| I'm doing pretty much the same thing, I also have the Frechet and Weibull
distributions (the Weibull having 3 parameters often beats the others). The results are plotted on a log graph with the data and even though the lines are not straight (as they would be using specifically developed graph paper) you can see which ones fit best, and you can still do stat tests to see which ones seem to be better fits.
On that matter, I have often found that rather than spending too much time calculating best-fit lines, you can get very useful information by looking at the river cross-sections, with its flood plains and restrictions (for flow data) or storm patterns (wind speed and direction, general origins of various winds, usual types of rainfall in the area...) for rainfall data.
This info will guide you better into choosing the better distribution. |
| With reference strictly to the original question, of how to construct Gumbel paper:
Mostafa's reference is a good one. The trick in Excel to make a Gumbel plot or a normal (or lognormal) plot is that you have to draw yourself the grid lines corresponding to the probability values. As explained by Stedinger et al., the idea is to get a linear plot of magnitude, x, vs. some function of cumulative probablity, F, or exceedance probability G = 1-F. In Stedinger's notation, the empirical value for G is q, the plotting position (e.g., by Cunnane). For the Gumbel distribution, the reduced variate, y = alpha(x-u) [u corresponds to Greek Xi in Stedinger Eqn. 18.2.14]. And for Gumbel, F 1-G = exp[-exp(-y)], easily inverted to find y as function of F or G. Letting q be the plotting position estimate of G, the inversion is Stedinger's Eqn. 18.3.9.
So, the procedure is:
1. Tabulate in ranked order, largest to smallest, the magnitudes and corresponding rank.
2. Compute q, the plotting position, e.g., by Cunnane position. 3. Use Eqn. 18.3.9 to compute the equivalent y. 4. Finally, plot magnitude x vs. y (theoretically, a linear relationship). The slope gives alpha and the intercept gives u. (Knowing alpha and u, you can also plot the fitted straight line found by the method of moments or other procedure.)
5. You may label the y-values with whatever the y-numbers are, but the idea of probability paper is to label them instead with their corresponding values of F or G, computed from the double-exponential relationship. And you have to draw the y-grid lines yourself.
The same method works just fine for constructing normal probability paper (or lognormal) in Excel. Here, magnitude = mean + z*stdev, a linear relationship, and z corresponds to a unique frequency, as in Stedinger Fig. 18.3.1. Use Excel's NORMINV for step 3 above and NORMDIST function for step 5 (to get the labels for the probability grid lines that correspond to the linear z-scale). This is just a bit more tedious than for the Gumbel, but you can construct a plot looking exactly like normal probability paper that you might purchase (or download -- see Paul's website). For a lognormal plot, ask Excel to plot magnitudes on a log scale. (Ditto for a log-Gumbel.)
Probably not as easy as it sounds, but it works for me. And once you have a template, it's easy to apply to different data. Maybe there's an Excel add-in to do all this. And there are likely graphics packages that make probability plots.
The advantage of this procedure over downloading the very handy and useful graph paper that Paul found on the Humboldt website is that using the latter means penciling in the data points on the paper (or "painting" data points onto the electronic graph paper background), whereas the Excel plot is electronic start to finish. But your Excel plot should look the same as if you'd used the graph paper from the Humboldt site.
And finally, speaking of brute force methods (i.e., pencil, paper and straight-edge), the Humboldt site reminds me of one of the most useful books I've ever owned, at least pre-Excel:
Carver, J.S., Graph Paper from your Computer or Copier, Fisher Books, Tucson, AZ, 3rd edition, 1996.
The idea is to photocopy the page you need, but the book also includes a floppy disk with .bmp files of most of the over 200 "sheets" of graph paper, including more log-cycles than you'd ever want to encounter, triangles, maps, polar coordinates, probability paper and almost every "ordinary" kind of graph paper you'd ever want. Of course, the .bmp files are only good as background or to print out, suffering the same disadvantage as the Humboldt website paper. Nonetheless, I love this book! |
| Hello, the general probability equation is
Q = Mean + Freq Factor *SD
As a result, the line, Q versus Freq Factor, is a "linear straight" line. To produce a Gumbel graphic paper, you need to produce a set of Gumbel Freq Factors for 2-, 5-, 10-, 25-, 50-, and 100-yr (The equation can be found on many textbooks). Let Q be plotted on the vertical axis, and Freq Factors be "linearly" plotted on the x-axis. Of course, you can convert the frequency factors (linear) back to their non-exceedance probabilities (nonlinear). If you prefer "Log-Gumbel", you may plot either Log (Q) or Q on the log-scale for the vertical axis.
The same procedure can be repeated for the Normal or Log-normal distribution when the Normal frequency factors, Z, are used for the x-axis. |
May 20, 2013