def stationarity_analysis(X, item_analysis_res):
res = pd.DataFrame(
[],
columns=[
'kpss(unit_sales)', 'kpss(boxcox(unit_sales))',
'kpss(unit_sales.diff(1))', 'kpss(unit_sales.diff(365))',
'kpss(unit_sales.diff(1).diff(365))', 'normal_test(unit_sales)',
'shapiro_test(unit_sales)', 'normal_test(boxcox(unit_sales))',
'shapiro_test(boxcox(unit sales))', 'normal_test(boxcox(resid))',
'shapiro_test(boxcox(resid))'
])
for k in item_analysis_res.keys():
item, _ = item_analysis_res[k]
x = X.loc[X['item_nbr'] == item, ['date', 'unit_sales']]
x.index = x['date']
x = x.drop(['date'], axis=1)
x_bc, x_bc_params = boxcox(x['unit_sales'])
daily = 365
decompose_x = seasonal_decompose(x['unit_sales'],
model='additive',
freq=daily)
alpha = 0.05
kpss_test = 'stationary' if kpss(x['unit_sales'],
regression='ct')[1] > alpha else '-'
kpss_test_bc = 'stationary' if kpss(
boxcox(x['unit_sales'])[0], regression='ct')[1] > alpha else '-'
kpss_test_diff1 = 'stationary' if kpss(
x['unit_sales'].diff(1), regression='ct')[1] > alpha else '-'
kpss_test_diff365 = 'stationary' if kpss(
x['unit_sales'].diff(365), regression='ct')[1] > alpha else '-'
kpss_test_diff1_365 = 'stationary' if kpss(
x['unit_sales'].diff(365), regression='ct')[1] > alpha else '-'
norm_test = 'normal' if normaltest(
x['unit_sales'])[1] >= alpha else '-'
shapiro_test = 'normal' if shapiro(
x['unit_sales'])[1] >= alpha else '-'
norm_test_bc = 'normal' if normaltest(boxcox(
x['unit_sales'])[0])[1] >= alpha else '-'
shapiro_test_bc = 'normal' if shapiro(boxcox(
x['unit_sales'])[0])[1] >= alpha else '-'
norm_test_resid_bc = 'normal' if normaltest(
yeojohnson(decompose_x.resid.dropna())[0])[1] >= alpha else '-'
shapiro_test_resid_bc = 'normal' if shapiro(
yeojohnson(decompose_x.resid.dropna())[0])[1] >= alpha else '-'
res.loc[k, :] = (kpss_test, kpss_test_bc, kpss_test_diff1,
kpss_test_diff365, kpss_test_diff1_365, norm_test,
shapiro_test, norm_test_bc, shapiro_test_bc,
norm_test_resid_bc, shapiro_test_resid_bc)
return res
def fit(self, X: pd.DataFrame, y: Optional[pd.Series] = None):
"""
Learns the optimal lambda for the Yeo-Johnson transformation.
Args:
X: pandas dataframe of shape = [n_samples, n_features]
The training input samples.
Can be the entire dataframe, not just the variables to transform.
y: It is not needed in this transformer. Defaults to None.
Alternatively takes Pandas Series.
Returns:
self
"""
# check input dataframe
X = super().fit(X)
self.lambda_dict_ = {}
# to avoid NumPy error
X[self.variables] = X[self.variables].astype("float")
for var in self.variables:
_, self.lambda_dict_[var] = stats.yeojohnson(X[var])
self.input_shape_ = X.shape
return self
def fit(self, X, y=None):
"""
Learns the numerical variables. Captures the optimal lambda for
the transformation.
Parameters
----------
X : pandas dataframe of shape = [n_samples, n_features]
The training input samples.
Can be the entire dataframe, not just seleted variables.
y : None
y is not needed in this transformer, yet the sklearn pipeline API
requires this parameter for checking. You can either leave it as None
or pass y.
"""
super().fit(X, y)
self.lambda_dict_ = {}
for var in self.variables:
X[var] = X[var].astype('float') # to avoid NumPy error
_, self.lambda_dict_[var] = stats.yeojohnson(X[var])
self.input_shape_ = X.shape
return self
def fit(self, X, y=None):
"""
Learns the optimal lambda for the Yeo-Johnson transformation.
Parameters
----------
X : pandas dataframe of shape = [n_samples, n_features]
The training input samples.
Can be the entire dataframe, not just the variables to transform.
y : None
y is not needed in this transformer. You can pass y or None.
"""
# check input dataframe
X = super().fit(X)
self.lambda_dict_ = {}
# to avoid NumPy error
X[self.variables] = X[self.variables].astype('float')
for var in self.variables:
_, self.lambda_dict_[var] = stats.yeojohnson(X[var])
self.input_shape_ = X.shape
return self
def transform(self, X: pd.DataFrame) -> pd.DataFrame:
"""
Apply the Yeo-Johnson transformation.
Parameters
----------
X: Pandas DataFrame of shape = [n_samples, n_features]
The data to be transformed.
Raises
------
TypeError
If the input is not a Pandas DataFrame
ValueError
- If the variable(s) contain null values
- If the df has different number of features than the df used in fit()
Returns
-------
X: pandas dataframe
The dataframe with the transformed variables.
"""
# check input dataframe and if class was fitted
X = super().transform(X)
for feature in self.variables_:
X[feature] = stats.yeojohnson(X[feature],
lmbda=self.lambda_dict_[feature])
return X
def transform(self, X):
"""
Applies the BoxCox trasformation.
Parameters
----------
X : pandas dataframe of shape = [n_samples, n_features]
The input samples.
Returns
-------
X_transformed : pandas dataframe of shape = [n_samples, n_features]
The dataframe with the transformed variables.
"""
# Check is fit had been called
check_is_fitted(self, ['input_shape_', 'lambda_dict_'])
if X.shape[1] != self.input_shape_[1]:
raise ValueError('Number of columns in dataset is different from training set used to fit the transformer')
X = X.copy()
for feature in self.variables:
X[feature] = stats.yeojohnson(X[feature], lmbda=self.lambda_dict_[feature])
return X
def gen_ts(self):
"""
generate data for trend changes
returns the new ts, season_ts, trend_ts, level_ts, change times for each trend and level, model param ts for each trend and level
"""
season_ = self.seasonal_data()
event_ = self.event_data()
trend_, self.trend_tbreak_, self.trend_thetat_, self.trend_theta_ = self._gen_ts(
'trend', self.trend_cpt, self.trend_noise_level,
self.trend_slow_bleed)
level_, self.level_tbreak_, self.level_thetat_, self.level_theta_ = self._gen_ts(
'level', self.level_cpt, self.level_noise_level,
self.level_slow_bleed)
if self.t_size > 0: # mixture
lambdas = np.random.uniform(low=-2.0, high=2.0, size=self.t_size)
print('TS mixture:: lambdas: ' + str(lambdas))
elif self.t_size == 0: # addtive
lambdas = [1.0]
print('TS is additive:: lambdas: ' + str(lambdas))
else: # mutiplicative
lambdas = np.random.uniform(low=1.5,
high=3.0,
size=int(np.abs(self.t_size)))
print('TS is multiplicative:: lambdas: ' + str(lambdas))
# additive components
self.event = 0 if event_ is None else event_
self.season = 0 if season_ is None else season_
self.trend = 0 if trend_ is None else trend_
self.level = 0 if level_ is None else level_
self.ts = self.event + self.season + self.trend + self.level
for lb in lambdas:
self.ts = sps.yeojohnson(self.ts, lmbda=lb)
def stabilize_variance(df):
if min(df) > 0:
print("Using Box-Cox Power Transformation...")
return boxcox(df)
else:
print("Using Yeo-Johnson Power Transformation...")
return yeojohnson(df)
def transform(self, X: dt.Frame):
XX = X.to_pandas().iloc[:, 0].values
is_na = np.isnan(XX) | np.array(XX <= -self._offset)
if not any(~is_na) or self._lmbda is None:
return X
ret = yeojohnson(self._offset + XX[~is_na], lmbda=self._lmbda) # apply transform with pre-computed lambda
XX[~is_na] = ret
return XX
def fit(self, X, y=None, **fitparams):
X = validate_dataframe(X)
self.lams = {}
for col in X.columns:
Xcol_float = X[col].astype(float)
_, lam = yeojohnson(Xcol_float)
self.lams[col] = lam
return self
def test_transformed_corr(self):
service_time = self.data["Service time"]
absenteeism_time = yeojohnson(
self.data["Absenteeism time in hours"].apply(float))[0]
pearson_corr = pearsonr(service_time, absenteeism_time)
self.assertAlmostEqual(pearson_corr[0], -0.042, places=2)
self.assertAlmostEqual(pearson_corr[1], 0.25, places=2)
def yeojohnson_transform(self):
l = self.numeric_features
to_numeric_features = []
for i in l:
self.data[f"{i}_yeo"] = yeojohnson(self.data[i])[0]
to_numeric_features.append(f"{i}_yeo")
self.transformed_numeric_features += to_numeric_features
def test_actual_results_power_transformer_yeo_johnson():
"""Test that the actual results are the expected ones."""
for standardize in [True, False]:
pt = PowerTransformer(method='yeo-johnson', standardize=standardize)
arr_actual = pt.fit_transform(X)
arr_desired = [yeojohnson(X[i].astype('float64'))[0] for i in range(3)]
if standardize:
arr_desired = StandardScaler().transform(arr_desired)
np.testing.assert_allclose(arr_actual, arr_desired, atol=1e-5, rtol=0.)
def trans(self):
#Remove Dupliactes
df = self.dataset.loc[:, ~self.dataset.columns.duplicated()]
#Remove Nan Values
remove_nan = self.dataset.replace(np.nan, 0)
#Check For The DataTypes
check_datatypes = remove_nan.dtypes
#To Get The Column Date Column Number and Name
Total_cloumns = remove_nan.columns
if 'Date' or 'Timestamp' in remove_nan.columns:
column_num = remove_nan.columns.get_loc("Date" or "Timestamp")
else:
print("no")
date_column = Total_cloumns[column_num]
#To Drop Date Column
drop = remove_nan.drop([date_column], axis=1)
#To Describe Entire Dataset
describe = drop.describe()
#To Pre=Process The Dataset
centered_scaled_data = preprocessing.scale(drop)
#To Convert 1d-array
dfconvert_array = drop.to_numpy()
y = sum(dfconvert_array.tolist(), [])
#Check If their are any negative or zero values
neg_count = len(list(filter(lambda x: (x < 0), y)))
pos_count = len(list(filter(lambda x: (x > 0), y)))
zero_count = len(list(filter(lambda x: (x <= 0), y)))
#Plot Before Yeo-Johnson Transformation
fig = plt.figure()
ax1 = fig.add_subplot(211)
prob = stats.probplot(y, dist=stats.norm, plot=ax1)
ax1.set_xlabel('')
ax1.set_title('Probplot against normal distribution')
#Plot After Yeo-Johnson Transformation
fig = plt.figure()
ax2 = fig.add_subplot(212)
xt, lmbda = stats.yeojohnson(y)
prob = stats.probplot(xt, dist=stats.norm, plot=ax2)
ax2.set_title('Probplot after Yeo-Johnson transformation')
#Skewness Before Transformation
skewness_before = skew(drop)
#Skewness After Transformation
skewness_after = skew(xt)
#Standardization
names = drop.columns
scaler = preprocessing.StandardScaler()
scaled_df = scaler.fit_transform(drop)
scaled_df = pd.DataFrame(scaled_df, columns=names)
#Normalization
x = drop.values
min_max_scaler = preprocessing.MinMaxScaler()
x_scaled = min_max_scaler.fit_transform(x)
df = pd.DataFrame(x_scaled)
return (df, remove_nan, check_datatypes, drop, describe,
centered_scaled_data, lmbda, skewness_before, skewness_after,
df)
def transform(self, X, **transformparams):
X = validate_dataframe(X)
X = X.copy()
new_col_list = []
for col in X.columns:
new_col = self.prefix + col + self.suffix
new_col_list.append(new_col)
Xcol_float = X[col].astype(float)
X[new_col] = yeojohnson(Xcol_float, lmbda=self.lams[col])
return X.loc[:, new_col_list]
def fit_transform(self, X: dt.Frame, y: np.array = None):
XX = X.to_pandas().iloc[:, 0].values
is_na = np.isnan(XX)
self._offset = -np.nanmin(XX) if np.nanmin(XX) < 0 else 0
self._offset += 1e-3
self._lmbda = None
if not any(~is_na):
return X
self._lmbda = yeojohnson(self._offset + XX[~is_na], lmbda=self._lmbda)[1] # compute lambda
return self.transform(X)
def _make_normal(self):
"""Normalize a list using Yeo-Johnson."""
if self.lofl == False:
is_normal = self.normal_test(self.lst, self.alpha)
if is_normal == False:
lst, self.lmbda = stats.yeojohnson(self.lst)
self.lst = lst.tolist()
else:
return
if self.lofl == True:
lst, self.lmbda = stats.yeojohnson(self.Make_Long(self.lst))
lst = lst.tolist()
self.lst = Capability.Brake_Down(lst=lst, n=self.n)
# normalised_lofl = []
# for sublist in self.lst:
# sublist, _ = stats.yeojohnson(sublist)
# normalised_lofl.append(list(sublist))
# self.lst = normalised_lofl
if self.lsl is not None:
lsl = stats.yeojohnson(self.lsl, self.lmbda).tolist()
lsl = round(lsl, 3)
if self.usl is not None:
usl = stats.yeojohnson(self.usl, self.lmbda).tolist()
usl = round(usl, 3)
self.lsl = min(lsl, usl)
self.usl = max(lsl, usl)
if self.target is not None:
self.target = stats.yeojohnson(self.target, self.lmbda).tolist()
self.target = round(self.target, 3)
try:
(self.usl - self.lsl) / 2
except ValueError:
self.midpoint = None
pass
else:
self.midpoint = (self.usl + self.lsl) / 2
self.midpoint = round(self.midpoint, 3)
def yeoJohnson(obs: 'pd.Series') -> 'pd.Series':
'''
Implement a Yeo Johnson Transformation of Data,
it is ideal for data with negative values.
Input:
:param obs: sequential data to analyze
Output:
obs_bc: Data transformed
lmbd: Optimal lambda
'''
obs_yj, lmbd = yeojohnson(obs)
return (obs_yj, lmbd)
def transform(self, X: dt.Frame):
XX = X.to_pandas().iloc[:, 0].values
is_na = np.isnan(XX) | np.array(XX <= -self._offset)
if not any(~is_na) or self._lmbda is None:
return X
ret = yeojohnson(
self._offset + XX[~is_na],
lmbda=self._lmbda) # apply transform with pre-computed lambda
XX[~is_na] = ret
XX = dt.Frame(XX)
# Don't leave inf/-inf
for i in range(XX.ncols):
XX.replace([math.inf, -math.inf], None)
return XX
def normal_tests(data, alpha=0.05):
yj = stats.yeojohnson(data)[0]
sq = np.sqrt(data)
cb = np.cbrt(data)
log = np.log(data)
return pd.DataFrame(
{
'Skewness': [
stats.skew(data),
stats.skew(sq),
stats.skew(cb),
stats.skew(log),
stats.skew(yj)
],
'Skewness Test': [
skewness_test(data, alpha=alpha),
skewness_test(sq, alpha=alpha),
skewness_test(cb, alpha=alpha),
skewness_test(log, alpha=alpha),
skewness_test(yj, alpha=alpha)
],
'Kurtosis': [
stats.kurtosis(data),
stats.kurtosis(sq),
stats.kurtosis(cb),
stats.kurtosis(log),
stats.kurtosis(yj)
],
'Kurtosis Test': [
kurtosis_test(data, alpha=alpha),
kurtosis_test(sq, alpha=alpha),
kurtosis_test(cb, alpha=alpha),
kurtosis_test(log, alpha=alpha),
kurtosis_test(yj, alpha=alpha)
],
'Normal Test': [
dagostino_test(data, alpha=alpha),
dagostino_test(sq, alpha=alpha),
dagostino_test(cb, alpha=alpha),
dagostino_test(log, alpha=alpha),
dagostino_test(yj, alpha=alpha)
]
},
index=['default', 'sqrt', 'cuberoot', 'log', 'yeojohnson'])
def transform_variable(var, transform_choice):
if transform_choice == 1:
if any(var < 0):
return np.log(var + (np.abs(np.min(var)) + 1))
elif 0 in var:
return np.log1p(var)
else:
return np.log(var)
elif transform_choice == 2:
if any(var < 0):
return np.sqrt(var + np.abs(np.min(var)))
else:
return np.sqrt(var)
elif transform_choice == 3:
x, _ = stats.yeojohnson(var)
return x
else:
raise ValueError('Transform option not supported')
def normalize(self, df, method="boxcox"):
"""
Normalizes each of the columns in the given dataframe according to the method
Inputs:
- df: dataframe with numeric columns
- method: string specifying the normalization method
Returns dataframe with normalized columns
"""
if method == "boxcox":
for column in df.columns:
df[column] = stats.boxcox(df[column])[0]
elif method == "log":
for column in df.columns:
df[column] = np.log(df[column])
elif method == "yeojohnson":
for column in df.columns:
df[column] = stats.yeojohnson(df[column])[0]
return df
def transform(self, X: pd.DataFrame) -> pd.DataFrame:
"""
Applies the Yeo-Johnson transformation.
Args:
X: Pandas DataFrame of shape = [n_samples, n_features]
The data to be transformed.
Returns:
The dataframe with the transformed variables.
"""
# check input dataframe and if class was fitted
X = super().transform(X)
for feature in self.variables:
X[feature] = stats.yeojohnson(X[feature],
lmbda=self.lambda_dict_[feature])
return X
def fit(self, X: pd.DataFrame, y: Optional[pd.Series] = None):
"""
Learn the optimal lambda for the Yeo-Johnson transformation.
Parameters
----------
X: pandas dataframe of shape = [n_samples, n_features]
The training input samples. Can be the entire dataframe, not just the
variables to transform.
y: pandas Series, default=None
It is not needed in this transformer. You can pass y or None.
Raises
------
TypeError
- If the input is not a Pandas DataFrame
- If any of the user provided variables are not numerical
ValueError
- If there are no numerical variables in the df or the df is empty
- If the variable(s) contain null values
Returns
-------
self
"""
# check input dataframe
X = super().fit(X)
self.lambda_dict_ = {}
# to avoid NumPy error
X[self.variables_] = X[self.variables_].astype("float")
for var in self.variables_:
_, self.lambda_dict_[var] = stats.yeojohnson(X[var])
self.n_features_in_ = X.shape[1]
return self
def transform(self, X):
# yeo-johnson
if self.yeojohnson:
# apply learned Yeo-Johnson transformation
for col in self.yj_lambdas_dict_.keys():
X[col + "_YeoJohnson"] = stats.yeojohnson(
X[col].values, lmbda=self.yj_lambdas_dict_[col]
)
# Box-Cox
if self.boxcox:
# apply learned Box-Cox transformation
for col in self.bc_zero_lambdas_dict_.keys():
try:
X[col + "_BoxCox"] = stats.boxcox(
X[col].values + 1, lmbda=self.bc_zero_lambdas_dict_[col]
)
except ValueError:
X[col + "_BoxCox"] = 0.
# apply learned Box-Cox transformation
for col in self.bc_neg_lambdas_dict_.keys():
try:
X[col + "_BoxCox"] = stats.boxcox(
X[col].values + np.abs(np.min(X[col].values)) + 1, lmbda=self.bc_neg_lambdas_dict_[col]
)
except ValueError:
X[col + "_BoxCox"] = 0.
# apply learned Box-Cox transformation
for col in self.bc_lambdas_dict_.keys():
try:
X[col + "_BoxCox"] = stats.boxcox(
X[col].values, lmbda=self.bc_lambdas_dict_[col]
)
except ValueError:
X[col + "_BoxCox"] = 0.
return X
def fit(self, X: pd.DataFrame, y: Optional[pd.Series] = None):
"""
Learn the optimal lambda for the Yeo-Johnson transformation.
Parameters
----------
X: pandas dataframe of shape = [n_samples, n_features]
The training input samples. Can be the entire dataframe, not just the
variables to transform.
y: pandas Series, default=None
It is not needed in this transformer. You can pass y or None.
"""
# check input dataframe
X = super()._fit_from_varlist(X)
self.lambda_dict_ = {}
for var in self.variables_:
_, self.lambda_dict_[var] = stats.yeojohnson(X[var])
return self
def transform_norm(data, type_transform="yeojohnson"):
mu = 0
std = 0
new_data = []
if type_transform == "box-cox":
#box-cox transformation
print("Minimum value:", min(data))
shift = 0
minimum = min(data)
if minimum < 0:
shift = round(abs(minimum))
posdata = [x + shift for x in data]
new_data, lmda = boxcox(posdata)
elif type_transform == "yeojohnson":
#Yeo-Johnson power transformation
new_data, lmbda = yeojohnson(data)
else:
raise ValueError("No such transformation")
mu, std = norm.fit(new_data)
return new_data, mu, std
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1.inset_locator import inset_axes
np.random.seed(1245)
# Generate some random variates and calculate Yeo-Johnson log-likelihood
# values for them for a range of ``lmbda`` values:
x = stats.loggamma.rvs(5, loc=10, size=1000)
lmbdas = np.linspace(-2, 10)
llf = np.zeros(lmbdas.shape, dtype=float)
for ii, lmbda in enumerate(lmbdas):
llf[ii] = stats.yeojohnson_llf(lmbda, x)
# Also find the optimal lmbda value with `yeojohnson`:
x_most_normal, lmbda_optimal = stats.yeojohnson(x)
# Plot the log-likelihood as function of lmbda. Add the optimal lmbda as a
# horizontal line to check that that's really the optimum:
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(lmbdas, llf, 'b.-')
ax.axhline(stats.yeojohnson_llf(lmbda_optimal, x), color='r')
ax.set_xlabel('lmbda parameter')
ax.set_ylabel('Yeo-Johnson log-likelihood')
# Now add some probability plots to show that where the log-likelihood is
# maximized the data transformed with `yeojohnson` looks closest to normal:
locs = [3, 10, 4] # 'lower left', 'center', 'lower right'
from scipy import stats import matplotlib.pyplot as plt # Generate some non-normally distributed data, and create a Yeo-Johnson plot: x = stats.loggamma.rvs(5, size=500) + 5 fig = plt.figure() ax = fig.add_subplot(111) prob = stats.yeojohnson_normplot(x, -20, 20, plot=ax) # Determine and plot the optimal ``lmbda`` to transform ``x`` and plot it in # the same plot: _, maxlog = stats.yeojohnson(x) ax.axvline(maxlog, color='r') plt.show()
def eda_numerical_variable(self, variable):
'''
Parameter:
variable: pass the variable for which EDA is required
provides basic statistcs, missing values, distribution, spread statistics,
Q-Q plot, Box plot, outliers using IQR, various variable transformations'''
c = variable
s = self.__df__[variable]
# 1. Basic Statistics
print ('Total Number of observations : ', len(s))
print ()
print ('Datatype :', (s.dtype))
print ()
printmd ('**<u>5 Point Summary :</u>**')
print (' Minimum :\t\t', s.min(), '\n 25th Percentile :\t', s.quantile(0.25),
'\n Median :\t\t', s.median(), '\n 75th Percentile :\t', s.quantile(0.75),
'\n Maximum :\t\t', s.max())
print ()
# 2. Missing values
printmd ('**<u>Missing Values :</u>**')
print (' Number :', s.isnull().sum())
print (' Percentage :', s.isnull().mean()*100, '%')
# 3. Histogram
printmd ('**<u>Variable distribution and Spread statistics :</u>**')
sns.distplot(s.dropna(), hist = True, fit = norm, kde = True)
plt.show()
# 4. Spread Statistics
print ('Skewness :' , s.skew())
print ('Kurtosis :', s.kurt())
print ()
# 5. Q-Q plot
printmd ('**<u>Normality Check :</u>**')
res = stats.probplot(s.dropna(), dist = 'norm', plot = plt)
plt.show()
# 6. Box plot to check the spread outliers
print ()
printmd ('**<u>Box Plot and Visual check for Outlier :</u>**')
sns.boxplot(s.dropna(), orient = 'v')
plt.show()
# 7. Get outliers. Here distance could be a user defined parameter which defaults to 1.5
print ()
printmd ('**<u>Outliers (using IQR):</u>**')
IQR = np.quantile(s, .75) - np.quantile(s, .25)
upper_boundary = np.quantile(s, .75) + 1.5 * IQR
lower_boundary = np.quantile(s, .25) - 1.5 * IQR
print (' Right end outliers :', np.sum(s>upper_boundary))
print (' Left end outliers :', np.sum(s < lower_boundary))
# 8. Various Variable Transformations
print ()
printmd (f'**<u>Explore various transformations for {c}</u>**')
print ()
print ('1. Logarithmic Transformation')
s_log = np.log(s)
normality_diagnostic(s_log)
print ('2. Exponential Transformation')
s_exp = np.exp(s)
normality_diagnostic(s_exp)
print ('3. Square Transformation')
s_sqr = np.square(s)
normality_diagnostic(s_sqr)
print ('4. Square-root Transformation')
s_sqrt = np.sqrt(s)
normality_diagnostic(s_sqrt)
print ('5. Box-Cox Transformation')
s_boxcox, lambda_param = stats.boxcox(s)
normality_diagnostic(s_boxcox)
print ('Optimal Lambda for Box-Cox transformation is :', lambda_param )
print ()
print ('6. Yeo Johnson Transformation')
s = s.astype('float')
s_yeojohnson, lambda_param = stats.yeojohnson(s)
normality_diagnostic(s_yeojohnson)
print ('Optimal Lambda for Yeo Johnson transformation is :', lambda_param )
print ()