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Two Concepts of Probability

Two Concepts of Probability. Statistical Relative frequency in repeated experiments. Inductive Subjective Based on incomplete information, judgment and logical reasoning Bayesian. Line Diagram. From Kottegoda and Rosso, 1997 p3. Dot diagram. From Kottegoda and Rosso, 1997 p4.

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Two Concepts of Probability

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  1. Two Concepts of Probability Statistical Relative frequency in repeated experiments Inductive Subjective Based on incomplete information, judgment and logical reasoning Bayesian

  2. Line Diagram From Kottegoda and Rosso, 1997 p3

  3. Dot diagram From Kottegoda and Rosso, 1997 p4

  4. Histogram of minimum annual flow in the Po river between 1918 and 1978

  5. Minimum annual flow in the Po river between 1918 and 1978 Alternative histogram axis scaling - Relative Frequency - Density

  6. Po River, Minimum annual flow cumulative relative frequency(number of values ≤ n)/n (KR p 8) qs=sort(q) n=length(q) crf=(0:(n-1))/n plot(qs,crf)

  7. Po River, Minimum annual flow Quantile plot (Q-plot) qs=sort(q) n=length(q) crf=(0:(n-1))/n plot(crf,qs) 75% Quantile or quartile Median Interquartile range IQR 25% Quantile or quartile

  8. pi qi Quantile Definition p x A quantile qi is the random variable value associated with a specific cumulative probability pi

  9. Numerical Quantities

  10. Helsel and Hirsch page 21

  11. Time Series Box Plot Median Box (Red Lines) enclose 50% of the values

  12. Box Plot Outliers: beyond 1.5*IQR Whiskers: 1.5*IQR or largest value Box: 25th %tile to 75th %tile Line: Median (50th %tile) - not the mean Note: The range shown by the box is called the “Inter-Quartile Range” or IQR. This is a robust measure of spread. It is insensitive to outliers since it is based purely on the rank of the values.

  13. Outliers Horizontal Line is the mean Compare change in mean and median between Aug-Sep. Note Skew in September Seasonality of Flow “Monthly Subseries Plot” - time series for each month Flow (cfs) Box Plots Flow (cfs)

  14. Scatter Plot - Flow v. Water Level

  15. Flow = f(Pumping) Causality? Co-effect? OR Pumping = f(Flow) Water Level = f(Pumping) Logical relationship Multiple Scatterplots

  16. Scatterplot - between raw x and y data Q-Q plot - between sorted x and y data Compares individual X and Y values Compares the distributions of X and Y

  17. pi qi Quantiles to compare to theoretical distribution Rank the data Theoretical distribution, e.g. Standard Normal x1 x2 x3 . . . xn qi is the distribution specific theoretical quantile associated with ranked data value xi

  18. QQ-plot for Log-Transformed Flows ln(xi) QQ-plot for Raw Flows qi Quantile-Quantile Plots xi qi Used as a basis for finding transformation to make the Raw flows Normally distributed.

  19. Quantile plots and Probability Plots Probability Plot Q-Q Plot Theoretical quantile axis is relabeled with corresponding probability values

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