Mixture distributions

Everitt, B.S. and Hand, D.J. (1981). Finite mixture distributions. Chapman and Hall, New York. 

Incubation Mixtures Distribution nanomoles per of radioactivity, mg protein, in % Recovery... 

Finite mixture distributions may be valuable when a distribution appears to result from mixing of somewhat distinct subpopulations, e.g., if there appear to be multiple modes in a distribution. 

Gibbons RD, Dorns E, Ostrow DG, et al. Mixture distributions in psychiatric research. Bioi Psychiatry 1984 19 935-961. 

For some empirical data sets for variability in a quantity, there can be one or more values that are either below a limit of detection of the measurement instrument or zero for structural reasons (e.g. zero consumption rate of a particular food by a particular person on a particular day). The former case is an example of missing data, whereas the latter is an example of a mixture distribution that is composed of one component that describes the frequency of zero values and a second component that might be a continuous distribution (e.g. consumption rates in this case). 

Zheng J, Frey HC (2004) Quantification of variability and uncertainty using mixture distributions Evaluation of sample size, mixing weights and separation between components. Risk Analysis, 24(3) 553-571. 

D. M. Titterington, A. F. M. Smith, and U. E. Makov, Statistical Analysis ofPinite Mixture Distributions. Wiley, Hoboken, NJ, 1985. 

True residual variance model Mixture distribution Mean value Between simulation CV(%) of parameter estimates ... 

The use of mixture models is not limited to identification of important subpopulations. A common assumption in modeling pharmacokinetic parameters is that the distribution of a random effect is log-normal, or approximately normal on a log-scale. Sometimes, the distribution of a random effect is heavy tailed and when examined on a log-scale, is skewed and not exactly normal. A mixture distribution can be used to account for the large skewness in the distribution. However, the mixture used in this way does not in any way imply the distribution consists of two populations, but acts solely to account for heavy tails in the distribution of the parameter. 

The first step towards inclusion of a mixture model in a PopPK analysis is often graphical examination of the histograms of the EBEs of the model parameters that are treated as random effects assuming no mixture is present in the distribution of the random effects (i.e., the model does not include a mixture distribution), with clear multimodality used as strong evidence that a mixture distribution should be included in the model. This approach is not ideal since in order to visually detect a bimodal distribution of two normally distributed random variates, the subpopulation means must be separated by at least two standard deviations (Schilling,... 

Everitt, B.S. An introduction to finite mixture distributions. Statistical Methods in Medical Research 1996 5 107-127. 

Improving the air/fuel mixture distribution with selected injectors has shown to be an effective way to significantly reduce HC, CO and NOx tailpipe emissions. 

Conditional and Joint Pmbability Distributions In addition to its graphical structure, a Bayesian network needs to be speerfied by the conditional probability distribution of eaeh node given its parents. Let A and D be variables of interest with a direct causal (parental) relationship in Example 11.6. This relationship can be represented by a conditional probabUity distribution P D A) which represents the probabilistic distribution of child node D given the information of parent node A. When both child and parent nodes arc discrete variables, a contingeney table can summarize the conditional probabdities for aU possible states given each of its parent node states. For continuous variables, a eonditional probability density function needs to be defined. For the combination of continuous and discrete nodes, a mixture distribution, for example, mixture normal distribution, will be required (Imoto et al., 2002). 

The substance does not function to provide one or more primary properties that would determine the use of the product or product mixture distributed in commerce, even though it may impart certain physicochemical characteristics to the product or product mixture of which it is a part and... 

Fuel /air metering Fuel /air mixing Fuel vaporisation Mixture distribution Fuel characteristics Exhaust gas recirculation... 

The influence of gas-liquid mixture distribution among parallel working reactors-polymerizers on... 

Final mixingLaminar mixing of the deformable and flowable crosslinking chemicals into the highly viscous basic mixture. Distributive mixing must be superimposed on the laminar mixing. 

Chromatography is one of the most widely used molecular separation techniques. The separation occurs because the compounds within a complex mixture distribute between two phases differently liquid and stationary phases in the case of hquid chromatography (LC) and gas and stationary phases in the case of gas chromatography (GC). Chromatographic chip devices aim to miniaturize chromatographic systems with the goal not only of size reduction but also of improved function. 

The primary function of chromatography is separation of a mixture of compounds into individual components. It is a dynamic separation system that is composed of two media, a stationary phase and a mobile phase. The stationary phase in most applications is a liquid supported on the surface of an inert solid support and is usually packed in a column. The mobile phase can be a gas, a liquid, or a supercritical fluid. Its purpose is to transport the sample through the column, a process known as elution. The components in a separation mixture distribute between these two phases to a different extent. A component that interacts... 

When the underlying distrihutions in the mixture that models the heterogeneous population of components are well separated, then maintenance cost savings are obtainable. Variation in the mixture distribution parameters tends to have a greater effect on the values of optimum inspection intervals than inspection and replacement costs and downtimes. 

Jiang R, Jardine AKS. 2007. An optimal burn-in preventive-replacement model associated with a mixture distribution. Quality and Reliability Engineering International 23, 83 93. 

Dutta et al. [54] used the unified approach to study mass transport between the channels of a PEM fuel cell with a serpentine flow field. Their model is three-dimensional and allows for multi-species transport. They studied the effect of flow channel width in the serpentine flow field on velocity distribution, gas mixture distribution and reactant consumption. Serpentine flow fields allow for a greater area for diffusion of the supply gases. Their results showed that for low humidity conditions, water transport is dominated by electro-osmotic effects, i.e., water flows from anode to cathode at the side of the cell closer to the gas channel inlet. At the outlet side of the cell, water transport is dominated by back diffusion, and it flows in the opposite direction. Thus the serpentine flow field allows for circulation of the water within the cell. 

Diebolt, X, and Robert, C.R (1994). Estimation of Finite Mixture Distributions through Bayesian Sampling. Journal of the Royal Statistical Society, Series B, 56, 363-375. 

Figure 15-9 Three ideal mixture distributions are shown, representing (a) well-mixed, (b) striated, and (c) completely segregated mixtures. A number of boxes are drawn over each distribution to represent possible sample size and orientations. These posited sample locations show how sample size and location can adversely affect the true measure of mixture quality.

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