# COMPUTER SCIENCE

#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)

#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

#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()

# transpose for variables by components listing

# 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})

# 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)

# for the unrotated maximum likelihood solution

# 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})

# 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

#wait, clean, friend group 2

# location, parking, view group 3

#How do you interpret this info?

#Compare with the choice model

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