Models 1- Probability Density Function (PDF) What is a PDF model? A mathematical equation that describes the frequency curve or probability distribution of a data set. Why modeling? It represents and summarizes the statistical distribution of the entire geologic phenomenon and helps in making predictions. Statistical characteristics and parameters can be derived and calculated easily from the model. Such parameters reflect the behavior of the geologic phenomenon. 1
Models 1- Probability Density Function (PDF) Types of models Normal distribution Gaussian (Gauss), Laplace, or Bell-shaped distribution. Values are symmetrically distributed around a central value. The mean is the most representative value of the distribution (i.e. use arithmetic average to estimate the unknown from a set of uncorrelated random variables ). The variance is the well-defined measure of the spread of observations. 2
Models 1- Probability Density Function (PDF) Types of models Non-normal distribution Lognormal distributions (natural or base-10 logarithms of measured observations). Values are asymmetrically distributed around a central value. The mean is not the most representative value of the distribution. (i.e. do not simply use arithmetic average to estimate the unknown from a set of uncorrelated random variables ). However, other averages or median value might work! 3
Models 2 - Probability Density Function (PDF): Example of a normal case Relative Frequency R. Freq. 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 72.5 77.5 Sample values (%) Relative Frequency 4
Models 3 - Cumulative Distribution Function (CDF): Example Example of a normal case CDF Model CDF (Probability) 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 72.5 77.5 Sample value (%) Exper. Cumulative distribution CDF 5
Models 4- Normality Tests Different procedures are used to test for normality. Some of these are listed below:» Normal-Probability graph papers» Chi-square test» Kolmogorov-Smirnov test» Shapiro-Wilks test 6
Models 5- Skewed Distributions What if a distribution is skewed? Use power transformation techniques to transform original data values into a defined power function. Why and how? Distribution of transformed data is much easier to describe than the distribution of original skewed data. Common power transforms: y p = ( z 1) / p y = Ln(z) 7
Models 6 - PDF of original data Relative Freq. Curve 0.3 0.25 Relative. Freq. 0.2 0.15 0.1 0.05 0 100 200 300 400 500 600 700 800 900 1000 Sample values (ppm) 8
Models 6 - PDF of log-transformed data R. Freq. for Log values R. Freq. 0.3 0.2 0.1 0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Logarithms 9
Models 6 - CDF of original data CDF CDF 1.50 1.00 0.50 0.00 100 200 300 400 500 600 700 800 900 1000 Sample value (ppm) Cumulative % CDF 10
Models 6 - CDF of log-transformed data CDF of Logs CDF (Prob.) 1.50 1.00 0.50 0.00 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Logartithms Cumulative % CDF 11
Models 7 - Other Types of Skewed distributions Lognormal Model 12
Models 7 - Other Types of Skewed distributions Exponential Model 13
Models 7 - Other Types of Skewed distributions Log-Pearson Type III Model 14
Models 7 - Other Types of Skewed distributions General Extreme Value (GEV) Model 15
Models 7 - Other Types of Skewed distributions Gumbel Model 16
Models 7 - Other Types of Skewed distributions Log-Gumbel Model 17
Models 7 - Other Types of Skewed distributions Weibull Model 18
Models 7 - Other Types of Skewed distributions Beta Model 19
Models 7 - Other Types of Skewed distributions Wakeby Model 20