Statistik untuk HidrologiANALISIS FREKUENSI

*ProbabilityA measure of how likely an event will occurA number expressing the ratio of favorable outcome to the all possible outcomes Probability is usually represented as P(.)P (getting a club from a deck of playing cards) = 13/52 = 0.25 = 25 %P (getting a 3 after rolling a dice) = 1/6

*Random VariableRandom variable: a quantity used to represent probabilistic uncertaintyIncremental precipitation Instantaneous streamflowWind velocityRandom variable (X) is described by a probability distributionProbability distribution is a set of probabilities associated with the values in a random variables sample space

*Sampling terminologySample: a finite set of observations x1, x2,.., xn of the random variableA sample comes from a hypothetical infinite population possessing constant statistical propertiesSample space: set of possible samples that can be drawn from a populationEvent: subset of a sample spaceExamplePopulation: streamflowSample space: instantaneous streamflow, annual maximum streamflow, daily average streamflow Sample: 100 observations of annual max. streamflowEvent: daily average streamflow > 100 cfs

*Types of samplingRandom sampling: the likelihood of selection of each member of the population is equal Pick any streamflow value from a population

Stratified sampling: Population is divided into groups, and then a random sampling is usedPick a streamflow value from annual maximum series.

Uniform sampling: Data are selected such that the points are uniformly far apart in time or spacePick steamflow values measured on Monday midnight

Convenience sampling: Data are collected according to the convenience of experimenter.Pick streamflow during summer

*Summary statisticsAlso called descriptive statisticsIf x1, x2, xn is a sample then

Mean, Variance, Standard deviation, Coeff. of variation, m for continuous data s2 for continuous data s for continuous data Also included in summary statistics are median, skewness, correlation coefficient,

*Graphical display Time Series plotsHistograms/Frequency distributionCumulative distribution functionsFlow duration curve

*Time series plotPlot of variable versus time (bar/line/points)Example. Annual maximum flow series

Colorado River near Austin

*HistogramPlots of bars whose height is the number ni, or fraction (ni/N), of data falling into one of several intervals of equal widthDividing the number of occurrences with the total number of points will give Probability Mass Function

*Probability density functionContinuous form of probability mass function is probability density functionpdf is the first derivative of a cumulative distribution function

*Using Excel to plot histograms1) Make sure Analysis Tookpak is added in Tools.This will add data analysis command in Tools2) Fill one column with the data, and another with the intervals (eg. for 50 cfs interval, fill 0,50,100,)3) Go to ToolsData AnalysisHistogram4) Organize the plot in a presentable form (change fonts, scale, color, etc.)

*Cumulative distribution functionCumulate the pdf to produce a cdfCdf describes the probability that a random variable is less than or equal to specified value of xP (Q 50000) = 0.8P (Q 25000) = 0.4

*Flow duration curveA cumulative frequency curve that shows the percentage of time that specified discharges are equaled or exceeded.

StepsArrange flows in chronological order Find the number of records (N)Sort the data from highest to lowest Rank the data (m=1 for the highest value and m=N for the lowest value)Compute exceedance probability for each value using the following formula

Plot p on x axis and Q (sorted) on y axis

*Flow duration curve in ExcelMedian flow

*Statistical analysisRegression analysisMass curve analysisFlood frequency analysisMany more which are beyond the scope of this class!

*Linear RegressionA technique to determine the relationship between two random variables.Relationship between discharge and velocity in a streamRelationship between discharge and water quality constituents A regression model is given by :yi = ith observation of the response (dependent variable)xi = ith observation of the explanatory (independent) variableb0 = interceptb1 = slopeei = random error or residual for the ith observationn = sample size

*Linear Regression in ExcelSteps: Prepare a scatter plotFit a trend line

Alternatively, one can use ToolsData AnalysisRegressionData are for Brazos River near Highbank, TX

*Coefficient of determination (R2)It is the proportion of observed y variation that can be explained by the simple linear regression modelTotal sum of squares, Ybar is the mean of yiError sum of squaresThe higher the value of R2, the more successful is the model in explaining y variation.If R2 is small, search for an alternative model (non linear or multiple regression model) that can more effectively explain y variation

Analisis Frekuensi HujanPengumpulan data curah hujanHitung hujan rata rata regional Analisis Frekuensi : a. Distribusi Normal b. Distribusi Log Normal c. Distribusi Gumbel d. Distribusi Log Pearson

Hujan RencanaR(T) = R rata-rata + Kt x S ( dev.standard) Kt = Faktor Frekuensi yang tergantung dg : - Jenis Distribusi Kemungkinan - Periode Ulang - Parameter statatistik

TahunJUMLAH DEBIT

( m3 / det )MaksimumMinimum19961446.730.001997798.3626.721998894.4351.861999989.570.002000798.3626.722001972.020.002002976.810.0020031970.0058.6020041400.0033.362005888.9933.362006780.36119.772007762.87107.95