Statistics andp lotting the distribution given the data

CE 264000 Data Analysis – Homework 4

Due November 22, 2013

Exercise 1: Pick a gauge time series USGS discharge gauge assigned to you in the table bellow.You will find the files are in the http://tulip.ccny.cuny.edu/CE26400/Labs/USGSdata directory.

• Import the time series into a spreadsheet program of your choice (Micro$oft Office/Excel or LibreOffice/Calc).

• Calculate the minimum, maximum, median and the first four moments (mean, standard deviation, skewness and kurtosis) of the time series

• Calculate the 10-base logarithm of the time series and compute the same properties (minimum, maximum, and the first four moments) for the resulting new data series.

Exercise 2: • Calculate and create an X-Y plot of the empirical cumulative distribution function of the

discharge time series.

• Plot the cumulative density function in normal and lognormal (along the x axis) coordinate spaces

• Calculate and create a bar chart of the probability mass density function of the 10-base logarithmic discharge time series by discretizing (binning) into 5-10 size categories (by slicing up the value ranges defined by the minimum and maximum). Try to maximize the round numbers in the category boundaries by selecting the starting “bin” and the binning size with exact numbers (e.g. log10 X =−1.0,−0.75, 0.50,…,2.0, 2.25, 3.0 )

Exercise 3: Fit logarithmic normal distribution to the empirical cumulative distribution function using the method of moments. In other words, estimate the normal distribution's parameters ( μ ,σ ) from the time series and plot it along with the CDF derived from the data in the previous exercise in different color or line style.

Calculate the probability mass function for the category bins that you used to discretize your data in the previous exercise (the difference of the CDF values at the upper and lower bounds of the bin). Plot the the fitted PMF against the empirical PMF.

Exercise 4: Fit Gamma distribution function using the shape and rate parameters given in the table for your time series and plot it (in different color or line style) along with the empirical and the lognormal cumulative distribution function. Calculate the probability mass function for the fitted gamma function as you did for the lognormal distribution.

Exercise 5: Summarize your work in a text document including the computed statistics and the figures you produced. You should pull out key attributes (station id – site_no, station name – station_nm, longitude – dec_long_va, latitude – dec_lat_va, altitude – alt_va, catchment area – drain_area_va) of your time series from the USGS_DischargeSitesNY.csv file in the same directory, where your time series was located.

First Name Last Name Filename Shape Rate Radwa Ahmed USGS_DailyDischarge03011020.csv 0.923430 0.011689 Ahmadu Bah USGS_DailyDischarge01531000.csv 0.733720 0.009838 Sergio Bracho USGS_DailyDischarge01335754.csv 1.875818 0.008446 Medwin Chiu USGS_DailyDischarge04231600.csv 1.250140 0.015626 Seth Danso USGS_DailyDischarge01350000.csv 0.723342 0.053229 Mira Elhawafri USGS_DailyDischarge01325000.csv 0.917743 0.014901 Carlos Filho USGS_DailyDischarge01315500.csv 1.583407 0.034813 Justin Gabel USGS_DailyDischarge04266500.csv 1.690770 0.044018 Ehren Gresehover USGS_DailyDischarge04227500.csv 0.903381 0.018630 Mostafa Haridi USGS_DailyDischarge04223000.csv 0.843928 0.023098 Jaret Horn USGS_DailyDischarge01503000.csv 0.983436 0.009488 Mideum Jung USGS_DailyDischarge04252500.csv 1.442422 0.068856 Hong Gi Moon USGS_DailyDischarge01321000.csv 0.953986 0.030165 Abu Moro USGS_DailyDischarge01512500.csv 0.977591 0.013965 Nelsyda Perez USGS_DailyDischarge01334500.csv 1.234457 0.044881 Kedarry Ransome USGS_DailyDischarge04269000.csv 1.644890 0.054368 Tyrone Shillingford USGS_DailyDischarge01421000.csv 1.079200 0.027393 Scott Stange USGS_DailyDischarge01426500.csv 0.993842 0.036449 Erika Villamarin USGS_DailyDischarge01420500.csv 1.051408 0.064590 Thomas Wen USGS_DailyDischarge01315000.csv 0.813469 0.096163 Jason Woo USGS_DailyDischarge01346000.csv 2.113566 0.054545 Peter Wytrwal USGS_DailyDischarge04263000.csv 1.528064 0.030526 Kevin Zea USGS_DailyDischarge01357500.csv 1.203781 0.007219 Yuan Zheng USGS_DailyDischarge04273500.csv 2.410184 0.096729 Sallem Ahmed USGS_DailyDischarge04262500.csv 1.291862 0.085697 Rowan Bado USGS_DailyDischarge04260500.csv 1.890853 0.015762 Benjamin Brandwein USGS_DailyDischarge01526500.csv 0.693463 0.017283 Cleff Egalite USGS_DailyDischarge01367500.csv 0.873177 0.046422 Mazen Elashrafi USGS_DailyDischarge01529500.csv 0.833884 0.063013 Rony Espinal USGS_DailyDischarge01318500.csv 1.511866 0.017872 Jairo Galarza Granda USGS_DailyDischarge01371500.csv 0.878546 0.027955 Rajvinder Kaur USGS_DailyDischarge01347000.csv 1.963423 0.024068 Marlon Landauro USGS_DailyDischarge04249000.csv 1.747992 0.008928 Noeliz Lee USGS_DailyDischarge04234000.csv 0.949727 0.177709 Jeanette Lopez USGS_DailyDischarge01336000.csv 2.498320 0.234872 Albert Munoz USGS_DailyDischarge01509000.csv 1.042391 0.072666 Berenice Oseguera USGS_DailyDischarge01372500.csv 0.881932 0.117412 Veronica Praino USGS_DailyDischarge04258000.csv 3.819987 0.215173 David Rauch USGS_DailyDischarge04270000.csv 2.742039 0.405824 Jermaine Reynolds USGS_DailyDischarge01362500.csv 1.479126 0.069039 Luzelenie Rojas USGS_DailyDischarge04275500.csv 1.212841 0.060671 John Salvo USGS_DailyDischarge01375000.csv 0.313407 0.030590 Peenija Thapa USGS_DailyDischarge01200000.csv 1.015546 0.112675 Hingkei Tsang USGS_DailyDischarge03014500.csv 0.989663 0.094614 Jah-Vin Vaughan USGS_DailyDischarge04242500.csv 0.981092 0.063831 Jiaming Ye USGS_DailyDischarge01413500.csv 0.901077 0.100100 Salman Zafar USGS_DailyDischarge01502500.csv 0.957847 0.038871