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Exceedance probability curves

While working on a project proposal to estimate drought risks of some countries in the coming years, I learned something about risk analysis: It's rather simple! But, you need to be cautious if you want to use this information for guessing what might happen in the future. In the context of anthropogenic climate change that usually means that you can't rely on exceedance probabilities from observed (and hence past) data. Because the underlying climate is changing. And even if that's not the case, your record may be over- or underestimating the occurrences of specific events, just by pure chance.

However, let me show you how to quickly derive exceedance probabilities from some river discharge data.

From the monthly discharge data, we estimate the cumulative distribution function (CDF) by assigning each sorted monthluy value a number starting from 0 for the lowest up to 1 for the highest monthly discharge value. The exceedance probability is simply 1 - CDF. Plotting the exceedance probability against the sorted monthly data, et voilĂ , your exceedance probability curve.

Here's the link to the CSV file elbe.csv and the Python code to process the data.

import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
from scipy import stats
import sys
import matplotlib
from matplotlib.ticker import ScalarFormatter, FormatStrFormatter, FixedLocator

matplotlib.style.use('bmh')

fnm  = 'elbe.csv'
raw = df['Elbe']
ser = raw.sort_values()

X = np.linspace(0.,max(ser.values)*1.1,100.)
s, loc, scale = stats.lognorm.fit(ser.values)

cum_dist = np.linspace(0.,1.,len(ser))
ser_cdf = pd.Series(cum_dist, index=ser)
ep = 1. - ser_cdf

fig = plt.figure(figsize=(11,8))

ax1 = plt.subplot(221)
ser_cdf.plot(ax=ax1,drawstyle='steps',label='data')
ax1.plot(X,stats.lognorm.cdf(X, s, loc, scale),label='lognormal')
ax1.set_xlabel('Discharge')
ax1.set_ylabel('CDF')
ax1.legend(loc=0,framealpha=0.5,fontsize=12)

ax2 = plt.subplot(223)
ax2.hist(ser.values, bins=21, normed=True, label='data')
ax2.plot(X,stats.lognorm.pdf(X, s, loc, scale),label='lognormal')
ax2.set_ylabel('Probability Density')
ax2.set_xlabel('Discharge')
ax2.legend(loc=0,framealpha=0.5,fontsize=12)

A = np.vstack([ep.values, np.ones(len(ep.values))]).T
[m, c], resid = np.linalg.lstsq(A, ser.values)[:2]
print m, c
r2 = 1 - resid / (len(ser.values) * np.var(ser.values))
print r2

ax3 = plt.subplot(222)
ax3.semilogx(100.*ep,ser.values,ls='',marker='o',label='data')
ax3.plot(100.*(1.-stats.lognorm.cdf(X, s, loc, scale)),X,label='lognormal')
minorLocator = FixedLocator([1,2,5,10,20,50,100])
ax3.xaxis.set_major_locator(minorLocator)
ax3.xaxis.set_major_formatter(ScalarFormatter())
ax3.xaxis.set_major_formatter(FormatStrFormatter("%d"))
ax3.set_xlim(1,100)
ax3.set_xlabel('Exceedance Probability (%)')
ax3.set_ylabel('Discharge')
ax3.invert_xaxis()
ax3.legend(loc=0,framealpha=0.5,fontsize=12)

ax4 = plt.subplot(224)
raw.plot(ax=ax4)
ax4.set_xlabel('time')
ax4.set_ylabel('Discharge')

plt.suptitle('Elbe discharge',y=1.01,fontsize=14)

plt.tight_layout()
plt.show()

fig.savefig('exceedance_curve_elbe.png',transparent=True,dpi=150,bbox_inches='tight')

 Tags: statistics, python, code, models 2016/02/20
My new job

That one went fast. And I hadn't seen that job coming, neither. But I've now found a new job at an organization supporting Least Developed Countries and Small Island Developing States in their efforts during climate negotiations. Here's the link to their website. And here's the good thing about it: I can go there by bike! That was (nearly) impossible at my former institute in Potsdam (though I did take the bike to get to and from the train station(s)).

For this reason, I had to change the title of this blog and I leave it open to new suggestions about what to write next. I'm a little bit sad just having 8 blog entries. Let me just think, maybe I come up with something great, next time...

 Tags: job 2015/02/20