MSDS Spring 2018
DATA 605 Fundamentals of Computational Mathematics
Jiadi Li
Final Exam: House Prices - Advanced Regression Techniques
. Pick one of the quantitative independent variables from the training data set (train.csv) , and define that variable as X. Make sure this variable is skewed to the right!
. Pick the dependent variable and define it as Y.
#import the dataset
house_price <- read.csv("https://raw.githubusercontent.com/xiaoxiaogao-DD/store/master/train.csv")
#overview
head(house_price)## Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape
## 1 1 60 RL 65 8450 Pave <NA> Reg
## 2 2 20 RL 80 9600 Pave <NA> Reg
## 3 3 60 RL 68 11250 Pave <NA> IR1
## 4 4 70 RL 60 9550 Pave <NA> IR1
## 5 5 60 RL 84 14260 Pave <NA> IR1
## 6 6 50 RL 85 14115 Pave <NA> IR1
## LandContour Utilities LotConfig LandSlope Neighborhood Condition1
## 1 Lvl AllPub Inside Gtl CollgCr Norm
## 2 Lvl AllPub FR2 Gtl Veenker Feedr
## 3 Lvl AllPub Inside Gtl CollgCr Norm
## 4 Lvl AllPub Corner Gtl Crawfor Norm
## 5 Lvl AllPub FR2 Gtl NoRidge Norm
## 6 Lvl AllPub Inside Gtl Mitchel Norm
## Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt
## 1 Norm 1Fam 2Story 7 5 2003
## 2 Norm 1Fam 1Story 6 8 1976
## 3 Norm 1Fam 2Story 7 5 2001
## 4 Norm 1Fam 2Story 7 5 1915
## 5 Norm 1Fam 2Story 8 5 2000
## 6 Norm 1Fam 1.5Fin 5 5 1993
## YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType
## 1 2003 Gable CompShg VinylSd VinylSd BrkFace
## 2 1976 Gable CompShg MetalSd MetalSd None
## 3 2002 Gable CompShg VinylSd VinylSd BrkFace
## 4 1970 Gable CompShg Wd Sdng Wd Shng None
## 5 2000 Gable CompShg VinylSd VinylSd BrkFace
## 6 1995 Gable CompShg VinylSd VinylSd None
## MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure
## 1 196 Gd TA PConc Gd TA No
## 2 0 TA TA CBlock Gd TA Gd
## 3 162 Gd TA PConc Gd TA Mn
## 4 0 TA TA BrkTil TA Gd No
## 5 350 Gd TA PConc Gd TA Av
## 6 0 TA TA Wood Gd TA No
## BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF
## 1 GLQ 706 Unf 0 150 856
## 2 ALQ 978 Unf 0 284 1262
## 3 GLQ 486 Unf 0 434 920
## 4 ALQ 216 Unf 0 540 756
## 5 GLQ 655 Unf 0 490 1145
## 6 GLQ 732 Unf 0 64 796
## Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF
## 1 GasA Ex Y SBrkr 856 854 0
## 2 GasA Ex Y SBrkr 1262 0 0
## 3 GasA Ex Y SBrkr 920 866 0
## 4 GasA Gd Y SBrkr 961 756 0
## 5 GasA Ex Y SBrkr 1145 1053 0
## 6 GasA Ex Y SBrkr 796 566 0
## GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr
## 1 1710 1 0 2 1 3
## 2 1262 0 1 2 0 3
## 3 1786 1 0 2 1 3
## 4 1717 1 0 1 0 3
## 5 2198 1 0 2 1 4
## 6 1362 1 0 1 1 1
## KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu
## 1 1 Gd 8 Typ 0 <NA>
## 2 1 TA 6 Typ 1 TA
## 3 1 Gd 6 Typ 1 TA
## 4 1 Gd 7 Typ 1 Gd
## 5 1 Gd 9 Typ 1 TA
## 6 1 TA 5 Typ 0 <NA>
## GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual
## 1 Attchd 2003 RFn 2 548 TA
## 2 Attchd 1976 RFn 2 460 TA
## 3 Attchd 2001 RFn 2 608 TA
## 4 Detchd 1998 Unf 3 642 TA
## 5 Attchd 2000 RFn 3 836 TA
## 6 Attchd 1993 Unf 2 480 TA
## GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch
## 1 TA Y 0 61 0 0
## 2 TA Y 298 0 0 0
## 3 TA Y 0 42 0 0
## 4 TA Y 0 35 272 0
## 5 TA Y 192 84 0 0
## 6 TA Y 40 30 0 320
## ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold
## 1 0 0 <NA> <NA> <NA> 0 2 2008
## 2 0 0 <NA> <NA> <NA> 0 5 2007
## 3 0 0 <NA> <NA> <NA> 0 9 2008
## 4 0 0 <NA> <NA> <NA> 0 2 2006
## 5 0 0 <NA> <NA> <NA> 0 12 2008
## 6 0 0 <NA> MnPrv Shed 700 10 2009
## SaleType SaleCondition SalePrice
## 1 WD Normal 208500
## 2 WD Normal 181500
## 3 WD Normal 223500
## 4 WD Abnorml 140000
## 5 WD Normal 250000
## 6 WD Normal 143000dim(house_price)## [1] 1460 81#pick SalePrice as the dependent variable, Y
Y <- house_price$SalePrice
summary(Y)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 34900 129975 163000 180921 214000 755000hist(Y,breaks = 80,main = "Histogram: the property's sale price in dollars")
#pick TotalBsmtSF (Total square feet of basement area) as the quantitative independent variable, X
X <- house_price$TotalBsmtSF
summary(X)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0 795.8 991.5 1057.4 1298.2 6110.0hist(X,breaks = 80,main = "Histogram: total square feet of basement area")
df <- data.frame(X,Y)Probability
Calculate as a minimum the below probabilities a through c. Assume the small letter “x” is estimated as the 1st quartile of the X variable, and the small letter “y” is estimated as the 1st quartile of the Y variable. Interpret the meaning of all probabilities. In addition, make a table of counts.
quantile(Y)## 0% 25% 50% 75% 100%
## 34900 129975 163000 214000 755000quantile(X)## 0% 25% 50% 75% 100%
## 0.00 795.75 991.50 1298.25 6110.00- P(X>x | Y>y)
Calculate the conditional probability of the basement area being greater than 795.75 \(ft^2\) given the sale price is greater than $129,975.00.
#prepare elements for calculating the probability
Y1 <- quantile(Y,0.25)
X1 <- quantile(X,0.25)
Y1_count <- nrow(subset(df, Y > Y1))
Y1_X1_count <- nrow(subset(df,(X > X1 & Y > Y1)))
Y1_X1_count / Y1_count## [1] 0.830137- P(X>x, Y>y)
Calculate the probability of the basement area being greater than 795.75 \(ft^2\) and sale price being greater than $129,975.00.
Y1_X1_count <- nrow(subset(df,(X > X1 & Y > Y1)))
Y1_X1_count / nrow(df)## [1] 0.6226027- P(X<x | Y<y)
Calculate the conditional probability of the basement area being less than 795.75 \(ft^2\) given the sale price is less than $129,975.00.
Y1_count <- nrow(subset(df, Y < Y1))
Y1_X1_count <- nrow(subset(df,(X < X1 & Y < Y1)))
Y1_X1_count / Y1_count## [1] 0.490411#prepare elements for making a table of counts
X3 <- quantile(X,0.75)
Y2 <- quantile(Y,0.5)
a1 <- nrow(subset(df,(Y <= Y2 & X <= X3)))
a2 <- nrow(subset(df,(Y > Y2 & X <= X3)))
a3 <- a1 + a2
b1 <- nrow(subset(df,(Y <= Y2 & X > X3)))
b2 <- nrow(subset(df,(Y > Y2 & X >X3)))
b3 <- b1 + b2
c1 <- a1 + b1
c2 <- a2 + b2
c3 <- c1 + c2
#table of counts:
matrix(c(a1,a2,a3,b1,b2,b3,c1,c2,c3),nrow = 3)## [,1] [,2] [,3]
## [1,] 696 36 732
## [2,] 399 329 728
## [3,] 1095 365 1460Does splitting the training data in this fashion make them independent? Let A be the new variable counting those observations above the 1st quartile for X, and let B be the new variable counting those observations above the 1st quartile for Y. Does P(AB)=P(A)P(B)? Check mathematically, and then evaluate by running a Chi Square test for association.
No, independence doesn’t change based on the size or selection of the whole dataset.
PA <- nrow(subset(df,X > X1))/nrow(df)
PB <- nrow(subset(df,Y > Y1))/nrow(df)
PAB <- nrow(subset(df,(X > X1 & Y > Y1)))/nrow(df)
PA*PB == PAB## [1] FALSEP(AB) \(\neq\) P(A)P(B).
Chi-square test for association:
chisq.test(matrix(c(a1,a2,b1,b2),ncol = 2))##
## Pearson's Chi-squared test with Yates' continuity correction
##
## data: matrix(c(a1, a2, b1, b2), ncol = 2)
## X-squared = 313.61, df = 1, p-value < 2.2e-16
Descriptive and Inferential Statistics
Provide univariate descriptive statistics and appropriate plots for the training data set. Provide a scatterplot of X and Y. Derive a correlation matrix for any THREE quantitative variables in the dataset. Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide a 92% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not?
summary(house_price)## Id MSSubClass MSZoning LotFrontage
## Min. : 1.0 Min. : 20.0 C (all): 10 Min. : 21.00
## 1st Qu.: 365.8 1st Qu.: 20.0 FV : 65 1st Qu.: 59.00
## Median : 730.5 Median : 50.0 RH : 16 Median : 69.00
## Mean : 730.5 Mean : 56.9 RL :1151 Mean : 70.05
## 3rd Qu.:1095.2 3rd Qu.: 70.0 RM : 218 3rd Qu.: 80.00
## Max. :1460.0 Max. :190.0 Max. :313.00
## NA's :259
## LotArea Street Alley LotShape LandContour
## Min. : 1300 Grvl: 6 Grvl: 50 IR1:484 Bnk: 63
## 1st Qu.: 7554 Pave:1454 Pave: 41 IR2: 41 HLS: 50
## Median : 9478 NA's:1369 IR3: 10 Low: 36
## Mean : 10517 Reg:925 Lvl:1311
## 3rd Qu.: 11602
## Max. :215245
##
## Utilities LotConfig LandSlope Neighborhood Condition1
## AllPub:1459 Corner : 263 Gtl:1382 NAmes :225 Norm :1260
## NoSeWa: 1 CulDSac: 94 Mod: 65 CollgCr:150 Feedr : 81
## FR2 : 47 Sev: 13 OldTown:113 Artery : 48
## FR3 : 4 Edwards:100 RRAn : 26
## Inside :1052 Somerst: 86 PosN : 19
## Gilbert: 79 RRAe : 11
## (Other):707 (Other): 15
## Condition2 BldgType HouseStyle OverallQual
## Norm :1445 1Fam :1220 1Story :726 Min. : 1.000
## Feedr : 6 2fmCon: 31 2Story :445 1st Qu.: 5.000
## Artery : 2 Duplex: 52 1.5Fin :154 Median : 6.000
## PosN : 2 Twnhs : 43 SLvl : 65 Mean : 6.099
## RRNn : 2 TwnhsE: 114 SFoyer : 37 3rd Qu.: 7.000
## PosA : 1 1.5Unf : 14 Max. :10.000
## (Other): 2 (Other): 19
## OverallCond YearBuilt YearRemodAdd RoofStyle
## Min. :1.000 Min. :1872 Min. :1950 Flat : 13
## 1st Qu.:5.000 1st Qu.:1954 1st Qu.:1967 Gable :1141
## Median :5.000 Median :1973 Median :1994 Gambrel: 11
## Mean :5.575 Mean :1971 Mean :1985 Hip : 286
## 3rd Qu.:6.000 3rd Qu.:2000 3rd Qu.:2004 Mansard: 7
## Max. :9.000 Max. :2010 Max. :2010 Shed : 2
##
## RoofMatl Exterior1st Exterior2nd MasVnrType MasVnrArea
## CompShg:1434 VinylSd:515 VinylSd:504 BrkCmn : 15 Min. : 0.0
## Tar&Grv: 11 HdBoard:222 MetalSd:214 BrkFace:445 1st Qu.: 0.0
## WdShngl: 6 MetalSd:220 HdBoard:207 None :864 Median : 0.0
## WdShake: 5 Wd Sdng:206 Wd Sdng:197 Stone :128 Mean : 103.7
## ClyTile: 1 Plywood:108 Plywood:142 NA's : 8 3rd Qu.: 166.0
## Membran: 1 CemntBd: 61 CmentBd: 60 Max. :1600.0
## (Other): 2 (Other):128 (Other):136 NA's :8
## ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure
## Ex: 52 Ex: 3 BrkTil:146 Ex :121 Fa : 45 Av :221
## Fa: 14 Fa: 28 CBlock:634 Fa : 35 Gd : 65 Gd :134
## Gd:488 Gd: 146 PConc :647 Gd :618 Po : 2 Mn :114
## TA:906 Po: 1 Slab : 24 TA :649 TA :1311 No :953
## TA:1282 Stone : 6 NA's: 37 NA's: 37 NA's: 38
## Wood : 3
##
## BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2
## ALQ :220 Min. : 0.0 ALQ : 19 Min. : 0.00
## BLQ :148 1st Qu.: 0.0 BLQ : 33 1st Qu.: 0.00
## GLQ :418 Median : 383.5 GLQ : 14 Median : 0.00
## LwQ : 74 Mean : 443.6 LwQ : 46 Mean : 46.55
## Rec :133 3rd Qu.: 712.2 Rec : 54 3rd Qu.: 0.00
## Unf :430 Max. :5644.0 Unf :1256 Max. :1474.00
## NA's: 37 NA's: 38
## BsmtUnfSF TotalBsmtSF Heating HeatingQC CentralAir
## Min. : 0.0 Min. : 0.0 Floor: 1 Ex:741 N: 95
## 1st Qu.: 223.0 1st Qu.: 795.8 GasA :1428 Fa: 49 Y:1365
## Median : 477.5 Median : 991.5 GasW : 18 Gd:241
## Mean : 567.2 Mean :1057.4 Grav : 7 Po: 1
## 3rd Qu.: 808.0 3rd Qu.:1298.2 OthW : 2 TA:428
## Max. :2336.0 Max. :6110.0 Wall : 4
##
## Electrical X1stFlrSF X2ndFlrSF LowQualFinSF
## FuseA: 94 Min. : 334 Min. : 0 Min. : 0.000
## FuseF: 27 1st Qu.: 882 1st Qu.: 0 1st Qu.: 0.000
## FuseP: 3 Median :1087 Median : 0 Median : 0.000
## Mix : 1 Mean :1163 Mean : 347 Mean : 5.845
## SBrkr:1334 3rd Qu.:1391 3rd Qu.: 728 3rd Qu.: 0.000
## NA's : 1 Max. :4692 Max. :2065 Max. :572.000
##
## GrLivArea BsmtFullBath BsmtHalfBath FullBath
## Min. : 334 Min. :0.0000 Min. :0.00000 Min. :0.000
## 1st Qu.:1130 1st Qu.:0.0000 1st Qu.:0.00000 1st Qu.:1.000
## Median :1464 Median :0.0000 Median :0.00000 Median :2.000
## Mean :1515 Mean :0.4253 Mean :0.05753 Mean :1.565
## 3rd Qu.:1777 3rd Qu.:1.0000 3rd Qu.:0.00000 3rd Qu.:2.000
## Max. :5642 Max. :3.0000 Max. :2.00000 Max. :3.000
##
## HalfBath BedroomAbvGr KitchenAbvGr KitchenQual
## Min. :0.0000 Min. :0.000 Min. :0.000 Ex:100
## 1st Qu.:0.0000 1st Qu.:2.000 1st Qu.:1.000 Fa: 39
## Median :0.0000 Median :3.000 Median :1.000 Gd:586
## Mean :0.3829 Mean :2.866 Mean :1.047 TA:735
## 3rd Qu.:1.0000 3rd Qu.:3.000 3rd Qu.:1.000
## Max. :2.0000 Max. :8.000 Max. :3.000
##
## TotRmsAbvGrd Functional Fireplaces FireplaceQu GarageType
## Min. : 2.000 Maj1: 14 Min. :0.000 Ex : 24 2Types : 6
## 1st Qu.: 5.000 Maj2: 5 1st Qu.:0.000 Fa : 33 Attchd :870
## Median : 6.000 Min1: 31 Median :1.000 Gd :380 Basment: 19
## Mean : 6.518 Min2: 34 Mean :0.613 Po : 20 BuiltIn: 88
## 3rd Qu.: 7.000 Mod : 15 3rd Qu.:1.000 TA :313 CarPort: 9
## Max. :14.000 Sev : 1 Max. :3.000 NA's:690 Detchd :387
## Typ :1360 NA's : 81
## GarageYrBlt GarageFinish GarageCars GarageArea GarageQual
## Min. :1900 Fin :352 Min. :0.000 Min. : 0.0 Ex : 3
## 1st Qu.:1961 RFn :422 1st Qu.:1.000 1st Qu.: 334.5 Fa : 48
## Median :1980 Unf :605 Median :2.000 Median : 480.0 Gd : 14
## Mean :1979 NA's: 81 Mean :1.767 Mean : 473.0 Po : 3
## 3rd Qu.:2002 3rd Qu.:2.000 3rd Qu.: 576.0 TA :1311
## Max. :2010 Max. :4.000 Max. :1418.0 NA's: 81
## NA's :81
## GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch
## Ex : 2 N: 90 Min. : 0.00 Min. : 0.00 Min. : 0.00
## Fa : 35 P: 30 1st Qu.: 0.00 1st Qu.: 0.00 1st Qu.: 0.00
## Gd : 9 Y:1340 Median : 0.00 Median : 25.00 Median : 0.00
## Po : 7 Mean : 94.24 Mean : 46.66 Mean : 21.95
## TA :1326 3rd Qu.:168.00 3rd Qu.: 68.00 3rd Qu.: 0.00
## NA's: 81 Max. :857.00 Max. :547.00 Max. :552.00
##
## X3SsnPorch ScreenPorch PoolArea PoolQC
## Min. : 0.00 Min. : 0.00 Min. : 0.000 Ex : 2
## 1st Qu.: 0.00 1st Qu.: 0.00 1st Qu.: 0.000 Fa : 2
## Median : 0.00 Median : 0.00 Median : 0.000 Gd : 3
## Mean : 3.41 Mean : 15.06 Mean : 2.759 NA's:1453
## 3rd Qu.: 0.00 3rd Qu.: 0.00 3rd Qu.: 0.000
## Max. :508.00 Max. :480.00 Max. :738.000
##
## Fence MiscFeature MiscVal MoSold
## GdPrv: 59 Gar2: 2 Min. : 0.00 Min. : 1.000
## GdWo : 54 Othr: 2 1st Qu.: 0.00 1st Qu.: 5.000
## MnPrv: 157 Shed: 49 Median : 0.00 Median : 6.000
## MnWw : 11 TenC: 1 Mean : 43.49 Mean : 6.322
## NA's :1179 NA's:1406 3rd Qu.: 0.00 3rd Qu.: 8.000
## Max. :15500.00 Max. :12.000
##
## YrSold SaleType SaleCondition SalePrice
## Min. :2006 WD :1267 Abnorml: 101 Min. : 34900
## 1st Qu.:2007 New : 122 AdjLand: 4 1st Qu.:129975
## Median :2008 COD : 43 Alloca : 12 Median :163000
## Mean :2008 ConLD : 9 Family : 20 Mean :180921
## 3rd Qu.:2009 ConLI : 5 Normal :1198 3rd Qu.:214000
## Max. :2010 ConLw : 5 Partial: 125 Max. :755000
## (Other): 9plot(X,Y)
sub_df <- data.frame(house_price$LotArea,house_price$GrLivArea,house_price$GarageArea)
cor(sub_df)## house_price.LotArea house_price.GrLivArea
## house_price.LotArea 1.0000000 0.2631162
## house_price.GrLivArea 0.2631162 1.0000000
## house_price.GarageArea 0.1804028 0.4689975
## house_price.GarageArea
## house_price.LotArea 0.1804028
## house_price.GrLivArea 0.4689975
## house_price.GarageArea 1.0000000cor.test(house_price$LotArea,house_price$GrLivArea,method = "pearson",conf.level = 0.92)##
## Pearson's product-moment correlation
##
## data: house_price$LotArea and house_price$GrLivArea
## t = 10.414, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 92 percent confidence interval:
## 0.2199359 0.3052674
## sample estimates:
## cor
## 0.2631162cor.test(house_price$LotArea,house_price$GarageArea,method = "pearson",conf.level = 0.92)##
## Pearson's product-moment correlation
##
## data: house_price$LotArea and house_price$GarageArea
## t = 7.0034, df = 1458, p-value = 3.803e-12
## alternative hypothesis: true correlation is not equal to 0
## 92 percent confidence interval:
## 0.1356921 0.2243801
## sample estimates:
## cor
## 0.1804028cor.test(house_price$GarageArea,house_price$GrLivArea,method = "pearson",conf.level = 0.92)##
## Pearson's product-moment correlation
##
## data: house_price$GarageArea and house_price$GrLivArea
## t = 20.276, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 92 percent confidence interval:
## 0.4324608 0.5039965
## sample estimates:
## cor
## 0.4689975All three confidence intervals have p-values less than 0.5 which means that the null hypothesis could be accepted.
For familywise error, it’s the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests. This problem can be avoid by ajusting the correlation test to a confident level of higher percentage.
Linear Algebra and Correlation
Invert your 3 x 3 correlation matrix from above. (This is known as the precision matrix and contains variance inflation factors on the diagonal.) Multiply the correlation matrix by the precision matrix, and then multiply the precision matrix by the correlation matrix. Conduct LU decomposition on the matrix.
cormatrix <- cor(sub_df)
cormatrix## house_price.LotArea house_price.GrLivArea
## house_price.LotArea 1.0000000 0.2631162
## house_price.GrLivArea 0.2631162 1.0000000
## house_price.GarageArea 0.1804028 0.4689975
## house_price.GarageArea
## house_price.LotArea 0.1804028
## house_price.GrLivArea 0.4689975
## house_price.GarageArea 1.0000000precisionmatrix <- solve(cormatrix)
precisionmatrix## house_price.LotArea house_price.GrLivArea
## house_price.LotArea 1.07920917 -0.2469705
## house_price.GrLivArea -0.24697046 1.3385010
## house_price.GarageArea -0.07886378 -0.5831994
## house_price.GarageArea
## house_price.LotArea -0.07886378
## house_price.GrLivArea -0.58319943
## house_price.GarageArea 1.28774631#multiply the correlation matrix by the precision matrix
round(cormatrix %*% precisionmatrix)## house_price.LotArea house_price.GrLivArea
## house_price.LotArea 1 0
## house_price.GrLivArea 0 1
## house_price.GarageArea 0 0
## house_price.GarageArea
## house_price.LotArea 0
## house_price.GrLivArea 0
## house_price.GarageArea 1#multiply the precision matrix by the correlation matrix
round(precisionmatrix %*% cormatrix)## house_price.LotArea house_price.GrLivArea
## house_price.LotArea 1 0
## house_price.GrLivArea 0 1
## house_price.GarageArea 0 0
## house_price.GarageArea
## house_price.LotArea 0
## house_price.GrLivArea 0
## house_price.GarageArea 1library(Matrix)## Warning: package 'Matrix' was built under R version 3.4.4#LU decomposition on correlation matrix
expand(lu(cormatrix))$L## 3 x 3 Matrix of class "dtrMatrix" (unitriangular)
## [,1] [,2] [,3]
## [1,] 1.0000000 . .
## [2,] 0.2631162 1.0000000 .
## [3,] 0.1804028 0.4528838 1.0000000expand(lu(cormatrix))$U## 3 x 3 Matrix of class "dtrMatrix"
## [,1] [,2] [,3]
## [1,] 1.0000000 0.2631162 0.1804028
## [2,] . 0.9307699 0.4215306
## [3,] . . 0.7765505#LU decomposition on precision matrix
expand(lu(precisionmatrix))$L## 3 x 3 Matrix of class "dtrMatrix" (unitriangular)
## [,1] [,2] [,3]
## [1,] 1.00000000 . .
## [2,] -0.22884393 1.00000000 .
## [3,] -0.07307553 -0.46899748 1.00000000expand(lu(precisionmatrix))$U## 3 x 3 Matrix of class "dtrMatrix"
## [,1] [,2] [,3]
## [1,] 1.07920917 -0.24697046 -0.07886378
## [2,] . 1.28198329 -0.60124693
## [3,] . . 1.00000000
Calculus-based Probability & Statistics Many times, it makes sense to fit a closed form distribution to data. For the first variable that you selected which is skewed to the right, shift it so that the minimum value is above zero as necessary. Then load the MASS package and run fitdistr to fit an exponential probability density function. (See https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/fitdistr.html ). Find the optimal value of ?? for this distribution, and then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, ??)). Plot a histogram and compare it with a histogram of your original variable. Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss.
library(MASS)## Warning: package 'MASS' was built under R version 3.4.4optimal_lambda <- fitdistr(house_price$TotalBsmtSF,"exponential")
optimal_lambda$estimate## rate
## 0.0009456896hist(rexp(1000,optimal_lambda$estimate),breaks = 200,main = "Fitted Exponential PDF",xlim = c(1,quantile(rexp(1000,optimal_lambda$estimate),0.99)))
hist(house_price$TotalBsmtSF,breaks = 400,main = "Observed Basement Area Size",xlim = c(1,quantile(house_price$TotalBsmtSF,0.99)))
#5th and 95th percentiles using CDF
qexp(0.05,rate = optimal_lambda$estimate,lower.tail = TRUE,log.p = FALSE)## [1] 54.23904qexp(0.95,rate = optimal_lambda$estimate,lower.tail = TRUE,log.p = FALSE)## [1] 3167.776#95% confidence interval from the empirical data (assuming normality)
Bsmt_mean <- mean(house_price$TotalBsmtSF)
Bsmt_sd <- sd(house_price$TotalBsmtSF)
qnorm(0.95,Bsmt_mean,Bsmt_sd)## [1] 1779.035#empirical 5th and 95th percentile of the data
quantile(house_price$TotalBsmtSF,c(0.05,0.95))## 5% 95%
## 519.3 1753.0The exponential simulated model doesn’t seem to be a good enough model for this set of house data since the range covers doesn’t fit the one generated for the actual data and relatively large bias exists.
Modeling
Build some type of multiple regression model and submit your model to the competition board. Provide your complete model summary and results with analysis. Report your Kaggle.com user name and score.
#select all the quantitative variables and eliminate the ones with low correlations (<30%)
quantitative <- data.frame(house_price$OverallQual,house_price$YearBuilt,house_price$YearRemodAdd,house_price$MasVnrArea,house_price$BsmtFinSF1,house_price$TotalBsmtSF,house_price$X1stFlrSF,house_price$X2ndFlrSF,house_price$GrLivArea,house_price$FullBath,house_price$TotRmsAbvGrd,house_price$Fireplaces,house_price$GarageCars,house_price$GarageArea,house_price$WoodDeckSF,house_price$OpenPorchSF,house_price$SalePrice)
#to save space, the result of the following command is not displayed
#cor(quantitative)
#create a linear regression model
m1 <- lm(house_price.SalePrice ~.,data = quantitative)
summary(m1)##
## Call:
## lm(formula = house_price.SalePrice ~ ., data = quantitative)
##
## Residuals:
## Min 1Q Median 3Q Max
## -512233 -17548 -1737 14681 283280
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.094e+06 1.268e+05 -8.627 < 2e-16 ***
## house_price.OverallQual 1.856e+04 1.174e+03 15.807 < 2e-16 ***
## house_price.YearBuilt 1.638e+02 4.978e+01 3.290 0.001028 **
## house_price.YearRemodAdd 3.564e+02 6.208e+01 5.741 1.15e-08 ***
## house_price.MasVnrArea 2.881e+01 6.159e+00 4.678 3.17e-06 ***
## house_price.BsmtFinSF1 1.725e+01 2.596e+00 6.646 4.26e-11 ***
## house_price.TotalBsmtSF 1.165e+01 4.298e+00 2.711 0.006796 **
## house_price.X1stFlrSF 2.618e+01 2.082e+01 1.257 0.208871
## house_price.X2ndFlrSF 1.753e+01 2.048e+01 0.856 0.392000
## house_price.GrLivArea 2.135e+01 2.035e+01 1.049 0.294370
## house_price.FullBath -1.489e+03 2.630e+03 -0.566 0.571228
## house_price.TotRmsAbvGrd 1.688e+03 1.089e+03 1.550 0.121402
## house_price.Fireplaces 7.888e+03 1.783e+03 4.423 1.05e-05 ***
## house_price.GarageCars 1.011e+04 2.960e+03 3.414 0.000659 ***
## house_price.GarageArea 1.040e+01 1.005e+01 1.035 0.301006
## house_price.WoodDeckSF 3.068e+01 8.129e+00 3.774 0.000167 ***
## house_price.OpenPorchSF 7.271e+00 1.572e+01 0.462 0.643861
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 36380 on 1435 degrees of freedom
## (8 observations deleted due to missingness)
## Multiple R-squared: 0.7918, Adjusted R-squared: 0.7894
## F-statistic: 341 on 16 and 1435 DF, p-value: < 2.2e-16#eliminate variables based on significant level
quantitative2 <- data.frame(house_price$OverallQual,house_price$YearRemodAdd,house_price$MasVnrArea,house_price$BsmtFinSF1,house_price$TotalBsmtSF,house_price$Fireplaces,house_price$GarageCars,house_price$WoodDeckSF,house_price$SalePrice)
colnames(quantitative2) <- c("OverallQual","YearRemodAdd","MasVnrArea","BsmtFinSF1","TotalBsmtSF","Fireplaces","GarageCars","WoodDeckSF","SalePrice")
#create a linear regression model
m2 <- lm(SalePrice ~.,data = quantitative2)
summary(m2)##
## Call:
## lm(formula = SalePrice ~ ., data = quantitative2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -407840 -21443 -2760 16410 363961
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -8.307e+05 1.210e+05 -6.867 9.70e-12 ***
## OverallQual 2.449e+04 1.183e+03 20.706 < 2e-16 ***
## YearRemodAdd 3.925e+02 6.256e+01 6.273 4.66e-10 ***
## MasVnrArea 4.651e+01 6.602e+00 7.045 2.85e-12 ***
## BsmtFinSF1 1.482e+01 2.752e+00 5.383 8.52e-08 ***
## TotalBsmtSF 2.504e+01 3.290e+00 7.611 4.89e-14 ***
## Fireplaces 1.551e+04 1.849e+03 8.389 < 2e-16 ***
## GarageCars 1.794e+04 1.820e+03 9.855 < 2e-16 ***
## WoodDeckSF 4.464e+01 8.848e+00 5.045 5.12e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 39960 on 1443 degrees of freedom
## (8 observations deleted due to missingness)
## Multiple R-squared: 0.7474, Adjusted R-squared: 0.746
## F-statistic: 533.6 on 8 and 1443 DF, p-value: < 2.2e-16hist(m2$residuals,breaks = 200)
qqnorm(m2$residuals)
qqline(m2$residuals)
This model, from the perspective of residuals, is not bad since the residuals are normally distributed, with all dependent variables being statistically significant.
The performance of the model will be evaluated by test data.
test_raw <- read.csv("https://raw.githubusercontent.com/xiaoxiaogao-DD/store/master/test.csv")
test_raw[complete.cases(test_raw),]## [1] Id MSSubClass MSZoning LotFrontage LotArea
## [6] Street Alley LotShape LandContour Utilities
## [11] LotConfig LandSlope Neighborhood Condition1 Condition2
## [16] BldgType HouseStyle OverallQual OverallCond YearBuilt
## [21] YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd
## [26] MasVnrType MasVnrArea ExterQual ExterCond Foundation
## [31] BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1
## [36] BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating
## [41] HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF
## [46] LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath
## [51] HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd
## [56] Functional Fireplaces FireplaceQu GarageType GarageYrBlt
## [61] GarageFinish GarageCars GarageArea GarageQual GarageCond
## [66] PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch
## [71] ScreenPorch PoolArea PoolQC Fence MiscFeature
## [76] MiscVal MoSold YrSold SaleType SaleCondition
## <0 rows> (or 0-length row.names)test_raw[is.na(test_raw)] <- 0## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generated
## Warning in `[<-.factor`(`*tmp*`, thisvar, value = 0): invalid factor level,
## NA generatedpred <- predict(m2,test_raw)
write.csv(as.data.frame(cbind(test_raw$Id,pred)),file = "kaggle_sub.csv",quote = FALSE,row.names = FALSE)Kaggle submission screenshot:
