CUNY 621 Homework 5
Joel Park
5/6/2018
Overview
In this homework assignment, you will explore, analyze and model a data set containing information on approximately 12,000 commercially available wines. The variables are mostly related to the chemical properties of the wine being sold. The response variable is the number of sample cases of wine that were purchased by wine distribution companies after sampling a wine. These cases would be used to provide tasting samples to restaurants and wine stores around the United States. The more sample cases purchased, the more likely is a wine to be sold at a high end restaurant. A large wine manufacturer is studying the data in order to predict the number of wine cases ordered based upon the wine characteristics. If the wine manufacturer can predict the number of cases, then that manufacturer will be able to adjust their wine offering to maximize sales.
Your objective is to build a count regression model to predict the number of cases of wine that will be sold given certain properties of the wine. HINT: Sometimes, the fact that a variable is missing is actually predictive of the target. You can only use the variables given to you (or variables that you derive from the variables provided). Below is a short description of the variables of interest in the data set:
Write-Up:
1. Data Exploration
As always, let us explore the training data set.
## TARGET FixedAcidity VolatileAcidity CitricAcid ResidualSugar Chlorides
## 1 3 3.2 1.160 -0.98 54.2 -0.567
## 2 3 4.5 0.160 -0.81 26.1 -0.425
## 3 5 7.1 2.640 -0.88 14.8 0.037
## 4 3 5.7 0.385 0.04 18.8 -0.425
## 5 4 8.0 0.330 -1.26 9.4 NA
## 6 0 11.3 0.320 0.59 2.2 0.556
## FreeSulfurDioxide TotalSulfurDioxide Density pH Sulphates Alcohol
## 1 NA 268 0.99280 3.33 -0.59 9.9
## 2 15 -327 1.02792 3.38 0.70 NA
## 3 214 142 0.99518 3.12 0.48 22.0
## 4 22 115 0.99640 2.24 1.83 6.2
## 5 -167 108 0.99457 3.12 1.77 13.7
## 6 -37 15 0.99940 3.20 1.29 15.4
## LabelAppeal AcidIndex STARS
## 1 0 8 2
## 2 -1 7 3
## 3 -1 8 3
## 4 -1 6 1
## 5 0 9 2
## 6 0 11 NA## TARGET FixedAcidity VolatileAcidity CitricAcid
## Min. :0.000 Min. :-18.100 Min. :-2.7900 Min. :-3.2400
## 1st Qu.:2.000 1st Qu.: 5.200 1st Qu.: 0.1300 1st Qu.: 0.0300
## Median :3.000 Median : 6.900 Median : 0.2800 Median : 0.3100
## Mean :3.029 Mean : 7.076 Mean : 0.3241 Mean : 0.3084
## 3rd Qu.:4.000 3rd Qu.: 9.500 3rd Qu.: 0.6400 3rd Qu.: 0.5800
## Max. :8.000 Max. : 34.400 Max. : 3.6800 Max. : 3.8600
##
## ResidualSugar Chlorides FreeSulfurDioxide TotalSulfurDioxide
## Min. :-127.800 Min. :-1.1710 Min. :-555.00 Min. :-823.0
## 1st Qu.: -2.000 1st Qu.:-0.0310 1st Qu.: 0.00 1st Qu.: 27.0
## Median : 3.900 Median : 0.0460 Median : 30.00 Median : 123.0
## Mean : 5.419 Mean : 0.0548 Mean : 30.85 Mean : 120.7
## 3rd Qu.: 15.900 3rd Qu.: 0.1530 3rd Qu.: 70.00 3rd Qu.: 208.0
## Max. : 141.150 Max. : 1.3510 Max. : 623.00 Max. :1057.0
## NA's :616 NA's :638 NA's :647 NA's :682
## Density pH Sulphates Alcohol
## Min. :0.8881 Min. :0.480 Min. :-3.1300 Min. :-4.70
## 1st Qu.:0.9877 1st Qu.:2.960 1st Qu.: 0.2800 1st Qu.: 9.00
## Median :0.9945 Median :3.200 Median : 0.5000 Median :10.40
## Mean :0.9942 Mean :3.208 Mean : 0.5271 Mean :10.49
## 3rd Qu.:1.0005 3rd Qu.:3.470 3rd Qu.: 0.8600 3rd Qu.:12.40
## Max. :1.0992 Max. :6.130 Max. : 4.2400 Max. :26.50
## NA's :395 NA's :1210 NA's :653
## LabelAppeal AcidIndex STARS
## Min. :-2.000000 Min. : 4.000 Min. :1.000
## 1st Qu.:-1.000000 1st Qu.: 7.000 1st Qu.:1.000
## Median : 0.000000 Median : 8.000 Median :2.000
## Mean :-0.009066 Mean : 7.773 Mean :2.042
## 3rd Qu.: 1.000000 3rd Qu.: 8.000 3rd Qu.:3.000
## Max. : 2.000000 Max. :17.000 Max. :4.000
## NA's :3359
The above shows that most of the data is more or less normally distributed. There are some values such as Sulphates, Chlorides, VolatileAcidity, CitricAcid, ResidualSugar, FixedAcid, FreeSulfurDioxide, and TotalSulfurDioxide that have negative values. After taking a closer look at these data points, it is likely that the creator of this dataset ended up standardizing the data. Because the data is already normally distributed and the test (eval) dataset is also likely structured in this way, we will not be making any adjustments to this data.
Unfortunately from the plots, there doesn’t appear to be any meaningful information that we can extract from this.
It would be interesting to see if there were any correlations between the independent variables to independent variables, and independent variables to the dependent variable.
## TARGET FixedAcidity VolatileAcidity
## TARGET 1.0000000000 -0.012538100 -0.0759978765
## FixedAcidity -0.0125380998 1.000000000 0.0190109733
## VolatileAcidity -0.0759978765 0.019010973 1.0000000000
## CitricAcid 0.0023450490 0.014000376 -0.0234315631
## ResidualSugar 0.0035195999 -0.015429391 0.0015279517
## Chlorides -0.0304301331 -0.006104447 0.0148489225
## FreeSulfurDioxide 0.0226398054 0.015438463 -0.0114408079
## TotalSulfurDioxide 0.0216020726 -0.023323485 -0.0007434083
## Density -0.0475989086 0.011574241 0.0130977690
## pH 0.0002198557 -0.004553886 0.0072030364
## Sulphates -0.0212203783 0.042229181 0.0015161001
## Alcohol 0.0737771084 -0.013085026 0.0002603082
## LabelAppeal 0.4979464796 0.011375965 -0.0202419713
## AcidIndex -0.1676430648 0.154167846 0.0250529742
## STARS 0.5546857223 -0.004937345 -0.0402432388
## CitricAcid ResidualSugar Chlorides
## TARGET 0.0023450490 0.003519600 -0.0304301331
## FixedAcidity 0.0140003760 -0.015429391 -0.0061044471
## VolatileAcidity -0.0234315631 0.001527952 0.0148489225
## CitricAcid 1.0000000000 -0.009843146 -0.0335608661
## ResidualSugar -0.0098431456 1.000000000 0.0041215692
## Chlorides -0.0335608661 0.004121569 1.0000000000
## FreeSulfurDioxide 0.0121132485 0.021959113 -0.0204924876
## TotalSulfurDioxide -0.0099174506 0.017030939 0.0004188605
## Density -0.0169919691 -0.007120841 0.0206724860
## pH -0.0007581304 0.017563769 -0.0179702278
## Sulphates -0.0144237270 -0.002705775 0.0026187777
## Alcohol 0.0169864284 -0.018943324 -0.0228849573
## LabelAppeal 0.0153315666 -0.004579308 -0.0063870237
## AcidIndex 0.0545838104 -0.020301890 -0.0017134096
## STARS 0.0071401699 0.019665541 -0.0063242568
## FreeSulfurDioxide TotalSulfurDioxide Density
## TARGET 0.022639805 0.0216020726 -0.047598909
## FixedAcidity 0.015438463 -0.0233234848 0.011574241
## VolatileAcidity -0.011440808 -0.0007434083 0.013097769
## CitricAcid 0.012113248 -0.0099174506 -0.016991969
## ResidualSugar 0.021959113 0.0170309394 -0.007120841
## Chlorides -0.020492488 0.0004188605 0.020672486
## FreeSulfurDioxide 1.000000000 0.0134616726 -0.008663509
## TotalSulfurDioxide 0.013461673 1.0000000000 0.023167955
## Density -0.008663509 0.0231679548 1.000000000
## pH -0.002008516 -0.0034227601 -0.002019229
## Sulphates 0.026829029 0.0025040509 -0.010609294
## Alcohol -0.023867458 -0.0168515467 -0.006128355
## LabelAppeal 0.014960087 -0.0027237419 -0.018094403
## AcidIndex -0.014733717 -0.0221292631 0.047778830
## STARS -0.015390398 0.0220949002 -0.028492455
## pH Sulphates Alcohol LabelAppeal
## TARGET 0.0002198557 -0.021220378 0.0737771084 0.4979464796
## FixedAcidity -0.0045538857 0.042229181 -0.0130850260 0.0113759650
## VolatileAcidity 0.0072030364 0.001516100 0.0002603082 -0.0202419713
## CitricAcid -0.0007581304 -0.014423727 0.0169864284 0.0153315666
## ResidualSugar 0.0175637691 -0.002705775 -0.0189433242 -0.0045793083
## Chlorides -0.0179702278 0.002618778 -0.0228849573 -0.0063870237
## FreeSulfurDioxide -0.0020085157 0.026829029 -0.0238674577 0.0149600871
## TotalSulfurDioxide -0.0034227601 0.002504051 -0.0168515467 -0.0027237419
## Density -0.0020192285 -0.010609294 -0.0061283546 -0.0180944026
## pH 1.0000000000 0.010449255 -0.0122034469 0.0002181758
## Sulphates 0.0104492547 1.000000000 0.0108443299 0.0037686996
## Alcohol -0.0122034469 0.010844330 1.0000000000 -0.0006449123
## LabelAppeal 0.0002181758 0.003768700 -0.0006449123 1.0000000000
## AcidIndex -0.0537128921 0.031071782 -0.0558919056 0.0103009840
## STARS -0.0044002985 -0.023135130 0.0648544864 0.3188970216
## AcidIndex STARS
## TARGET -0.16764306 0.554685722
## FixedAcidity 0.15416785 -0.004937345
## VolatileAcidity 0.02505297 -0.040243239
## CitricAcid 0.05458381 0.007140170
## ResidualSugar -0.02030189 0.019665541
## Chlorides -0.00171341 -0.006324257
## FreeSulfurDioxide -0.01473372 -0.015390398
## TotalSulfurDioxide -0.02212926 0.022094900
## Density 0.04777883 -0.028492455
## pH -0.05371289 -0.004400299
## Sulphates 0.03107178 -0.023135130
## Alcohol -0.05589191 0.064854486
## LabelAppeal 0.01030098 0.318897022
## AcidIndex 1.00000000 -0.095482582
## STARS -0.09548258 1.000000000
Not surprisingly, STARS and LabelAppeal appear to have a fairly strong positive correlation with the number of cases sold.
Are there any missing values?
## STARS Sulphates TotalSulfurDioxide
## 3359 1210 682
## Alcohol FreeSulfurDioxide Chlorides
## 653 647 638
## ResidualSugar pH AcidIndex
## 616 395 0
## LabelAppeal Density CitricAcid
## 0 0 0
## VolatileAcidity FixedAcidity TARGET
## 0 0 0
There’s a couple of missing data points that needs to be imputed and corrected. pH, ResidualSugar,Chlorides,FreeSulfurDioxide, Alcohol, TotalSulfurDioxide, Sulphates, STARS.
2. Data Preparation
Let’s prepare the data. We will first impute the missing data points and then create some buckets for some of the terms in the dataset. Given that in the previous section, we have observed that the variables appear for the most part normally distributed, we will NOT be performing any transformations to the dataset.
Given that the histogram distribution of the missing data points all (for the most part) appear to have normal distribution, we will impute the mean to all of the missing data points.
## TARGET FixedAcidity VolatileAcidity CitricAcid ResidualSugar Chlorides
## 1 3 3.2 1.160 -0.98 54.2 -0.56700000
## 2 3 4.5 0.160 -0.81 26.1 -0.42500000
## 3 5 7.1 2.640 -0.88 14.8 0.03700000
## 4 3 5.7 0.385 0.04 18.8 -0.42500000
## 5 4 8.0 0.330 -1.26 9.4 0.05482249
## 6 0 11.3 0.320 0.59 2.2 0.55600000
## FreeSulfurDioxide TotalSulfurDioxide Density pH Sulphates Alcohol
## 1 30.84557 268 0.99280 3.33 -0.59 9.90000
## 2 15.00000 -327 1.02792 3.38 0.70 10.48924
## 3 214.00000 142 0.99518 3.12 0.48 22.00000
## 4 22.00000 115 0.99640 2.24 1.83 6.20000
## 5 -167.00000 108 0.99457 3.12 1.77 13.70000
## 6 -37.00000 15 0.99940 3.20 1.29 15.40000
## LabelAppeal AcidIndex STARS
## 1 0 8 2.000000
## 2 -1 7 3.000000
## 3 -1 8 3.000000
## 4 -1 6 1.000000
## 5 0 9 2.000000
## 6 0 11 2.041755We will not perform any transformations at the moment given that the predictor variables appear to be generally normally distributed (with the exception of the target variable).
Although important information can be lost in a bucket transformation, we will perform two bucket transformations. The LabelAppeal data is a marketing score indicating the appeal of label design for consumers. High numbers suggest customers like the label design. Negative numbers suggests customers don’t like the design. A score greater than 0, 0, and less than zero suggests a positive, neutral, and negative appeal respectively. Therefore, we will be creating three buckets with the LabelAppeal.
All wine tends to be acidic. However, from prior experiences and knowledge, the degree of acidity depends on the wine. In this particular exercise, we will separate the pH of wine to acidic if the pH is greater than 3 and very acidic if less than 3. Interestingly, the gastric acid in the stomach is on average between a pH of 1.5 to 3.5. (Reference: https://en.wikipedia.org/wiki/Gastric_acid). So could wine in this range be good? Regardless, let’s prepare the data for this transformation.
## labelappeal
## NEGATIVE NEUTRAL POSITIVE
## 3640 5617 3538## pH
## ACIDIC VERY ACIDIC
## 9442 33533. Build Models
Given that the response variable is given as a count, we will be investigating different models including the Poisson, Negative Binomial, simple Linear Regression, and hybrid models. From these models, we will determine which model would potentially best predict the evaluation data set.
Poisson Regression Model 1
## Start: AIC=50483.41
## TARGET ~ FixedAcidity + VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + Density +
## pH + Sulphates + Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - FixedAcidity 1 18512 50482
## - ResidualSugar 1 18512 50482
## <none> 18511 50483
## - Density 1 18516 50487
## - CitricAcid 1 18517 50487
## - pH 1 18521 50491
## - Sulphates 1 18522 50492
## - Chlorides 1 18525 50495
## - Alcohol 1 18527 50497
## - FreeSulfurDioxide 1 18528 50498
## - TotalSulfurDioxide 1 18534 50504
## - VolatileAcidity 1 18573 50543
## - AcidIndex 1 19328 51298
## - STARS 1 19554 51524
## - LabelAppeal 1 19614 51584
##
## Step: AIC=50481.7
## TARGET ~ VolatileAcidity + CitricAcid + ResidualSugar + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + pH + Sulphates +
## Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - ResidualSugar 1 18513 50481
## <none> 18512 50482
## - Density 1 18517 50485
## - CitricAcid 1 18517 50485
## - pH 1 18522 50490
## - Sulphates 1 18523 50491
## - Chlorides 1 18525 50493
## - Alcohol 1 18527 50495
## - FreeSulfurDioxide 1 18528 50496
## - TotalSulfurDioxide 1 18534 50502
## - VolatileAcidity 1 18573 50541
## - AcidIndex 1 19356 51324
## - STARS 1 19554 51522
## - LabelAppeal 1 19615 51583
##
## Step: AIC=50480.65
## TARGET ~ VolatileAcidity + CitricAcid + Chlorides + FreeSulfurDioxide +
## TotalSulfurDioxide + Density + pH + Sulphates + Alcohol +
## LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## <none> 18513 50481
## - CitricAcid 1 18518 50484
## - Density 1 18518 50484
## - pH 1 18522 50489
## - Sulphates 1 18524 50490
## - Chlorides 1 18526 50492
## - Alcohol 1 18528 50494
## - FreeSulfurDioxide 1 18529 50495
## - TotalSulfurDioxide 1 18535 50501
## - VolatileAcidity 1 18574 50540
## - AcidIndex 1 19357 51323
## - STARS 1 19556 51522
## - LabelAppeal 1 19616 51582##
## Call:
## glm(formula = TARGET ~ VolatileAcidity + CitricAcid + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + pH + Sulphates +
## Alcohol + LabelAppeal + AcidIndex + STARS, family = poisson,
## data = train)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -3.5135 -0.5169 0.2084 0.6350 2.5697
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.045e+00 1.957e-01 10.451 < 2e-16 ***
## VolatileAcidity -5.106e-02 6.491e-03 -7.866 3.68e-15 ***
## CitricAcid 1.336e-02 5.892e-03 2.267 0.023405 *
## Chlorides -6.056e-02 1.645e-02 -3.681 0.000232 ***
## FreeSulfurDioxide 1.422e-04 3.513e-05 4.047 5.19e-05 ***
## TotalSulfurDioxide 1.078e-04 2.268e-05 4.753 2.00e-06 ***
## Density -4.366e-01 1.921e-01 -2.273 0.023025 *
## pH -2.401e-02 7.637e-03 -3.143 0.001671 **
## Sulphates -1.910e-02 5.737e-03 -3.328 0.000873 ***
## Alcohol 5.503e-03 1.410e-03 3.904 9.47e-05 ***
## LabelAppeal 1.996e-01 6.014e-03 33.190 < 2e-16 ***
## AcidIndex -1.236e-01 4.409e-03 -28.031 < 2e-16 ***
## STARS 2.113e-01 6.491e-03 32.554 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 22861 on 12794 degrees of freedom
## Residual deviance: 18513 on 12782 degrees of freedom
## AIC: 50481
##
## Number of Fisher Scoring iterations: 5All of these variables appear to have minor but have statistically significant impact. Overall, it does appear that the more citric, sulfur dioxide containing, alcohol, label appeal, and stars that the observation had, the more likely the case was going to sell. Likewise, the less acid, more dense, more sulphates, and more chlorides, the less likely the wine was going to sell cases.
Poisson Regression Model 2 - Zero-inflated
Reference: https://stats.idre.ucla.edu/r/dae/zip/
“Zero-inflated poisson regression is used to model count data that has an excess of zero counts. Further, theory suggests that the excess zeros are generated by a separate process from the count values and that the excess zeros can be modeled independently. Thus, the zip model has two parts, a poisson count model and the logit model for predicting excess zeros.”
Reference: https://www.theanalysisfactor.com/zero-inflated-poisson-models-for-count-outcomes/
“The Poisson distribution assumes that each count is the result of each Poisson process - a random process that says each counted event is independent and equally likely. If this count variable is used as the outcome of a regression model, we can use Poisson regression to estimate how predictors affect the number of times the devent occurred.”
“But sometimes, it’s just a matter of having too many zeros that a Poisson would predict. In this case, a better solution is often the Zero-Inflated Poisson (ZIP) model…ZIP models assume that some zeros occurred by a Poisson process…The ZIP model fits, simultaneously, two separate regression models. One is a logistic model that models the probability of being eligible for a non-zero count. The other models the size of that count.”
“Both models use the same predictor variables, but estimate their coefficients separately. So the predictors can have vastly different effects on the two processes.”
Below is the second model utilizing ZIP.
##
## Call:
## zeroinfl(formula = TARGET ~ ., data = train)
##
## Pearson residuals:
## Min 1Q Median 3Q Max
## -2.0500 -0.3576 0.1641 0.5043 4.3466
##
## Count model coefficients (poisson with log link):
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.432e+00 2.065e-01 6.935 4.06e-12 ***
## FixedAcidity 2.548e-04 8.578e-04 0.297 0.766467
## VolatileAcidity -1.314e-02 6.869e-03 -1.913 0.055696 .
## CitricAcid -9.139e-05 6.143e-03 -0.015 0.988131
## ResidualSugar -9.652e-05 1.623e-04 -0.595 0.551929
## Chlorides -1.783e-02 1.729e-02 -1.031 0.302499
## FreeSulfurDioxide 3.019e-05 3.602e-05 0.838 0.401912
## TotalSulfurDioxide -3.123e-05 2.299e-05 -1.358 0.174465
## Density -3.266e-01 2.025e-01 -1.612 0.106880
## pH 7.043e-03 8.037e-03 0.876 0.380857
## Sulphates -4.662e-04 6.053e-03 -0.077 0.938609
## Alcohol 7.702e-03 1.467e-03 5.248 1.53e-07 ***
## LabelAppeal 2.516e-01 6.373e-03 39.476 < 2e-16 ***
## AcidIndex -1.694e-02 4.987e-03 -3.396 0.000683 ***
## STARS 9.941e-02 6.456e-03 15.399 < 2e-16 ***
##
## Zero-inflation model coefficients (binomial with logit link):
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -5.8375100 1.0427097 -5.598 2.16e-08 ***
## FixedAcidity 0.0031792 0.0043351 0.733 0.46334
## VolatileAcidity 0.2443276 0.0347530 7.030 2.06e-12 ***
## CitricAcid -0.0823095 0.0315877 -2.606 0.00917 **
## ResidualSugar -0.0014124 0.0008243 -1.713 0.08665 .
## Chlorides 0.2707215 0.0872970 3.101 0.00193 **
## FreeSulfurDioxide -0.0007069 0.0001856 -3.808 0.00014 ***
## TotalSulfurDioxide -0.0008624 0.0001195 -7.216 5.34e-13 ***
## Density 0.8840376 1.0245676 0.863 0.38823
## pH 0.1960739 0.0408076 4.805 1.55e-06 ***
## Sulphates 0.1209183 0.0307322 3.935 8.33e-05 ***
## Alcohol 0.0096226 0.0074735 1.288 0.19790
## LabelAppeal 0.3110249 0.0330563 9.409 < 2e-16 ***
## AcidIndex 0.4854658 0.0198008 24.518 < 2e-16 ***
## STARS -0.6286823 0.0378754 -16.599 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Number of iterations in BFGS optimization: 37
## Log-likelihood: -2.254e+04 on 30 Df## $count
## (Intercept) FixedAcidity VolatileAcidity
## 1.432365e+00 2.547780e-04 -1.314284e-02
## CitricAcid ResidualSugar Chlorides
## -9.138811e-05 -9.652085e-05 -1.783056e-02
## FreeSulfurDioxide TotalSulfurDioxide Density
## 3.019450e-05 -3.122704e-05 -3.265780e-01
## pH Sulphates Alcohol
## 7.042765e-03 -4.661626e-04 7.701558e-03
## LabelAppeal AcidIndex STARS
## 2.515711e-01 -1.693781e-02 9.940990e-02
##
## $zero
## (Intercept) FixedAcidity VolatileAcidity
## -5.8375099785 0.0031791813 0.2443276030
## CitricAcid ResidualSugar Chlorides
## -0.0823095335 -0.0014123831 0.2707215386
## FreeSulfurDioxide TotalSulfurDioxide Density
## -0.0007069038 -0.0008624352 0.8840376227
## pH Sulphates Alcohol
## 0.1960739330 0.1209182605 0.0096225512
## LabelAppeal AcidIndex STARS
## 0.3110249237 0.4854658254 -0.6286822992It is interesting to compare the zero inflated Poisson model to the original Poisson model. There does appear to be some differences, but even more interesting is the differences in direction (positive and negative) of the independent variables. For instance, the density of the wine is positive in $count, but negative in $zero (any many other variables). It is unclear why this is, but we will certainly keep this in the back of our mind.
Is the zero-inflated model better than the standard Poisson regression model? We will perform the Vuong test between the two models.
## Vuong Non-Nested Hypothesis Test-Statistic:
## (test-statistic is asymptotically distributed N(0,1) under the
## null that the models are indistinguishible)
## -------------------------------------------------------------
## Vuong z-statistic H_A p-value
## Raw -38.67164 model2 > model1 < 2.22e-16
## AIC-corrected -38.42710 model2 > model1 < 2.22e-16
## BIC-corrected -37.51535 model2 > model1 < 2.22e-16“The Vuong test compares the zero-inflated model with an ordinary Poisson regression model.” The test statistic is significant indicating that the zero-inflated model is superior to the standard Poisson model.
Negative Binomial Model 3
Reference: http://data.library.virginia.edu/getting-started-with-negative-binomial-regression-modeling/
“A Poisson distribution is parameterized by \(\lambda\), which happens to be both its mean and variance. While convenient, it’s not often realistic. A distribution of counts will usually have a variance that’s not equal to its mean. When we see this happen with data that we assume is (or hope) is Poisson distributed, we say we have under- or overdispersion, depending on if the variance is smaller or larger than the mean. Performing Poisson regression on count data that exhibits this behavior results in a model that doesn’t fit well.”
“One approach that addresses this issue is the Negative Binomial Regresion. The negative binomial distribution describes the probabilities of the occurrence of whole numbers greater than or equal to 0. Unlike the Poisson distribution, the variance and the mean are not equivalent. This suggests it might serve as a useful approximation for modeling counts with variability different from its mean. The variance of a negative binomial distribution is a function of its mean and has an additional parameter, k, called the dispersion parameter. Say our count is a random variable Y from a negative binomial distribution, when the variance of Y is:”
\[var(Y) = \mu + \mu^2lk\]
“As the dispersion parameter gets larger and larger, the variance converges to the same value as the mean, and the negative binomial turns into a Poisson distribution.”
## [1] "TARGET mean: 3"## [1] "TARGET variance: 3.711"So there appears to be a slight overdispersion with the variance greater than the mean. Let’s apply a negative binomial model for model 3.
##
## Call:
## glm.nb(formula = TARGET ~ ., data = train, init.theta = 38671.03175,
## link = log)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -3.5116 -0.5144 0.2080 0.6344 2.5663
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.045e+00 1.957e-01 10.447 < 2e-16 ***
## FixedAcidity -4.444e-04 8.195e-04 -0.542 0.587593
## VolatileAcidity -5.097e-02 6.492e-03 -7.851 4.13e-15 ***
## CitricAcid 1.343e-02 5.892e-03 2.279 0.022638 *
## ResidualSugar 1.489e-04 1.545e-04 0.964 0.335073
## Chlorides -6.058e-02 1.645e-02 -3.683 0.000231 ***
## FreeSulfurDioxide 1.420e-04 3.513e-05 4.041 5.32e-05 ***
## TotalSulfurDioxide 1.072e-04 2.268e-05 4.727 2.28e-06 ***
## Density -4.364e-01 1.921e-01 -2.272 0.023078 *
## pH -2.411e-02 7.639e-03 -3.156 0.001599 **
## Sulphates -1.901e-02 5.738e-03 -3.312 0.000925 ***
## Alcohol 5.528e-03 1.410e-03 3.920 8.85e-05 ***
## LabelAppeal 1.996e-01 6.015e-03 33.178 < 2e-16 ***
## AcidIndex -1.232e-01 4.461e-03 -27.615 < 2e-16 ***
## STARS 2.112e-01 6.492e-03 32.541 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Negative Binomial(38671.03) family taken to be 1)
##
## Null deviance: 22860 on 12794 degrees of freedom
## Residual deviance: 18511 on 12780 degrees of freedom
## AIC: 50486
##
## Number of Fisher Scoring iterations: 1
##
##
## Theta: 38671
## Std. Err.: 59727
## Warning while fitting theta: iteration limit reached
##
## 2 x log-likelihood: -50453.53It is interesting to note that the beta values for the independent variables in the negative binomial model is the same as the beta values in the Poisson regression model.
Negative Binomial Model 4
Reference: http://rstudio-pubs-static.s3.amazonaws.com/2516_d204dc109a2c44d58899afb418ae3885.html
The negative binomial model appears to have many multiple statistically significant values. Let’s see if we can improve the model by using the bucket data with pH and Label Appeal instead.
##
## Call:
## glm.nb(formula = TARGET ~ ., data = train2, init.theta = 37640.68841,
## link = log)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -3.3086 -0.5466 0.1984 0.6401 2.8234
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.652e+00 1.942e-01 8.504 < 2e-16 ***
## FixedAcidity -5.261e-04 8.196e-04 -0.642 0.520967
## VolatileAcidity -5.203e-02 6.491e-03 -8.015 1.10e-15 ***
## CitricAcid 1.320e-02 5.892e-03 2.240 0.025072 *
## ResidualSugar 1.765e-04 1.544e-04 1.143 0.252976
## Chlorides -5.978e-02 1.644e-02 -3.637 0.000276 ***
## FreeSulfurDioxide 1.425e-04 3.512e-05 4.057 4.96e-05 ***
## TotalSulfurDioxide 1.053e-04 2.269e-05 4.640 3.48e-06 ***
## Density -4.059e-01 1.920e-01 -2.113 0.034561 *
## Sulphates -1.895e-02 5.735e-03 -3.305 0.000951 ***
## Alcohol 5.607e-03 1.409e-03 3.980 6.88e-05 ***
## AcidIndex -1.216e-01 4.454e-03 -27.296 < 2e-16 ***
## STARS 2.171e-01 6.477e-03 33.517 < 2e-16 ***
## LabelAppealNEUTRAL 2.881e-01 1.378e-02 20.908 < 2e-16 ***
## LabelAppealPOSITIVE 4.663e-01 1.479e-02 31.524 < 2e-16 ***
## pHVERY ACIDIC 1.502e-02 1.158e-02 1.296 0.194824
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Negative Binomial(37640.69) family taken to be 1)
##
## Null deviance: 22860 on 12794 degrees of freedom
## Residual deviance: 18595 on 12779 degrees of freedom
## AIC: 50572
##
## Number of Fisher Scoring iterations: 1
##
##
## Theta: 37641
## Std. Err.: 58626
## Warning while fitting theta: iteration limit reached
##
## 2 x log-likelihood: -50537.63Not surprisingly, this negative binomial model performed slightly worse than the previous negative binomial model. I suspect that the transformation had lost some key information. (Again, notable is that the beta values of the independent variables are similar to the beta values of the prior negative binomial model.)
Linear Regression Model 5
Instead of utilizing a count regression to perform the prediction, let’s see if a backward stepwise simple linear regression using OLS will provide the best (or perhaps similar) model.
## Start: AIC=12584.82
## TARGET ~ FixedAcidity + VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + Density +
## pH + Sulphates + Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Sum of Sq RSS AIC
## - FixedAcidity 1 0.7 34134 12583
## - ResidualSugar 1 3.3 34137 12584
## <none> 34134 12585
## - Density 1 14.9 34149 12588
## - CitricAcid 1 15.2 34149 12588
## - pH 1 23.8 34157 12592
## - Sulphates 1 31.0 34165 12594
## - Chlorides 1 46.6 34180 12600
## - FreeSulfurDioxide 1 49.1 34183 12601
## - Alcohol 1 62.4 34196 12606
## - TotalSulfurDioxide 1 63.2 34197 12606
## - VolatileAcidity 1 191.6 34325 12654
## - AcidIndex 1 2294.9 36429 13415
## - LabelAppeal 1 3401.0 37535 13798
## - STARS 1 3581.2 37715 13859
##
## Step: AIC=12583.07
## TARGET ~ VolatileAcidity + CitricAcid + ResidualSugar + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + pH + Sulphates +
## Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Sum of Sq RSS AIC
## - ResidualSugar 1 3.4 34138 12582
## <none> 34134 12583
## - Density 1 14.9 34149 12587
## - CitricAcid 1 15.2 34149 12587
## - pH 1 23.8 34158 12590
## - Sulphates 1 31.3 34166 12593
## - Chlorides 1 46.5 34181 12598
## - FreeSulfurDioxide 1 49.0 34183 12599
## - Alcohol 1 62.4 34197 12604
## - TotalSulfurDioxide 1 63.4 34198 12605
## - VolatileAcidity 1 191.7 34326 12653
## - AcidIndex 1 2382.4 36517 13444
## - LabelAppeal 1 3402.3 37537 13797
## - STARS 1 3580.6 37715 13857
##
## Step: AIC=12582.33
## TARGET ~ VolatileAcidity + CitricAcid + Chlorides + FreeSulfurDioxide +
## TotalSulfurDioxide + Density + pH + Sulphates + Alcohol +
## LabelAppeal + AcidIndex + STARS
##
## Df Sum of Sq RSS AIC
## <none> 34138 12582
## - Density 1 14.9 34153 12586
## - CitricAcid 1 15.1 34153 12586
## - pH 1 23.6 34161 12589
## - Sulphates 1 31.4 34169 12592
## - Chlorides 1 46.6 34184 12598
## - FreeSulfurDioxide 1 49.4 34187 12599
## - Alcohol 1 61.9 34200 12604
## - TotalSulfurDioxide 1 64.0 34202 12604
## - VolatileAcidity 1 192.0 34330 12652
## - AcidIndex 1 2383.5 36521 13444
## - LabelAppeal 1 3402.0 37540 13796
## - STARS 1 3584.4 37722 13858##
## Call:
## lm(formula = TARGET ~ VolatileAcidity + CitricAcid + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + pH + Sulphates +
## Alcohol + LabelAppeal + AcidIndex + STARS, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.2763 -0.7375 0.3684 1.1259 4.3324
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.450e+00 5.543e-01 9.833 < 2e-16 ***
## VolatileAcidity -1.566e-01 1.847e-02 -8.478 < 2e-16 ***
## CitricAcid 3.995e-02 1.681e-02 2.377 0.017483 *
## Chlorides -1.947e-01 4.660e-02 -4.179 2.95e-05 ***
## FreeSulfurDioxide 4.295e-04 9.986e-05 4.301 1.71e-05 ***
## TotalSulfurDioxide 3.141e-04 6.415e-05 4.896 9.91e-07 ***
## Density -1.287e+00 5.453e-01 -2.359 0.018319 *
## pH -6.432e-02 2.164e-02 -2.973 0.002959 **
## Sulphates -5.591e-02 1.630e-02 -3.429 0.000607 ***
## Alcohol 1.921e-02 3.990e-03 4.814 1.50e-06 ***
## LabelAppeal 6.043e-01 1.693e-02 35.690 < 2e-16 ***
## AcidIndex -3.301e-01 1.105e-02 -29.873 < 2e-16 ***
## STARS 7.155e-01 1.953e-02 36.634 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.634 on 12782 degrees of freedom
## Multiple R-squared: 0.281, Adjusted R-squared: 0.2803
## F-statistic: 416.2 on 12 and 12782 DF, p-value: < 2.2e-16The model performs reasonably well with an adjusted R-squared value of 0.2803. Let us fit one more model with a model utilizing the bucketed data with a backwards stepwise approach. As of note, the beta values for the independent variables are different than the previous models (but the direction remains the same compared to the prior negative binomial model.)
## Start: AIC=12717.39
## TARGET ~ FixedAcidity + VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + Density +
## Sulphates + Alcohol + AcidIndex + STARS + LabelAppeal + pH
##
## Df Sum of Sq RSS AIC
## - FixedAcidity 1 0.8 34485 12716
## - pH 1 3.9 34488 12717
## - ResidualSugar 1 3.9 34488 12717
## <none> 34484 12717
## - Density 1 13.8 34497 12720
## - CitricAcid 1 15.3 34499 12721
## - Sulphates 1 31.8 34516 12727
## - Chlorides 1 45.5 34529 12732
## - FreeSulfurDioxide 1 49.6 34533 12734
## - TotalSulfurDioxide 1 60.1 34544 12738
## - Alcohol 1 61.5 34545 12738
## - VolatileAcidity 1 197.7 34681 12789
## - AcidIndex 1 2250.1 36734 13524
## - LabelAppeal 2 3069.2 37553 13804
## - STARS 1 3744.0 38228 14034
##
## Step: AIC=12715.68
## TARGET ~ VolatileAcidity + CitricAcid + ResidualSugar + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + Sulphates +
## Alcohol + AcidIndex + STARS + LabelAppeal + pH
##
## Df Sum of Sq RSS AIC
## - pH 1 3.9 34488 12715
## - ResidualSugar 1 4.0 34489 12715
## <none> 34485 12716
## - Density 1 13.8 34498 12719
## - CitricAcid 1 15.3 34500 12719
## - Sulphates 1 32.1 34517 12726
## - Chlorides 1 45.5 34530 12730
## - FreeSulfurDioxide 1 49.4 34534 12732
## - TotalSulfurDioxide 1 60.3 34545 12736
## - Alcohol 1 61.5 34546 12736
## - VolatileAcidity 1 197.9 34682 12787
## - AcidIndex 1 2337.5 36822 13553
## - LabelAppeal 2 3070.4 37555 13803
## - STARS 1 3743.3 38228 14032
##
## Step: AIC=12715.13
## TARGET ~ VolatileAcidity + CitricAcid + ResidualSugar + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + Sulphates +
## Alcohol + AcidIndex + STARS + LabelAppeal
##
## Df Sum of Sq RSS AIC
## - ResidualSugar 1 3.9 34492 12715
## <none> 34488 12715
## - Density 1 13.9 34502 12718
## - CitricAcid 1 15.3 34504 12719
## - Sulphates 1 32.1 34521 12725
## - Chlorides 1 45.2 34534 12730
## - FreeSulfurDioxide 1 49.2 34538 12731
## - TotalSulfurDioxide 1 60.4 34549 12736
## - Alcohol 1 61.5 34550 12736
## - VolatileAcidity 1 198.5 34687 12786
## - AcidIndex 1 2333.9 36822 13551
## - LabelAppeal 2 3068.1 37557 13802
## - STARS 1 3747.3 38236 14033
##
## Step: AIC=12714.58
## TARGET ~ VolatileAcidity + CitricAcid + Chlorides + FreeSulfurDioxide +
## TotalSulfurDioxide + Density + Sulphates + Alcohol + AcidIndex +
## STARS + LabelAppeal
##
## Df Sum of Sq RSS AIC
## <none> 34492 12715
## - Density 1 13.8 34506 12718
## - CitricAcid 1 15.2 34508 12718
## - Sulphates 1 32.3 34525 12725
## - Chlorides 1 45.3 34538 12729
## - FreeSulfurDioxide 1 49.7 34542 12731
## - Alcohol 1 61.0 34553 12735
## - TotalSulfurDioxide 1 61.0 34553 12735
## - VolatileAcidity 1 198.7 34691 12786
## - AcidIndex 1 2335.2 36828 13551
## - LabelAppeal 2 3067.0 37559 13800
## - STARS 1 3751.6 38244 14034##
## Call:
## lm(formula = TARGET ~ VolatileAcidity + CitricAcid + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + Sulphates +
## Alcohol + AcidIndex + STARS + LabelAppeal, data = train2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.9093 -0.7744 0.3638 1.1413 4.8406
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.444e+00 5.530e-01 8.037 1.00e-15 ***
## VolatileAcidity -1.593e-01 1.856e-02 -8.582 < 2e-16 ***
## CitricAcid 4.013e-02 1.690e-02 2.375 0.017553 *
## Chlorides -1.919e-01 4.684e-02 -4.098 4.20e-05 ***
## FreeSulfurDioxide 4.307e-04 1.004e-04 4.291 1.79e-05 ***
## TotalSulfurDioxide 3.066e-04 6.448e-05 4.755 2.01e-06 ***
## Density -1.240e+00 5.481e-01 -2.262 0.023703 *
## Sulphates -5.669e-02 1.639e-02 -3.459 0.000544 ***
## Alcohol 1.906e-02 4.011e-03 4.753 2.03e-06 ***
## AcidIndex -3.261e-01 1.109e-02 -29.417 < 2e-16 ***
## STARS 7.309e-01 1.960e-02 37.286 < 2e-16 ***
## LabelAppealNEUTRAL 7.137e-01 3.530e-02 20.219 < 2e-16 ***
## LabelAppealPOSITIVE 1.361e+00 4.042e-02 33.670 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.643 on 12782 degrees of freedom
## Multiple R-squared: 0.2735, Adjusted R-squared: 0.2728
## F-statistic: 401 on 12 and 12782 DF, p-value: < 2.2e-16This model not surprisingly (from the bucket data) performed slightly worse.
Hybrid Model - Logistic/Negative Binomial Regression 7
Before we move on, it may be beneficial to consider splitting the model into two models: 1) a logistic model to determine if a case was sold or not sold, and 2) and if cases were sold, how many cases were sold. Let’s create the 7th model and see if this improves prediction.
## Start: AIC=11970.36
## case_sold ~ (TARGET + FixedAcidity + VolatileAcidity + CitricAcid +
## ResidualSugar + Chlorides + FreeSulfurDioxide + TotalSulfurDioxide +
## Density + pH + Sulphates + Alcohol + LabelAppeal + AcidIndex +
## STARS) - TARGET
##
## Df Deviance AIC
## - Alcohol 1 11941 11969
## - FixedAcidity 1 11941 11969
## - Density 1 11942 11970
## <none> 11940 11970
## - ResidualSugar 1 11943 11971
## - CitricAcid 1 11946 11974
## - Chlorides 1 11950 11978
## - FreeSulfurDioxide 1 11956 11984
## - Sulphates 1 11957 11985
## - LabelAppeal 1 11961 11989
## - pH 1 11964 11992
## - TotalSulfurDioxide 1 11989 12017
## - VolatileAcidity 1 11997 12025
## - STARS 1 12246 12274
## - AcidIndex 1 12646 12674
##
## Step: AIC=11968.81
## case_sold ~ FixedAcidity + VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + Density +
## pH + Sulphates + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - FixedAcidity 1 11941 11967
## - Density 1 11942 11968
## <none> 11941 11969
## - ResidualSugar 1 11944 11970
## - CitricAcid 1 11946 11972
## - Chlorides 1 11950 11976
## - FreeSulfurDioxide 1 11957 11983
## - Sulphates 1 11957 11983
## - LabelAppeal 1 11962 11988
## - pH 1 11964 11990
## - TotalSulfurDioxide 1 11990 12016
## - VolatileAcidity 1 11997 12023
## - STARS 1 12246 12272
## - AcidIndex 1 12646 12672
##
## Step: AIC=11967.27
## case_sold ~ VolatileAcidity + CitricAcid + ResidualSugar + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + pH + Sulphates +
## LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - Density 1 11942 11966
## <none> 11941 11967
## - ResidualSugar 1 11944 11968
## - CitricAcid 1 11947 11971
## - Chlorides 1 11951 11975
## - FreeSulfurDioxide 1 11957 11981
## - Sulphates 1 11958 11982
## - LabelAppeal 1 11962 11986
## - pH 1 11965 11989
## - TotalSulfurDioxide 1 11990 12014
## - VolatileAcidity 1 11998 12022
## - STARS 1 12247 12271
## - AcidIndex 1 12679 12703
##
## Step: AIC=11966.52
## case_sold ~ VolatileAcidity + CitricAcid + ResidualSugar + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + pH + Sulphates +
## LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## <none> 11942 11966
## - ResidualSugar 1 11945 11967
## - CitricAcid 1 11948 11970
## - Chlorides 1 11952 11974
## - FreeSulfurDioxide 1 11958 11980
## - Sulphates 1 11959 11981
## - LabelAppeal 1 11963 11985
## - pH 1 11966 11988
## - TotalSulfurDioxide 1 11991 12013
## - VolatileAcidity 1 11999 12021
## - STARS 1 12248 12270
## - AcidIndex 1 12684 12706##
## Call:
## glm(formula = case_sold ~ VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + pH +
## Sulphates + LabelAppeal + AcidIndex + STARS, family = "binomial",
## data = train_log)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.5458 0.3422 0.5377 0.6985 2.2987
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.2393464 0.1937075 21.885 < 2e-16 ***
## VolatileAcidity -0.2206034 0.0293638 -7.513 5.79e-14 ***
## CitricAcid 0.0629051 0.0266872 2.357 0.01842 *
## ResidualSugar 0.0011449 0.0006960 1.645 0.09995 .
## Chlorides -0.2296164 0.0736767 -3.117 0.00183 **
## FreeSulfurDioxide 0.0006271 0.0001583 3.962 7.44e-05 ***
## TotalSulfurDioxide 0.0007080 0.0001017 6.961 3.38e-12 ***
## pH -0.1671440 0.0343027 -4.873 1.10e-06 ***
## Sulphates -0.1054237 0.0258674 -4.076 4.59e-05 ***
## LabelAppeal -0.1201902 0.0266018 -4.518 6.24e-06 ***
## AcidIndex -0.4364384 0.0168384 -25.919 < 2e-16 ***
## STARS 0.5688977 0.0336740 16.894 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 13276 on 12794 degrees of freedom
## Residual deviance: 11943 on 12783 degrees of freedom
## AIC: 11967
##
## Number of Fisher Scoring iterations: 4Now that we have created this logistic model, let us take all of the data points and create a negative binomial model from the resulting data that had more than 0 cases sold.
## Start: AIC=33869.3
## TARGET ~ FixedAcidity + VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + Density +
## pH + Sulphates + Alcohol + LabelAppeal + AcidIndex + STARS +
## case_sold
##
##
## Step: AIC=33869.3
## TARGET ~ FixedAcidity + VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + Density +
## pH + Sulphates + Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - Sulphates 1 1897.2 33867
## - CitricAcid 1 1897.2 33867
## - FixedAcidity 1 1897.2 33867
## - ResidualSugar 1 1897.4 33868
## - FreeSulfurDioxide 1 1897.7 33868
## - pH 1 1898.1 33868
## - Chlorides 1 1898.4 33869
## - TotalSulfurDioxide 1 1898.7 33869
## <none> 1897.2 33869
## - Density 1 1899.3 33869
## - VolatileAcidity 1 1900.0 33870
## - AcidIndex 1 1908.6 33879
## - Alcohol 1 1922.6 33893
## - STARS 1 2113.8 34084
## - LabelAppeal 1 3294.8 35265
##
## Step: AIC=33867.3
## TARGET ~ FixedAcidity + VolatileAcidity + CitricAcid + ResidualSugar +
## Chlorides + FreeSulfurDioxide + TotalSulfurDioxide + Density +
## pH + Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - CitricAcid 1 1897.2 33865
## - FixedAcidity 1 1897.2 33865
## - ResidualSugar 1 1897.4 33866
## - FreeSulfurDioxide 1 1897.7 33866
## - pH 1 1898.1 33866
## - Chlorides 1 1898.4 33867
## - TotalSulfurDioxide 1 1898.7 33867
## <none> 1897.2 33867
## - Density 1 1899.3 33867
## - VolatileAcidity 1 1900.0 33868
## - AcidIndex 1 1908.6 33877
## - Alcohol 1 1922.6 33891
## - STARS 1 2113.8 34082
## - LabelAppeal 1 3294.8 35263
##
## Step: AIC=33865.31
## TARGET ~ FixedAcidity + VolatileAcidity + ResidualSugar + Chlorides +
## FreeSulfurDioxide + TotalSulfurDioxide + Density + pH + Alcohol +
## LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - FixedAcidity 1 1897.2 33863
## - ResidualSugar 1 1897.4 33864
## - FreeSulfurDioxide 1 1897.7 33864
## - pH 1 1898.1 33864
## - Chlorides 1 1898.4 33865
## - TotalSulfurDioxide 1 1898.7 33865
## <none> 1897.2 33865
## - Density 1 1899.3 33865
## - VolatileAcidity 1 1900.1 33866
## - AcidIndex 1 1908.6 33875
## - Alcohol 1 1922.6 33889
## - STARS 1 2113.8 34080
## - LabelAppeal 1 3295.0 35261
##
## Step: AIC=33863.38
## TARGET ~ VolatileAcidity + ResidualSugar + Chlorides + FreeSulfurDioxide +
## TotalSulfurDioxide + Density + pH + Alcohol + LabelAppeal +
## AcidIndex + STARS
##
## Df Deviance AIC
## - ResidualSugar 1 1897.5 33862
## - FreeSulfurDioxide 1 1897.8 33862
## - pH 1 1898.2 33862
## - Chlorides 1 1898.5 33863
## - TotalSulfurDioxide 1 1898.8 33863
## <none> 1897.2 33863
## - Density 1 1899.3 33863
## - VolatileAcidity 1 1900.1 33864
## - AcidIndex 1 1908.6 33873
## - Alcohol 1 1922.7 33887
## - STARS 1 2114.0 34078
## - LabelAppeal 1 3295.1 35259
##
## Step: AIC=33861.64
## TARGET ~ VolatileAcidity + Chlorides + FreeSulfurDioxide + TotalSulfurDioxide +
## Density + pH + Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - FreeSulfurDioxide 1 1898.0 33860
## - pH 1 1898.4 33861
## - Chlorides 1 1898.7 33861
## - TotalSulfurDioxide 1 1899.1 33861
## <none> 1897.5 33862
## - Density 1 1899.6 33862
## - VolatileAcidity 1 1900.4 33863
## - AcidIndex 1 1908.9 33871
## - Alcohol 1 1923.0 33885
## - STARS 1 2114.1 34076
## - LabelAppeal 1 3295.3 35257
##
## Step: AIC=33860.17
## TARGET ~ VolatileAcidity + Chlorides + TotalSulfurDioxide + Density +
## pH + Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - pH 1 1898.9 33859
## - Chlorides 1 1899.3 33859
## - TotalSulfurDioxide 1 1899.6 33860
## <none> 1898.0 33860
## - Density 1 1900.1 33860
## - VolatileAcidity 1 1900.9 33861
## - AcidIndex 1 1909.5 33870
## - Alcohol 1 1923.5 33884
## - STARS 1 2114.2 34074
## - LabelAppeal 1 3297.1 35257
##
## Step: AIC=33859.08
## TARGET ~ VolatileAcidity + Chlorides + TotalSulfurDioxide + Density +
## Alcohol + LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - Chlorides 1 1900.3 33858
## - TotalSulfurDioxide 1 1900.5 33859
## <none> 1898.9 33859
## - Density 1 1901.0 33859
## - VolatileAcidity 1 1901.8 33860
## - AcidIndex 1 1911.0 33869
## - Alcohol 1 1924.2 33882
## - STARS 1 2115.2 34073
## - LabelAppeal 1 3299.5 35258
##
## Step: AIC=33858.4
## TARGET ~ VolatileAcidity + TotalSulfurDioxide + Density + Alcohol +
## LabelAppeal + AcidIndex + STARS
##
## Df Deviance AIC
## - TotalSulfurDioxide 1 1901.8 33858
## <none> 1900.3 33858
## - Density 1 1902.4 33859
## - VolatileAcidity 1 1903.2 33859
## - AcidIndex 1 1912.4 33868
## - Alcohol 1 1925.9 33882
## - STARS 1 2116.6 34073
## - LabelAppeal 1 3300.3 35256
##
## Step: AIC=33857.97
## TARGET ~ VolatileAcidity + Density + Alcohol + LabelAppeal +
## AcidIndex + STARS
##
## Df Deviance AIC
## <none> 1901.8 33858
## - Density 1 1904.0 33858
## - VolatileAcidity 1 1904.7 33859
## - AcidIndex 1 1913.8 33868
## - Alcohol 1 1927.6 33882
## - STARS 1 2118.1 34072
## - LabelAppeal 1 3303.1 35257##
## Call:
## glm.nb(formula = TARGET ~ VolatileAcidity + Density + Alcohol +
## LabelAppeal + AcidIndex + STARS, data = yes_sold, init.theta = 336414.4343,
## link = log)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.66877 -0.32091 0.00247 0.24759 1.64333
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.461377 0.194972 7.495 6.61e-14 ***
## VolatileAcidity -0.011055 0.006553 -1.687 0.09159 .
## Density -0.286880 0.193062 -1.486 0.13729
## Alcohol 0.007129 0.001403 5.081 3.75e-07 ***
## LabelAppeal 0.227464 0.006082 37.400 < 2e-16 ***
## AcidIndex -0.016071 0.004664 -3.445 0.00057 ***
## STARS 0.092876 0.006300 14.743 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Negative Binomial(336414.4) family taken to be 1)
##
## Null deviance: 4227.4 on 10060 degrees of freedom
## Residual deviance: 1901.8 on 10054 degrees of freedom
## AIC: 33860
##
## Number of Fisher Scoring iterations: 1
##
##
## Theta: 336414
## Std. Err.: 593623
## Warning while fitting theta: iteration limit reached
##
## 2 x log-likelihood: -33843.97This model appears from an AIC perspective to be superior compare to the direct negative binomial model. We will take this into consideration when selecting our models.
4. Select Models
Knowing that the mean and variance are not the same, the Poisson regression may not be the most optimal model to use. The mean and variance TARGET value are 3 and 3.711 respectively, which suggests that there is overdispersion. Because of this information, the negative binomial regression models will be a better choice than the Poisson regression models.
From our discussion in the previous section, any data transformation that converts continuous data into discrete or categorical data will inevitably lose information. And not surprisingly, the AIC of the second negative binomial model performed worse than the first negative binomial model. Likewise, with the stepwise linear regression model, the model with the bucketed model performed worse than its counterpart. Therefore, we are left with two models, the 1st negative binomial model with the 1st stepwise linear regression model to compare.
Because the negative binomial model reports its results with an AIC and log likelihood, and the stepwise linear regression model reports its results with adjusted R-squared values, direct comparision can be ambiguous. We will instead perform a cross validation and determine how well each model performs.
25 cross-validation iterations were performed and averaged, and an accuracy and mean squared error was calculated for the negative binomial model.
## [1] "Accuracy for the Negative Binomial Model: 23.184%"## [1] "Mean Squared Error for Negative Binomial Model: 0.383"Let’s perform a similar iteration with the linear regression model.
## [1] "Accuracy for the Linear Regression Model: 25.865%"## [1] "Mean Squared Error for Linear Regression Model: 2.668"The overall accuracies do not differ a tremendous amount, however, the mean squared error is overall better with the negative binomial model than for the linear regression model. Before we continue on, let us take a look at the 7th model utilizing a hybrid logistic and negative binomial regression.
Before we evaluate our model, we need to determine a cutoff probability value that will help us balance the sensitivity and specificity of the model.
## cutoff sens spec acc
## 1 0.1 1.0000000 0.00364520 0.5018226
## 2 0.2 0.9990050 0.01336574 0.5061854
## 3 0.3 0.9963516 0.03888214 0.5176169
## 4 0.4 0.9930348 0.06804374 0.5305393
## 5 0.5 0.9844113 0.14580802 0.5651096
## 6 0.6 0.9538972 0.25030377 0.6021005
## 7 0.7 0.8626866 0.42041312 0.6415498
## 8 0.8 0.6421227 0.66221142 0.6521671
## 9 0.9 0.2089552 0.96719320 0.5880742
It appears that accuracy is highest when we attempt to balance out the sensitivity and specificity. That cutoff appears to be approximately 0.8 (which we will use for our logistic model).
## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1
## 0 545 1079
## 1 278 1936
##
## Accuracy : 0.6464
## 95% CI : (0.6311, 0.6616)
## No Information Rate : 0.7856
## P-Value [Acc > NIR] : 1
##
## Kappa : 0.2248
## Mcnemar's Test P-Value : <2e-16
##
## Sensitivity : 0.6421
## Specificity : 0.6622
## Pos Pred Value : 0.8744
## Neg Pred Value : 0.3356
## Prevalence : 0.7856
## Detection Rate : 0.5044
## Detection Prevalence : 0.5769
## Balanced Accuracy : 0.6522
##
## 'Positive' Class : 1
## ## Sensitivity Specificity Pos Pred Value
## 0.6421227 0.6622114 0.8744354
## Neg Pred Value Precision Recall
## 0.3355911 0.8744354 0.6421227
## F1 Prevalence Detection Rate
## 0.7404858 0.7855654 0.5044294
## Detection Prevalence Balanced Accuracy
## 0.5768629 0.6521671
## Area under the curve: 0.723525 iterations were performed and averaged for the modified negative binomial model (after use with the logistic model).
## [1] "Accuracy for the Modified Negative Binomial Model: 30.3%"## [1] "Mean Squared Error for the Modified Negative Binomial Regression Model: 0.054"It appears that this model works the best with a mean squared error of 0.054. Given this number, we will be using this for our model prediction.
Prediction
Let’s create a prediction for the evaluation dataset. Let’s first check to see if any NULL values exist in the testing dataset.
## [1] TRUEWhich values are missing? And for these values, let us impute mean values for each column.
## STARS Sulphates TotalSulfurDioxide
## 3359 1210 682
## Alcohol FreeSulfurDioxide Chlorides
## 653 647 638
## ResidualSugar pH AcidIndex
## 616 395 0
## LabelAppeal Density CitricAcid
## 0 0 0
## VolatileAcidity FixedAcidity
## 0 0
## FixedAcidity VolatileAcidity CitricAcid ResidualSugar Chlorides
## 1 3.2 1.160 -0.98 54.2 -0.56700000
## 2 4.5 0.160 -0.81 26.1 -0.42500000
## 3 7.1 2.640 -0.88 14.8 0.03700000
## 4 5.7 0.385 0.04 18.8 -0.42500000
## 5 8.0 0.330 -1.26 9.4 0.05482249
## 6 11.3 0.320 0.59 2.2 0.55600000
## FreeSulfurDioxide TotalSulfurDioxide Density pH Sulphates Alcohol
## 1 30.84557 268 0.99280 3.33 -0.59 9.90000
## 2 15.00000 -327 1.02792 3.38 0.70 10.48924
## 3 214.00000 142 0.99518 3.12 0.48 22.00000
## 4 22.00000 115 0.99640 2.24 1.83 6.20000
## 5 -167.00000 108 0.99457 3.12 1.77 13.70000
## 6 -37.00000 15 0.99940 3.20 1.29 15.40000
## LabelAppeal AcidIndex STARS
## 1 0 8 2.000000
## 2 -1 7 3.000000
## 3 -1 8 3.000000
## 4 -1 6 1.000000
## 5 0 9 2.000000
## 6 0 11 2.041755Now that all of the data has been imputed, let us create a prediction for the number of cases that were sold for the evaluation dataset.
## 1 2 3 4 5 6 7 8 9 10 11 12
## 4 3 3 3 0 0 4 5 4 5 4 0
## 13 14 15 16 17 18 19 20 21 22 23 24
## 5 7 0 0 0 4 0 3 3 3 0 0
## 25 26 27 28 29 30 31 32 33 34 35 36
## 4 3 4 6 0 0 4 0 4 0 0 0
## 37 38 39 40 41 42 43 44 45 46 47 48
## 0 4 0 0 0 5 5 0 0 4 4 3
## 49 50 51 52 53 54 55 56 57 58 59 60
## 0 0 0 0 0 0 3 0 3 0 4 0
## 61 62 63 64 65 66 67 68 69 70 71 72
## 2 0 4 0 0 0 6 3 3 5 4 0
## 73 74 75 76 77 78 79 80 81 82 83 84
## 4 0 0 2 2 0 0 0 5 0 4 0
## 85 86 87 88 89 90 91 92 93 94 95 96
## 0 5 4 5 0 3 3 0 0 3 3 0
## 97 98 99 100 101 102 103 104 105 106 107 108
## 3 0 0 0 0 3 3 3 4 3 0 0
## 109 110 111 112 113 114 115 116 117 118 119 120
## 0 0 0 0 3 0 5 6 5 0 0 0
## 121 122 123 124 125 126 127 128 129 130 131 132
## 0 3 0 3 3 3 3 2 0 4 5 0
## 133 134 135 136 137 138 139 140 141 142 143 144
## 4 4 0 3 0 4 4 0 4 0 0 0
## 145 146 147 148 149 150 151 152 153 154 155 156
## 0 0 0 0 5 5 4 0 4 3 0 0
## 157 158 159 160 161 162 163 164 165 166 167 168
## 4 4 3 3 5 0 6 4 0 4 6 4
## 169 170 171 172 173 174 175 176 177 178 179 180
## 0 0 0 0 0 4 0 3 0 0 2 5
## 181 182 183 184 185 186 187 188 189 190 191 192
## 7 4 3 0 0 4 5 0 3 4 3 4
## 193 194 195 196 197 198 199 200 201 202 203 204
## 0 5 4 0 3 0 5 0 0 0 4 3
## 205 206 207 208 209 210 211 212 213 214 215 216
## 4 5 0 0 3 4 4 0 4 2 5 4
## 217 218 219 220 221 222 223 224 225 226 227 228
## 0 0 4 0 3 3 2 3 0 0 0 0
## 229 230 231 232 233 234 235 236 237 238 239 240
## 4 4 6 4 4 0 0 0 4 0 4 0
## 241 242 243 244 245 246 247 248 249 250 251 252
## 4 0 6 5 0 3 6 3 5 3 0 6
## 253 254 255 256 257 258 259 260 261 262 263 264
## 0 2 3 0 5 0 5 4 0 4 5 3
## 265 266 267 268 269 270 271 272 273 274 275 276
## 3 0 4 4 4 2 4 0 0 6 4 0
## 277 278 279 280 281 282 283 284 285 286 287 288
## 3 4 4 0 0 0 0 3 4 0 4 0
## 289 290 291 292 293 294 295 296 297 298 299 300
## 0 0 3 0 0 4 0 3 0 5 2 4
## 301 302 303 304 305 306 307 308 309 310 311 312
## 4 4 5 0 0 6 5 0 0 0 4 4
## 313 314 315 316 317 318 319 320 321 322 323 324
## 0 0 5 4 4 0 4 0 0 0 3 0
## 325 326 327 328 329 330 331 332 333 334 335 336
## 0 4 0 3 0 0 4 0 0 3 0 0
## 337 338 339 340 341 342 343 344 345 346 347 348
## 4 4 0 5 0 3 4 4 7 2 4 0
## 349 350 351 352 353 354 355 356 357 358 359 360
## 0 5 3 4 0 0 0 0 4 3 0 3
## 361 362 363 364 365 366 367 368 369 370 371 372
## 0 4 3 3 0 7 0 5 0 0 0 3
## 373 374 375 376 377 378 379 380 381 382 383 384
## 0 3 0 5 0 4 0 0 0 4 0 0
## 385 386 387 388 389 390 391 392 393 394 395 396
## 4 0 3 0 3 0 4 0 5 4 3 0
## 397 398 399 400 401 402 403 404 405 406 407 408
## 3 4 0 4 6 4 4 0 0 4 4 3
## 409 410 411 412 413 414 415 416 417 418 419 420
## 0 4 3 0 4 0 0 3 3 4 5 4
## 421 422 423 424 425 426 427 428 429 430 431 432
## 5 0 0 4 0 0 0 0 0 0 5 4
## 433 434 435 436 437 438 439 440 441 442 443 444
## 0 0 4 4 0 0 2 4 0 0 3 4
## 445 446 447 448 449 450 451 452 453 454 455 456
## 4 4 0 6 3 2 5 0 3 6 4 0
## 457 458 459 460 461 462 463 464 465 466 467 468
## 4 2 5 0 0 5 4 0 4 0 3 0
## 469 470 471 472 473 474 475 476 477 478 479 480
## 0 5 0 6 0 4 0 5 0 0 4 0
## 481 482 483 484 485 486 487 488 489 490 491 492
## 0 3 3 4 0 3 3 3 0 4 0 0
## 493 494 495 496 497 498 499 500 501 502 503 504
## 0 0 6 4 4 0 0 4 3 0 0 5
## 505 506 507 508 509 510 511 512 513 514 515 516
## 4 3 4 0 5 0 4 5 3 0 0 0
## 517 518 519 520 521 522 523 524 525 526 527 528
## 0 0 5 0 5 2 0 0 5 0 0 3
## 529 530 531 532 533 534 535 536 537 538 539 540
## 3 0 4 4 0 3 0 4 5 4 0 5
## 541 542 543 544 545 546 547 548 549 550 551 552
## 3 4 6 4 5 0 3 0 0 2 7 0
## 553 554 555 556 557 558 559 560 561 562 563 564
## 0 3 0 0 6 3 0 0 5 0 5 0
## 565 566 567 568 569 570 571 572 573 574 575 576
## 0 4 0 3 0 0 4 3 5 4 0 0
## 577 578 579 580 581 582 583 584 585 586 587 588
## 3 0 0 0 4 0 3 0 0 4 0 4
## 589 590 591 592 593 594 595 596 597 598 599 600
## 5 0 4 3 2 5 0 6 0 4 5 4
## 601 602 603 604 605 606 607 608 609 610 611 612
## 0 0 3 5 4 4 3 3 4 3 0 0
## 613 614 615 616 617 618 619 620 621 622 623 624
## 0 5 4 4 4 5 0 5 3 0 3 0
## 625 626 627 628 629 630 631 632 633 634 635 636
## 0 3 0 0 3 7 4 0 4 6 0 4
## 637 638 639 640 641 642 643 644 645 646 647 648
## 0 0 8 5 4 5 4 0 3 0 0 0
## 649 650 651 652 653 654 655 656 657 658 659 660
## 4 5 0 0 3 4 3 0 3 3 0 6
## 661 662 663 664 665 666 667 668 669 670 671 672
## 0 0 3 4 3 0 5 5 0 0 0 0
## 673 674 675 676 677 678 679 680 681 682 683 684
## 4 3 3 4 5 0 5 3 0 5 0 0
## 685 686 687 688 689 690 691 692 693 694 695 696
## 6 0 0 4 4 0 5 0 0 4 4 3
## 697 698 699 700 701 702 703 704 705 706 707 708
## 3 0 0 0 0 0 0 0 0 0 5 5
## 709 710 711 712 713 714 715 716 717 718 719 720
## 0 4 4 0 4 0 0 4 5 4 4 0
## 721 722 723 724 725 726 727 728 729 730 731 732
## 0 0 5 3 3 5 4 0 4 4 0 4
## 733 734 735 736 737 738 739 740 741 742 743 744
## 0 5 0 5 4 5 0 0 3 4 0 4
## 745 746 747 748 749 750 751 752 753 754 755 756
## 4 0 4 0 3 3 0 0 5 3 4 0
## 757 758 759 760 761 762 763 764 765 766 767 768
## 4 6 2 5 3 4 0 0 0 0 0 0
## 769 770 771 772 773 774 775 776 777 778 779 780
## 5 3 0 3 4 3 0 4 3 4 0 0
## 781 782 783 784 785 786 787 788 789 790 791 792
## 3 3 0 0 6 4 0 4 4 4 4 2
## 793 794 795 796 797 798 799 800 801 802 803 804
## 0 2 0 0 0 5 5 0 4 6 4 0
## 805 806 807 808 809 810 811 812 813 814 815 816
## 4 0 0 3 4 0 3 4 4 0 0 4
## 817 818 819 820 821 822 823 824 825 826 827 828
## 0 4 0 0 0 0 0 0 0 0 5 0
## 829 830 831 832 833 834 835 836 837 838 839 840
## 0 3 4 4 5 0 4 6 6 6 0 4
## 841 842 843 844 845 846 847 848 849 850 851 852
## 4 0 0 3 4 5 0 0 0 0 0 4
## 853 854 855 856 857 858 859 860 861 862 863 864
## 0 0 0 4 5 0 6 3 4 0 4 0
## 865 866 867 868 869 870 871 872 873 874 875 876
## 5 6 0 3 4 3 0 0 4 3 0 3
## 877 878 879 880 881 882 883 884 885 886 887 888
## 0 0 4 0 4 3 4 5 0 3 5 0
## 889 890 891 892 893 894 895 896 897 898 899 900
## 4 0 4 3 0 4 3 3 0 0 4 4
## 901 902 903 904 905 906 907 908 909 910 911 912
## 4 0 3 5 3 3 4 2 2 4 0 4
## 913 914 915 916 917 918 919 920 921 922 923 924
## 4 4 4 4 4 0 3 4 4 5 3 0
## 925 926 927 928 929 930 931 932 933 934 935 936
## 3 4 4 2 0 0 5 4 4 0 4 4
## 937 938 939 940 941 942 943 944 945 946 947 948
## 7 4 0 3 4 0 5 4 0 3 0 0
## 949 950 951 952 953 954 955 956 957 958 959 960
## 0 4 0 0 0 0 5 5 5 4 5 4
## 961 962 963 964 965 966 967 968 969 970 971 972
## 3 0 6 4 0 3 3 4 0 3 0 5
## 973 974 975 976 977 978 979 980 981 982 983 984
## 0 0 0 3 5 4 7 4 6 5 5 4
## 985 986 987 988 989 990 991 992 993 994 995 996
## 4 0 5 3 4 5 3 4 3 4 0 3
## 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008
## 4 0 3 4 5 5 4 3 0 5 6 4
## 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020
## 4 4 5 4 0 3 0 0 2 4 0 5
## 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032
## 5 3 5 4 5 3 4 0 4 6 3 3
## 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044
## 3 0 0 4 0 4 2 4 3 4 0 4
## 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056
## 4 3 4 0 4 0 4 4 3 3 0 0
## 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068
## 0 4 0 5 5 3 0 4 3 4 6 0
## 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080
## 0 5 5 0 4 4 3 0 3 4 0 4
## 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092
## 3 0 0 7 0 0 0 0 0 0 5 0
## 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104
## 0 4 4 3 4 0 7 0 7 5 4 0
## 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116
## 3 5 4 0 0 0 0 0 0 2 4 0
## 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128
## 0 5 5 3 0 3 0 0 4 3 5 5
## 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140
## 6 3 4 3 0 0 6 4 0 5 0 4
## 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152
## 5 0 2 0 4 0 0 5 0 0 4 0
## 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164
## 0 5 0 4 4 3 4 0 3 0 0 4
## 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176
## 0 4 0 0 0 5 0 4 0 5 3 4
## 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188
## 0 5 0 0 0 0 0 0 0 4 0 0
## 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200
## 4 4 4 0 0 0 4 0 0 4 5 0
## 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212
## 5 6 4 0 4 4 3 3 0 4 4 5
## 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224
## 0 4 0 0 3 5 5 6 5 4 5 0
## 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236
## 3 0 4 3 0 0 5 0 4 0 0 0
## 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248
## 5 0 3 5 4 3 4 0 5 0 0 3
## 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260
## 3 4 0 0 0 4 4 0 0 4 0 0
## 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272
## 0 0 4 4 0 0 0 0 3 0 4 5
## 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284
## 0 2 4 4 4 5 0 4 5 0 0 0
## 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296
## 4 0 0 3 0 4 0 4 2 4 3 0
## 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308
## 4 0 3 5 0 3 0 0 0 0 5 3
## 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320
## 6 4 4 5 0 0 4 4 0 0 4 0
## 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332
## 0 4 5 4 0 3 0 0 2 3 4 3
## 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344
## 0 0 0 3 3 0 0 0 0 0 0 0
## 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356
## 0 2 5 0 0 0 4 0 4 4 3 3
## 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368
## 0 0 0 0 3 4 5 5 0 7 4 0
## 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380
## 4 4 5 0 4 0 5 0 0 3 0 0
## 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392
## 5 0 0 5 3 4 4 5 0 0 4 0
## 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404
## 0 0 4 0 0 3 0 5 4 5 5 5
## 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416
## 4 4 0 5 4 3 4 0 0 3 4 0
## 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428
## 4 0 0 0 3 4 0 0 5 0 0 0
## 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440
## 0 5 0 2 4 3 5 4 4 2 5 0
## 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452
## 0 4 0 0 4 3 0 0 4 5 4 0
## 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464
## 0 7 0 6 5 0 4 5 0 0 0 3
## 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476
## 0 3 0 5 4 5 3 0 5 4 3 4
## 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488
## 6 0 0 4 0 6 5 0 0 0 0 3
## 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500
## 5 0 0 0 0 3 0 0 2 0 5 0
## 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512
## 0 0 2 0 4 4 0 4 0 5 0 0
## 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524
## 3 4 0 4 0 0 4 3 0 3 0 4
## 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536
## 6 0 0 4 3 0 2 4 0 0 4 4
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## 3 0 0 0 0 7 0 0 5 0 0 3
## 10897 10898 10899 10900 10901 10902 10903 10904 10905 10906 10907 10908
## 3 0 4 4 3 0 4 5 0 4 0 3
## 10909 10910 10911 10912 10913 10914 10915 10916 10917 10918 10919 10920
## 0 6 4 5 4 4 4 0 0 4 5 0
## 10921 10922 10923 10924 10925 10926 10927 10928 10929 10930 10931 10932
## 0 3 0 4 4 0 4 0 0 0 0 5
## 10933 10934 10935 10936 10937 10938 10939 10940 10941 10942 10943 10944
## 4 4 6 0 3 3 4 5 0 0 0 5
## 10945 10946 10947 10948 10949 10950 10951 10952 10953 10954 10955 10956
## 4 0 4 4 6 5 0 0 0 3 4 4
## 10957 10958 10959 10960 10961 10962 10963 10964 10965 10966 10967 10968
## 5 5 0 4 0 0 0 5 3 0 0 4
## 10969 10970 10971 10972 10973 10974 10975 10976 10977 10978 10979 10980
## 0 5 5 0 3 5 3 5 4 0 0 4
## 10981 10982 10983 10984 10985 10986 10987 10988 10989 10990 10991 10992
## 5 5 2 3 5 0 3 4 5 0 0 0
## 10993 10994 10995 10996 10997 10998 10999 11000 11001 11002 11003 11004
## 3 4 0 4 5 5 3 0 4 3 0 4
## 11005 11006 11007 11008 11009 11010 11011 11012 11013 11014 11015 11016
## 6 0 0 0 3 0 0 6 0 0 3 0
## 11017 11018 11019 11020 11021 11022 11023 11024 11025 11026 11027 11028
## 4 0 6 4 5 4 0 0 6 0 4 0
## 11029 11030 11031 11032 11033 11034 11035 11036 11037 11038 11039 11040
## 0 0 0 2 5 0 5 5 4 3 7 0
## 11041 11042 11043 11044 11045 11046 11047 11048 11049 11050 11051 11052
## 6 4 0 2 0 0 3 0 4 4 0 7
## 11053 11054 11055 11056 11057 11058 11059 11060 11061 11062 11063 11064
## 5 0 6 5 5 0 0 0 0 0 6 5
## 11065 11066 11067 11068 11069 11070 11071 11072 11073 11074 11075 11076
## 0 5 3 0 4 3 5 0 3 0 0 0
## 11077 11078 11079 11080 11081 11082 11083 11084 11085 11086 11087 11088
## 2 0 0 4 0 0 4 5 0 0 0 4
## 11089 11090 11091 11092 11093 11094 11095 11096 11097 11098 11099 11100
## 5 5 0 5 0 0 4 3 5 0 4 5
## 11101 11102 11103 11104 11105 11106 11107 11108 11109 11110 11111 11112
## 5 4 4 3 0 4 2 0 6 3 4 4
## 11113 11114 11115 11116 11117 11118 11119 11120 11121 11122 11123 11124
## 0 0 0 0 5 4 0 5 2 0 4 4
## 11125 11126 11127 11128 11129 11130 11131 11132 11133 11134 11135 11136
## 0 3 3 4 0 4 0 0 4 0 6 4
## 11137 11138 11139 11140 11141 11142 11143 11144 11145 11146 11147 11148
## 5 4 0 0 0 0 6 0 3 3 3 0
## 11149 11150 11151 11152 11153 11154 11155 11156 11157 11158 11159 11160
## 0 0 5 3 4 7 0 4 5 0 4 5
## 11161 11162 11163 11164 11165 11166 11167 11168 11169 11170 11171 11172
## 4 4 4 3 7 0 0 0 4 5 0 0
## 11173 11174 11175 11176 11177 11178 11179 11180 11181 11182 11183 11184
## 3 0 4 3 0 0 3 5 0 3 4 0
## 11185 11186 11187 11188 11189 11190 11191 11192 11193 11194 11195 11196
## 4 0 0 0 0 6 3 4 5 0 4 4
## 11197 11198 11199 11200 11201 11202 11203 11204 11205 11206 11207 11208
## 5 4 3 6 6 0 0 0 0 4 0 4
## 11209 11210 11211 11212 11213 11214 11215 11216 11217 11218 11219 11220
## 4 3 5 0 5 5 4 0 0 4 3 0
## 11221 11222 11223 11224 11225 11226 11227 11228 11229 11230 11231 11232
## 0 4 4 0 0 2 3 0 4 0 0 0
## 11233 11234 11235 11236 11237 11238 11239 11240 11241 11242 11243 11244
## 3 4 5 2 3 4 0 3 3 3 0 4
## 11245 11246 11247 11248 11249 11250 11251 11252 11253 11254 11255 11256
## 4 4 0 0 4 0 4 0 3 0 0 0
## 11257 11258 11259 11260 11261 11262 11263 11264 11265 11266 11267 11268
## 0 3 3 0 2 5 2 3 3 4 0 5
## 11269 11270 11271 11272 11273 11274 11275 11276 11277 11278 11279 11280
## 3 0 3 0 4 5 0 3 0 3 5 0
## 11281 11282 11283 11284 11285 11286 11287 11288 11289 11290 11291 11292
## 0 4 5 0 0 0 0 0 0 0 3 0
## 11293 11294 11295 11296 11297 11298 11299 11300 11301 11302 11303 11304
## 0 0 4 2 3 0 0 4 0 3 5 6
## 11305 11306 11307 11308 11309 11310 11311 11312 11313 11314 11315 11316
## 4 0 4 3 6 0 4 0 0 4 4 6
## 11317 11318 11319 11320 11321 11322 11323 11324 11325 11326 11327 11328
## 4 0 4 0 0 4 4 3 4 0 0 4
## 11329 11330 11331 11332 11333 11334 11335 11336 11337 11338 11339 11340
## 3 4 0 0 6 5 5 0 0 5 0 0
## 11341 11342 11343 11344 11345 11346 11347 11348 11349 11350 11351 11352
## 0 3 0 0 3 3 0 3 4 3 0 4
## 11353 11354 11355 11356 11357 11358 11359 11360 11361 11362 11363 11364
## 0 3 4 3 2 0 4 0 0 3 3 4
## 11365 11366 11367 11368 11369 11370 11371 11372 11373 11374 11375 11376
## 0 0 3 0 4 0 4 0 0 5 0 0
## 11377 11378 11379 11380 11381 11382 11383 11384 11385 11386 11387 11388
## 3 5 5 4 0 0 4 0 4 0 0 6
## 11389 11390 11391 11392 11393 11394 11395 11396 11397 11398 11399 11400
## 0 0 3 0 5 5 0 5 0 0 2 3
## 11401 11402 11403 11404 11405 11406 11407 11408 11409 11410 11411 11412
## 0 0 5 5 0 2 0 0 0 0 5 0
## 11413 11414 11415 11416 11417 11418 11419 11420 11421 11422 11423 11424
## 3 4 6 0 0 5 3 3 2 3 0 5
## 11425 11426 11427 11428 11429 11430 11431 11432 11433 11434 11435 11436
## 4 3 3 3 0 4 5 4 5 4 5 4
## 11437 11438 11439 11440 11441 11442 11443 11444 11445 11446 11447 11448
## 4 3 3 7 0 4 0 0 3 3 4 2
## 11449 11450 11451 11452 11453 11454 11455 11456 11457 11458 11459 11460
## 0 0 0 0 4 4 5 0 4 0 6 3
## 11461 11462 11463 11464 11465 11466 11467 11468 11469 11470 11471 11472
## 0 0 6 4 0 0 0 0 0 5 4 3
## 11473 11474 11475 11476 11477 11478 11479 11480 11481 11482 11483 11484
## 0 0 0 0 6 5 4 0 0 5 4 4
## 11485 11486 11487 11488 11489 11490 11491 11492 11493 11494 11495 11496
## 0 0 0 0 0 3 2 0 4 4 0 4
## 11497 11498 11499 11500 11501 11502 11503 11504 11505 11506 11507 11508
## 0 4 4 0 3 0 0 4 0 3 0 5
## 11509 11510 11511 11512 11513 11514 11515 11516 11517 11518 11519 11520
## 0 7 2 4 0 5 2 5 6 0 6 4
## 11521 11522 11523 11524 11525 11526 11527 11528 11529 11530 11531 11532
## 4 0 0 3 3 5 0 3 0 3 3 3
## 11533 11534 11535 11536 11537 11538 11539 11540 11541 11542 11543 11544
## 0 0 4 0 3 4 4 5 4 4 4 4
## 11545 11546 11547 11548 11549 11550 11551 11552 11553 11554 11555 11556
## 2 3 0 0 0 0 4 0 0 3 4 4
## 11557 11558 11559 11560 11561 11562 11563 11564 11565 11566 11567 11568
## 3 4 0 0 0 3 4 4 4 0 4 0
## 11569 11570 11571 11572 11573 11574 11575 11576 11577 11578 11579 11580
## 0 0 0 0 4 0 0 3 0 5 4 0
## 11581 11582 11583 11584 11585 11586 11587 11588 11589 11590 11591 11592
## 4 0 3 4 0 2 4 5 4 3 4 3
## 11593 11594 11595 11596 11597 11598 11599 11600 11601 11602 11603 11604
## 0 0 3 4 4 3 4 0 5 0 4 0
## 11605 11606 11607 11608 11609 11610 11611 11612 11613 11614 11615 11616
## 0 5 4 0 0 0 0 0 4 5 0 0
## 11617 11618 11619 11620 11621 11622 11623 11624 11625 11626 11627 11628
## 6 5 0 3 0 0 0 3 4 0 0 0
## 11629 11630 11631 11632 11633 11634 11635 11636 11637 11638 11639 11640
## 6 4 4 4 4 3 2 5 0 3 0 5
## 11641 11642 11643 11644 11645 11646 11647 11648 11649 11650 11651 11652
## 4 0 0 0 0 4 0 2 0 4 0 4
## 11653 11654 11655 11656 11657 11658 11659 11660 11661 11662 11663 11664
## 5 0 0 4 4 0 0 0 3 4 4 0
## 11665 11666 11667 11668 11669 11670 11671 11672 11673 11674 11675 11676
## 3 6 0 3 0 4 3 3 0 0 0 4
## 11677 11678 11679 11680 11681 11682 11683 11684 11685 11686 11687 11688
## 0 0 4 5 0 3 3 4 0 4 4 0
## 11689 11690 11691 11692 11693 11694 11695 11696 11697 11698 11699 11700
## 3 4 4 0 4 0 4 3 5 4 5 4
## 11701 11702 11703 11704 11705 11706 11707 11708 11709 11710 11711 11712
## 0 3 0 6 0 3 4 0 0 0 0 0
## 11713 11714 11715 11716 11717 11718 11719 11720 11721 11722 11723 11724
## 0 4 4 0 6 4 0 0 0 3 2 5
## 11725 11726 11727 11728 11729 11730 11731 11732 11733 11734 11735 11736
## 0 4 3 0 5 5 0 5 6 0 3 3
## 11737 11738 11739 11740 11741 11742 11743 11744 11745 11746 11747 11748
## 0 0 5 3 0 0 5 4 5 3 5 5
## 11749 11750 11751 11752 11753 11754 11755 11756 11757 11758 11759 11760
## 2 4 4 3 4 0 0 4 0 4 5 4
## 11761 11762 11763 11764 11765 11766 11767 11768 11769 11770 11771 11772
## 4 0 0 5 5 3 3 6 5 0 0 0
## 11773 11774 11775 11776 11777 11778 11779 11780 11781 11782 11783 11784
## 4 5 0 0 2 4 0 4 4 6 3 4
## 11785 11786 11787 11788 11789 11790 11791 11792 11793 11794 11795 11796
## 4 0 4 0 7 3 4 5 6 3 4 4
## 11797 11798 11799 11800 11801 11802 11803 11804 11805 11806 11807 11808
## 4 4 0 4 0 0 5 3 3 0 4 0
## 11809 11810 11811 11812 11813 11814 11815 11816 11817 11818 11819 11820
## 0 3 4 0 3 4 0 4 5 4 4 4
## 11821 11822 11823 11824 11825 11826 11827 11828 11829 11830 11831 11832
## 5 3 0 4 6 0 4 5 5 5 4 0
## 11833 11834 11835 11836 11837 11838 11839 11840 11841 11842 11843 11844
## 0 7 4 0 0 0 0 4 4 4 4 4
## 11845 11846 11847 11848 11849 11850 11851 11852 11853 11854 11855 11856
## 0 0 5 0 4 6 4 3 0 0 0 4
## 11857 11858 11859 11860 11861 11862 11863 11864 11865 11866 11867 11868
## 3 5 4 3 0 0 0 0 0 0 0 5
## 11869 11870 11871 11872 11873 11874 11875 11876 11877 11878 11879 11880
## 3 4 0 0 0 4 0 4 0 4 6 6
## 11881 11882 11883 11884 11885 11886 11887 11888 11889 11890 11891 11892
## 5 0 4 3 0 3 3 0 0 5 4 0
## 11893 11894 11895 11896 11897 11898 11899 11900 11901 11902 11903 11904
## 0 0 4 3 4 0 4 0 0 0 3 4
## 11905 11906 11907 11908 11909 11910 11911 11912 11913 11914 11915 11916
## 0 3 0 3 0 0 5 4 4 0 0 0
## 11917 11918 11919 11920 11921 11922 11923 11924 11925 11926 11927 11928
## 0 0 3 0 4 4 4 3 3 0 4 0
## 11929 11930 11931 11932 11933 11934 11935 11936 11937 11938 11939 11940
## 6 3 0 4 4 0 4 0 4 4 5 3
## 11941 11942 11943 11944 11945 11946 11947 11948 11949 11950 11951 11952
## 4 0 3 0 4 5 0 4 6 5 0 3
## 11953 11954 11955 11956 11957 11958 11959 11960 11961 11962 11963 11964
## 4 3 0 0 3 2 3 4 3 4 0 0
## 11965 11966 11967 11968 11969 11970 11971 11972 11973 11974 11975 11976
## 4 5 4 0 3 2 5 0 4 3 0 0
## 11977 11978 11979 11980 11981 11982 11983 11984 11985 11986 11987 11988
## 0 3 4 3 6 6 6 6 4 0 0 0
## 11989 11990 11991 11992 11993 11994 11995 11996 11997 11998 11999 12000
## 5 0 0 4 4 3 4 3 0 0 3 0
## 12001 12002 12003 12004 12005 12006 12007 12008 12009 12010 12011 12012
## 0 0 4 2 0 5 4 0 0 0 0 3
## 12013 12014 12015 12016 12017 12018 12019 12020 12021 12022 12023 12024
## 3 4 3 0 4 0 2 4 4 5 4 0
## 12025 12026 12027 12028 12029 12030 12031 12032 12033 12034 12035 12036
## 0 4 4 0 0 4 0 4 0 0 4 6
## 12037 12038 12039 12040 12041 12042 12043 12044 12045 12046 12047 12048
## 0 4 0 0 4 4 0 3 0 4 3 4
## 12049 12050 12051 12052 12053 12054 12055 12056 12057 12058 12059 12060
## 7 0 4 0 2 3 0 4 5 0 0 0
## 12061 12062 12063 12064 12065 12066 12067 12068 12069 12070 12071 12072
## 3 5 5 4 5 4 5 4 0 5 0 0
## 12073 12074 12075 12076 12077 12078 12079 12080 12081 12082 12083 12084
## 7 4 3 0 0 4 3 0 3 4 5 4
## 12085 12086 12087 12088 12089 12090 12091 12092 12093 12094 12095 12096
## 4 0 4 0 3 3 0 3 0 3 0 0
## 12097 12098 12099 12100 12101 12102 12103 12104 12105 12106 12107 12108
## 3 3 0 0 0 5 4 5 4 0 0 3
## 12109 12110 12111 12112 12113 12114 12115 12116 12117 12118 12119 12120
## 6 0 4 5 3 3 2 5 0 5 4 0
## 12121 12122 12123 12124 12125 12126 12127 12128 12129 12130 12131 12132
## 0 4 0 4 4 0 3 4 4 3 0 0
## 12133 12134 12135 12136 12137 12138 12139 12140 12141 12142 12143 12144
## 3 0 4 0 3 4 3 0 2 0 3 5
## 12145 12146 12147 12148 12149 12150 12151 12152 12153 12154 12155 12156
## 5 0 0 5 0 4 4 4 0 0 4 0
## 12157 12158 12159 12160 12161 12162 12163 12164 12165 12166 12167 12168
## 5 3 4 6 0 5 0 5 0 0 5 6
## 12169 12170 12171 12172 12173 12174 12175 12176 12177 12178 12179 12180
## 3 5 0 0 5 4 4 3 3 4 3 0
## 12181 12182 12183 12184 12185 12186 12187 12188 12189 12190 12191 12192
## 4 5 4 2 5 0 0 0 6 0 3 0
## 12193 12194 12195 12196 12197 12198 12199 12200 12201 12202 12203 12204
## 4 0 0 0 0 4 4 2 2 3 0 0
## 12205 12206 12207 12208 12209 12210 12211 12212 12213 12214 12215 12216
## 4 3 3 4 0 0 0 0 0 3 6 0
## 12217 12218 12219 12220 12221 12222 12223 12224 12225 12226 12227 12228
## 0 0 4 3 0 3 0 0 4 0 3 4
## 12229 12230 12231 12232 12233 12234 12235 12236 12237 12238 12239 12240
## 0 4 5 0 5 3 0 3 0 3 0 0
## 12241 12242 12243 12244 12245 12246 12247 12248 12249 12250 12251 12252
## 5 4 0 4 0 4 0 0 0 5 5 3
## 12253 12254 12255 12256 12257 12258 12259 12260 12261 12262 12263 12264
## 3 4 4 5 5 4 4 3 4 5 0 3
## 12265 12266 12267 12268 12269 12270 12271 12272 12273 12274 12275 12276
## 5 0 0 2 0 0 0 0 3 3 5 0
## 12277 12278 12279 12280 12281 12282 12283 12284 12285 12286 12287 12288
## 4 6 3 0 4 0 7 0 0 0 0 5
## 12289 12290 12291 12292 12293 12294 12295 12296 12297 12298 12299 12300
## 0 0 0 5 3 0 4 0 5 0 4 4
## 12301 12302 12303 12304 12305 12306 12307 12308 12309 12310 12311 12312
## 3 2 0 0 0 5 0 4 7 4 0 4
## 12313 12314 12315 12316 12317 12318 12319 12320 12321 12322 12323 12324
## 0 0 3 0 4 0 3 3 0 4 4 4
## 12325 12326 12327 12328 12329 12330 12331 12332 12333 12334 12335 12336
## 4 0 5 5 0 0 4 5 5 0 0 0
## 12337 12338 12339 12340 12341 12342 12343 12344 12345 12346 12347 12348
## 4 3 0 5 3 6 0 0 3 3 5 3
## 12349 12350 12351 12352 12353 12354 12355 12356 12357 12358 12359 12360
## 4 0 7 0 0 0 4 0 4 5 0 0
## 12361 12362 12363 12364 12365 12366 12367 12368 12369 12370 12371 12372
## 3 0 5 0 0 3 0 4 0 4 4 4
## 12373 12374 12375 12376 12377 12378 12379 12380 12381 12382 12383 12384
## 0 6 0 3 4 4 5 0 0 0 0 3
## 12385 12386 12387 12388 12389 12390 12391 12392 12393 12394 12395 12396
## 4 0 5 2 0 3 4 0 4 3 0 0
## 12397 12398 12399 12400 12401 12402 12403 12404 12405 12406 12407 12408
## 3 0 0 3 5 0 4 5 3 6 6 0
## 12409 12410 12411 12412 12413 12414 12415 12416 12417 12418 12419 12420
## 5 5 0 0 0 0 3 0 0 5 5 3
## 12421 12422 12423 12424 12425 12426 12427 12428 12429 12430 12431 12432
## 3 5 4 4 5 3 0 4 4 0 0 0
## 12433 12434 12435 12436 12437 12438 12439 12440 12441 12442 12443 12444
## 4 4 5 5 4 0 0 6 4 5 0 2
## 12445 12446 12447 12448 12449 12450 12451 12452 12453 12454 12455 12456
## 0 0 0 0 0 4 0 3 0 0 4 5
## 12457 12458 12459 12460 12461 12462 12463 12464 12465 12466 12467 12468
## 0 4 0 3 0 3 0 0 4 3 4 0
## 12469 12470 12471 12472 12473 12474 12475 12476 12477 12478 12479 12480
## 0 0 3 0 0 4 5 0 3 0 6 6
## 12481 12482 12483 12484 12485 12486 12487 12488 12489 12490 12491 12492
## 0 0 5 0 5 4 4 0 5 4 5 2
## 12493 12494 12495 12496 12497 12498 12499 12500 12501 12502 12503 12504
## 4 3 5 3 4 3 0 0 4 0 3 2
## 12505 12506 12507 12508 12509 12510 12511 12512 12513 12514 12515 12516
## 5 5 0 0 3 0 4 0 8 0 0 3
## 12517 12518 12519 12520 12521 12522 12523 12524 12525 12526 12527 12528
## 3 4 4 3 4 0 2 4 0 0 5 3
## 12529 12530 12531 12532 12533 12534 12535 12536 12537 12538 12539 12540
## 3 0 4 4 3 0 0 5 4 0 0 0
## 12541 12542 12543 12544 12545 12546 12547 12548 12549 12550 12551 12552
## 0 4 4 5 5 3 0 5 3 0 0 3
## 12553 12554 12555 12556 12557 12558 12559 12560 12561 12562 12563 12564
## 0 4 0 0 6 0 0 4 0 4 6 5
## 12565 12566 12567 12568 12569 12570 12571 12572 12573 12574 12575 12576
## 0 3 0 4 3 4 0 0 0 0 0 5
## 12577 12578 12579 12580 12581 12582 12583 12584 12585 12586 12587 12588
## 4 3 0 0 0 0 0 0 0 0 4 5
## 12589 12590 12591 12592 12593 12594 12595 12596 12597 12598 12599 12600
## 5 0 4 3 4 5 4 0 0 4 0 5
## 12601 12602 12603 12604 12605 12606 12607 12608 12609 12610 12611 12612
## 4 3 5 4 4 5 0 4 0 4 3 0
## 12613 12614 12615 12616 12617 12618 12619 12620 12621 12622 12623 12624
## 0 0 0 3 3 0 0 0 4 3 0 0
## 12625 12626 12627 12628 12629 12630 12631 12632 12633 12634 12635 12636
## 3 3 0 0 5 4 3 0 5 0 0 4
## 12637 12638 12639 12640 12641 12642 12643 12644 12645 12646 12647 12648
## 7 4 0 0 4 5 2 3 0 0 3 0
## 12649 12650 12651 12652 12653 12654 12655 12656 12657 12658 12659 12660
## 3 4 5 3 2 3 6 0 0 5 0 0
## 12661 12662 12663 12664 12665 12666 12667 12668 12669 12670 12671 12672
## 0 3 5 3 2 3 5 0 2 3 4 0
## 12673 12674 12675 12676 12677 12678 12679 12680 12681 12682 12683 12684
## 3 0 4 7 0 0 0 0 4 4 4 0
## 12685 12686 12687 12688 12689 12690 12691 12692 12693 12694 12695 12696
## 5 0 3 0 5 0 4 0 0 5 0 4
## 12697 12698 12699 12700 12701 12702 12703 12704 12705 12706 12707 12708
## 7 4 0 0 3 3 5 5 0 3 4 3
## 12709 12710 12711 12712 12713 12714 12715 12716 12717 12718 12719 12720
## 0 5 0 3 3 0 3 0 0 0 3 4
## 12721 12722 12723 12724 12725 12726 12727 12728 12729 12730 12731 12732
## 5 0 3 4 0 0 4 4 0 0 3 0
## 12733 12734 12735 12736 12737 12738 12739 12740 12741 12742 12743 12744
## 0 3 0 5 0 0 5 3 0 4 0 3
## 12745 12746 12747 12748 12749 12750 12751 12752 12753 12754 12755 12756
## 0 0 0 0 6 4 4 5 4 4 4 3
## 12757 12758 12759 12760 12761 12762 12763 12764 12765 12766 12767 12768
## 5 0 0 0 4 0 5 0 4 4 0 5
## 12769 12770 12771 12772 12773 12774 12775 12776 12777 12778 12779 12780
## 0 0 3 0 0 4 4 4 4 4 3 3
## 12781 12782 12783 12784 12785 12786 12787 12788 12789 12790 12791 12792
## 3 0 3 0 6 0 0 3 3 0 5 3
## 12793 12794 12795
## 4 3 4Summary of our predictions:
## test_ans_2
## 0 2 3 4 5 6 7 8
## 5493 268 1998 2972 1507 424 128 5
So above is what my predictions are with the evaluation dataset. Interestingly, this model is having difficulty identifying observations that have only sold one case. Regardless, this hybrid model is likely to the most accurate in regards to predicting the number of cases sold.
Appendix
# Loading necessary packages
list.of.packages <- c("ggplot2", "knitr", "kableExtra", "reshape2", "dplyr","tidyr","corrplot", "caret", "glmnet","leaps", "pROC","car", "DataExplorer", "pscl", "MASS")
new.packages <- list.of.packages[!(list.of.packages %in% installed.packages()[,"Package"])]
if(length(new.packages)) install.packages(new.packages)
lapply(list.of.packages, require, character.only = TRUE)
# Loading training and testing data sets.
url_train <- 'https://raw.githubusercontent.com/jcp9010/MSDA/master/CUNY%20621/Homework%205/wine-training-data.csv'
url_eval <- 'https://raw.githubusercontent.com/jcp9010/MSDA/master/CUNY%20621/Homework%205/wine-evaluation-data.csv'
train <- read.csv(url_train, header=TRUE)
eval <- read.csv(url_eval, header=TRUE)
# Get rid of the index column
train <- train[,-1]
eval <- train[,-1]
head(train)
# Exploratory Data Analysis
summary(train)
out <- split_columns(train)
plot_histogram(out$continuous)
plot_scatterplot(train, 'TARGET', position='jitter')
# Correlation
a <- cor(train, method="pearson", use="complete.obs")
a
corrplot(a, method="circle")
# Missing Values?
rev(sort(colSums(sapply(train, is.na))))
plot_missing(train)
# Imputing means to NA data
# Reference: https://stackoverflow.com/questions/25835643/replace-missing-values-with-column-mean
NA2mean <- function(x) replace(x, is.na(x), mean(x, na.rm = TRUE))
train <- replace(train, TRUE, lapply(train, NA2mean))
head(train)
# Bucket data sets
labelappeal <- ifelse(train$LabelAppeal > 0, 'POSITIVE', ifelse(train$LabelAppeal == 0, 'NEUTRAL', 'NEGATIVE'))
pH <- ifelse(train$pH >= 3, 'ACIDIC', 'VERY ACIDIC')
train2 <- train[,!names(train) %in% c('pH','LabelAppeal')]
train2$LabelAppeal <- labelappeal
train2$pH <- pH
table(labelappeal)
table(pH)
# Building models
poisson_model_1 <- glm(TARGET ~ ., family=poisson, data=train)
poisson_model_1 <- step(poisson_model_1, direction="backward")
summary(poisson_model_1)
poisson_model_2 <- zeroinfl(TARGET ~ ., data = train)
summary(poisson_model_2)
poisson_model_2$coefficients
# Vuong
vuong(poisson_model_1, poisson_model_2)
# Mean/Variance
print(paste0("TARGET mean: ", round(mean(train$TARGET, 3))))
print(paste0("TARGET variance: ", round(var(train$TARGET),3)))
negative_binomial_3 <- glm.nb(TARGET ~ ., data=train)
summary(negative_binomial_3)
negative_binomial_4 <- glm.nb(TARGET ~ ., data=train2)
summary(negative_binomial_4)
lm_5 <- lm(TARGET ~ ., train)
lm_5 <- step(lm_5, direction='backward')
summary(lm_5)
lm6 <- step(lm(TARGET ~ ., train2), direction='backward')
summary(lm6)
train_log <- train
train_log$case_sold <- ifelse(train_log$TARGET > 0, 1, 0)
log_lm_7 <- glm(case_sold ~ . - TARGET, family="binomial", data=train_log)
log_lm_7 <- step(log_lm_7, direction="backward")
summary(log_lm_7)
yes_sold <- subset(train_log, train_log$case_sold == 1)
negative_binomial_7 <- glm.nb(TARGET ~ ., data=yes_sold)
negative_binomial_7 <- step(negative_binomial_7, direction="backward")
summary(negative_binomial_7)
# Define Mean Squared Errors function
# Reference: https://stats.stackexchange.com/questions/107643/how-to-get-the-value-of-mean-squared-error-in-a-linear-regression-in-r
mse <- function(sm)
mean(sm$residuals^2)
# Negative Binomial 3
avg_count <- c()
for (j in 1:25){
Train <- createDataPartition(train$TARGET, p=0.7, list=FALSE)
training <- train[Train, ]
testing <- train[-Train, ]
pred <- round(predict(negative_binomial_3, newdata=testing, type="response"),0)
total_right <- 0
for (i in 1:length(pred)){
if (pred[i] == testing$TARGET){
total_right <- total_right + 1
}
}
avg_count[j] <- total_right/length(pred)
}
print(paste0("Accuracy for the Negative Binomial Model: ", round(sum(avg_count)/length(avg_count) * 100,3), "%"))
print(paste0("Mean Squared Error for Negative Binomial Model: ", round(mse(negative_binomial_3),3)))
# Linear Regression
avg_count <- c()
for (j in 1:25){
Train <- createDataPartition(train$TARGET, p=0.7, list=FALSE)
training <- train[Train, ]
testing <- train[-Train, ]
pred <- round(predict(lm_5, newdata=testing, type="response"),0)
total_right <- 0
for (i in 1:length(pred)){
if (pred[i] == testing$TARGET){
total_right <- total_right + 1
}
}
avg_count[j] <- total_right/length(pred)
}
print(paste0("Accuracy for the Linear Regression Model: ", round(sum(avg_count)/length(avg_count) * 100,3), "%"))
print(paste0("Mean Squared Error for Linear Regression Model: ", round(mse(lm_5),3)))
# Sensitivity/Specificity/Accuracy
Train <- createDataPartition(train_log$case_sold, p=0.7, list=FALSE)
training <- train_log[Train, ]
testing <- train_log[-Train, ]
pred <- round(predict(log_lm_7, newdata=testing, type="response"), 3)
Cutoff <- seq(.1, .9, by=0.1)
cutoff <- c()
sens <- c()
spec <- c()
acc <- c()
for (i in 1:length(Cutoff)) {
tmp_pred <- ifelse(pred > Cutoff[i], 1, 0)
ans <- confusionMatrix(data=tmp_pred, testing$case_sold, positive='1')
cutoff <- c(cutoff, Cutoff[i])
sens <- c(sens, ans$byClass['Sensitivity'][[1]])
spec <- c(spec, ans$byClass['Specificity'][[1]])
acc <- c(acc, ans$byClass['Balanced Accuracy'][[1]])
}
cutoff_df <- as.data.frame(cbind(cutoff, sens, spec, acc))
cutoff_df
ggplot(cutoff_df, aes(x=cutoff)) +
geom_line(aes(y = sens, colour = "Sensitivity")) +
geom_line(aes(y = spec, colour = "Specificity")) +
geom_line(aes(y = acc, colour = "Accuracy")) +
xlab("Cutoff Values") + ylab("")
# Confusion Matrix
pred1 <- ifelse(pred > 0.8, 1, 0)
#Confusion Matrix
ans <- confusionMatrix(data=pred1, testing$case_sold, positive='1')
ans
ans$byClass
plot(roc(testing$case_sold, pred), main="ROC Curve from pROC Package")
# Please note that the X axis is in Specificity (as opposed to 1 - Specificity in the above function)
# Area Under the Curve
auc(roc(testing$case_sold, pred))
# Hybrid Model
avg_count <- c()
for (j in 1:25){
Train <- createDataPartition(train$TARGET, p=0.7, list=FALSE)
training <- train[Train, ]
testing <- train[-Train, ]
pred <- round(predict(negative_binomial_7, newdata=testing, type="response"),0)
total_right <- 0
for (i in 1:length(pred)){
if (pred[i] == testing$TARGET){
total_right <- total_right + 1
}
}
avg_count[j] <- total_right/length(pred)
}
print(paste0("Accuracy for the Modified Negative Binomial Model: ", round(sum(avg_count)/length(avg_count) * 100,3), "%"))
print(paste0("Mean Squared Error for the Modified Negative Binomial Regression Model: ", round(mse(negative_binomial_7),3)))
# Any Missing Values for eval
anyNA(eval)
# Imputing values for eval
rev(sort(colSums(sapply(eval, is.na))))
plot_missing(eval)
eval <- replace(eval, TRUE, lapply(eval, NA2mean))
head(eval)
# Predicting the eval dataset
test_ans <- round(predict(log_lm_7, newdata=eval, type="response"), 3)
test_ans <- ifelse(test_ans > 0.8, 1, 0)
test_ans_2 <- ifelse(test_ans == 1, round(predict(negative_binomial_7, newdata=eval, type="response"), 0), 0)
test_ans_2
# Summary of the predictions
table(test_ans_2)
hist(test_ans_2, col="green", main="Histogram of the Predictions", xlab="Number of Cases Sold")
# You may un-comment this section if you would like to download a csv file of the predictions
# write.csv(test_ans_2, file="WinePrediction.csv")