def ci_mean(m_sample, s_sample, ndof_eff, alpha=0.95, verbose=True):
"""
Calculates a confidence interval at the 'alpha' significance level
for the sample mean of a normally-distributed random variable with
sample mean 'm_sample', sample standard deviation 's_sample' and
effective degrees of freedom 'ndof_eff'.
References
----------
TODO
Example
-------
TODO
"""
# z-score of the CI associated with the given significance level 'alpha'.
# Standard normal curve is symmetric about the mean (0), therefore take
# only the upper CI.
zs = norm.interval(alpha)[1]
# Lower (upper) 100 * alpha % confidence interval.
std_err = s_sample/np.sqrt(ndof_eff)
xl = m_sample - zs*std_err
xu = m_sample + zs*std_err
if verbose:
print("")
print("Sample mean CI (xl,xu): (%.3f,%.3f)"%(xl, xu))
print("")
return (xl,xu)
def ci_mean(m_sample, s_sample, ndof_eff, alpha=0.95, verbose=True): """ Calculates a confidence interval at the 'alpha' significance level for the sample mean of a normally-distributed random variable with sample mean 'm_sample', sample standard deviation 's_sample' and effective degrees of freedom 'ndof_eff'. References ---------- TODO Example ------- TODO """ # z-score of the CI associated with the given significance level 'alpha'. # Standard normal curve is symmetric about the mean (0), therefore take # only the upper CI. zs = norm.interval(alpha)[1] # Lower (upper) 100 * alpha % confidence interval. std_err = s_sample/np.sqrt(ndof_eff) xl = m_sample - zs*std_err xu = m_sample + zs*std_err if verbose: print "" print "Sample mean CI (xl,xu): (%.3f,%.3f)"%(xl, xu) print "" return (xl,xu)
def rci_fisher(r, ndof_eff, alpha=0.95, verbose=True):
"""
Calculate a confidence interval for the Pearson correlation coefficient r
between two series 'x' and 'y' using the Fisher's transformation method.
References
----------
Cox (2008): Speaking Stata: Correlation with confidence, or Fisher’s z revisited.
The Stata Journal (2008) 8, Number 3, pp. 413-439.
https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient
Example
-------
TODO
"""
## OBS: equivalent to the form z_r = 0.5*np.log((1+r)/(1-r))
## which is also commonly found in the literature.
z_r = np.arctanh(r) # Transform r to a standard normal variable z_r.
se_r = 1./np.sqrt(ndof_eff-3) # Standard error of the transformed distribution.
# z-score of the CI associated with the given significance level 'alpha'.
# Standard normal curve is symmetric about the mean (0), therefore take
# only the upper CI.
zs = norm.interval(alpha)[1]
# Lower (upper) 100 * alpha % confidence interval.
z_xl = z_r - zs*se_r
z_xu = z_r + zs*se_r
## Use the inverse transformation to convert intervals
## in the z-scale back to the r-scale.
xl = np.tanh(z_xl)
xu = np.tanh(z_xu)
if verbose:
print("")
print("Fisher transform CI (xl,xu): (%.3f,%.3f)"%(xl, xu))
print("")
return (xl,xu)
def rci_fisher(r, ndof_eff, alpha=0.95, verbose=True): """ Calculate a confidence interval for the Pearson correlation coefficient r between two series 'x' and 'y' using the Fisher's transformation method. References ---------- Cox (2008): Speaking Stata: Correlation with confidence, or Fisher’s z revisited. The Stata Journal (2008) 8, Number 3, pp. 413-439. https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient Example ------- TODO """ ## OBS: equivalent to the form z_r = 0.5*np.log((1+r)/(1-r)) ## which is also commonly found in the literature. z_r = np.arctanh(r) # Transform r to a standard normal variable z_r. se_r = 1./np.sqrt(ndof_eff-3) # Standard error of the transformed distribution. # z-score of the CI associated with the given significance level 'alpha'. # Standard normal curve is symmetric about the mean (0), therefore take # only the upper CI. zs = norm.interval(alpha)[1] # Lower (upper) 100 * alpha % confidence interval. z_xl = z_r - zs*se_r z_xu = z_r + zs*se_r ## Use the inverse transformation to convert intervals ## in the z-scale back to the r-scale. xl = np.tanh(z_xl) xu = np.tanh(z_xu) if verbose: print "" print "Fisher transform CI (xl,xu): (%.3f,%.3f)"%(xl, xu) print "" return (xl,xu)
data=FilteredSystem).fit()
CVEffect_Models = [CV_Model_CVEffect, F_CV_Model_CVEffect]
Datasets = ['Complete', 'Filtered']
SystemFitted = [System2Fit, FilteredSystem]
## Analyze distribution of Ln(CV)
for i in range(2):
D = SystemFitted[i]['LogCV'].values
D.sort()
D_bar = np.mean(D)
S_D = np.std(D, ddof=1)
N_D = len(D)
## Kernel density estimation (Gaussian kernel)
KernelEstimator = np.zeros(N_D)
NormalIQR = np.abs(norm.interval(0.25, 0, 1)).sum()
DataIQR = np.abs(np.quantile(D, 0.75)) - np.abs(np.quantile(D, 0.25))
KernelHalfWidth = 0.9 * N_D**(-1 / 5) * S_D
for Value in D:
KernelEstimator += norm.pdf(D - Value, 0, KernelHalfWidth * 2)
KernelEstimator = KernelEstimator / N_D
## Histogram and density distribution
TheoreticalDistribution = norm.pdf(D, D_bar, S_D)
Figure, Axes = plt.subplots(1, 1, figsize=(5.5, 4.5), dpi=100)
Histogram = Axes.hist(D,
density=True,
bins=20,
edgecolor=(0, 0, 1),
color=(1, 1, 1),
label='Histogram')
def PermutationTest(x,y,NRepetition=45**2,SignificanceLevel=0.05):
# Analyze data
x_bar = np.mean(x)
y_bar = np.mean(y)
d = x_bar - y_bar
XData = pd.DataFrame({'Values':x,'Group':'Control'},index=range(len(x)))
YData = pd.DataFrame({'Values':y,'Group':'Test'},index=range(len(y)))
Pool = XData.append(YData,ignore_index=True)
N = len(Pool)
D = np.zeros(NRepetition)
for i in range(NRepetition):
n = np.random.randint(1,N-1)
SampleA = Pool.sample(n)
SampleB = Pool.drop(SampleA.index)
D[i] = SampleA['Values'].mean() - SampleB['Values'].mean()
# Analyze distribution of D
from scipy.stats.distributions import norm
D.sort()
D_bar = np.mean(D)
S_D = np.std(D,ddof=1)
N_D = len(D)
# Kernel density estimation (Gaussian kernel)
KernelEstimator = np.zeros(N_D)
NormalIQR = np.abs(norm.interval(0.25,0,1)).sum()
DataIQR = np.abs(np.quantile(D,0.75)) - np.abs(np.quantile(D,0.25))
KernelHalfWidth = 0.9 * N_D ** (-1 / 5) * S_D
for Value in D:
KernelEstimator += norm.pdf(D - Value, 0, KernelHalfWidth * 2)
KernelEstimator = KernelEstimator / N_D
## Histogram and density distribution
TheoreticalDistribution = norm.pdf(D, D_bar, S_D)
Figure, Axes = plt.subplots(1, 1, figsize=(5.5, 4.5), dpi=100)
Histogram = Axes.hist(D, density=True, bins=20, edgecolor=(0, 0, 1), color=(1, 1, 1), label='Histogram')
Axes.plot(D, KernelEstimator, color=(1, 0, 0), label='Kernel Density')
Axes.plot(D, TheoreticalDistribution, linestyle='--', color=(0, 0, 0), label='Normal Distribution')
plt.xlabel('D values')
plt.ylabel('Density (-)')
plt.legend(loc='upper center', ncol=3, bbox_to_anchor=(0.5, 1.15), prop={'size':10})
plt.show()
plt.close(Figure)
EmpiricalQuantiles = np.arange(0.5, N_D + 0.5) / N_D
MinValue = np.quantile(D,SignificanceLevel / 2)
MaxValue = np.quantile(D,1 - SignificanceLevel / 2)
RejectionRange = np.array([[-np.inf,MinValue],[MaxValue,np.inf]])
Figure, Axes = plt.subplots(1, 1, figsize=(5.5, 4.5), dpi=100)
Histogram = Axes.hist(D, density=True, bins=20, edgecolor=(0, 0, 1), color=(1, 1, 1), label='Histogram')
Axes.fill_between([min(D),MinValue], [max(Histogram[0]),max(Histogram[0])], color=(0, 0, 0), alpha=0.1)
Axes.fill_between([max(D),MaxValue], [max(Histogram[0]),max(Histogram[0])], color=(0, 0, 0), alpha=0.1, label='Rejection range')
Axes.plot([d,d], [0,max(Histogram[0])], color=(1, 0, 0), label='Actual difference')
plt.xlabel('D values')
plt.ylabel('Density (-)')
plt.legend(loc='upper center', ncol=3, bbox_to_anchor=(0.5, 1.15), prop={'size':10})
plt.show()
plt.close(Figure)
p = len(D[abs(D)>=abs(d)]) / len(D)
# Z = (D - D_bar) / S_D
# z_d = (d - D_bar) / S_D
# TheoreticalQuantiles = norm.cdf(Z)
#
# # Compute range of Z values
# D_zmin = D_bar - 10 * S_D
# D_zmax = D_bar + 10 * S_D
#
# Step = 0.001
# x = np.arange(D.min(), D.max(), Step) # range of x in spec
# y = norm.pdf(x, D_bar, S_D)
#
# x_all = np.arange(D_zmin, D_zmax, Step) # entire range of x, both in and out of spec
# # y_all = norm.pdf(x_all, D_bar, S_D)
#
# x_all = np.arange(-10, 10, Step) # entire range of x, both in and out of spec
# y_all = norm.pdf(x_all, 0, 1)
# y_d = norm.pdf(z_d, 0, 1)
#
# y_sorted = np.zeros(len(y_all))
# y_sorted += y_all
# y_sorted.sort()
#
# CI = 0.95
# y_area = 0
# i = 1
# while y_area / y_all.sum() < CI:
# y_area += y_sorted[-i]
# i += 1
# z_CI = i / 2 * Step
#
# # Entire range of x, both in and out of spec
# x_CI = np.arange(-z_CI, z_CI, Step)
# y_CI = norm.pdf(x_CI, 0, 1)
#
# # Plot in data space
# Figure, Axes = plt.subplots(1, 1, figsize=(5.5, 4.5), dpi=100)
# Axes.fill_between(x_CI, y_CI, 0, alpha=0.15, color=(0, 0, 0), label=str(int(0.95 * 100)) + '% CI')
# Axes.plot([z_d,z_d], [0,y_d], color=(0, 0, 1), label='Difference Observed')
# Axes.plot(x_all, y_all, color=(1, 0, 0), label='Normal distribution')
# Axes.set_xlabel('Z values')
# # plt.xlim([D_bar - 4.2 * S_D, D_bar + 4.2 * S_D])
# plt.xlim([-5, 5])
# plt.ylim([0, 0.45])
# plt.legend(loc='upper center', ncol=3, bbox_to_anchor=(0.5, 1.15))
# # plt.show()
# plt.close()
#
# # Plot in data space
# d_CI = (z_CI + D_bar) * S_D
# dx_CI = np.arange(-d_CI, d_CI, Step)
# dy_CI = norm.pdf(dx_CI, D_bar, S_D)
# d_y = norm.pdf(d, D_bar, S_D)
#
# Figure, Axes = plt.subplots(1, 1, figsize=(5.5, 4.5), dpi=100)
# Axes.fill_between(dx_CI, dy_CI, 0, alpha=0.15, color=(0, 0, 0), label=str(int(0.95 * 100)) + '% CI')
# Axes.plot([d, d], [0, d_y], color=(0, 0, 1), label='Difference Observed')
# Axes.plot(D, TheoreticalDistribution, color=(1, 0, 0), label='Normal distribution')
# Axes.set_xlabel('D values')
# plt.ylim([0,max(TheoreticalDistribution)*1.05])
# plt.legend(loc='upper center', ncol=3, bbox_to_anchor=(0.5, 1.15))
# plt.show()
return d, RejectionRange, p
