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Probability Plotting - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

Bhupendra Narayan Mandal University Mathematical Statistics

The main points i the stochastic hydrology are listed below:Probability Plotting, Graphical Construction, Normal Distribution Table, Arithmetic Scale Plot, Transformation Plot, Plotting Position, Empirical Methods, California Method, Formulae for Exceedence Probability

Typology: Study notes

2012/2013

Uploaded on 04/20/2013

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Graphical construction:

• Graphical construction is done by transforming the

arithmetic scale to probability scale so that a

straight line is obtained when cumulative

distribution function is plotted.

• The transformation technique is explained with the

normal distribution.

• Consider the coordinates from the standardized

normal distribution table.

Probability Plotting

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Graphical construction:

Graphical construction is done by transforming the arithmetic scale to probability scale so that a straight line is obtained when cumulative distribution function is plotted.
The transformation technique is explained with the normal distribution.
Consider the coordinates from the standardized normal distribution table. 3

Probability Plotting

4 z 0 2 4 6 8 0 0 0.008 0.016 0.0239 0. 0.1 0.0398 0.0478 0.0557 0.0636 0. 0.2 0.0793 0.0871 0.0948 0.1026 0. 0.3 0.1179 0.1255 0.1331 0.1406 0. 0.4 0.1554 0.1628 0.17 0.1772 0. 0.5 0.1915 0.1985 0.2054 0.2123 0. 0.6 0.2257 0.2324 0.2389 0.2454 0. 0.7 0.258 0.2642 0.2704 0.2764 0. 0.8 0.2881 0.2939 0.2995 0.3051 0. 0.9 0.3159 0.3212 0.3264 0.3315 0. 1 0.3413 0.3461 0.3508 0.3554 0.

Normal Distribution Tables

z (Partial tables shown here)

Arithmetic scale plot: 6

Probability Plotting

-‐ -‐ -‐ 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Z F(Z)

0.01 0.05 0.1 0.2 (^) 0.4 0.5 0.6 0.7 0.8 0.9 (^) 0.95 0. Transformation plot: 7

Probability Plotting

F(Z) (probability scale) F(Z) (arithmetic scale)

The purpose is to check if a data set fits the particular distribution.
The plot can be used for interpolation, extrapolation and comparison purposes.
The plot can be used for estimating magnitudes with specified return periods.
Extrapolation must be attempted only when a reasonable fit is assured for the distribution. 9

Probability Plotting

PLOTTING POSITION

Arrange the given series of data in descending order.
Assign a order number to each of the data (termed as rank of the data).
Let n is the total no. of values to be plotted and m is the rank of a value, the exceedence probability (p) of the m th largest value is obtained by various plotting position formulae.
The return period (T) of the event is calculated by T = 1/p
Plot magnitude of event verses the exceedence probability (p) or 1 – p or the return period T. 12

Plotting Position

Formulae for exceedence probability: California Method: Limitations

Produces a probability of 100% for m = n 13

Plotting Position

( (^) m )

m

p X x

n

Hazen s formula: Chegodayev s formula: 15

Plotting Position

( )

m

p X x

n

( )

m

p X x

n

Weibull s formula:

Most commonly used method.
If n values are distributed uniformly between 0 and 100 percent probability, then there must be n+ intervals, n–1 between the data points and 2 at the ends.
Indicates a return period T one year longer than the period of record for the largest value. 16

Plotting Position

( )

m

p X x

n

Cunnane (1978) studied the various available plotting position methods based on unbiasedness and minimum variance criteria.
If large number of equally sized samples are plotted, the average of the plotted points for each value of m lie on the theoretical distribution line.
Minimum variance plotting minimizes the variance of the plotted points about the theoretical line. 18

Plotting Position

Ref: Cunnane, C., Unbiased plotting positions – a review, J,Hydrol., Vol. 37, pp.205-222, 1978

For normally distributed data, Blom s plotting position formula (b = 3/8) is commonly used.
For Extreme Value Type I distribution, the Gringorten formula (b = 0.44) is used.
All the relationships give similar values near the center of the distribution but may vary near the tails considerably.
Predicting extreme events depend on the tails of the distribution. 19

Plotting Position

The observations should be equally spaced on the frequency scale
The plotting position should be analytically simple and easy to use. 21

Plotting Position

Ref: Gumbel, E. J., 1958. Statistics of Extreme Values, Columbia University Press, New York

The Weibull plotting position formula meets all the 5 of the above criteria. 1.All the observations can be plotted since the plotting positions range from 1/(n+1) (which is greater than 0) to n/(n+1) (which is less than 1). 2.The relationship lies between (m – 1)/n and m/n for all values of m and n. 22

Probability Plotting - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

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