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HYDROLOGIC STATISTICS
1. Summary Statistics (Moments: Product and L-moments)
2. Distributional
(Magnitude and
Frequency) Analysis
3. Nonparametric
Statistics (Introduction to Hypothesis Testing)
a) Trend Testing
b) Rank Sum Test
Effects of urbanization on flood
peaks (1956-1980) on Waller
Creek??????
Frequency Distribution-->the mean and
beyond . . . .
PROBABILITY DISTRIBUTIONS
• Discrete and Continuous
Random Variables
• Cumulative Distribution
Function (cdf)
• Statistical Expectation
• Quantiles
– median, quartiles,
interquartile range
– plotting position estimators
• Plotting Positions
– expressed as functions
– have parameters
• Quantile Functions
1. order data x1 x2 ... xn
2. rank’em 1, 2, ..., n (i is rank)
3. F(x) = i-0.40/n+0.2
Cunnane plotting-positions
F(x) = i/n+1
Weibull plotting-positions
MORE PLOTTING POSITION STUFF
PLOTTING POSITIONS
1. order data x1 x2 ... xn
2. rank’em 1, 2, ..., n (i is rank)
3. F(x) = nonexceedance
probability or just the percentile.
4. 1-F(x) = exceedance probability
GENERAL FORMULA
1-F(x) = (i-a) / (n+1-2*a)
The true probability associated with the
largest (and smallest) observation is a
random variable with mean 1/(n+1) and a
standard deviation of nearly 1/(n+1).
Hence, all plotting position formula give
crude estimates of the unknown
probabilities associated with largest and
smallest events.
Cunnane plotting-positions (a=0.40)
F(x) = (i-0.40)/(n+0.2)
“approx. quantile unbiased”
Weibull plotting-positions (a=0)
F(x) = i/(n+1)
“unbiased [F(x)] for all distributions”
Hazen plotting-positions (a=0.50)
F(x) = (i-0.5)/n
“long legacy”
Blom plotting-positions (a=0.375)
F(x) = (i-3/8)/(n+1/4)
“optimal for normal distribution”
http://pubs.usgs.gov/twri/twri4a3/
See chapter 2
Comal Springs Daily Mean Flow
Comal Springs Daily Mean Flow
(Flow) Duration Curves--I
• Simple, yet highly informative graphical summaries of the
variability of a (daily) time series--Streamflow (flowduration)
• An FDC is a graph plotting the magnitude of a variable Q
verses fraction of time the Q does not exceed a specified
value [Q(F)]. The fraction of time can be thought of as
probability and cumulative fraction of time is termed
nonexceedance probability (F).
• The probability refers to the frequency or probability of
nonexceedance (or exceedance) in a “suitably long” period
of time rather than probability of exceedance on a specific
time interval (daily).
(Flow) Duration Curves--II
• Area under the curve is equal to the average for the period.
• Other statistics or statistical concepts visible include: median, quartiles,
other percentiles, variability, and skewness. Steeper curves are
associated with increasingly variable data.
• The slopes and changes in the slope of the curves can be important
diagnostics of streamflow conditions in a watershed.
(Flow) Duration Curves--III
Duration
curves for
neighboring
stations yield
valuable
insights into
hydrologic or
hydrogeologic
processes
(Flow) Duration Curves--IV
For natural streams
Slope of FDC for upper end is determined by regional climate and characteristics
of large precipitation events.
Slope of the lower end is determined by geology, soils, topography.
Slope of the upper end is relatively flat where snowmelt is the principal cause of
floods and for large streams where floods are caused by long duration storms.
Flashy watersheds and watersheds effected by short duration storms have steep
upper ends.
A flat lower end slope usually indicates that flows come from significant storage
in ground water aquifers or frequency precipitation inputs.
SUMMARY STATISTICS
1.
Product Moments (PMs)
2.
L-moments—
seen already, but
will study in detail
later in the semester.
See powers-”product”
Theoretical PMs---->
E[ ] = Expectation operator
In terms of PDF
In terms of quantile function
SUMMARY STATISTICS
Sample PMs---->
Biased Estimators
SUMMARY STATISTICS
1. Summary Statistics
The uniformly minimum
unbiased estimator of the
standard deviation.
PM Boundness!!!
Careful in hydrologic data
sets.
NONPARAMETRIC STATISTICS
Nonparametric statistics (NP) are a branch of statistics
based on the ranking or ranks of the data rather than the
data values themselves. This fact has many desirable
properties in hydrologic data analysis because data sets are
often highly variable, measured with large error, censored,
contaminated, and a host of other problems.
• NP require fewer assumptions about the distribution
generating the data. The normal or bell-shape curve
assumption is NOT required.
• NP are easier than classical statistics to apply.
• NP are remarkably(?) straightforward to understand.
NONPARAMETRIC STATISTICS
• NP can be used in situations that normal theory or classical
statistics can not.
• NP seem to sacrifice too much information. This is NOT the
case. More often than not, NP are only slightly less efficient
than classical statistics when distributions are normal. NP
can be absurbly more efficient than classical statistics.
• NP are robust in the presence of outliers, contaminated data,
censored data, highly skewed data and so on.
Hollander, M., and Wolfe, D.A., 1973, Nonparametric
statistical methods: John Wiley Inc., New York, 503 p.
NP STATISTICS—Trend Testing
Trend Testing—that is the testing for temporal (time) trends—
in data might be the most common use of NP in physical
hydrology. Therefore, we’ll use trend testing as a starting
point for introduction.
Trend Testing = Relation Testing = Independence Testing
KENDALL’S TAU
Kendall’s Tau—NP Trend Testing
• We have n bivariate observations (X1,Y1), . . . , (Xn,Yn).
• We want to test whether there is a relation between the X’s
and the Y’s. We can not test for cause and effects—very
important to remember.
• We assume that each data pair are mutually independent and
each pair is derived from the same population.
Kendall’s Tau—NP Trend Testing
Define Kendall’s Tau by
t = 2*Prob{(X1-X2)(Y1-Y2) > 0} - 1
t = 0 if X’s and Y’s are unrelated because half of the time
the X differences and Y differences would have the same
sign.
t = 2 * (1/2) - 1 = 0
-1  t  1
1. For each 1  i < j  n
calculate x(Xi,Xj,Yi,Yj)
x(a,b,c,d) = score for . . .
1 if (a-b)(c-d) > 0
0 if (a-b)(c-d) = 0
-1 if (a-b)(c-d) < 0
Kendall’s Tau—NP Trend Testing
2. Sum up ones and minus ones and calculate the sum (K):
K = S(i=1,n-1)S(j=i+1,n){x(a,b,c,d)}
There are n*(n-1)/2 terms to compute.
3. Compute t = 2K/[n*(n-1)], which is known as Kendall’s
Rank Correlation Coefficient or simply “Kendall’s Tau”
t estimates the probability parameter:
Prob{(X1-X2)(Y1-Y2) > 0} = (t+1)/2
t will generally be lower than values of the traditional correlation
coefficient for linear associations of equal strength. “Strong” linear
correlations of r > 0.9 correspond to t > 0.7. t measures all monotonic
correlations (linear or nonlinear), and does not change with monotonic
power transformations of X and/or Y [for example, log(X)].
Kendall’s Tau—NP Trend Testing
4.
Hypothesis Testing—We know that inherent randomness will produce a
range of t differing from zero. If we know the distribution of t, hence
K under conditions in which t = 0, we can perform a test by specifying
some error or some tolerance in being right or wrong about whether the
data is independent.
Start with hypothesis, the Null Hypothesis, Ho, that the data is
independent at the a level of significance, then
a = a1 + a2 often it is taken that a1 = a2
reject Ho(t = 0) if K  k(a2,n) or K  -k(a1,n)
INDEPENDENT
accept Ha(t ≠ 0) if K < k(a2,n) or K > -k(a1,n)
DEPENDENT
k is the null distribution of K, which we will investigate in more detail.
We can also test whether t > 0, which means positive correlation
between X and Y or whether t < 0 (negative correlation.)
Kendall’s Tau—NP Trend Testing
t > 0 at the a significant level
reject Ho(t = 0)
if K  k(a,n)
accept Ha(t > 0) if K < k(a,n)
t < 0 at the a significant level
reject Ho(t = 0)
if K  -k(a,n)
accept Ha(t < 0) if K > -k(a,n)
CIRCULAR STATISTICS
Circular statistics are used to quantify the time of
occurrence of hydrologic variables on a circle—typically on a
yearly basis.
• Successive samples of circular statistic results
• The math :(
• Really comprehensive analysis
Circular Statistics—see BOX 4-3
Circular statistics are used to quantify the time of
occurrence of hydrologic variables on a circle—
typically on a yearly basis.
Two values require calculation:
1. Average Time of Occurrence (Angle of the Mean)
- analogous to the arithmetic mean
2. Index of Seasonality
- analogous to the standard deviation
The average hydrologic quantity (say a monthly value) is considered to
be a vector quantity. Length is proportional to the amount and
direction (angle) of the time of the value.
Circular Statistics
1. Average Time of Occurrence (Angle of the Mean)
a) Time through the year (or other interval) is
represented on a circle with (usually) each month
assigned an angle.
Think of the sin/cos terms as weight factors.
a) Resultant Angle Prime: fR’ = atan(S/C)
b) Resultant Angle (deal with quadrant):
fR = fR’
if(S > 0 and C > 0)
fR = fR’+180 if(C < 0)
fR = fR’+360 if(S < 0 and C > 0)
But other conversions
are sometimes needed
depending upon the
output of the atan
function.
Circular Statistics
c) Resultant Angle (deal with quadrant):
In the Perl
$PHI = ( ($Sterm > 0 and $Cterm > 0) language
or
($Sterm > 0 and $Cterm < 0) ) ?
$PHIp : $PHIp+360;
fR = fR’
fR = fR’+360
if[(S > 0 and C > 0) or (S < 0 and C < 0)]
2. Index of Seasonality (IS)
PR = sqrt(S2 + C2)
IS = PR / (Total of Xm Values)
Circular Statistics
List of examples of hydrologic
variables on which circular
statistics would be useful:
Example: Total Rainfall = 36 inches
------------------------------------------------Season
Rainfall
sin
cos
------------------------------------------------Spring (Mar.31;DoY=90)
4.00
0.9998 0.0215
Summer(Jun.30;DoY=181) 16.00
.0258 -.9997
Fall (Sept.30;DoY=273) 11.00
-.9999 -.0129
Winter(Dec.31;DoY=365)
5.00
.0000 1.0000
------------------------------------------------S = -6.587; C = -11.05; f’=atan(S/C)=> 30.8 degrees
f = 30.8 + 180 = 211 degrees
PR = 12.87; IS = 12.87/36 = 0.357
Circular
Statistics
for
08155500
Barton
Springs at
Austin,
Texas
• 1978 to 2003
• Vector lengths are short
• No definitive angle
Are these observations
consistent with your
expectation?
Circular
Statistics
for
08158000
Colorado
River at
Austin,
Texas
• 1899 to 2003
• Vector lengths are
moderately long.
• Concentration of angle
near end of September to
(through?) November.
Are these observations
consistent with your
expectation?
Circular
Statistics
for
08169000
Comal
River at
New
Braunfels,
Texas
• 1933 to 2002
• Vector lengths are short
• No definitive angle--but
perhaps more in January
through March?
Circular
Statistics
for
08169000
Comal
River at
New
Braunfels,
Texas
Circular
Statistics
for
08169000
Comal
River at
New
Braunfels,
Texas
Extensive
Circular
Statistics