Principal Components And Factor Analysis Using Python Code In 40 Hours
#This assignment is a tutorial so have a bit of fun with it
#If you would like to explore some additional options give it a try
#Goal is to provide some meaningful info to the restaurant owner
#Some notes below
#Both PCA and FA provide useful summary info for multivariate data, but
#all of the original variables are needed for their calculation, so
#the big question is can we use them to find a subset of variables to
#predict overall score?
#Also,trying to give meaningful labels to components is really hard.
#When the variables are on different scales you need to work with the
#correlation matrix. For this assignment they are on same scale so
#we will work with the raw data.
#PCA only helps if the original variables are correlated, if they
#are independent PCA will not help.
#Approach takes two steps
#First step find the dimensionality of the data, that is the
#number of original variables to be retained
#Second step find which ones, more on this below
# import packages for this example
import pandas as pd
import numpy as np # arrays and math functions
import matplotlib.pyplot as plt # static plotting
from sklearn.decomposition import PCA, FactorAnalysis
import statsmodels.formula.api as smf # R-like model specification
#Set some display options
pd.set_option(‘display.notebook_repr_html’, False)
pd.set_option(‘display.max_columns’, 40)
pd.set_option(‘display.max_rows’, 10)
pd.set_option(‘display.width’, 120)
#Read in the restaurant dataset
food_df = pd.read_csv(‘C:/Users/Jahee Koo/Desktop/MSPA/2018_Winter_410_regression/HW03 PCA/FACTOR1.csv’)
#A good step to take is to convert all variable names to lower case
food_df.columns = [s.lower() for s in food_df.columns]
print(food_df)
print(”)
print(‘—– Summary of Input Data —–‘)
print(”)
# show the object is a DataFrame
print(‘Object type: ‘, type(food_df))
# show number of observations in the DataFrame
print(‘Number of observations: ‘, len(food_df))
# show variable names
print(‘Variable names: ‘, food_df.columns)
# show descriptive statistics
print(food_df.describe())
# show a portion of the beginning of the DataFrame
print(food_df.head())
#look at correlation structure
cdata = food_df.loc[:,[‘overall’,’taste’,’temp’,’freshness’,’wait’,’clean’,’friend’,’location’,’parking’,’view’]]
corr = cdata[cdata.columns].corr()
print(corr)
#Use the correlation matrix to help provide advice to the restaurant owner
#Look at four different models and compare them
#Which model do you think is best and why?
#Model 1 full regression model
#Model 2 select my reduced regression model taste, wait and location
#Model 3 Full PCA model
#Model 4 Reduced PCA model with parking, taste and clean
#Model 5 FA model
#First find the PCA
#Second find the FA
#Run the models
#Compare the models and show VIF for each model
#PCA
print(”)
print(‘—– Principal Component Analysis —–‘)
print(”)
pca_data = food_df.loc[:,[‘taste’,’temp’,’freshness’,’wait’,’clean’,’friend’,’location’,’parking’,’view’]]
pca = PCA()
P = pca.fit(pca_data)
print(pca_data)
np.set_printoptions(threshold=np.inf)
np.around([pca.components_], decimals=3)
#Note per Everett p209 pick the three variables with the largest
#absolute coefficient on the component not already picked
#So, choose parking, taste and clean for the PCA variable reduction model
# show summary of pca solution
pca_explained_variance = pca.explained_variance_ratio_
print(‘Proportion of variance explained:’, pca_explained_variance)
# note that principal components analysis corresponds
# to finding eigenvalues and eigenvectors of the pca_data
pca_data_cormat = np.corrcoef(pca_data.T)
eigenvalues, eigenvectors = np.linalg.eig(pca_data_cormat)
np.around([eigenvalues], decimals=3)
print(‘Linear algebra demonstration: Proportion of variance explained: ‘,
eigenvalues/eigenvalues.sum())
np.around([eigenvectors], decimals=3)
# show the plot for the pricipal component analysis
plt.bar(np.arange(len(pca_explained_variance)), pca_explained_variance,
color = ‘grey’, alpha = 0.5, align = ‘center’)
plt.title(‘PCA Proportion of Total Variance’)
plt.show()
# show a scree plot
d = {‘eigenvalues’: eigenvalues }
df1 = pd.DataFrame(data=d)
df2 =pd.Series([1,2,3,4,5,6,7,8,9])
#df2 = {‘factors’: factors}
# merge eigenvalues with # of factors
result = pd.concat([df1, df2], axis=1, join_axes=[df2.index])
print (result)
def scat(dataframe,var1,var2):
dataframe[var2].plot()
plt.title(‘Scree Plot’)
plt.xlabel(‘# of factors’)
plt.ylabel(‘Eigenvalues’)
scat(result,’0′,’eigenvalues’)
plt.show()
# provide partial listing of variable loadings on principal components
# transpose for variables by components listing
pca_loadings = pca.components_.T
# provide full formatted listing of loadings for first three components
# print loadings while rounding to three digits
# and suppress printing of very small numbers
# but do not suppress printing of zeroes
np.set_printoptions(precision = 3, suppress = True,
formatter={‘float’: ‘{: 0.3f}’.format})
print(pca_loadings[:,0:3])
# compute full set of principal components (scores)
C = pca.transform(pca_data)
print(C)
# add first three principal component scores to the original data frame
pca_data[‘pca1’] = C[:,0]
pca_data[‘pca2’] = C[:,1]
pca_data[‘pca3’] = C[:,2]
print(pca_data)
# add first three principal component scores to the food_df
food_df[‘pca1’] = C[:,0]
food_df[‘pca2’] = C[:,1]
food_df[‘pca3’] = C[:,2]
print(food_df)
# explore relationships between pairs of principal components
# working with the first three components only
pca_scores = pca_data.loc[:,[‘pca1′,’pca2’, ‘pca3’]]
pca_model_cormat = \
np.corrcoef(pca_scores.as_matrix().transpose()).round(decimals=3)
print(pca_model_cormat)
#Looks like that worked
#Factor Analysis
print(”)
print(‘—– Factor Analysis (Unrotated) —–‘)
print(”)
# assume three factors will be sufficient
# this is an unrotated orthogonal solution
# maximum likelihood estimation is employed
# for best results set tolerance low and max iterations high
fa = FactorAnalysis(n_components = 3, tol=1e-8, max_iter=1000000)
#the unrotated solution
fa.fit(pca_data)
# retrieve the factor loadings as an array of arrays
# transpose for variables by factors listing of loadings
fa_loadings = fa.components_.T
print(fa_loadings)
# show the loadings of the variables on the factors
# for the unrotated maximum likelihood solution
# print loadings while rounding to three digits
# and suppress printing of very small numbers
# but do not suppress printing of zeroes
np.set_printoptions(precision = 3, suppress = True,
formatter={‘float’: ‘{: 0.3f}’.format})
print(fa_loadings)
# compute full set of factor scores
F = fa.transform(pca_data)
print(F)
# add factor scores to the original data frame
food_df[‘fa1’] = F[:,0]
food_df[‘fa2’] = F[:,1]
food_df[‘fa3’] = F[:,2]
print(food_df)
#Look at five different models and compare them
#Which model do you think is best and why?
#Model 1 full regression model
#Model 2 select my reduced regression model taste, wait and location
#Model 3 Full PCA model
#Model 4 Reduced PCA model with parking, taste and clean
#Model 5 FA model
#Run the Models
#Model 1 full model
regress_model_fit = smf.ols(formula = ‘overall~taste+temp+freshness+wait+clean+friend+location+parking+view’, data = food_df).fit()
# summary of model fit
print(regress_model_fit.summary())
#Model 2
#Note, Model 2 is a choice from looking at the correlation, you may choose a
#different selection for this if you like, just explain why
regress_model_fit = smf.ols(formula = ‘overall~taste+wait+location’, data = food_df).fit()
# summary of model fit
print(regress_model_fit.summary())
#Model 3
#regress the response overall on principal components
pca_model_fit = smf.ols(formula = ‘overall~pca1+pca2+pca3’, data = food_df).fit()
# summary of model fit
print(pca_model_fit.summary())
#Model 4
#regress the response overall on principal components
pca_model_fit = smf.ols(formula = ‘overall~parking+taste+clean’, data = food_df).fit()
# summary of model fit
print(pca_model_fit.summary())
#Model 5
#regress the response overall on factor scores
fa_model_fit = smf.ols(formula = ‘overall~fa1+fa2+fa3’, data = food_df).fit()
# summary of model fit
print(fa_model_fit.summary())
#next look at VIF to see what the full, choice, PCA and FA models did
# Break into left and right hand side; y and X then find VIF for each model
import statsmodels.formula.api as sm
from patsy import dmatrices
from statsmodels.stats.outliers_influence import variance_inflation_factor
y = food_df.loc[:,[‘overall’]]
X = food_df.loc[:,[‘taste’,’temp’,’freshness’,’wait’,’clean’,’friend’,’location’,’parking’,’view’]]
y, X = dmatrices(‘overall ~ taste+temp+freshness+wait+clean+friend+location+parking+view ‘, data=food_df, return_type=”dataframe”)
# For each Xi, calculate VIF
vif = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]
print(”)
print(‘—– VIF for Full Regression Model —–‘)
print(”)
print(vif)
#VIF for choice model
y = food_df.loc[:,[‘overall’]]
X = food_df.loc[:,[‘taste’,’clean’,’location’]]
y, X = dmatrices(‘overall ~ taste+clean+location ‘, data=food_df, return_type=”dataframe”)
vif = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]
print(”)
print(‘—– VIF for Choice Model —–‘)
print(”)
print(vif)
#VIF for PCA
y = food_df.loc[:,[‘overall’]]
X = food_df.loc[:,[‘pca1′,’pca2′,’pca3’]]
y, X = dmatrices(‘overall ~ pca1+pca2+pca3 ‘, data=food_df, return_type=”dataframe”)
vif = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]
print(”)
print(‘—– VIF for PCA Model —–‘)
print(”)
print(vif)
#VIF for FA
y = food_df.loc[:,[‘overall’]]
X = food_df.loc[:,[‘fa1′,’fa2′,’fa3’]]
y, X = dmatrices(‘overall ~ fa1+fa2+fa3 ‘, data=food_df, return_type=”dataframe”)
vif = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]
print(”)
print(‘—– VIF for FA Model —–‘)
print(”)
print(vif)
#Which model do you like best and why?
#For the full regression model sum the coefficients for each three variable
#grouping, taste, temp freshness group 1
# location, parking, view group 3
#How do you interpret this info?
#Compare with the choice model
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