Stanford Stats 191¶

Introduction¶

This is a re-creation of the Stanford Stats 191 course (see https://web.stanford.edu/class/stats191/), using Python eco-system tools, instead of R. This is lecture "Transformations and Weighted Least Squares"

Initial Notebook Setup¶

watermark documents the current Python and package environment, black is my preferred Python formatter

In [2]:

%load_ext watermark

In [3]:

%load_ext lab_black

In [4]:

%matplotlib inline

All import got here (not all are used in this Notebook)

In [43]:

import pandas as pd
import numpy as np
import seaborn as sns

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

from scipy import stats
from statsmodels.formula.api import ols
from statsmodels.formula.api import rlm
from statsmodels.formula.api import mixedlm
from statsmodels.formula.api import wls
import statsmodels.api as sm

from patsy import dmatrices

from patsy.contrasts import Treatment
from patsy.contrasts import Sum
from patsy.contrasts import Diff


import warnings

Nonlinear Models¶

We look at a dataset where a linear model is not appropriate

Load and Explore Dataset¶

In [6]:

data = pd.read_csv('../data/bacteria.txt', sep='\t')

In [7]:

data.head()

Out[7]:

	t	N_t
0	1	355
1	2	211
2	3	197
3	4	166
4	5	142

In [8]:

fig, ax = plt.subplots(figsize=(8, 8))

ax.plot(data['t'], data['N_t'], 'd')
ax.set_xlabel('Time')
ax.set_ylabel('Count')

Out[8]:

Text(0, 0.5, 'Count')

No description has been provided for this image

Fit a Linear Model¶

In [9]:

res1 = ols('N_t ~ t', data=data).fit()
res1.summary()

D:\Anaconda3\envs\ac5-py37\lib\site-packages\scipy\stats\stats.py:1394: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=15
  "anyway, n=%i" % int(n))

Out[9]:

OLS Regression Results
Dep. Variable:	N_t	R-squared:	0.823
Model:	OLS	Adj. R-squared:	0.810
Method:	Least Squares	F-statistic:	60.62
Date:	Tue, 07 Apr 2020	Prob (F-statistic):	3.01e-06
Time:	16:42:55	Log-Likelihood:	-76.216
No. Observations:	15	AIC:	156.4
Df Residuals:	13	BIC:	157.8
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	259.5810	22.730	11.420	0.000	210.476	308.686
t	-19.4643	2.500	-7.786	0.000	-24.865	-14.063

Omnibus:	14.123	Durbin-Watson:	0.803
Prob(Omnibus):	0.001	Jarque-Bera (JB):	10.490
Skew:	1.648	Prob(JB):	0.00527
Kurtosis:	5.432	Cond. No.	19.3

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Visualize the fit of our linear model to the data

In [10]:

fig, ax = plt.subplots(figsize=(8, 8))

ax.plot(data['t'], data['N_t'], 'd', label='Data')
ax.plot(
    data['t'],
    res1.predict(),
    'r-',
    label='Linear Prediction',
)

ax.set_xlabel('Time')
ax.set_ylabel('Count')

ax.legend(loc='best')

Out[10]:

<matplotlib.legend.Legend at 0x1f6a725e9e8>

Regression Diagnostics¶

The STATS191 lecture notes have quartet of plots that can be used to assess linear model quality. We define a function to show these, given the RegressionResults object from a ols model fit

In [77]:

def show_regression_diagnostic_plots(res1):
    '''
    show_regression_diagnostic_plots: show four plots to assess validity of linear regression
    
    Refer to https://web.stanford.edu/class/stats191/notebooks/Transformations_WLS.html for examples
    that guided this function
    
    Parameters
    res1: statsmodels RegressionResults value
    
    Returns
    matplotlib Figure object containing plots (one in each Axes object linked to Figure object)
    
    Notes
    In a Jupyter Notebook, calling this function will display a Figure, and also return a Figure
    Just a naked call will result in the plots being displayed twice.  To prevent this, assign
    the function call return to a variable
    '''

    fig, ax = plt.subplots(
        nrows=2,
        ncols=2,
        figsize=(10, 10),
        constrained_layout=True,
    )

    # create a qq plot, to assess how (if) the residuals are distributed
    axr1c2 = ax[0, 1]
    sm.qqplot(res1.resid, line='s', ax=axr1c2)
    axr1c2.set_title('Normal QQ Plot')

    # plot residuals against  fitted value, to see if residuals are correlated with exog variable
    # (i.e. are not normally distributed)
    axr1c1 = ax[0, 0]
    axr1c1.plot(res1.predict(), res1.resid, 'rd')
    axr1c1.set_xlabel('Fitted Values')
    axr1c1.set_ylabel('Residuals')
    axr1c1.axhline(0, color='k')
    axr1c1.set_title('Residuals vs Fitted')

    # plot standardizerd residuals against leverage
    axr2c2 = ax[1, 1]
    influence = res1.get_influence()
    hat = influence.hat_matrix_diag
    standardized_residuals_int = (
        influence.resid_studentized_internal
    )
    axr2c2.plot(hat, standardized_residuals_int, 'd')
    axr2c2.set_xlabel('Leverage')
    axr2c2.set_ylabel('Standardised Residuals')
    axr2c2.set_title('Residuals vs Leverage')

    axr2c1 = ax[1, 0]
    standardized_residuals_ext = (
        influence.resid_studentized_external
    )
    zz = np.sqrt(np.abs(standardized_residuals_ext))
    axr2c1.plot(res1.predict(), zz, 'rd', label='Data')
    axr2c1.set_xlabel('Fitted Values')
    axr2c1.set_ylabel(
        'sqrt(abs(Externally Studentized Residual))'
    )
    axr2c1.set_title('Scale-Location')

    return fig


# end

In [76]:

fig = show_regression_diagnostic_plots(res1)

We see that residuals are clearly not distributed uniformly (Residuals vs Fitted graph).

Further, if we look at the statsmodels influence and leverage graphs, we see a clear outlier

In [13]:

fig, ax = plt.subplots(figsize=(6, 6))

_ = sm.graphics.influence_plot(res1, ax=ax)

In [14]:

fig, ax = plt.subplots(figsize=(6, 6))

_ = sm.graphics.plot_leverage_resid2(res1, ax=ax)

Fit Exponential¶

At this stage we decide that a better model might be that of exponential decay, where the log of N_t might be linear in time.

Note that we can use numpy functions inside the ols formula

In [15]:

res2 = ols('np.log(N_t) ~ t', data=data).fit()
res2.summary()

D:\Anaconda3\envs\ac5-py37\lib\site-packages\scipy\stats\stats.py:1394: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=15
  "anyway, n=%i" % int(n))

Out[15]:

OLS Regression Results
Dep. Variable:	np.log(N_t)	R-squared:	0.988
Model:	OLS	Adj. R-squared:	0.987
Method:	Least Squares	F-statistic:	1104.
Date:	Tue, 07 Apr 2020	Prob (F-statistic):	5.86e-14
Time:	16:42:57	Log-Likelihood:	12.896
No. Observations:	15	AIC:	-21.79
Df Residuals:	13	BIC:	-20.38
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	5.9732	0.060	99.922	0.000	5.844	6.102
t	-0.2184	0.007	-33.222	0.000	-0.233	-0.204

Omnibus:	0.024	Durbin-Watson:	2.659
Prob(Omnibus):	0.988	Jarque-Bera (JB):	0.204
Skew:	-0.074	Prob(JB):	0.903
Kurtosis:	2.448	Cond. No.	19.3

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Assess Exponential Model¶

Plotting the model predictions against actuals in two different ways, we can see we have a much better fit (also the R^2 value is much closer to 1.0)

In [16]:

fig, ax = plt.subplots(figsize=(6, 6))

ax.plot(data['t'], data['N_t'], 'o', label='Data')
ax.plot(
    data['t'],
    np.exp(res2.predict()),
    'g-',
    label='Exp. Fit',
)
ax.plot(data['t'], res1.predict(), 'r-', label='Linear Fit')
ax.legend(loc='best')
ax.set_xlabel('Time')
ax.set_ylabel('Count')

Out[16]:

Text(0, 0.5, 'Count')

In [17]:

fig, ax = plt.subplots(figsize=(6, 6))

ax.plot(data['t'], np.log(data['N_t']), 'o', label='Data')
ax.plot(data['t'], (res2.predict()), 'g-', label='Exp. Fit')
ax.plot(
    data['t'],
    np.log(res1.predict()),
    'r-',
    label='Linear Fit',
)
ax.legend(loc='best')
ax.set_xlabel('Time')
ax.set_ylabel('Log Count (ln)')

D:\Anaconda3\envs\ac5-py37\lib\site-packages\ipykernel_launcher.py:7: RuntimeWarning: invalid value encountered in log
  import sys

Out[17]:

Text(0, 0.5, 'Log Count (ln)')

The Diagnostic Graphs look better too (especially Residuals vs Fitted)

In [18]:

fig = show_regression_diagnostic_plots(res2)

In [19]:

fig, ax = plt.subplots(figsize=(6, 6))

_ = sm.graphics.influence_plot(res2, ax=ax)

In [20]:

fig, ax = plt.subplots(figsize=(6, 6))

_ = sm.graphics.plot_leverage_resid2(res2, ax=ax)

Non-Constant Variance¶

We now look at a dataset, with an error term that is not constant from observation to observation (see below for description of the dataset)

Variable	Description

Read and Explore Dataset¶

In [21]:

data2 = pd.read_csv('../data/education1975.txt', sep='\t')

In [22]:

data2.head()

Out[22]:

	STATE	Y	X1	X2	X3	Region
0	ME	235	3944	325	508	1
1	NH	231	4578	323	564	1
2	VT	270	4011	328	322	1
3	MA	261	5233	305	846	1
4	RI	300	4780	303	871	1

Annoyingly, we have trailing spaces in the column names, so lets fix those

In [23]:

data2.columns

Out[23]:

Index(['STATE ', 'Y ', 'X1 ', 'X2 ', 'X3 ', 'Region  '], dtype='object')

In [24]:

new_names = [n.strip() for n in data2.columns]
data2.columns = new_names
data2.head(1)

Out[24]:

	STATE	Y	X1	X2	X3	Region
0	ME	235	3944	325	508	1

In [25]:

data2.describe()

Out[25]:

	Y	X1	X2	X3	Region
count	50.000000	50.000000	50.000000	50.000000	50.000000
mean	284.600000	4675.120000	325.740000	657.800000	2.660000
std	61.340136	644.506254	19.423119	145.016396	1.061574
min	208.000000	3448.000000	287.000000	322.000000	1.000000
25%	234.250000	4137.250000	310.750000	546.750000	2.000000
50%	269.500000	4706.000000	324.500000	662.500000	3.000000
75%	316.750000	5054.250000	333.000000	782.250000	3.750000
max	546.000000	5889.000000	386.000000	909.000000	4.000000

Fit Linear Model¶

In [26]:

res3 = ols('Y ~ X1 + X2 + X3', data=data2).fit()
res3.summary()

Out[26]:

OLS Regression Results
Dep. Variable:	Y	R-squared:	0.591
Model:	OLS	Adj. R-squared:	0.565
Method:	Least Squares	F-statistic:	22.19
Date:	Tue, 07 Apr 2020	Prob (F-statistic):	4.94e-09
Time:	16:43:02	Log-Likelihood:	-253.89
No. Observations:	50	AIC:	515.8
Df Residuals:	46	BIC:	523.4
Df Model:	3
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	-556.5680	123.195	-4.518	0.000	-804.547	-308.589
X1	0.0724	0.012	6.239	0.000	0.049	0.096
X2	1.5521	0.315	4.932	0.000	0.919	2.185
X3	-0.0043	0.051	-0.083	0.934	-0.108	0.099

Omnibus:	0.719	Durbin-Watson:	2.149
Prob(Omnibus):	0.698	Jarque-Bera (JB):	0.760
Skew:	0.264	Prob(JB):	0.684
Kurtosis:	2.705	Cond. No.	1.03e+05

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.03e+05. This might indicate that there are
strong multicollinearity or other numerical problems.

Assess Model¶

The Residuals vs Fitted graph shows a clear increase in residual, as the fitted value increases (i.e. variance of the error term is not constant)

In [27]:

fig = show_regression_diagnostic_plots(res3)

Further, we appear to have an outlier observation (the State of Alaska, code AK)

In [28]:

fig, ax = plt.subplots(figsize=(6, 6))

_ = sm.graphics.influence_plot(res3, ax=ax)

In [29]:

data2.iloc[48]

Out[29]:

STATE      AK 
Y          546
X1        5613
X2         386
X3         484
Region       4
Name: 48, dtype: object

In [30]:

fig, ax = plt.subplots(figsize=(6, 6))

_ = sm.graphics.plot_leverage_resid2(res3, ax=ax)

We now visualize the residuals (standardized) by region, and it appears that each region has a different variance of the error term

In [78]:

fig, ax = plt.subplots(figsize=(6, 6))

influence = res3.get_influence()
standardized_residuals_int = (
    influence.resid_studentized_internal
)
sns.boxplot(
    'Region', standardized_residuals_int, data=data2, ax=ax
)
ax.set_ylabel('Standardized Residuals')
ax.set_title('Standardized Residuals by Region')

Out[78]:

Text(0.5, 1.0, 'Standardized Residuals by Region')

Annoyingly, we find the state codes also have a trailing space, so fix these

In [34]:

'*' + data2['STATE'].iloc[48] + '*'

Out[34]:

'*AK *'

In [35]:

data2['STATE'] = data2['STATE'].str.strip()
'*' + data2['STATE'].iloc[48] + '*'

Out[35]:

'*AK*'

We now clean the dataset, by excluding the state of Alaska, and making a copy of the DataFrame.

We then fit a linear model again

In [36]:

data2_clean = data2[data2['STATE'] != 'AK'].copy()
print(f'Clean data items {len(data2_clean)}')
res4 = ols('Y ~ X1 + X2 + X3', data=data2_clean).fit()
res4.summary()

Clean data items 49

Out[36]:

OLS Regression Results
Dep. Variable:	Y	R-squared:	0.497
Model:	OLS	Adj. R-squared:	0.463
Method:	Least Squares	F-statistic:	14.80
Date:	Tue, 07 Apr 2020	Prob (F-statistic):	7.65e-07
Time:	17:05:11	Log-Likelihood:	-242.77
No. Observations:	49	AIC:	493.5
Df Residuals:	45	BIC:	501.1
Df Model:	3
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	-277.5773	132.423	-2.096	0.042	-544.291	-10.864
X1	0.0483	0.012	3.976	0.000	0.024	0.073
X2	0.8869	0.331	2.678	0.010	0.220	1.554
X3	0.0668	0.049	1.354	0.183	-0.033	0.166

Omnibus:	0.883	Durbin-Watson:	1.946
Prob(Omnibus):	0.643	Jarque-Bera (JB):	0.932
Skew:	0.289	Prob(JB):	0.628
Kurtosis:	2.652	Cond. No.	1.23e+05

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.23e+05. This might indicate that there are
strong multicollinearity or other numerical problems.

Assess Model¶

Cleaning the data didn't fix our non_constant variance residuals

In [39]:

_ = show_regression_diagnostic_plots(res4)

Residuals by Region still show inter-Region difference

In [40]:

fig, ax = plt.subplots(figsize=(6, 6))

influence = res4.get_influence()
standardized_residuals_int = (
    influence.resid_studentized_internal
)
sns.boxplot(
    'Region',
    standardized_residuals_int,
    data=data2_clean,
    ax=ax,
)
ax.set_title('Standardized Residuals by Region')

Out[40]:

Text(0.5, 1.0, 'Standardized Residuals by Region')

Weighted Least Squares¶

We define a new model, where we standardized variance by weighting with value of Y (assumes a model of error term as N(0,eps * Y). We assume observations within each Region has a constant variance error term

weights array will hold the (1/variance) of the error term (in our model)

In [47]:

weights = np.zeros(len(data2_clean))

Get list of unique Region codes

In [50]:

regions = data2_clean['Region'].unique()
regions

Out[50]:

array([1, 2, 3, 4], dtype=int64)

For each Region, compute an estimate of the variance of the error term for that region, and fill in the weights for all observations from that region

In [52]:

for r in regions:
    region_mask = data2_clean['Region'] == r
    region_count = sum(region_mask)

    value = sum(res4.resid[region_mask] ** 2) / sum(
        region_mask
    )
    value = 1.0 / value
    weights[region_mask] = value
    print(value)

# end for

0.0006891262889463299
0.0004103443401877075
0.00400908850190694
0.0010521627801787171

Note that the value of the weights above agree with the STATS191 lecture notes

Fit Model with Weights¶

In [59]:

res5 = wls(
    'Y ~ X1 + X2 + X3', data=data2_clean, weights=weights
).fit()
res5.summary()

Out[59]:

WLS Regression Results
Dep. Variable:	Y	R-squared:	0.757
Model:	WLS	Adj. R-squared:	0.741
Method:	Least Squares	F-statistic:	46.75
Date:	Tue, 07 Apr 2020	Prob (F-statistic):	7.10e-14
Time:	18:22:54	Log-Likelihood:	-231.58
No. Observations:	49	AIC:	471.2
Df Residuals:	45	BIC:	478.7
Df Model:	3
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	-315.5170	78.178	-4.036	0.000	-472.975	-158.059
X1	0.0623	0.008	7.916	0.000	0.046	0.078
X2	0.8743	0.200	4.366	0.000	0.471	1.278
X3	0.0294	0.034	0.857	0.396	-0.040	0.098

Omnibus:	6.395	Durbin-Watson:	2.023
Prob(Omnibus):	0.041	Jarque-Bera (JB):	2.405
Skew:	0.147	Prob(JB):	0.300
Kurtosis:	1.955	Cond. No.	1.05e+05

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.05e+05. This might indicate that there are
strong multicollinearity or other numerical problems.

The results above agree with the STATS191 results in the first two decimal places (I suspect the warning:

The condition number is large, 1.05e+05. This might indicate that there are strong multicollinearity or other numerical problems.

should be heeded

To check the weights, you can get them via the model attribute of the RegressionResults

In [79]:

res5.model.weights[0:5]

Out[79]:

array([0.00068913, 0.00068913, 0.00068913, 0.00068913, 0.00068913])

It appears that after fitting a Weighted Least Squares model, Pearson Residuals are a better way of standardising residuals. The boxplot below shows that we have accounted for much of the difference between Regions in our model, but not all.

In [67]:

fig, ax = plt.subplots(figsize=(6, 6))


sns.boxplot(
    'Region', res5.resid_pearson, data=data2_clean, ax=ax
)
ax.set_title('Pearson Residuals by Region')
ax.set_ylabel('Pearson Residuals')

Out[67]:

Text(0, 0.5, 'Pearson Residuals')

The Residuals vs Fitted plots are very much the same

In [89]:

def show_regression_diagnostic_plots_wls(res1):
    '''
    show_regression_diagnostic_plots_wls: show two plots to assess validity of linear regression
    
    Refer to https://web.stanford.edu/class/stats191/notebooks/Transformations_WLS.html for examples
    that guided this function
    
    Parameters
    res1: statsmodels RegressionResults value
    
    Returns
    matplotlib Figure object containing plots (one in each Axes object linked to Figure object)
    
    Notes
    In a Jupyter Notebook, calling this function will display a Figure, and also return a Figure
    Just a naked call will result in the plots being displayed twice.  To prevent this, assign
    the function call return to a variable
    
    This is intended to be called after a wls().fit() call.  No influence plots will be created,
    because these are not supported by statsmodels 
    
    Quote: The influence and outlier statistics have not been extended to other models [including wls]
    mostly for a lack of reference to the statistical literature that explicitly handles 
    this case, and for not knowing of a reference implementation in another package that 
    could be used for the unit tests.
    '''

    fig, ax = plt.subplots(
        nrows=1,
        ncols=2,
        figsize=(10, 5),
        constrained_layout=True,
    )

    # create a qq plot, to assess how (if) the residuals are distributed
    axr1c2 = ax[1]
    sm.qqplot(res1.resid, line='s', ax=axr1c2)
    axr1c2.set_title('Normal QQ Plot')

    # plot residuals against  fitted value, to see if residuals are correlated with exog variable
    # (i.e. are not normally distributed)
    axr1c1 = ax[0]
    axr1c1.plot(res1.predict(), res1.resid, 'rd')
    axr1c1.set_xlabel('Fitted Values')
    axr1c1.set_ylabel('Residuals')
    axr1c1.axhline(0, color='k')
    axr1c1.set_title('Residuals vs Fitted')

    return fig


# end

In [90]:

_ = show_regression_diagnostic_plots_wls(res5)

Environment¶

In [60]:

%watermark -h -iv
%watermark

seaborn     0.9.0
matplotlib  3.0.2
scipy       1.1.0
numpy       1.15.4
statsmodels 0.9.0
pandas      1.0.0
host name: DESKTOP-SODFUN6
2020-04-06T21:19:53+10:00

CPython 3.7.1
IPython 7.2.0

compiler   : MSC v.1915 64 bit (AMD64)
system     : Windows
release    : 10
machine    : AMD64
processor  : Intel64 Family 6 Model 94 Stepping 3, GenuineIntel
CPU cores  : 8
interpreter: 64bit

In [46]:

sm.show_versions()

INSTALLED VERSIONS
------------------
Python: 3.7.1.final.0

Statsmodels
===========

Installed: 0.9.0 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\statsmodels)

Required Dependencies
=====================

cython: 0.29.2 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\Cython)
numpy: 1.15.4 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\numpy)
scipy: 1.1.0 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\scipy)
pandas: 1.0.0 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\pandas)
    dateutil: 2.7.5 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\dateutil)
patsy: 0.5.1 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\patsy)

Optional Dependencies
=====================

matplotlib: 3.0.2 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\matplotlib)
    backend: module://ipykernel.pylab.backend_inline 
cvxopt: Not installed
joblib: 0.13.2 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\joblib)

Developer Tools
================

IPython: 7.2.0 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\IPython)
    jinja2: 2.10.1 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\jinja2)
sphinx: 1.8.2 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\sphinx)
    pygments: 2.3.1 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\pygments)
nose: 1.3.7 (D:\Anaconda3\envs\ac5-py37\lib\site-packages\nose)
pytest: 5.0.1 (D:\Anaconda3\envs\ac5-py37\lib\site-packages)
virtualenv: Not installed

In [ ]:

Stanford STATS191 in Python, Lecture 9 : Transformations and Weighted Least Squares

Stanford Stats 191¶

Introduction¶

Initial Notebook Setup¶

Nonlinear Models¶

Load and Explore Dataset¶

Fit a Linear Model¶

Regression Diagnostics¶

Fit Exponential¶

Assess Exponential Model¶

Non-Constant Variance¶

Read and Explore Dataset¶

Fit Linear Model¶

Assess Model¶

Assess Model¶

Weighted Least Squares¶

Fit Model with Weights¶

Environment¶

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