Reply to the comment on data analysis in the shot noise limit. 1. Single parameter estimation with Poisson and normal probability density functions

1990 ◽  
Vol 62 (19) ◽  
pp. 2142-2144 ◽  
Author(s):  
Stephen E. Bialkowski
Keyword(s):  
Parameter Estimation ◽  
Data Analysis ◽  
Probability Density ◽  
Shot Noise ◽  
Probability Density Functions ◽  
Single Parameter ◽  
Density Functions ◽  
Normal Probability ◽  
Noise Limit ◽  
Shot Noise Limit

Quantile Analysis: A Method for Characterizing Data Distributions

1988 ◽  
Vol 42 (8) ◽  
pp. 1512-1520 ◽  
Author(s):  
Robert A. Lodder ◽  
Gary M. Hieftje
Keyword(s):  
Probability Density ◽  
Statistical Methods ◽  
Probability Density Functions ◽  
Density Functions ◽  
Single Test ◽  
Test Statistic ◽  
Data Set ◽  
Normal Probability Plot ◽  
Probability Plot ◽  
Normal Probability

Analyzing distributions of data represents a common problem in chemistry. Quantile-quantile (QQ) plots provide a useful way to attack this problem. These graphs are often used in the form of the normal probability plot, to determine whether the residuals from a fitting process are randomly distributed and therefore whether an assumed model fits the data at hand. By comparing the integrals of two probability density functions in a single plot, QQ plotting methods are able to capture the location, scale, and skew of a data set. This procedure provides more information to the analyst than do classical statistical methods that rely on a single test statistic for distribution comparisons.

Density Function Optimization for the Homogenized Energy Model of Shape Memory Alloys

2011 ◽  
Author(s):  
John H. Crews ◽  
Ralph C. Smith
Keyword(s):  
Shape Memory Alloys ◽  
Shape Memory ◽  
Probability Density ◽  
Energy Model ◽  
Probability Density Functions ◽  
Density Functions ◽  
Normal Probability ◽  
Homogenized Energy Model ◽  
Log Normal ◽  
Normal Probability Density

In this paper, we present two methods for optimizing the density functions in the homogenized energy model (HEM) of shape memory alloys (SMA). The density functions incorporate the polycrystalline behavior of SMA by accounting for material inhomogeneities and localized interaction effects. One method represents the underlying densities for the relative stress and interaction stress as log-normal and normal probability density functions, respectively. The optimal parameters in the underlying densities are found using a genetic algorithm. A second method represents the densities as a linear parameterization of log-normal and normal probability density functions. The optimization algorithm determines the optimal weights of the underlying densities. For both cases, the macroscopic model is integrated over the localized constitutive behavior using these densities. In addition, the estimation of model parameters using experimental data is described. Both optimized models accurately and efficiently quantify the SMA’s hysteretic dependence on stress and temperature, making the model suitable for use in real-time control algorithms.

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