Final Project
Antonio Bayquen
December 12, 2018
1. Probability. 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.
5 points a. P(X>x | Y>y) b. P(X>x, Y>y) c. P(Xy)
(Hint: P(X>3d quartile of x values | Y> 1st quartile of y values.)
X2 = c(7.4,
6.4,
8.5,
9.5,
11.8,
8.8,
8.4,
5.1,
11.4,
15.1,
12.6,
8.0,
10.3,
10.4,
9.5,
9.5,
15.1,
6.6,
15.4,
8.2)
x = quantile(X2, 0.75)
x## 75%
## 11.5Y2 = c(20.8,
14.6,
18.0,
7.3,
19.4,
13.5,
14.7,
15.3,
12.6,
13.0,
13.1,
10.3,
14.9,
14.8,
16.2,
15.7,
16.3,
11.5,
12.2,
11.8)
y = quantile(Y2, 0.25)
y## 25%
## 12.5 count = 0
ty = Y2 > y
for (i in 1:20)
{
if (ty[i] == TRUE)
{
if (X2[i] > x)
{
count = count + 1
}
}
}
count## [1] 4 Pab = count/20
Pab## [1] 0.2 #conditional probability of P(X>x | Y>y)
Pc = Pab/y
Pc## 25%
## 0.016#joint probabiilty of P(X>x, Y>y)
Pab## [1] 0.2 count2 = 0
for (i in 1:20)
{
if (ty[i] == TRUE)
{
if (X2[i] < x)
{
count2 = count2 + 1
}
}
}
count2## [1] 11 Pab2 = count2/20
Pab2## [1] 0.55 #conditional probability of P(X<x | Y>y)
Pc2 = Pab2/y
Pc2## 25%
## 0.0445 points. In addition, make a table of counts as shown below.
x/y —————<=3d quartile >3d quartile Total
<=1st quartile———- 4 ——— 11 ——– 15
>1 st quartile———- 1 ———- 4 ——— 5
Total——————— 5 ——— 15 ——– 20
#y - <=3d quartile and x <=1st quartile
ty = Y2 <= y
cnt1 = 0
for (i in 1:20)
{
if (ty[i] == TRUE)
{
if (X2[i] <= x)
{
cnt1 = cnt1 + 1
}
}
}
cnt1## [1] 4 cnt2 = 0
for (i in 1:20)
{
if (ty[i] == TRUE)
{
if (X2[i] > x)
{
cnt2 = cnt2 + 1
}
}
}
cnt2## [1] 1 ty = Y2 > y
cnt3 = 0
for (i in 1:20)
{
if (ty[i] == TRUE)
{
if (X2[i] <= x)
{
cnt3 = cnt3 + 1
}
}
}
cnt3## [1] 11 cnt4 = 0
for (i in 1:20)
{
if (ty[i] == TRUE)
{
if (X2[i] > x)
{
cnt4 = cnt4 + 1
}
}
}
cnt4## [1] 45 points. 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.
x = quantile(X2, 0.25)
y = quantile(Y2, 0.25)
A = sum(X2>x)
A## [1] 15B = sum(Y2>y)
B## [1] 15y## 25%
## 12.5ty = Y2 > y
for (i in 1:20)
{
if (ty[i] == TRUE)
{
if (X2[i] > x)
{
count = count + 1
}
}
}
count## [1] 16Pab = count/20
Pab## [1] 0.8Pab == (A/20)*(B/20)## [1] FALSE#perform the Chi Square Test for assoication
rows = 20
Matriz = matrix(c(X2, Y2),
nrow=rows,
byrow=TRUE)
chisq.test(Matriz)##
## Pearson's Chi-squared test
##
## data: Matriz
## X-squared = 10.588, df = 19, p-value = 0.9369Alternative hypothesis (H1): X and Y observations are dependent
2. Descriptive and Inferential Statistics. Provide univariate descriptive statistics and appropriate plots for the training data set. Provide a scatterplot matrix for at least two of the independent variables and the dependent variable. 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 80% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not?
#getwd()
library(stats)
library(ggplot2)
library(psych)##
## Attaching package: 'psych'## The following objects are masked from 'package:ggplot2':
##
## %+%, alphaTraindata <- read.csv(file="C:/Users/ajb2/Documents/training.csv", header=TRUE, sep=",", stringsAsFactors = FALSE)
train_subset <- cbind(Traindata$MSSubClass, Traindata$LotArea, Traindata$OverallQual, Traindata$OverallCond, Traindata$YearBuilt, Traindata$MasVnrArea, Traindata$BsmtFinSF1, Traindata$BsmtFinSF2, Traindata$BsmtUnfSF, Traindata$X1stFlrSF, Traindata$X2ndFlrSF, Traindata$BedroomAbvGr, Traindata$KitchenAbvGr, Traindata$GarageYrBlt, Traindata$PoolArea)
head(Traindata,10)## 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
## 7 7 20 RL 75 10084 Pave <NA> Reg
## 8 8 60 RL NA 10382 Pave <NA> IR1
## 9 9 50 RM 51 6120 Pave <NA> Reg
## 10 10 190 RL 50 7420 Pave <NA> Reg
## 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
## 7 Lvl AllPub Inside Gtl Somerst Norm
## 8 Lvl AllPub Corner Gtl NWAmes PosN
## 9 Lvl AllPub Inside Gtl OldTown Artery
## 10 Lvl AllPub Corner Gtl BrkSide Artery
## 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
## 7 Norm 1Fam 1Story 8 5 2004
## 8 Norm 1Fam 2Story 7 6 1973
## 9 Norm 1Fam 1.5Fin 7 5 1931
## 10 Artery 2fmCon 1.5Unf 5 6 1939
## 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
## 7 2005 Gable CompShg VinylSd VinylSd Stone
## 8 1973 Gable CompShg HdBoard HdBoard Stone
## 9 1950 Gable CompShg BrkFace Wd Shng None
## 10 1950 Gable CompShg MetalSd MetalSd None
## MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond
## 1 196 Gd TA PConc Gd TA
## 2 0 TA TA CBlock Gd TA
## 3 162 Gd TA PConc Gd TA
## 4 0 TA TA BrkTil TA Gd
## 5 350 Gd TA PConc Gd TA
## 6 0 TA TA Wood Gd TA
## 7 186 Gd TA PConc Ex TA
## 8 240 TA TA CBlock Gd TA
## 9 0 TA TA BrkTil TA TA
## 10 0 TA TA BrkTil TA TA
## BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF
## 1 No GLQ 706 Unf 0 150
## 2 Gd ALQ 978 Unf 0 284
## 3 Mn GLQ 486 Unf 0 434
## 4 No ALQ 216 Unf 0 540
## 5 Av GLQ 655 Unf 0 490
## 6 No GLQ 732 Unf 0 64
## 7 Av GLQ 1369 Unf 0 317
## 8 Mn ALQ 859 BLQ 32 216
## 9 No Unf 0 Unf 0 952
## 10 No GLQ 851 Unf 0 140
## TotalBsmtSF Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF
## 1 856 GasA Ex Y SBrkr 856 854
## 2 1262 GasA Ex Y SBrkr 1262 0
## 3 920 GasA Ex Y SBrkr 920 866
## 4 756 GasA Gd Y SBrkr 961 756
## 5 1145 GasA Ex Y SBrkr 1145 1053
## 6 796 GasA Ex Y SBrkr 796 566
## 7 1686 GasA Ex Y SBrkr 1694 0
## 8 1107 GasA Ex Y SBrkr 1107 983
## 9 952 GasA Gd Y FuseF 1022 752
## 10 991 GasA Ex Y SBrkr 1077 0
## LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath
## 1 0 1710 1 0 2 1
## 2 0 1262 0 1 2 0
## 3 0 1786 1 0 2 1
## 4 0 1717 1 0 1 0
## 5 0 2198 1 0 2 1
## 6 0 1362 1 0 1 1
## 7 0 1694 1 0 2 0
## 8 0 2090 1 0 2 1
## 9 0 1774 0 0 2 0
## 10 0 1077 1 0 1 0
## BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd Functional
## 1 3 1 Gd 8 Typ
## 2 3 1 TA 6 Typ
## 3 3 1 Gd 6 Typ
## 4 3 1 Gd 7 Typ
## 5 4 1 Gd 9 Typ
## 6 1 1 TA 5 Typ
## 7 3 1 Gd 7 Typ
## 8 3 1 TA 7 Typ
## 9 2 2 TA 8 Min1
## 10 2 2 TA 5 Typ
## Fireplaces FireplaceQu GarageType GarageYrBlt GarageFinish GarageCars
## 1 0 <NA> Attchd 2003 RFn 2
## 2 1 TA Attchd 1976 RFn 2
## 3 1 TA Attchd 2001 RFn 2
## 4 1 Gd Detchd 1998 Unf 3
## 5 1 TA Attchd 2000 RFn 3
## 6 0 <NA> Attchd 1993 Unf 2
## 7 1 Gd Attchd 2004 RFn 2
## 8 2 TA Attchd 1973 RFn 2
## 9 2 TA Detchd 1931 Unf 2
## 10 2 TA Attchd 1939 RFn 1
## GarageArea GarageQual GarageCond PavedDrive WoodDeckSF OpenPorchSF
## 1 548 TA TA Y 0 61
## 2 460 TA TA Y 298 0
## 3 608 TA TA Y 0 42
## 4 642 TA TA Y 0 35
## 5 836 TA TA Y 192 84
## 6 480 TA TA Y 40 30
## 7 636 TA TA Y 255 57
## 8 484 TA TA Y 235 204
## 9 468 Fa TA Y 90 0
## 10 205 Gd TA Y 0 4
## EnclosedPorch X3SsnPorch ScreenPorch PoolArea PoolQC Fence MiscFeature
## 1 0 0 0 0 <NA> <NA> <NA>
## 2 0 0 0 0 <NA> <NA> <NA>
## 3 0 0 0 0 <NA> <NA> <NA>
## 4 272 0 0 0 <NA> <NA> <NA>
## 5 0 0 0 0 <NA> <NA> <NA>
## 6 0 320 0 0 <NA> MnPrv Shed
## 7 0 0 0 0 <NA> <NA> <NA>
## 8 228 0 0 0 <NA> <NA> Shed
## 9 205 0 0 0 <NA> <NA> <NA>
## 10 0 0 0 0 <NA> <NA> <NA>
## MiscVal MoSold YrSold SaleType SaleCondition SalePrice
## 1 0 2 2008 WD Normal 208500
## 2 0 5 2007 WD Normal 181500
## 3 0 9 2008 WD Normal 223500
## 4 0 2 2006 WD Abnorml 140000
## 5 0 12 2008 WD Normal 250000
## 6 700 10 2009 WD Normal 143000
## 7 0 8 2007 WD Normal 307000
## 8 350 11 2009 WD Normal 200000
## 9 0 4 2008 WD Abnorml 129900
## 10 0 1 2008 WD Normal 118000describe(Traindata)## Warning in describe(Traindata): NAs introduced by coercion
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## -Inf## vars n mean sd median trimmed mad
## Id 1 1460 730.50 421.61 730.5 730.50 541.15
## MSSubClass 2 1460 56.90 42.30 50.0 49.15 44.48
## MSZoning* 3 1460 NaN NA NA NaN NA
## LotFrontage 4 1201 70.05 24.28 69.0 68.94 16.31
## LotArea 5 1460 10516.83 9981.26 9478.5 9563.28 2962.23
## Street* 6 1460 NaN NA NA NaN NA
## Alley* 7 91 NaN NA NA NaN NA
## LotShape* 8 1460 NaN NA NA NaN NA
## LandContour* 9 1460 NaN NA NA NaN NA
## Utilities* 10 1460 NaN NA NA NaN NA
## LotConfig* 11 1460 NaN NA NA NaN NA
## LandSlope* 12 1460 NaN NA NA NaN NA
## Neighborhood* 13 1460 NaN NA NA NaN NA
## Condition1* 14 1460 NaN NA NA NaN NA
## Condition2* 15 1460 NaN NA NA NaN NA
## BldgType* 16 1460 NaN NA NA NaN NA
## HouseStyle* 17 1460 NaN NA NA NaN NA
## OverallQual 18 1460 6.10 1.38 6.0 6.08 1.48
## OverallCond 19 1460 5.58 1.11 5.0 5.48 0.00
## YearBuilt 20 1460 1971.27 30.20 1973.0 1974.13 37.06
## YearRemodAdd 21 1460 1984.87 20.65 1994.0 1986.37 19.27
## RoofStyle* 22 1460 NaN NA NA NaN NA
## RoofMatl* 23 1460 NaN NA NA NaN NA
## Exterior1st* 24 1460 NaN NA NA NaN NA
## Exterior2nd* 25 1460 NaN NA NA NaN NA
## MasVnrType* 26 1452 NaN NA NA NaN NA
## MasVnrArea 27 1452 103.69 181.07 0.0 63.15 0.00
## ExterQual* 28 1460 NaN NA NA NaN NA
## ExterCond* 29 1460 NaN NA NA NaN NA
## Foundation* 30 1460 NaN NA NA NaN NA
## BsmtQual* 31 1423 NaN NA NA NaN NA
## BsmtCond* 32 1423 NaN NA NA NaN NA
## BsmtExposure* 33 1422 NaN NA NA NaN NA
## BsmtFinType1* 34 1423 NaN NA NA NaN NA
## BsmtFinSF1 35 1460 443.64 456.10 383.5 386.08 568.58
## BsmtFinType2* 36 1422 NaN NA NA NaN NA
## BsmtFinSF2 37 1460 46.55 161.32 0.0 1.38 0.00
## BsmtUnfSF 38 1460 567.24 441.87 477.5 519.29 426.99
## TotalBsmtSF 39 1460 1057.43 438.71 991.5 1036.70 347.67
## Heating* 40 1460 NaN NA NA NaN NA
## HeatingQC* 41 1460 NaN NA NA NaN NA
## CentralAir* 42 1460 NaN NA NA NaN NA
## Electrical* 43 1459 NaN NA NA NaN NA
## X1stFlrSF 44 1460 1162.63 386.59 1087.0 1129.99 347.67
## X2ndFlrSF 45 1460 346.99 436.53 0.0 285.36 0.00
## LowQualFinSF 46 1460 5.84 48.62 0.0 0.00 0.00
## GrLivArea 47 1460 1515.46 525.48 1464.0 1467.67 483.33
## BsmtFullBath 48 1460 0.43 0.52 0.0 0.39 0.00
## BsmtHalfBath 49 1460 0.06 0.24 0.0 0.00 0.00
## FullBath 50 1460 1.57 0.55 2.0 1.56 0.00
## HalfBath 51 1460 0.38 0.50 0.0 0.34 0.00
## BedroomAbvGr 52 1460 2.87 0.82 3.0 2.85 0.00
## KitchenAbvGr 53 1460 1.05 0.22 1.0 1.00 0.00
## KitchenQual* 54 1460 NaN NA NA NaN NA
## TotRmsAbvGrd 55 1460 6.52 1.63 6.0 6.41 1.48
## Functional* 56 1460 NaN NA NA NaN NA
## Fireplaces 57 1460 0.61 0.64 1.0 0.53 1.48
## FireplaceQu* 58 770 NaN NA NA NaN NA
## GarageType* 59 1379 NaN NA NA NaN NA
## GarageYrBlt 60 1379 1978.51 24.69 1980.0 1981.07 31.13
## GarageFinish* 61 1379 NaN NA NA NaN NA
## GarageCars 62 1460 1.77 0.75 2.0 1.77 0.00
## GarageArea 63 1460 472.98 213.80 480.0 469.81 177.91
## GarageQual* 64 1379 NaN NA NA NaN NA
## GarageCond* 65 1379 NaN NA NA NaN NA
## PavedDrive* 66 1460 NaN NA NA NaN NA
## WoodDeckSF 67 1460 94.24 125.34 0.0 71.76 0.00
## OpenPorchSF 68 1460 46.66 66.26 25.0 33.23 37.06
## EnclosedPorch 69 1460 21.95 61.12 0.0 3.87 0.00
## X3SsnPorch 70 1460 3.41 29.32 0.0 0.00 0.00
## ScreenPorch 71 1460 15.06 55.76 0.0 0.00 0.00
## PoolArea 72 1460 2.76 40.18 0.0 0.00 0.00
## PoolQC* 73 7 NaN NA NA NaN NA
## Fence* 74 281 NaN NA NA NaN NA
## MiscFeature* 75 54 NaN NA NA NaN NA
## MiscVal 76 1460 43.49 496.12 0.0 0.00 0.00
## MoSold 77 1460 6.32 2.70 6.0 6.25 2.97
## YrSold 78 1460 2007.82 1.33 2008.0 2007.77 1.48
## SaleType* 79 1460 NaN NA NA NaN NA
## SaleCondition* 80 1460 NaN NA NA NaN NA
## SalePrice 81 1460 180921.20 79442.50 163000.0 170783.29 56338.80
## min max range skew kurtosis se
## Id 1 1460 1459 0.00 -1.20 11.03
## MSSubClass 20 190 170 1.40 1.56 1.11
## MSZoning* Inf -Inf -Inf NA NA NA
## LotFrontage 21 313 292 2.16 17.34 0.70
## LotArea 1300 215245 213945 12.18 202.26 261.22
## Street* Inf -Inf -Inf NA NA NA
## Alley* Inf -Inf -Inf NA NA NA
## LotShape* Inf -Inf -Inf NA NA NA
## LandContour* Inf -Inf -Inf NA NA NA
## Utilities* Inf -Inf -Inf NA NA NA
## LotConfig* Inf -Inf -Inf NA NA NA
## LandSlope* Inf -Inf -Inf NA NA NA
## Neighborhood* Inf -Inf -Inf NA NA NA
## Condition1* Inf -Inf -Inf NA NA NA
## Condition2* Inf -Inf -Inf NA NA NA
## BldgType* Inf -Inf -Inf NA NA NA
## HouseStyle* Inf -Inf -Inf NA NA NA
## OverallQual 1 10 9 0.22 0.09 0.04
## OverallCond 1 9 8 0.69 1.09 0.03
## YearBuilt 1872 2010 138 -0.61 -0.45 0.79
## YearRemodAdd 1950 2010 60 -0.50 -1.27 0.54
## RoofStyle* Inf -Inf -Inf NA NA NA
## RoofMatl* Inf -Inf -Inf NA NA NA
## Exterior1st* Inf -Inf -Inf NA NA NA
## Exterior2nd* Inf -Inf -Inf NA NA NA
## MasVnrType* Inf -Inf -Inf NA NA NA
## MasVnrArea 0 1600 1600 2.66 10.03 4.75
## ExterQual* Inf -Inf -Inf NA NA NA
## ExterCond* Inf -Inf -Inf NA NA NA
## Foundation* Inf -Inf -Inf NA NA NA
## BsmtQual* Inf -Inf -Inf NA NA NA
## BsmtCond* Inf -Inf -Inf NA NA NA
## BsmtExposure* Inf -Inf -Inf NA NA NA
## BsmtFinType1* Inf -Inf -Inf NA NA NA
## BsmtFinSF1 0 5644 5644 1.68 11.06 11.94
## BsmtFinType2* Inf -Inf -Inf NA NA NA
## BsmtFinSF2 0 1474 1474 4.25 20.01 4.22
## BsmtUnfSF 0 2336 2336 0.92 0.46 11.56
## TotalBsmtSF 0 6110 6110 1.52 13.18 11.48
## Heating* Inf -Inf -Inf NA NA NA
## HeatingQC* Inf -Inf -Inf NA NA NA
## CentralAir* Inf -Inf -Inf NA NA NA
## Electrical* Inf -Inf -Inf NA NA NA
## X1stFlrSF 334 4692 4358 1.37 5.71 10.12
## X2ndFlrSF 0 2065 2065 0.81 -0.56 11.42
## LowQualFinSF 0 572 572 8.99 82.83 1.27
## GrLivArea 334 5642 5308 1.36 4.86 13.75
## BsmtFullBath 0 3 3 0.59 -0.84 0.01
## BsmtHalfBath 0 2 2 4.09 16.31 0.01
## FullBath 0 3 3 0.04 -0.86 0.01
## HalfBath 0 2 2 0.67 -1.08 0.01
## BedroomAbvGr 0 8 8 0.21 2.21 0.02
## KitchenAbvGr 0 3 3 4.48 21.42 0.01
## KitchenQual* Inf -Inf -Inf NA NA NA
## TotRmsAbvGrd 2 14 12 0.67 0.87 0.04
## Functional* Inf -Inf -Inf NA NA NA
## Fireplaces 0 3 3 0.65 -0.22 0.02
## FireplaceQu* Inf -Inf -Inf NA NA NA
## GarageType* Inf -Inf -Inf NA NA NA
## GarageYrBlt 1900 2010 110 -0.65 -0.42 0.66
## GarageFinish* Inf -Inf -Inf NA NA NA
## GarageCars 0 4 4 -0.34 0.21 0.02
## GarageArea 0 1418 1418 0.18 0.90 5.60
## GarageQual* Inf -Inf -Inf NA NA NA
## GarageCond* Inf -Inf -Inf NA NA NA
## PavedDrive* Inf -Inf -Inf NA NA NA
## WoodDeckSF 0 857 857 1.54 2.97 3.28
## OpenPorchSF 0 547 547 2.36 8.44 1.73
## EnclosedPorch 0 552 552 3.08 10.37 1.60
## X3SsnPorch 0 508 508 10.28 123.06 0.77
## ScreenPorch 0 480 480 4.11 18.34 1.46
## PoolArea 0 738 738 14.80 222.19 1.05
## PoolQC* Inf -Inf -Inf NA NA NA
## Fence* Inf -Inf -Inf NA NA NA
## MiscFeature* Inf -Inf -Inf NA NA NA
## MiscVal 0 15500 15500 24.43 697.64 12.98
## MoSold 1 12 11 0.21 -0.41 0.07
## YrSold 2006 2010 4 0.10 -1.19 0.03
## SaleType* Inf -Inf -Inf NA NA NA
## SaleCondition* Inf -Inf -Inf NA NA NA
## SalePrice 34900 755000 720100 1.88 6.50 2079.11ggplot(Traindata, aes(x=SalePrice))+
geom_bar(stat="bin", fill="steelblue")+
theme_minimal()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot(Traindata, aes(x=log(SalePrice)))+
geom_bar(stat="bin", fill="steelblue")+
theme_minimal()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
housevars <- cbind(Traindata$LotArea, Traindata$TotRmsAbvGrd, Traindata$OverallQual, Traindata$X1stFlrSF, Traindata$SalePrice)
summary(housevars)## V1 V2 V3 V4
## Min. : 1300 Min. : 2.000 Min. : 1.000 Min. : 334
## 1st Qu.: 7554 1st Qu.: 5.000 1st Qu.: 5.000 1st Qu.: 882
## Median : 9478 Median : 6.000 Median : 6.000 Median :1087
## Mean : 10517 Mean : 6.518 Mean : 6.099 Mean :1163
## 3rd Qu.: 11602 3rd Qu.: 7.000 3rd Qu.: 7.000 3rd Qu.:1391
## Max. :215245 Max. :14.000 Max. :10.000 Max. :4692
## V5
## Min. : 34900
## 1st Qu.:129975
## Median :163000
## Mean :180921
## 3rd Qu.:214000
## Max. :755000colnames(housevars) = c("LotArea", "TotRmsAbvGrd", "OverallQual", "X1stFlrSF" , "SalePrice")
plot(Traindata$TotRmsAbvGrd,Traindata$LotArea,main='Scatterplot', xlab="Total Rooms Above Ground", ylab="Lot Area", col=2)
abline(lm(Traindata$TotRmsAbvGrd~Traindata$LotArea),col=4)
plot(Traindata$TotRmsAbvGrd,Traindata$SalePrice, main='Scatterplot', xlab="Total Rooms Above Ground", ylab="Sale Price", col=6)
abline(lm(Traindata$TotRmsAbvGrd~Traindata$SalePrice),col=3)
plot(Traindata$LotArea,Traindata$SalePrice, main='Scatterplot', xlab="Lot Area", ylab="Sale Price", col=5, xlim=c(0,20000))
abline(lm(Traindata$LotArea~Traindata$SalePrice),col=1)
plot(Traindata$OverallQual,Traindata$SalePrice, main='Scatterplot', xlab="Overall Quality", ylab="Sale Price", col=3)
abline(lm(Traindata$OverallQual~Traindata$SalePrice),col=1)
cor(housevars)## LotArea TotRmsAbvGrd OverallQual X1stFlrSF SalePrice
## LotArea 1.0000000 0.1900148 0.1058057 0.2994746 0.2638434
## TotRmsAbvGrd 0.1900148 1.0000000 0.4274523 0.4095160 0.5337232
## OverallQual 0.1058057 0.4274523 1.0000000 0.4762238 0.7909816
## X1stFlrSF 0.2994746 0.4095160 0.4762238 1.0000000 0.6058522
## SalePrice 0.2638434 0.5337232 0.7909816 0.6058522 1.0000000cor.test(Traindata$LotArea, Traindata$SalePrice, conf.level = 0.8)##
## Pearson's product-moment correlation
##
## data: Traindata$LotArea and Traindata$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(Traindata$TotRmsAbvGrd, Traindata$SalePrice, conf.level = 0.8)##
## Pearson's product-moment correlation
##
## data: Traindata$TotRmsAbvGrd and Traindata$SalePrice
## t = 24.099, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.5092841 0.5573021
## sample estimates:
## cor
## 0.5337232cor.test(Traindata$OverallQual, Traindata$SalePrice, conf.level = 0.8)##
## Pearson's product-moment correlation
##
## data: Traindata$OverallQual and Traindata$SalePrice
## t = 49.364, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.7780752 0.8032204
## sample estimates:
## cor
## 0.7909816Would you be worried about familywise error? Why or why not?
Yes, I will be. To be more confident that we did not reject the null hypothesis erroneously (type I error), I will restrict my range to a confidence level of 90 or even 95%.
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(Matrix)
chosen_vars <- cbind(Traindata$TotRmsAbvGrd, Traindata$OverallQual, Traindata$X1stFlrSF)
corr_mat = cor(chosen_vars)
corr_mat## [,1] [,2] [,3]
## [1,] 1.0000000 0.4274523 0.4095160
## [2,] 0.4274523 1.0000000 0.4762238
## [3,] 0.4095160 0.4762238 1.0000000det(corr_mat)## [1] 0.5895166precision_mat = solve(corr_mat)
precision_mat## [,1] [,2] [,3]
## [1,] 1.3116015 -0.3942740 -0.3493591
## [2,] -0.3942740 1.4118290 -0.5108851
## [3,] -0.3493591 -0.5108851 1.3863638id_mat = corr_mat %*% precision_mat
id_mat## [,1] [,2] [,3]
## [1,] 1 2.775558e-17 0
## [2,] 0 1.000000e+00 0
## [3,] 0 -1.110223e-16 1id2_mat = precision_mat %*% corr_mat
id2_mat## [,1] [,2] [,3]
## [1,] 1.000000e+00 1.110223e-16 0.000000e+00
## [2,] -8.326673e-17 1.000000e+00 -1.110223e-16
## [3,] 0.000000e+00 0.000000e+00 1.000000e+00ex <- expand(lu(t(corr_mat)))
L <- ex$L
P <- ex$P
C <- U <- ex$U
C[lower.tri(U)] <- L[lower.tri(L)]
print(C)## 3 x 3 Matrix of class "dgeMatrix"
## [,1] [,2] [,3]
## [1,] 1.0000000 0.4274523 0.4095160
## [2,] 0.4274523 0.8172845 0.3011753
## [3,] 0.4095160 0.3685073 0.7213114Calculus-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 \(\lambda\) for this distribution, and then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, \(\lambda\)). 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.
require(MASS)## Loading required package: MASShist(Traindata$SalePrice/100000, main="histogram",xlab="Sales Price (in $100,000)",freq=FALSE)
boxplot(Traindata$SalePrice/100000, col='blue',main="Sale Price")
fitdistr(Traindata$SalePrice/100000,"exponential")## rate
## 0.55272684
## (0.01446552)# data generation, $\lambda$ = 0.55
ex <- rexp(1000, rate = 0.55) # generate some exponential distribution
hist(ex, main="histogram",xlab="Sales Price (in $100,000)",freq=FALSE)
fiftile = pexp(0.094, rate = 0.55, log=FALSE)
fiftile## [1] 0.05038629#5th percentile is 0.094 ($9,400 houses)
Ninefiftile = pexp(5.45, rate = 0.55, log=FALSE)
Ninefiftile## [1] 0.9500883#95th percentile is 5.45 ($545,000 houses)
n = 1460
me <- qnorm(0.975)*(sd(Traindata$SalePrice/100000)/sqrt(n))
mean(Traindata$SalePrice/100000) - me## [1] 1.768462mean(Traindata$SalePrice/100000) + me## [1] 1.849962emp_fif = quantile(Traindata$SalePrice/100000,0.05)
emp_fif## 5%
## 0.88emp_Ninefif = quantile(Traindata$SalePrice/100000,0.95)
emp_Ninefif## 95%
## 3.261Modeling. 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.
#To buillt a simplified and more realistic model, pick only dependendent variables that have statistical significance of p>0.05
#attach(Traindata)
train_subset <- cbind(Traindata$MSSubClass, Traindata$LotArea, Traindata$OverallQual, Traindata$OverallCond, Traindata$YearBuilt, Traindata$MasVnrArea, Traindata$BsmtFinSF1, Traindata$X1stFlrSF, Traindata$X2ndFlrSF, Traindata$BedroomAbvGr, Traindata$GarageYrBlt)
head(train_subset)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
## [1,] 60 8450 7 5 2003 196 706 856 854 3 2003
## [2,] 20 9600 6 8 1976 0 978 1262 0 3 1976
## [3,] 60 11250 7 5 2001 162 486 920 866 3 2001
## [4,] 70 9550 7 5 1915 0 216 961 756 3 1998
## [5,] 60 14260 8 5 2000 350 655 1145 1053 4 2000
## [6,] 50 14115 5 5 1993 0 732 796 566 1 1993train_model <- lm(Traindata$SalePrice ~ Traindata$MSSubClass + Traindata$LotArea + Traindata$OverallQual + Traindata$OverallCond + Traindata$YearBuilt + Traindata$MasVnrArea + Traindata$BsmtFinSF1 + Traindata$X1stFlrSF + Traindata$X2ndFlrSF + Traindata$BedroomAbvGr + Traindata$GarageYrBlt)
summary(train_model)##
## Call:
## lm(formula = Traindata$SalePrice ~ Traindata$MSSubClass + Traindata$LotArea +
## Traindata$OverallQual + Traindata$OverallCond + Traindata$YearBuilt +
## Traindata$MasVnrArea + Traindata$BsmtFinSF1 + Traindata$X1stFlrSF +
## Traindata$X2ndFlrSF + Traindata$BedroomAbvGr + Traindata$GarageYrBlt)
##
## Residuals:
## Min 1Q Median 3Q Max
## -549022 -16701 -2070 13787 254449
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.174e+06 1.036e+05 -11.332 < 2e-16 ***
## Traindata$MSSubClass -2.128e+02 2.667e+01 -7.978 3.12e-15 ***
## Traindata$LotArea 5.480e-01 1.030e-01 5.322 1.20e-07 ***
## Traindata$OverallQual 2.217e+04 1.178e+03 18.817 < 2e-16 ***
## Traindata$OverallCond 5.851e+03 1.019e+03 5.740 1.17e-08 ***
## Traindata$YearBuilt 3.591e+02 6.609e+01 5.433 6.56e-08 ***
## Traindata$MasVnrArea 3.100e+01 6.103e+00 5.080 4.30e-07 ***
## Traindata$BsmtFinSF1 1.672e+01 2.508e+00 6.667 3.77e-11 ***
## Traindata$X1stFlrSF 7.152e+01 3.963e+00 18.049 < 2e-16 ***
## Traindata$X2ndFlrSF 6.236e+01 3.473e+00 17.953 < 2e-16 ***
## Traindata$BedroomAbvGr -7.713e+03 1.631e+03 -4.729 2.49e-06 ***
## Traindata$GarageYrBlt 1.983e+02 7.194e+01 2.756 0.00594 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 36310 on 1359 degrees of freedom
## (89 observations deleted due to missingness)
## Multiple R-squared: 0.7898, Adjusted R-squared: 0.7881
## F-statistic: 464.2 on 11 and 1359 DF, p-value: < 2.2e-16#Load the test dataset
Testdata <- read.csv(file="C:/Users/ajb2/Documents/test.csv", header=TRUE, sep=",", stringsAsFactors = FALSE)
test_subset <- data.frame(Testdata$MSSubClass, Testdata$LotArea, Testdata$OverallQual, Testdata$OverallCond, Testdata$YearBuilt, Testdata$MasVnrArea, Testdata$BsmtFinSF1, Testdata$X1stFlrSF, Testdata$X2ndFlrSF, Testdata$BedroomAbvGr, Testdata$GarageYrBlt)
head(test_subset)## Testdata.MSSubClass Testdata.LotArea Testdata.OverallQual
## 1 20 11622 5
## 2 20 14267 6
## 3 60 13830 5
## 4 60 9978 6
## 5 120 5005 8
## 6 60 10000 6
## Testdata.OverallCond Testdata.YearBuilt Testdata.MasVnrArea
## 1 6 1961 0
## 2 6 1958 108
## 3 5 1997 0
## 4 6 1998 20
## 5 5 1992 0
## 6 5 1993 0
## Testdata.BsmtFinSF1 Testdata.X1stFlrSF Testdata.X2ndFlrSF
## 1 468 896 0
## 2 923 1329 0
## 3 791 928 701
## 4 602 926 678
## 5 263 1280 0
## 6 0 763 892
## Testdata.BedroomAbvGr Testdata.GarageYrBlt
## 1 2 1961
## 2 3 1958
## 3 3 1997
## 4 3 1998
## 5 2 1992
## 6 3 1993#predict home prices result based on multiple regession model built
test_results = predict(train_model, newdata=test_subset, type="response")## Warning: 'newdata' had 1459 rows but variables found have 1460 rowssummary(test_results)## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## -1710 133947 177903 185182 225864 709022 89#compare test dataset sale price prediction by model to that generated in the training dataset
hist(test_results, col='green', main="Predicted Test Data Sale Price")
hist(Traindata$SalePrice, col='green', main="Training Data Sale Price")
boxplot(test_results, col='pink',main="Predicted Test Data Sale Price")
boxplot(Traindata$SalePrice, col='pink',main="Training Sale Price")
#write.csv(test_results, file = "test_results.csv")