OOhannessian_FinalProject
Ohannes (Hovig) Ohannessian
12/4/2018
Problem 1
Assign X4 to X and Y4 to Y.
Y <- c(20.2, 18.6, 22.6, 11.4, 23.6, 24.0, 26.0, 26.8, 19.7, 22.7, 16.8, 20.2, 21.7, 20.9, 26.9, 16.3, 19.9, 15.5, 26.5, 21.7)
X <- c(9.3, 12.4, 19.9, 6.9, -1.0, 10.6, 6.4, 10.6, 1.2, 7.7, 15.5, 6.9, 13.7, 3.7, 4.4, 11.5, 4.2, 13.9, 12.9, 1.2)
(x <- quantile(X,0.75))## 75%
## 12.525(y <- quantile(Y,0.25))## 25%
## 19.425Probability.
- Calculate as a minimum the below probabilities a through c. Assume the small letter “x” is estimated as the 3d 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.
- \(P(X>x~ | ~Y>y)\)
length(X[X>x & Y>y])/length(Y[Y>y])## [1] 0.2\[P(X>x~ | ~Y>y) = \frac{P(X>x~, ~ Y>y)}{P(Y >y)} = \frac{P(X>x) P(Y>y)}{P(Y>y)} = 0.2\]
- \(P(X>x~, ~Y>y)\)
length(X[X>x & Y>y])/length(Y)## [1] 0.15\[P(X>x~, ~Y>y) = \frac{P(X>x~ | ~ Y>y)}{P(Y >y)} = 0.15\]
- \(P(X<x~ | ~Y>y)\)
length(X[X<x & Y>y])/length(Y[Y>y])## [1] 0.8\[P(X<x~ | ~Y>y) = \frac{P(X<x~, ~ Y>y)}{P(Y >y)} = \frac{P(<x) P(Y>y)}{P(Y>y)} = 0.8\]
- In addition, make a table of counts as shown below.
library(knitr)
a0 <- "<= 1st quartile"
a1 <- length(X[Y<=y & X<=x])
a2 <- length(X[Y>y & X<=x])
a3 <- a1 + a2
b0 <- "> 1st quartile"
b1 <- length(X[Y<=y & X>x])
b2 <- length(X[Y>y & X>x])
b3 <- b1 + b2
c0 <- "Total"
c1 <- a1 + b1
c2 <- a2 + b2
c3 <- c1 + c2
rownames = c()
colnames = c("y/x", "<= 3rd quartile", "> 3rd quartile", "Total")
kable(matrix(c(a0,a1,a2,a3,b0,b1,b2,b3,c0,c1,c2,c3),nrow = 3, byrow = TRUE, dimnames = list(rownames, colnames)))| y/x | <= 3rd quartile | > 3rd quartile | Total |
|---|---|---|---|
| <= 1st quartile | 3 | 12 | 15 |
| > 1st quartile | 2 | 3 | 5 |
| Total | 5 | 15 | 20 |
- Does 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.
A <- X > x
B <- Y > y
probability_AB <- sum(A[B])
cat("P(AB) = ", probability_AB/length(X), "\n")## P(AB) = 0.15probability_A <- sum(A)/length(X)
probability_B <- sum(B)/length(X)
cat("P(A)P(B) = ", probability_A*probability_B, "\n")## P(A)P(B) = 0.1875We conclude that \(P\left(AB\right) \neq P\left(A\right)P\left(B\right)\) => A and B are not independent.
- Chi-Square test on A, B for association:
ptest <- chisq.test(rbind(x,y))
print(ptest)##
## Chi-squared test for given probabilities
##
## data: rbind(x, y)
## X-squared = 1.4901, df = 1, p-value = 0.2222The p-value indicates the two variables are statistically dependent.
Problem 2
Descriptive and Inferential Statistics.
- Providing univariate descriptive statistics and appropriate plots for the training data set.
library(ggplot2)
library(corrplot)## corrplot 0.84 loadedtrain <- read.csv('https://raw.githubusercontent.com/hovig/MSDS_CUNY/master/DATA605/train.csv')
test <- read.csv('https://raw.githubusercontent.com/hovig/MSDS_CUNY/master/DATA605/test.csv')
cat("Training dataset count of rows and columns: ", dim(train))## Training dataset count of rows and columns: 1460 81cat("Testing dataset count of rows and columns: ", dim(test))## Testing dataset count of rows and columns: 1459 80head(train)## 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 143000summary(train)## 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): 9- Provide a scatterplot matrix for at least two of the independent variables and the dependent variable.
df <- subset(train, select = c("LotArea", "SalePrice"))
head(df)## LotArea SalePrice
## 1 8450 208500
## 2 9600 181500
## 3 11250 223500
## 4 9550 140000
## 5 14260 250000
## 6 14115 143000ggplot(df, aes(x = LotArea)) + geom_histogram(binwidth=550, color="blue")
ggplot(df, aes(x = SalePrice)) + geom_histogram(binwidth = 1500, color="blue")
summary(df$LotArea)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1300 7554 9478 10517 11602 215245ggplot(df, aes(x = df$LotArea, y=df$SalePrice)) + geom_point(size=2, shape = 18, color="blue") +
labs(title = "SalePrice vs. LotArea", x = "LotArea", y = "SalePrice")
ggplot(train, aes(YearBuilt)) + geom_histogram(color="darkblue", fill="lightblue")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot(train, aes(GrLivArea)) + geom_histogram(color="darkblue", fill="lightblue")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot(train, aes(SalePrice)) + geom_histogram(color="darkblue", fill="lightblue")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot(train, aes(HouseStyle)) + geom_bar(color="darkblue", fill="lightblue")
ggplot(train, aes(BldgType)) + geom_bar(color="darkblue", fill="lightblue")
ggplot(train, aes(MSZoning)) + geom_bar(color="darkblue", fill="lightblue")
- Derive a correlation matrix for any THREE quantitative variables in the dataset.
df_matrix <- train[c("LotArea", "GrLivArea", "SalePrice")]
corr_matrix <- cor(df_matrix, use="complete.obs", method="kendall")
corrplot(corr_matrix, method = "circle")
- Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide a 80% confidence interval.
cor.test(train$LotArea, train$SalePrice, conf.level = 0.8)##
## Pearson's product-moment correlation
##
## data: train$LotArea and train$SalePrice
## t = 10.445, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.2323391 0.2947946
## sample estimates:
## cor
## 0.2638434cor.test(train$GrLivArea, train$SalePrice, conf.level = 0.8)##
## Pearson's product-moment correlation
##
## data: train$GrLivArea and train$SalePrice
## t = 38.348, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.6915087 0.7249450
## sample estimates:
## cor
## 0.7086245- Discuss the meaning of your analysis. Would you be worried about family wise error? Why or why not?
Concluding the correlation test, a fmaily wise error should be considering SalePrice & GrLivArea that has a better a corrolation result than SalePrice & LotArea.
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.
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(Matrix)
pre_matrix <- solve(corr_matrix)
round(pre_matrix %*% corr_matrix)## LotArea GrLivArea SalePrice
## LotArea 1 0 0
## GrLivArea 0 1 0
## SalePrice 0 0 1round(corr_matrix %*% pre_matrix)## LotArea GrLivArea SalePrice
## LotArea 1 0 0
## GrLivArea 0 1 0
## SalePrice 0 0 1expand(lu(pre_matrix))## $L
## 3 x 3 Matrix of class "dtrMatrix" (unitriangular)
## [,1] [,2] [,3]
## [1,] 1.0000000 . .
## [2,] -0.1899995 1.0000000 .
## [3,] -0.2108257 -0.5439421 1.0000000
##
## $U
## 3 x 3 Matrix of class "dtrMatrix"
## [,1] [,2] [,3]
## [1,] 1.1417147 -0.2169252 -0.2407028
## [2,] . 1.4201985 -0.7725058
## [3,] . . 1.0000000
##
## $P
## 3 x 3 sparse Matrix of class "pMatrix"
##
## [1,] | . .
## [2,] . | .
## [3,] . . |Calculus-Based Probability & Statistics.
Many times, it makes sense to fit a closed form distribution to data. Select a variable in the Kaggle.com training dataset that is skewed to the right, shift it so that the minimum value is absolutely above zero if necessary. Then load the MASS package and run fitdistr to fit an exponential probability density function. (See https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/fitdistr.html ). Find the optimal value of for this distribution, and then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, )). Plot a histogram and compare it with a histogram of your original variable. Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss.
library(MASS)
X <- df$LotArea
Y <- df$SalePrice
(exp_params = fitdistr(X, "exponential"))## rate
## 9.508570e-05
## (2.488507e-06)(lambda = as.double(exp_params$estimate))## [1] 9.50857e-05print(lambda)## [1] 9.50857e-05exponential_distribution = rexp(1000, lambda)par(mfrow=c(1, 2))
hist(X, main="X (Lot Area)")
hist(exponential_distribution, main="Exponential Distr for X")
quantile(exponential_distribution, probs=c(0.05, 0.95))## 5% 95%
## 719.1411 31836.7502sd_X = sd(X); mean_X = mean(X); length_X = length(X)
err = qnorm(0.975)*sd_X/sqrt(length_X)
cat("A 95% confidence interval for Lot Area is [", mean_X - err, ",", mean_X + err, "]") ## A 95% confidence interval for Lot Area is [ 10004.84 , 11028.81 ]quantile(X, probs=c(0.05, 0.95))## 5% 95%
## 3311.70 17401.15The LotArea follows an exponential distribution that does not fit the observed data very well.
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.
regression <- lm(SalePrice ~ LotArea + LotFrontage + BldgType + OverallQual + BsmtQual + GarageArea + GarageYrBlt + GrLivArea + TotalBsmtSF, data = train)
summary(regression)##
## Call:
## lm(formula = SalePrice ~ LotArea + LotFrontage + BldgType + OverallQual +
## BsmtQual + GarageArea + GarageYrBlt + GrLivArea + TotalBsmtSF,
## data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -488971 -17498 -706 15316 261798
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.962e+05 1.460e+05 -2.028 0.042807 *
## LotArea 7.263e-01 1.618e-01 4.488 7.94e-06 ***
## LotFrontage -1.650e+02 6.425e+01 -2.569 0.010338 *
## BldgType2fmCon -1.904e+04 8.886e+03 -2.142 0.032395 *
## BldgTypeDuplex -2.797e+04 8.208e+03 -3.407 0.000680 ***
## BldgTypeTwnhs -3.152e+04 7.306e+03 -4.313 1.75e-05 ***
## BldgTypeTwnhsE -1.708e+04 4.922e+03 -3.471 0.000539 ***
## OverallQual 2.095e+04 1.463e+03 14.321 < 2e-16 ***
## BsmtQualFa -6.976e+04 9.359e+03 -7.453 1.86e-13 ***
## BsmtQualGd -5.056e+04 4.601e+03 -10.988 < 2e-16 ***
## BsmtQualTA -6.244e+04 5.921e+03 -10.545 < 2e-16 ***
## GarageArea 3.940e+01 9.245e+00 4.262 2.20e-05 ***
## GarageYrBlt 1.510e+02 7.417e+01 2.036 0.042034 *
## GrLivArea 4.606e+01 3.134e+00 14.696 < 2e-16 ***
## TotalBsmtSF 1.997e+01 3.831e+00 5.214 2.21e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 38490 on 1088 degrees of freedom
## (357 observations deleted due to missingness)
## Multiple R-squared: 0.7893, Adjusted R-squared: 0.7866
## F-statistic: 291.1 on 14 and 1088 DF, p-value: < 2.2e-16new_regression <- update(regression, . ~ . - GarageYrBlt)
summary(new_regression)##
## Call:
## lm(formula = SalePrice ~ LotArea + LotFrontage + BldgType + OverallQual +
## BsmtQual + GarageArea + GrLivArea + TotalBsmtSF, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -486529 -16814 -782 15762 264867
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.242e+04 1.061e+04 1.171 0.241819
## LotArea 7.101e-01 1.587e-01 4.475 8.39e-06 ***
## LotFrontage -1.676e+02 6.158e+01 -2.722 0.006577 **
## BldgType2fmCon -1.787e+04 7.715e+03 -2.316 0.020731 *
## BldgTypeDuplex -2.079e+04 6.845e+03 -3.037 0.002441 **
## BldgTypeTwnhs -2.740e+04 6.779e+03 -4.042 5.66e-05 ***
## BldgTypeTwnhsE -1.685e+04 4.791e+03 -3.516 0.000455 ***
## OverallQual 2.021e+04 1.364e+03 14.815 < 2e-16 ***
## BsmtQualFa -7.946e+04 8.559e+03 -9.284 < 2e-16 ***
## BsmtQualGd -5.292e+04 4.494e+03 -11.776 < 2e-16 ***
## BsmtQualTA -7.070e+04 5.382e+03 -13.135 < 2e-16 ***
## GarageArea 4.269e+01 6.763e+00 6.312 3.91e-10 ***
## GrLivArea 4.410e+01 2.906e+00 15.175 < 2e-16 ***
## TotalBsmtSF 2.124e+01 3.705e+00 5.733 1.26e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 38070 on 1156 degrees of freedom
## (290 observations deleted due to missingness)
## Multiple R-squared: 0.7941, Adjusted R-squared: 0.7918
## F-statistic: 343 on 13 and 1156 DF, p-value: < 2.2e-16plot(fitted(new_regression), resid(new_regression))
qqnorm(resid(new_regression))
SalePrice <- predict(new_regression, newdata = test)
Id <- 1461:2919
prediction <- data.frame(Id, SalePrice)
head(prediction)## Id SalePrice
## 1 1461 127023.8
## 2 1462 159688.1
## 3 1463 170089.1
## 4 1464 167458.5
## 5 1465 205908.6
## 6 1466 183259.6summary(prediction)## Id SalePrice
## Min. :1461 Min. : -2620
## 1st Qu.:1826 1st Qu.:127314
## Median :2190 Median :162729
## Mean :2190 Mean :178581
## 3rd Qu.:2554 3rd Qu.:214093
## Max. :2919 Max. :603124
## NA's :265prediction$SalePrice[is.na(prediction$SalePrice)] <- 0
write.csv(prediction, "/Users/hovig/Downloads/kaggle_submission.csv",row.names=F)