QUESTIONS

  1. What is a joint probability? What is a marginal probability?

    A joint probability is the probability of an outcome consisting of a pair of values occurring simultaneously.

    A marginal probability is obtained by summing up the values of a joint probability over all values of one of the variables.

  2. What is a joint density function? Give an example.

    A joint density function is the analytic expression of a joint probability distribution.

    An example of a joint density function is the bivariate normal distribution.

  3. What is a conditional probability? How is it used in regression analysis?

    A conditional probability is the ratio of joint and marginal probabilities.

    The conditional normal probability distribution is used in regression theory to determine the slope of the regression line.

  4. Define covariance.

    The covariance is a second moment of a joint probability distribution in which both x and y enter into the formulation.

  5. What is a correlation coefficient?

    The correlation coefficient is a dimensionless value defined as the ratio of the covariance of (x,y) to the product of the standard deviations of x and y. The correlation coefficient is a measure of the linear dependence between predictor and criterion variables.

  6. What is the difference between correlation and regression?

    Regression evaluates the parameters of the prediction equation relating a criterion variable to one or more predictor variables. Correlation provides a measure of the goodness of fit of the regression. Therefore, whereas regression provides the parameters of the prediction equation, correlation describes its quality.

  7. Describe briefly the index-flood method for regional analysis of flood frequency.

    The index-flood method consists of developing two regional curves. The first curve depicts the mean annual flood versus catchment area (or any other set of catchment characteristics). The second curve shows peak flow ratio vs frequency. Using these two curves, a flood frequency curve can be developed for any catchment in the region. The procedure consists of: (1) measuring the catchment area from maps, (2) using the first curve to obtain the mean annual flood based on catchment area, (3) using the second curve to obtain peak flow ratios for selected frequencies, (4) calculating the peak flows for each frequency based mean annual flood and peak flow ratios, and (5) plotting peak flows versus frequencies.

PROBLEMS

  1. Using ONLINE TWOD CORRELATION, calculate the correlation coefficient of the following joint distribution of quarterly flows (expressed as mean values in each class) in streams A and B:


    Stream A
    (ac-ft)    		 1000 2000 3000 4000 5000
    Stream B
    (ac-ft)
    1000 0.07 0.03 0.02 0.00 0.00
    2000 0.03 0.08 0.04 0.03 0.00
    3000 0.02 0.04 0.08 0.05 0.02
    4000 0.00 0.04 0.08 0.11 0.06
    5000 0.00 0.00 0.03 0.08 0.09

    Variance sx = 1260

    Variance sy = 1295.029

    Covariance sx,y = 1111400

    Correlation coefficient rx,y = 0.681

  2. Develop a spreadsheet to calculate the regression constants, correlation coefficient, and standard error of estimate of a series of paired flow values X and Y. Test your program using the data of Example 7-2 in the text.

    TBD

  3. Using the spreadsheet developed in Problem 7-2, calculate the regression constants, correlation coefficient, and standard error of estimate for the following paired low-flow series (annual minima):


    Stream X
    (m3/s)    		 Stream Y
    (m3/s)
    50 65
    66 76
    32 45
    78 95
    12 18
    34 50
    23 31
    50 64
    43 67
    89 99
    76 89
    22 33

    Verify with ONLINE REGRESSION11.

    Using the spreadsheet developed in Problem 7-2, the linear regression of the low-flow series of streams X and Y (Eq. 7-24) leads to:

    Y = α + βX, in which α = 11.09, and β = 1.04. ANSWER.

    The correlation coefficient is: r = 0.984. ANSWER.

    The standard error of estimate is: se = 4.85.  ANSWER.

    The results are confirmed by running ONLINE REGRESSION11.

  4. Modify the spreadsheet developed in Problem 7-2 to calculate the regression constants to fit a power function of the following form (Eq. 7-51):

    Qp = cAn

    in which Qp = peak discharge; A = drainage area; c and m are coefficient and exponent, respectively. Using the spreadsheet, fit a power function to the following data:


    Peak Discharge
    (m3/s)    		 Drainage Area
    (km2)
    124 25
    254 46
    378 78
    101 22
    678 99
    540 89
    490 83
    267 52
    350 73

    Verify with ONLINE REGRESSION12.

    Using the spreadsheet, the nonlinear regression of peak discharge on drainage area (Eq. 7-51) for the given data set leads to:

    Qp = c Am, in which c = 2.882 and m = 1.154.  ANSWER.

    The results are confirmed by running ONLINE REGRESSION12.

  5. ONLINE REGRESSION13 solves the two-predictor-variable linear regression problem (Eq. 7-35). Use this program to determine the regression constants for the following data set:


    Y
    Time of Concentration
    (min)    		 X1
    Hydraulic Length
    (m)    		 X2
    Catchment Slope
    (m/m)
    89 3245 0.008
    75 2567 0.011
    57 2783 0.009
    34 1234 0.015
    101 5345 0.006
    121 5329 0.007
    68 3002 0.008
    79 2976 0.010
    25 1034 0.018
    59 2984 0.010
    96 3892 0.007
    12 534 0.020

    The linear regression of time of concentration (Y) on hydraulic length (X1) and catchment slope (X2) (Eq. 7-35) leads to:

    Y = α + β1X1 + β2X2

    in which α = 42.466, β1 = 0.0155, and β2 = -1830.41. ANSWER.

    The results are confirmed by running ONLINE REGRESSION13.

  6. Use ONLINE REGRESSION14 to solve the two-predictor-variable nonlinear regression problem of Eq. 7-48, for the data of Problem 7-5.

    The nonlinear regression of time of concentration (Y) on hydraulic length (X1) and catchment slope (X2) (Eq. 7-48) leads to:

    Y = a X1b1 X2b2

    in which a = 0.0386, b1 = 1.1768, and b2 = 0.4064. ANSWER.

    The results are confirmed by running ONLINE REGRESSION14.

  7. The median Qi/Q2.33 ratios (i = frequency) for 10 stations have been found to be 1.95 for the 10-y frequency and 2.45 for the 50-y frequency. Use the index-flood method to calculate the 25-y flood for a point in a stream having a 340-km2 catchment and a mean annual flood given by the following formula:

    Q 2.33 = 3.93 A 0.75

    in which Q = flood discharge in cubic meters per second, and A = drainage area in square kilometers.

    With drainage area: A = 340 km2, the mean annual flood is:

    Q 2.33 = 3.93 A 0.75 = 311 m3/s.

    The two median ratios are: Q 10 / Q 2.33 = 1.95; and Q 50 / Q 2.33 = 2.45.

    These two median ratios are plotted on Gumbel paper and a straight line is drawn throughthe two points.

    The 25-y median ratio read from this graph is: Q 25 / Q 2.33 = 2.25.

    Therefore, the 25-y flood discharge is:

    Q 25 = 2.25 × 311 = 700 m3/s. ANSWER.

  8. Modify the spreadsheet developed in Problem 7-2 to calculate the regression constants and correlation coefficient to fit intensity-duration-frequency rainfall data. Test your spreadsheet using the data of Example 7-4 in the text. Verify with ONLINE REGRESSION15.

    a = 155.702, b = 23.5632.

    The results are confirmed by running ONLINE REGRESSION15.

  9. Using ONLINE REGRESSION15 for a hyperbolic regression, calculate the regression constants a and b (Eq. 7-55) for the following 25-y frequency rainfall data:


    Duration (min) 5 10 15 30 60 120 180
    Intensity (mm/h) 15.5 7.5 6.5 4.5 3.5 2.5 1.5

    Eq. 7-55 is:

                 a
    i =   _________
             (tr + b)

    in which i = rainfall intensity, tr = rainfall duration.

    The results of ONLINE REGRESSION15 are:

    a = 329.82, and b = 30.76. ANSWER.

http://enghydro.sdsu.edu
150902 13:00

Documents in Portable Document Format (PDF) require Adobe Acrobat Reader 5.0 or higher to view; download Adobe Acrobat Reader.