal distributions instead of a single normal distribution. Additionally, the paper analyzes the components of the mixtures in order to contrast the smaller and larger capitalization portfolios. In doing so, it is shown that the portfolios behave similarly during periods of low volatility, but quite differently during periods of high volatility.

Frequency distribution of sward height of kikuyu (Pennisetum clandestinum) grass pastures intensively grazed by dairy cows

Swards that are grazed continuously by cattle can become heterogeneous in structure, such that frequencies of height measurements have a skewed distribution. This structure has been best described by the double-normal, gamma and Weibull distributions rather than by the more common single-normal distribution. Limited analysis of the frequency distribution of sward height under dairy cattle grazing management is available. The current paper reports the frequency distribution of sward height under intensive dairy cow grazing of kikuyu (Pennisetum clandestinum) in a subtropical environment. Frequencies of sward height were collected during an experiment designed to investigate the effects of compressed sward height and grazing duration on grazing behaviour of dairy cows. There were two compressed sward heights (10 and 13 cm; Earle and McGowan 1979) and five grazing durations (1, 2, 4, 8 and 15 h). The frequency of height measurements taken were then fitted to four distributions – single-normal, log-normal, gamma and Weibull – to determine if treatments influenced the structure of the sward as grazing duration progressed. The frequency of sward height measurement was best described by the log-normal distribution. However, no one distribution described the frequency of height measurements consistently as grazing duration progressed, as determined by Akaike information criteria. The present analysis did not provide unequivocal evidence for either bimodal or unimodal distributions describing the height measurements for 10- and 13-cm compressed sward height treatments as grazing duration progressed.

The Turbulent Warm Ionized Medium: Emission Measure Distribution and MHD Simulations

We present an analysis of the distribution of H-alpha emission measures for the warm ionized medium (WIM) of the Galaxy using data from the Wisconsin H-Alpha Mapper (WHAM) Northern Sky Survey. Our sample is restricted to Galactic latitudes |b| > 10. We removed sightlines intersecting nineteen high-latititude classical H II regions, leaving only sightlines that sample the diffuse WIM. The distribution of EM sin |b| for the full sample is poorly characterized by a single normal distribution, but is extraordinarily well fit by a lognormal distribution, with = 0.146 +/- 0.001 and standard deviation 0.190 +/- 0.001. drops from 0.260 +/- 0.002 at Galactic latitude 10<|b|<30 to 0.038 +/- 0.002 at Galactic latitude 60<|b|<90. The distribution may widen slightly at low Galactic latitude. We compare the observed EM distribution function to the predictions of three-dimensional magnetohydrodynamic simulations of isothermal turbulence within a non-stratified interstellar medium. We find that the distribution of EM sin |b| is well described by models of mildy supersonic turbulence with a sonic Mach number of ~1.4-2.4. The distribution is weakly sensitive to the magnetic field strength. The model also successfully predicts the distribution of dispersion measures of pulsars and H-alpha line profiles. In the best fitting model, the turbulent WIM occupies a vertical path length of 400-500 pc within the 1.0-1.8 kpc scale height of the layer. The WIM gas has a lognormal distribution of densities with a most probable electron density n_{pk} = 0.03 cm^{-3}. We also discuss the implications of these results for interpreting the filling factor, the power requirement, and the magnetic field of the WIM.

Identification of age‐structured models: Cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.

SBS/Pani · DBSA mixture plasticized with DOP and NCLS – Effect of the plasticizers on the probability density of volume resistivity measurements

Plasticized mixtures of styrene–butadiene–styrene block copolymer (SBS) and Polyaniline (Pani) were prepared in a Haake internal mixer. Two different plasticizers were used: dioctyl phthalate (DOP) and cashew nut shell liquid (CNSL). Pani and plasticizers were characterized by FTIR and the resistive behavior of plasticized mixtures was investigated along the electrification time. It is shown that obtained experimental data are subject to deterministic dynamic fluctuations that cannot be described by single normal probability distribution functions (PDF) along the time. The PDF analysis shows that obtained PDFs must be described as sums of at least three distinct Gaussian distributions with different areas. It is also shown that the Gaussian component with larger area may provide better representation of the measured volume resistivity values.

Mixture Models and Subpopulation Classification: A Pharmacokinetic Simulation Study and Application to Metoprolol CYP2D6 Phenotype

Mixture models are applied in population pharmacometrics to characterize underlying population distributions that are not adequately approximated by a single normal or lognormal distribution. In addition to obtaining individualized maximum a posteriori Bayesian post hoc parameter estimates, the subpopulation to which an individual was classified can be determined. However, the accuracy of the classification of subjects to subpopulations is not well studied. We investigated the impact of several factors on the accuracy of classification in mixture models applied to pharmacokinetics using a simulation strategy. The availability of actual subject data allowed us to evaluate mixture model classification in a potentially common application, namely, the classification of clearance into poor metabolizer (PM) or extensive metabolizer (EM) subgroups with the known phenotype status in subjects receiving metoprolol. The factors explored in the simulation study were the magnitude of difference between the clearances in two subpopulations, the between subject variability in clearance, the mixing-fraction, and the population sample size. Populations were simulated at various levels of the above factors and analyzed with a mixture model using NONMEM. The population pharmacokinetics of metoprolol were modeled with the EM/PM phenotype as a known covariate, and without the phenotype covariate using a mixture model. Within the range of scenarios studied, the proportion of subjects classified into the correct subpopulation was high. The simulation-estimation study suggests that a greater separation between two subpopulations, a smaller variability in the parameter distribution, a larger sample size, and a smaller size subpopulation tend to be associated with a greater accuracy of subpopulation classification when a mixture model is applied to pharmacokinetic data. In a population pharmacokinetic analysis of metoprolol, a drug that undergoes polymorphic metabolism, it was possible to correctly identify phenotype status using a mixture model.

SMIXTURE: strategy for mixture model clustering of multivariate images

AbstractSMIXTURE, a novel strategy for mixture model clustering of multivariate images, has been developed. Most other clustering approaches require good guesses of the number of components (clusters) and the initial statistical parameters. In our approach, the initial parameters are determined by agglomerative clustering on homogenous regions, identified by region growing segmentation. SMIXTURE can be used in both a normal situation of mixture modeling, where the density of a cluster is modeled by a single normal distribution; and in a more complex situation, where the density of a single cluster is a mixture of several normal sub‐clusters. The method has proven to be very robust to noise/outliers, overlapping clusters, is reasonably fast and is suitable for moderate to large images. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Bias of the structural quasi-score estimator of a measurement error model under misspecification of the regressor distribution

In a structural measurement error model the structural quasi-score (SQS) estimator is based on the distribution of the latent regressor variable. If this distribution is misspecified, the SQS estimator is (asymptotically) biased. Two types of misspecification are considered. Both assume that the statistician erroneously adopts a normal distribution as his model for the regressor distribution. In the first type of misspecification, the true model consists of a mixture of normal distributions which cluster around a single normal distribution, in the second type, the true distribution is a normal distribution admixed with a second normal distribution of low weight. In both cases of misspecification, the bias, of course, tends to zero when the size of misspecification tends to zero. However, in the first case the bias goes to zero in a flat way so that small deviations from the true model lead to a negligible bias, whereas in the second case the bias is noticeable even for small deviations from the true model.

Incidence and distribution of first-episode mania by age: results from a 35-year study

Few epidemiological studies have investigated incidence by age or age at onset distributions for mania or bipolar disorder. The current study aimed to determine these in a defined area in south-east London, over a 35-year period. All cases of first-episode mania presenting to psychiatric services in Camberwell, south-east London, between 1965 and 1999 were identified. Incidence rates by age, using 5-year age-at-onset bands, were estimated and the structure of the age-at-onset distribution for first-episode mania was investigated using finite mixture distributions (admixture analysis). The incidence of DSM-IV bipolar I disorder (BP I), first manic episode peaked in early adult life (16.38/100,000 population per year in the 21-25 years band) with a much smaller peak in mid-life. A two-component normal mixture distribution fitted age at onset better than either a single normal distribution or a three-component mixture, implying the existence of early and later onset subgroups. The early onset group had a stronger family history of bipolar disorder, and showed more acute, severe and atypical symptoms during their first manic episode. The incidence of mania peaks in early adult life but there is clear evidence of early and later onset subgroups which may represent different forms of disorder.

A quantitative trait locus mixture model that avoids spurious LOD score peaks.

In standard interval mapping of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. At any given location in the genome, the evidence of a putative QTL is measured by the likelihood ratio of the mixture model compared to a single normal distribution (the LOD score). This approach can occasionally produce spurious LOD score peaks in regions of low genotype information (e.g., widely spaced markers), especially if the phenotype distribution deviates markedly from a normal distribution. Such peaks are not indicative of a QTL effect; rather, they are caused by the fact that a mixture of normals always produces a better fit than a single normal distribution. In this study, a mixture model for QTL mapping that avoids the problems of such spurious LOD score peaks is presented.