def norm_cdf_plot(sample, alpha): #Plots the normal theoretical cdf compared to the empirical one
mean = mou.mean_sample(sample)
std = mou.std_sample(sample)
step = abs(min(sample) - max(sample))/100
x = np.arange(min(sample), max(sample) + step, step)
cdf = st.norm.cdf(x, mean, std)
n = len(sample)
y = np.arange(1, n+1)/n
F1 = []
F2 = []
for i in range(0, len(sample)):
e = (((mt.log(2/alpha))/(2*n))**0.5)
F1.append(y[i] - e)
F2.append(y[i] + e)
plt.plot(x, cdf, color = 'dimgray', label = 'Theoretical CDF')
plt.scatter(sorted(sample), y, label='Empirical CDF')
plt.plot(sorted(sample), F1, linestyle='--', color='red', alpha = 0.8, lw = 0.9, label = 'Dvoretzky–Kiefer–Wolfowitz Confidence Bands')
plt.plot(sorted(sample), F2, linestyle='--', color='red', alpha = 0.8, lw = 0.9)
plt.ylabel('Cumulative Distribution Function')
plt.xlabel('Observed Data')
plt.legend()
plt.show()
return cdf
def norm_qq_plot(sample, alpha): #plots the quantile-quantie plot for the given data
y = np.arange(1, len(sample)+1)/(len(sample)+1)
mean = mou.mean_sample(sample)
std = mou.std_sample(sample)
theo_qq = phi_inverse(y)
x = theo_qq*std + mean
#Kolmogorov-Smirnov Test for getting the confidence interval
K = (-0.5*mt.log(alpha/2))**0.5
M = (len(sample)**2/(2*len(sample)))**0.5
CI_qq_high = []
CI_qq_low = []
for prob in y:
F1 = prob - K/M
F2 = prob + K/M
s_low = phi_inverse(F1)
s_high = phi_inverse(F2)
CI_qq_low.append(s_low*std + mean)
CI_qq_high.append(s_high*std + mean)
sns.regplot(x, sorted(sample), ci = None, line_kws={'color':'dimgray','label':'Regression Line'})
plt.plot(sorted(sample), CI_qq_low, linestyle='--', color='red', alpha = 1, lw = 0.8, label = 'Kolmogorov-Smirnov Confidence Bands')
plt.legend()
plt.plot(sorted(sample), CI_qq_high, linestyle='--', color='red', alpha = 1, lw = 0.8)
plt.xlabel('Theoretical Normal Quantiles')
plt.ylabel('Sample Quantiles')
plt.show()
def norm_pdf_plot(sample): #Plots the normal pdf with normalized histogram
mean = mou.mean_sample(sample)
std = mou.std_sample(sample)
step = abs(min(sample) - max(sample))/100
x = np.arange(min(sample), max(sample) + step, step)
pdf = st.norm.pdf(x, mean, std)
plt.plot(x, pdf, color = 'dimgray', label = 'Theoretical PDF')
plt.legend()
mou.hist(sample, 'sturges', dens_norm = True)
return pdf
def chi_square_test(sample, alpha):
m = mt.ceil(1 + 3.322 * mt.log(len(sample), 10))
k = 2
f = m - 1 - k
c = chi2.ppf(1 - alpha, f)
h = (max(sample) - min(sample)) / m
mean = mou.mean_sample(sample)
std = mou.std_sample(sample)
class_ = [min(sample)]
for i in range(0, m):
class_.append(class_[i] + h)
e = []
for i in range(1, len(class_)):
if i == 1:
e.append(npd.phi(class_[i], mean, std) * len(sample))
elif i == (len(class_)):
e.append((1 - npd.phi(class_[i - 1], mean, std)) * len(sample))
else:
e.append((npd.phi(class_[i], mean, std) -
npd.phi(class_[i - 1], mean, std)) * len(sample))
n = []
t = 0
i = 1
sample = sorted(sample)
sample.append(max(sample) + 1)
for j in range(0, len(sample) - 1):
if sample[j + 1] > class_[i]:
n.append((j + 1) - t)
t = j + 1
i = i + 1
test_1 = []
for i in range(0, len(n)):
test_1.append(((e[i] - n[i])**2) / e[i])
statistics = sum(test_1)
if statistics < c:
print(
'Chi-Square Statistics = {}\nP-Value = {}\nConsidering that Chi-Square Statistics < P-Value, with a confidence level of {}%, the normal distribution is acceptable.'
.format(statistics, c, (1 - alpha) * 100))
elif statistics > c:
print(
'Chi-Square Statistics = {}\nP-Value = {}\nConsidering that Chi-Square Statistics > P-Value, with a confidence level of {}%, the normal distribution is not acceptable.'
.format(statistics, c, (1 - alpha) * 100))
def lognorm_pdf_plot(sample):
ln_x = []
for data in sample:
ln_x.append(mt.log(data))
mean = mou.mean_sample(sample)
std = mou.std_sample(ln_x)
step = abs(min(sample) - max(sample)) / 100
x = np.arange(min(sample), max(sample) + step, step)
pdf = st.lognorm.pdf(x, s=std, scale=mean, loc=0)
plt.plot(x, pdf, color='dimgray', label='Theoretical PDF')
plt.legend()
mou.hist(sample, 'sturges', dens_norm=True)
print(pdf)
return pdf
def lognorm_qq_plot(
sample, alpha): #plots the quantile-quantie plot for the given data
y = np.arange(1, len(sample) + 1) / (len(sample) + 1)
ln_x = []
for data in sample:
ln_x.append(mt.log(data))
mean = mou.mean_sample(sample)
std = mou.std_sample(ln_x)
theo_qq = st.lognorm.ppf(y, s=std, loc=0, scale=mean)
x = theo_qq
#Kolmogorov-Smirnov Test for getting the confidence interval
K = (-0.5 * mt.log(alpha / 2))**0.5
M = (len(sample)**2 / (2 * len(sample)))**0.5
CI_qq_high = []
CI_qq_low = []
for prob in y:
F1 = prob - K / M
F2 = prob + K / M
s_low = st.lognorm.ppf(F1, s=std, loc=0, scale=mean)
s_high = st.lognorm.ppf(F2, s=std, loc=0, scale=mean)
CI_qq_low.append(s_low)
CI_qq_high.append(s_high)
sns.regplot(x,
sorted(sample),
ci=None,
line_kws={
'color': 'dimgray',
'label': 'Regression Line'
})
plt.plot(sorted(sample),
CI_qq_low,
linestyle='--',
color='red',
alpha=1,
lw=0.8,
label='Kolmogorov-Smirnov Confidence Bands')
plt.legend()
plt.plot(sorted(sample),
CI_qq_high,
linestyle='--',
color='red',
alpha=1,
lw=0.8)
plt.xlabel('Theoretical Lognormal Quantiles')
plt.ylabel('Sample Quantiles')
plt.show()
def ks_test(sample, alpha):
sort_sample = sorted(sample)
n = len(sort_sample)
S = np.arange(1, n + 1)/n
ln_x = []
for data in sample:
ln_x.append(mt.log(data))
mean = mou.mean_sample(sample)
std = mou.std_sample(ln_x)
F = lnpd.phi(sort_sample, mean, std)
D = max(abs(F-S))
D_ks = st.ksone.ppf(1 - alpha/2, n)
if D < D_ks:
print('Kolmogorov-Smirnov Statistics = {}\nP-Value = {}\nConsidering that K-S Test Statistics < P-Value, with a confidence level of {}%, the normal distribution is acceptable.'.format(D, D_ks, (1-alpha)*100))
elif D > D_ks:
print('Kolmogorov-Smirnov Statistics = {}\nP-Value = {}\nConsidering that K-S Test Statistics > P-Value, with a confidence level of {}%, the normal distribution is not acceptable.'.format(D, D_ks, (1-alpha)*100))
def mean_norm(sample, alpha):
n = len(sample)
std = mou.std_sample(sample)
k = st.norm.ppf(1 - alpha/2)
e = k*std/(n**0.5)
print("The population mean with a confidence level of {}% is {} \u00B1 {}.".format(int((1-alpha)*100), mou.mean_sample(sample), e))