def construct_IDF(self):
if self.ci:
bts = {}
for col in self.reformatted_ams.columns:
mams = []
for i in range(self.number_bootstrap):
bootsams = np.random.choice(
self.reformatted_ams[col].values, replace=True, size=len(self.reformatted_ams))
fit = gev.fit(bootsams)
mams.append(
gev.isf(self.quantiles, c=fit[0], loc=fit[1], scale=fit[2]))
bts[col] = np.asarray(mams)
p_lo = ((1.0-self.alpha)/2.0) * 100
p_up = (self.alpha+((1.0-self.alpha)/2.0)) * 100
for col in self.reformatted_ams.columns:
lower = np.apply_along_axis(np.percentile, 0, bts[col], p_lo)
upper = np.apply_along_axis(np.percentile, 0, bts[col], p_up)
median = np.apply_along_axis(
np.percentile, 0, bts[col], 50)
self.idf[col] = np.append(lower, np.append(median, upper))
else:
for col in self.reformatted_ams.columns:
fit = gev.fit(self.reformatted_ams[col])
self.idf[col] = gev.isf(self.quantiles, c=fit[0],
loc=fit[1], scale=fit[2])
def extreme_values(weighted_residuals, confidence_interval):
'''
This function uses extreme value theory to calculate the number of
standard deviations away from the mean at which we should expect to bracket
*all* of our n data points at a certain confidence level.
It then uses that value to identify which (if any) of the data points
lie outside that region, and calculates the corresponding probabilities
of finding a data point at least that many standard deviations away.
Parameters
----------
weighted_residuals : array of floats
Array of residuals weighted by the square root of their
variances wr_i = r_i/sqrt(var_i)
confidence_interval : float
Probability at which all the weighted residuals lie
within the confidence bounds
Returns
-------
confidence_bound : float
Number of standard deviations at which we should expect to encompass
all data at the user-defined confidence interval.
indices : array of floats
Indices of weighted residuals exceeding the confidence_interval
defined by the user
probabilities : array of floats
The probabilities that the extreme data point of the distribution lies
further from the mean than the observed position wr_i for each i in
the "indices" output array.
'''
n = len(weighted_residuals)
mean = norm.isf(1./n)
# good approximation for > 10 data points
scale = 0.8/np.power(np.log(n), 1./2.)
# good approximation for > 10 data points
c = 0.33/np.power(np.log(n), 3./4.)
# We now need a 1-tailed probability from the given confidence_interval
# p_total = 1. - confidence_interval = p_upper + p_lower - p_upper*p_lower
# p_total = 1. - confidence_interval = 2p - p^2, therefore:
p = 1. - np.sqrt(confidence_interval)
confidence_bound = genextreme.isf(p, c, loc=mean, scale=scale)
indices = [i for i, r in enumerate(weighted_residuals)
if np.abs(r) > confidence_bound]
# Convert back to 2-tailed probabilities
probabilities = (1.
- np.power(genextreme.sf(np.abs(weighted_residuals[indices]),
c, loc=mean, scale=scale) - 1., 2.))
return confidence_bound, indices, probabilities
def extreme_values(weighted_residuals, confidence_interval):
'''
This function uses extreme value theory to calculate the number of
standard deviations away from the mean at which we should expect to bracket
*all* of our n data points at a certain confidence level.
It then uses that value to identify which (if any) of the data points
lie outside that region, and calculates the corresponding probabilities
of finding a data point at least that many standard deviations away.
Parameters
----------
weighted_residuals : array of floats
Array of residuals weighted by the square root of their
variances wr_i = r_i/sqrt(var_i)
confidence_interval : float
Probability at which all the weighted residuals lie
within the confidence bounds
Returns
-------
confidence_bound : float
Number of standard deviations at which we should expect to encompass all
data at the user-defined confidence interval.
indices : array of floats
Indices of weighted residuals exceeding the confidence_interval
defined by the user
probabilities : array of floats
The probabilities that the extreme data point of the distribution lies
further from the mean than the observed position wr_i for each i in
the "indices" output array.
'''
n=len(weighted_residuals)
mean = norm.isf(1./n)
scale = 0.8/np.power(np.log(n), 1./2.) # good approximation for > 10 data points
c = 0.33/np.power(np.log(n), 3./4.) # good approximation for > 10 data points
# We now need a 1-tailed probability from the given confidence_interval
# p_total = 1. - confidence_interval = p_upper + p_lower - p_upper*p_lower
# p_total = 1. - confidence_interval = 2p - p^2, therefore:
p = 1. - np.sqrt(confidence_interval)
confidence_bound = genextreme.isf(p, c, loc=mean, scale=scale)
indices = [i for i, r in enumerate(weighted_residuals) if np.abs(r) > confidence_bound]
probabilities = 1. - np.power(genextreme.sf(np.abs(weighted_residuals[indices]), c, loc=mean, scale=scale) - 1., 2.) # Convert back to 2-tailed probabilities
return confidence_bound, indices, probabilities
def plot_return_values(annual_max, station_id):
fig, axes = plt.subplots(figsize=(20,6))
T=np.r_[1:500]
mle = genextreme.fit(sorted(annual_max), 0)
mu = mle[1]
sigma = mle[2]
xi = mle[0]
# print "The mean, sigma, and shape parameters are %s, %s, and %s, resp." % (mu, sigma, xi)
sT = genextreme.isf(1./T, 0, mu, sigma)
axes.semilogx(T, sT, 'r'), hold
N=np.r_[1:len(annual_max)+1];
Nmax=max(N);
axes.plot(Nmax/N, sorted(annual_max)[::-1], 'bo')
title = station_id
axes.set_title(title)
axes.set_xlabel('Return Period (yrs)')
axes.set_ylabel('Wind Speed (m/s)')
axes.grid(True)
def plot_return_values(annual_max, station_id):
fig, axes = plt.subplots(figsize=(20, 6))
T = np.r_[1:500]
mle = genextreme.fit(sorted(annual_max), 0)
mu = mle[1]
sigma = mle[2]
xi = mle[0]
# print "The mean, sigma, and shape parameters are %s, %s, and %s, resp." % (mu, sigma, xi)
sT = genextreme.isf(1. / T, 0, mu, sigma)
axes.semilogx(T, sT, 'r'), hold
N = np.r_[1:len(annual_max) + 1]
Nmax = max(N)
axes.plot(Nmax / N, sorted(annual_max)[::-1], 'bo')
title = station_id
axes.set_title(title)
axes.set_xlabel('Return Period (yrs)')
axes.set_ylabel('Wind Speed (m/s)')
axes.grid(True)
# <codecell>
Image(url='http://tidesandcurrents.noaa.gov/est/images/color_legend.png')
# <markdowncell>
# <script type="text/javascript">
# $('div.input').show();
# </script>
# <codecell>
fig = plt.figure(figsize=(20,6))
axes = fig.add_axes([0.1, 0.1, 0.8, 0.8])
T=np.r_[1:250]
sT = genextreme.isf(1./T, 0, mu, sigma)
axes.semilogx(T, sT, 'r'), hold
N=np.r_[1:len(annual_max_levels)+1];
Nmax=max(N);
axes.plot(Nmax/N, sorted(annual_max_levels)[::-1], 'bo')
title = s["long_name"][0]
axes.set_title(title)
axes.set_xlabel('Return Period (yrs)')
axes.set_ylabel('Meters above MHHW')
axes.set_xticklabels([0,1,10,100,1000])
axes.set_xlim([0,260])
axes.set_ylim([0,1.8])
axes.grid(True)
# <markdowncell>
sigma_wis = mle_wis[2]
xi_wis = mle_wis[0]
print "The mean, sigma, and shape parameters are %s, %s, and %s, resp." % (
mu_wis, sigma_wis, xi_wis)
# <markdowncell>
# ### Return Value Plot
# <codecell>
fig, axes = plt.subplots(2, 1, figsize=(20, 12))
# fig = plt.figure()
# axes = fig.add_axes([0.1, 0.1, 0.8, 0.8])
T = np.r_[1:500]
sT = genextreme.isf(1. / T, 0, mu, sigma)
axes[0].semilogx(T, sT, 'r'), hold
N = np.r_[1:len(annual_max) + 1]
Nmax = max(N)
axes[0].plot(Nmax / N, sorted(annual_max)[::-1], 'bo')
title = station_longName
axes[0].set_title(title)
axes[0].set_xlabel('Return Period (yrs)')
axes[0].set_ylabel('Significant Wave Height (m)')
axes[0].grid(True)
sT_wis = genextreme.isf(1. / T, 0, mu_wis, sigma_wis)
axes[1].semilogx(T, sT_wis, 'r'), hold
N = np.r_[1:len(wis_maximums) + 1]
Nmax = max(N)
axes[1].plot(Nmax / N, sorted(wis_maximums)[::-1], 'bo')
mu_wis = mle_wis[1]
sigma_wis = mle_wis[2]
xi_wis = mle_wis[0]
print "The mean, sigma, and shape parameters are %s, %s, and %s, resp." % (mu_wis, sigma_wis, xi_wis)
# <markdowncell>
# ### Return Value Plot
# <codecell>
fig, axes = plt.subplots(2, 1, figsize=(20,12))
# fig = plt.figure()
# axes = fig.add_axes([0.1, 0.1, 0.8, 0.8])
T=np.r_[1:500]
sT = genextreme.isf(1./T, 0, mu, sigma)
axes[0].semilogx(T, sT, 'r'), hold
N=np.r_[1:len(annual_max)+1];
Nmax=max(N);
axes[0].plot(Nmax/N, sorted(annual_max)[::-1], 'bo')
title = station_longName
axes[0].set_title(title)
axes[0].set_xlabel('Return Period (yrs)')
axes[0].set_ylabel('Significant Wave Height (m)')
axes[0].grid(True)
sT_wis = genextreme.isf(1./T, 0, mu_wis, sigma_wis)
axes[1].semilogx(T, sT_wis, 'r'), hold
N=np.r_[1:len(wis_maximums)+1];
Nmax=max(N);
axes[1].plot(Nmax/N, sorted(wis_maximums)[::-1], 'bo')
