Math 605: Final Assignment
David Apolinar
12/8/2019
Row
Problem 1
Using R, generate a random variable X that has 10,000 random uniform numbers from 1 to N, where N can be any number of your choosing greater than or equal to 6. Then generate a random variable Y that has 10,000 random normal numbers with a mean of \(\mu =\sigma =(N+1)/2\)
set.seed(223)
MAX=8
X=runif(10000,min=1,max=MAX)
x <- median(X)
Y=rnorm(10000,mean=(MAX+1)/2,sd=(MAX+1)/2)
y <- quantile(Y,.25)
# median value for x
x## [1] 4.506875# first quantile for y
y## 25%
## 1.412316all.data <- data.frame(cbind(X,Y))
head(all.data)## X Y
## 1 6.770470 7.509567
## 2 3.284944 10.529317
## 3 2.057999 10.055423
## 4 4.522908 6.825093
## 5 4.223694 -2.546282
## 6 7.249178 7.524893summary(all.data)## X Y
## Min. :1.001 Min. :-13.978
## 1st Qu.:2.767 1st Qu.: 1.412
## Median :4.507 Median : 4.487
## Mean :4.497 Mean : 4.480
## 3rd Qu.:6.236 3rd Qu.: 7.535
## Max. :7.998 Max. : 22.674hist(X)
hist(Y)
Probability. Calculate as a minimum the below probabilities a through c. Assume the small letter “x” is estimated as the median 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 | X>y) b. P(X>x, Y>y) c. P(X<x | X>y)
Reading this, X should be 50% since it is mostly equally distributed according to the histo graph above.
- P(X>x | X>y)
P(A|B) = P(A and B)/P(B) P(X>x|Y>y) = P(X>x and X>y)/P(X>y)
first quantile for y 4.824869 median for x 5.544612
X.greater.x.and.X.greater.y <- nrow(subset(all.data, all.data$X > x & all.data$X > y))/nrow(all.data)
X.greater.x.and.X.greater.y## [1] 0.5X.greater.y <- nrow(subset(all.data, all.data$X > y))/nrow(all.data)
X.greater.y## [1] 0.9416# This is the probability that X is greater than the median of 5.55 and X is greater than 4.82, given that X is greater than 4.82
X.greater.x.and.X.greater.y/X.greater.y## [1] 0.531011- P(X>x, Y>y)
nrow(subset(all.data, all.data$X > x & all.data$Y > y))/nrow(all.data)## [1] 0.3762- P(X<x | X>y)
P (X is less than 5.55 given that X is greater than 4.83)
#nrow(subset(all.data, all.data$X < x & all.data$X > y) )/nrow(all.data)
data.X.greater.Y <- subset(all.data,all.data$X > y)
nrow(data.X.greater.Y)## [1] 9416# number of items where X > y
nrow(subset(data.X.greater.Y,data.X.greater.Y$X < x))## [1] 4416# of these X greater than y, how many are less than the median, x
nrow(subset(data.X.greater.Y,X < x & X>y))/nrow(all.data)## [1] 0.44165 points. Investigate whether P(X>x and Y>y)=P(X>x)P(Y>y) by building a table and evaluating the marginal and joint probabilities.
X.greater.x <- subset(all.data, X > x)
Y.greater.y <- subset(all.data, Y > y)
X.less.x <- subset(all.data, X < x)
Y.less.y <- subset(all.data, Y < y)
df <- matrix(c(nrow(X.greater.x), nrow(X.less.x),sum(nrow(X.greater.x), nrow(X.less.x)), nrow(Y.greater.y ), nrow(Y.less.y),sum(nrow(Y.greater.y ), nrow(Y.less.y)),sum(nrow(X.greater.x),nrow(Y.greater.y )),sum(nrow(X.less.x),nrow(Y.less.y)),20000), 3, 3,
byrow = TRUE )
colnames(df) <- c("X>x", "X<x", "total")
row.names(df) <- c("Y<y", "Y>y", "total")
df/20000## X>x X<x total
## Y<y 0.250 0.250 0.5
## Y>y 0.375 0.125 0.5
## total 0.625 0.375 1.0# P(X>x and Y>y)
P.Xgreaterx.and.Ygreatery <- nrow(subset(all.data, X> x & Y>y))/10000
#P(X>x)
P.Xgreater.x <- nrow(X.greater.x)/10000
#P(Y>y)
P.Ygreater.y <- nrow(Y.greater.y)/10000
round(P.Xgreater.x * P.Ygreater.y,3)## [1] 0.375round(P.Xgreaterx.and.Ygreatery,3)## [1] 0.376# the numbers are pretty similar, if we round them to 2 decimals, they are equal
# P(X>x and Y>y)=P(X>x)P(Y>y)
round(P.Xgreater.x * P.Ygreater.y,2)## [1] 0.38round(P.Xgreaterx.and.Ygreatery,2)## [1] 0.38round(P.Xgreater.x * P.Ygreater.y,2)==round(P.Xgreaterx.and.Ygreatery,2)## [1] TRUE5 points. Check to see if independence holds by using Fisher’s Exact Test and the Chi Square Test. What is the difference between the two? Which is most appropriate?
Chisquare is used when the data is larger. If I try to se the fisher test, I will get the following error:
Error in fisher.test(df) : FEXACT error 501. The hash table key cannot be computed because the largest key is larger than the largest representable int. The algorithm cannot proceed. Reduce the workspace, consider using ‘simulate.p.value=TRUE’ or another algorithm.
Therefore, the chi-sqaure test makes sense in this scenario due to the table size.
chisq.test(df)##
## Pearson's Chi-squared test
##
## data: df
## X-squared = 1333.3, df = 4, p-value < 2.2e-16fisher.test(df,simulate.p.value=TRUE)##
## Fisher's Exact Test for Count Data with simulated p-value (based
## on 2000 replicates)
##
## data: df
## p-value = 0.0004998
## alternative hypothesis: two.sidedProblem 2
Data Load
# Data Load
train_file = "https://raw.githubusercontent.com/dapolloxp/R-Projects/master/train.csv"
test_file = "https://raw.githubusercontent.com/dapolloxp/R-Projects/master/test.csv"
pullCSVFile <- function (url)
{
if(url.exists(url))
{
expr <- tryCatch(
{
message(paste("CSV File ", url, " exists"))
message("Downloading CSV File from github...\nThis may take a while....")
out_file <- getURL(url)
csv_file <- read.csv(text = out_file)
},
error = function(err)
{
print(paste("Couldn't download file: ", err))
},
finally = {
return(csv_file)
}
)
}
else
{
print(paste("Could not location CSV file at: ", url))
break()
}
}
train_set <- pullCSVFile(train_file)## CSV File https://raw.githubusercontent.com/dapolloxp/R-Projects/master/train.csv exists## Downloading CSV File from github...
## This may take a while....test_set <- pullCSVFile(test_file)## CSV File https://raw.githubusercontent.com/dapolloxp/R-Projects/master/test.csv exists
## Downloading CSV File from github...
## This may take a while....head(train_set)## 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 143000head(test_set)## Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape
## 1 1461 20 RH 80 11622 Pave <NA> Reg
## 2 1462 20 RL 81 14267 Pave <NA> IR1
## 3 1463 60 RL 74 13830 Pave <NA> IR1
## 4 1464 60 RL 78 9978 Pave <NA> IR1
## 5 1465 120 RL 43 5005 Pave <NA> IR1
## 6 1466 60 RL 75 10000 Pave <NA> IR1
## LandContour Utilities LotConfig LandSlope Neighborhood Condition1
## 1 Lvl AllPub Inside Gtl NAmes Feedr
## 2 Lvl AllPub Corner Gtl NAmes Norm
## 3 Lvl AllPub Inside Gtl Gilbert Norm
## 4 Lvl AllPub Inside Gtl Gilbert Norm
## 5 HLS AllPub Inside Gtl StoneBr Norm
## 6 Lvl AllPub Corner Gtl Gilbert Norm
## Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt
## 1 Norm 1Fam 1Story 5 6 1961
## 2 Norm 1Fam 1Story 6 6 1958
## 3 Norm 1Fam 2Story 5 5 1997
## 4 Norm 1Fam 2Story 6 6 1998
## 5 Norm TwnhsE 1Story 8 5 1992
## 6 Norm 1Fam 2Story 6 5 1993
## YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType
## 1 1961 Gable CompShg VinylSd VinylSd None
## 2 1958 Hip CompShg Wd Sdng Wd Sdng BrkFace
## 3 1998 Gable CompShg VinylSd VinylSd None
## 4 1998 Gable CompShg VinylSd VinylSd BrkFace
## 5 1992 Gable CompShg HdBoard HdBoard None
## 6 1994 Gable CompShg HdBoard HdBoard None
## MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure
## 1 0 TA TA CBlock TA TA No
## 2 108 TA TA CBlock TA TA No
## 3 0 TA TA PConc Gd TA No
## 4 20 TA TA PConc TA TA No
## 5 0 Gd TA PConc Gd TA No
## 6 0 TA TA PConc Gd TA No
## BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF
## 1 Rec 468 LwQ 144 270 882
## 2 ALQ 923 Unf 0 406 1329
## 3 GLQ 791 Unf 0 137 928
## 4 GLQ 602 Unf 0 324 926
## 5 ALQ 263 Unf 0 1017 1280
## 6 Unf 0 Unf 0 763 763
## Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF
## 1 GasA TA Y SBrkr 896 0 0
## 2 GasA TA Y SBrkr 1329 0 0
## 3 GasA Gd Y SBrkr 928 701 0
## 4 GasA Ex Y SBrkr 926 678 0
## 5 GasA Ex Y SBrkr 1280 0 0
## 6 GasA Gd Y SBrkr 763 892 0
## GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr
## 1 896 0 0 1 0 2
## 2 1329 0 0 1 1 3
## 3 1629 0 0 2 1 3
## 4 1604 0 0 2 1 3
## 5 1280 0 0 2 0 2
## 6 1655 0 0 2 1 3
## KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu
## 1 1 TA 5 Typ 0 <NA>
## 2 1 Gd 6 Typ 0 <NA>
## 3 1 TA 6 Typ 1 TA
## 4 1 Gd 7 Typ 1 Gd
## 5 1 Gd 5 Typ 0 <NA>
## 6 1 TA 7 Typ 1 TA
## GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual
## 1 Attchd 1961 Unf 1 730 TA
## 2 Attchd 1958 Unf 1 312 TA
## 3 Attchd 1997 Fin 2 482 TA
## 4 Attchd 1998 Fin 2 470 TA
## 5 Attchd 1992 RFn 2 506 TA
## 6 Attchd 1993 Fin 2 440 TA
## GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch
## 1 TA Y 140 0 0 0
## 2 TA Y 393 36 0 0
## 3 TA Y 212 34 0 0
## 4 TA Y 360 36 0 0
## 5 TA Y 0 82 0 0
## 6 TA Y 157 84 0 0
## ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold
## 1 120 0 <NA> MnPrv <NA> 0 6 2010
## 2 0 0 <NA> <NA> Gar2 12500 6 2010
## 3 0 0 <NA> MnPrv <NA> 0 3 2010
## 4 0 0 <NA> <NA> <NA> 0 6 2010
## 5 144 0 <NA> <NA> <NA> 0 1 2010
## 6 0 0 <NA> <NA> <NA> 0 4 2010
## SaleType SaleCondition
## 1 WD Normal
## 2 WD Normal
## 3 WD Normal
## 4 WD Normal
## 5 WD Normal
## 6 WD Normal5 points. 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 an 80% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not?
For this problem, I choose the following variables:
- Total Basement Area
- Total Garage Area
- Greater Living Area
- SalePrice
Below is the summary for the entire set of variables in the training set. I have choosen to do a scatter for the following variables:
summary(train_set)## 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#plot(train_set$GrLivArea, train_set$SalePrice, main="Square Footage vs. Price", xlab="Square Footage", ylab="Sales Price") + Living Areas vs Sales Price
ggplot(train_set, aes(x=GrLivArea,y=SalePrice)) + geom_point(color="blue") +ylim(0,800000) + geom_smooth(method=lm,color="darkred")
ggplot(train_set, aes(x=TotalBsmtSF)) + geom_histogram(color="blue", bins = 100)
Total Basement Square Footage vs Sales Price
ggplot(train_set, aes(x=TotalBsmtSF,y=SalePrice)) + geom_point(color="red") +ylim(0,800000) + geom_smooth(method=lm)
ggplot(train_set, aes(x=TotalBsmtSF)) + geom_histogram(color="red", bins = 100)
Total Garage Area vs Sales Price
ggplot(train_set, aes(x=GarageArea,y=SalePrice)) + geom_point(color="green") +ylim(0,800000) + geom_smooth(method=lm)
ggplot(train_set, aes(x=GarageArea)) + geom_histogram(color="green", bins = 100)
## Correlation Data and Plot
correlation_set <- train_set %>% select("TotalBsmtSF", "GarageArea","GrLivArea", "SalePrice")
cor.value <- cor(correlation_set)
corrplot.mixed(cor.value )
Model 1 - Comparing Square Area vs Sales Price
model1 <- lm(SalePrice ~ GrLivArea, train_set)
summary(model1)##
## Call:
## lm(formula = SalePrice ~ GrLivArea, data = train_set)
##
## Residuals:
## Min 1Q Median 3Q Max
## -462999 -29800 -1124 21957 339832
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18569.026 4480.755 4.144 3.61e-05 ***
## GrLivArea 107.130 2.794 38.348 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 56070 on 1458 degrees of freedom
## Multiple R-squared: 0.5021, Adjusted R-squared: 0.5018
## F-statistic: 1471 on 1 and 1458 DF, p-value: < 2.2e-16Model 2 - Comparing Total Basement Square Area vs Sales Price
model2 <- lm(SalePrice ~ TotalBsmtSF, train_set)
summary(model2)##
## Call:
## lm(formula = SalePrice ~ TotalBsmtSF, data = train_set)
##
## Residuals:
## Min 1Q Median 3Q Max
## -582310 -39612 -14095 33315 420018
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 63430.629 4286.892 14.80 <2e-16 ***
## TotalBsmtSF 111.110 3.745 29.67 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 62750 on 1458 degrees of freedom
## Multiple R-squared: 0.3765, Adjusted R-squared: 0.3761
## F-statistic: 880.3 on 1 and 1458 DF, p-value: < 2.2e-16Model 3 - Comparing Total Garage Area vs Sales Price
model3 <- lm(SalePrice ~ GarageArea, train_set)
summary(model3)##
## Call:
## lm(formula = SalePrice ~ GarageArea, data = train_set)
##
## Residuals:
## Min 1Q Median 3Q Max
## -279451 -33024 -5045 24479 490913
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 71357.421 3949.003 18.07 <2e-16 ***
## GarageArea 231.646 7.608 30.45 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 62140 on 1458 degrees of freedom
## Multiple R-squared: 0.3887, Adjusted R-squared: 0.3882
## F-statistic: 927 on 1 and 1458 DF, p-value: < 2.2e-16Hypothesis Testing
Square Area vs Sales Price
The correlation between garage square footage and the sales price is 71%. This value alone provides the best correlation between the quantitive variables and the target variable. true correlation is not equal to 0
cor.test(formula= ~GrLivArea +SalePrice, conf.level = .80, data=train_set)##
## Pearson's product-moment correlation
##
## data: GrLivArea and 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
## 0.7086245Total Basement SF vs Sales Price
The correlation between garage square footage and the sales price is only 61%, true correlation is not equal to 0
cor.test(formula= ~TotalBsmtSF +SalePrice, conf.level = .80, data=train_set)##
## Pearson's product-moment correlation
##
## data: TotalBsmtSF and SalePrice
## t = 29.671, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.5922142 0.6340846
## sample estimates:
## cor
## 0.6135806Garage SF vs Sales Price
The correlation between garage square footage and the sales price is only 62%, the true correlation is not equal to 0.
cor.test(formula= ~GarageArea +SalePrice, conf.level = .80, data=train_set)##
## Pearson's product-moment correlation
##
## data: GarageArea and SalePrice
## t = 30.446, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.6024756 0.6435283
## sample estimates:
## cor
## 0.6234314Analysis and familywise errors
Due to the low correlation values for each of the selected variables, I would not be concerned with familywise errors (making Type 1 errors).
Linear Algebra
5 points. Linear Algebra and Correlation. Invert your 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.
Inverting the matrix using the R built-in solve function
# Inverting the matrix using the R built-in solve function
precession.matrix <- solve(cor.value)
precession.matrix## TotalBsmtSF GarageArea GrLivArea SalePrice
## TotalBsmtSF 1.65207569 -0.27914378 -0.05133909 -0.8032744
## GarageArea -0.27914378 1.68692624 -0.08097502 -0.8230248
## GrLivArea -0.05133909 -0.08097502 2.01512843 -1.3459863
## SalePrice -0.80327436 -0.82302478 -1.34598629 2.9597719Multiply the correlation matrix by the precision matrix
# Multiply the correlation matrix by the precision matrix
corr.times.precession <- cor.value %*% precession.matrix
round(corr.times.precession ,4)## TotalBsmtSF GarageArea GrLivArea SalePrice
## TotalBsmtSF 1 0 0 0
## GarageArea 0 1 0 0
## GrLivArea 0 0 1 0
## SalePrice 0 0 0 1Conduct LU decomposition on the matrix.
library('matrixcalc')
LU <- lu.decomposition(cor.value)
# LU L
LU$L## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 0.0000000 0.0000000 0
## [2,] 0.4866655 1.0000000 0.0000000 0
## [3,] 0.4548682 0.3244797 1.0000000 0
## [4,] 0.6135806 0.4256308 0.4547601 1# LU U
LU$U## [,1] [,2] [,3] [,4]
## [1,] 1 0.4866655 0.4548682 0.6135806
## [2,] 0 0.7631567 0.2476288 0.3248230
## [3,] 0 0.0000000 0.7127444 0.3241277
## [4,] 0 0.0000000 0.0000000 0.3378639# Check if we get the original
cor.value## TotalBsmtSF GarageArea GrLivArea SalePrice
## TotalBsmtSF 1.0000000 0.4866655 0.4548682 0.6135806
## GarageArea 0.4866655 1.0000000 0.4689975 0.6234314
## GrLivArea 0.4548682 0.4689975 1.0000000 0.7086245
## SalePrice 0.6135806 0.6234314 0.7086245 1.0000000LU$L %*% LU$U## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 0.4866655 0.4548682 0.6135806
## [2,] 0.4866655 1.0000000 0.4689975 0.6234314
## [3,] 0.4548682 0.4689975 1.0000000 0.7086245
## [4,] 0.6135806 0.6234314 0.7086245 1.0000000round(LU$L %*% LU$U,4) == round(cor.value,4)## TotalBsmtSF GarageArea GrLivArea SalePrice
## TotalBsmtSF TRUE TRUE TRUE TRUE
## GarageArea TRUE TRUE TRUE TRUE
## GrLivArea TRUE TRUE TRUE TRUE
## SalePrice TRUE TRUE TRUE TRUE5 points. 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.
For this exercise, I decided to to choose garage area as it is slighly skewed to the right. I installed the moments library as it has a function to measure skewness. In this case, our skew is 0.18, which implies a slighlt rigth skew.
# Load the mass libraries
library('MASS')##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectlibrary('moments') # https://cran.r-project.org/web/packages/moments/moments.pdf
# check amount of skewness
moments::skewness(train_set$GarageArea)## [1] 0.1797959# Then load the MASS package and run fitdistr to fit an exponential probability density function.
lamda.rate <- fitdistr(train_set$GarageArea, densfun = "exponential")
lamda.rate$estimate## rate
## 0.002114254# then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, $\lambda $))
lamda.thousand.samples <- rexp(1000, lamda.rate$estimate)
# Plot a histogram and compare it with a histogram of your original variable. Histogram of Optimal Lamda
ggplot(as.data.frame(lamda.thousand.samples), aes(x=lamda.thousand.samples)) + geom_histogram(color="blue", bins = 100) + labs("Lamda Histogram")
Histogram of Garage data
ggplot(train_set, aes(x=GarageArea)) + geom_histogram(color="green", bins = 100) + labs("Lamda Histogram")
Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF).
I calculated the formula for CI manually as many of the libraries did not work on my version of R.
# Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF).
quantile(ecdf(lamda.thousand.samples), c(0.05,0.95))## 5% 95%
## 26.63478 1421.46595# Also generate a 95% confidence interval from the empirical data, assuming normality.
se <- sd(train_set$GarageArea) / sqrt((length(train_set$GarageArea)))
lower.bound <- mean(train_set$GarageArea) - 1.96 * se
upper.bound <- mean(train_set$GarageArea) + 1.96 * se
c(lower.bound,upper.bound)## [1] 462.0129 483.9474# Finally, provide the empirical 5th percentile and 95th percentile of the data.
quantile(train_set$GarageArea, c(0.05,0.95))## 5% 95%
## 0.0 850.110 points. Modeling. Build some type of multiple regression model and submit your model to the competition board. Provide your complete model summary and results with analysis. Report your Kaggle.com user name and score.
library(MASS)
library(dplyr)
library(tidyr)##
## Attaching package: 'tidyr'## The following object is masked from 'package:RCurl':
##
## complete# Fit the full model
full.model <- lm(SalePrice ~ CentralAir + GrLivArea + Fireplaces +GarageCars+GarageFinish + WoodDeckSF+ Heating+ CentralAir+TotalBsmtSF+OverallCond+BldgType+Utilities+LotShape+LandSlope+HouseStyle+Neighborhood,data =train_set)
# Prediction (Linear Prediction)
prediction <- predict(full.model, newdata = test_set[,-1])
prediction.df <- data.frame(Id = test_set$Id, SalePrice = prediction)
prediction.df <- prediction.df %>% mutate(SalePrice = replace_na(SalePrice,163000))
write.csv(prediction.df , file = "prediction_linear.csv", row.names = FALSE)Random Forest
YouTube Video:
https://www.youtube.com/watch?v=J1gxjLpCVt4
library(randomForest)## randomForest 4.6-14## Type rfNews() to see new features/changes/bug fixes.##
## Attaching package: 'randomForest'## The following object is masked from 'package:ggplot2':
##
## margin## The following object is masked from 'package:dplyr':
##
## combine## The following object is masked from 'package:gridExtra':
##
## combinelibrary(randomForestExplainer)
forest <- randomForest(SalePrice ~ CentralAir + GrLivArea + Fireplaces +GarageCars+GarageFinish + WoodDeckSF+ Heating+ CentralAir+TotalBsmtSF+OverallCond+BldgType+Utilities+LotShape+LandSlope+HouseStyle+Neighborhood , data = train_set[,-1], localImp = TRUE, na.action = na.roughfix)
forest##
## Call:
## randomForest(formula = SalePrice ~ CentralAir + GrLivArea + Fireplaces + GarageCars + GarageFinish + WoodDeckSF + Heating + CentralAir + TotalBsmtSF + OverallCond + BldgType + Utilities + LotShape + LandSlope + HouseStyle + Neighborhood, data = train_set[, -1], localImp = TRUE, na.action = na.roughfix)
## Type of random forest: regression
## Number of trees: 500
## No. of variables tried at each split: 5
##
## Mean of squared residuals: 903863289
## % Var explained: 85.67#forest$predicted
importance(forest)## %IncMSE IncNodePurity
## CentralAir 16.117915 5.347314e+10
## GrLivArea 50.822620 2.121055e+12
## Fireplaces 15.712075 3.189821e+11
## GarageCars 27.075063 1.455789e+12
## GarageFinish 23.539672 5.179643e+11
## WoodDeckSF 8.231356 2.220690e+11
## Heating -3.439319 7.980295e+09
## TotalBsmtSF 35.436334 1.472188e+12
## OverallCond 17.789019 1.265622e+11
## BldgType 18.990226 8.730312e+10
## Utilities 0.000000 5.583130e+08
## LotShape 6.724943 8.164638e+10
## LandSlope 3.030324 3.477353e+10
## HouseStyle 19.615356 1.799720e+11
## Neighborhood 49.131628 2.351230e+12varImpPlot(forest)
test_set$SalePrice <- 0
new_data <- rbind(train_set,test_set)
n.d <- new_data[new_data[,1] > 1460,]
prediction2.df <- predict(forest, n.d[,-1])
prediction2.df## 1461 1462 1463 1464 1465 1466 1467
## 121887.02 150420.97 184256.57 186023.14 196019.92 182497.47 174420.89
## 1468 1469 1470 1471 1472 1473 1474
## 176564.26 174338.52 127943.66 168396.87 98753.81 89960.66 150878.64
## 1475 1476 1477 1478 1479 1480 1481
## 128541.59 357953.92 245883.56 327831.42 299257.41 492561.40 380048.47
## 1482 1483 1484 1485 1486 1487 1488
## 199146.53 178420.91 162906.56 166348.87 202811.63 369071.25 239098.19
## 1489 1490 1491 1492 1493 1494 1495
## 191604.44 223105.22 185957.51 111782.80 175939.91 299490.76 292889.73
## 1496 1497 1498 1499 1500 1501 1502
## 244917.29 199322.17 150674.04 151371.87 159237.16 175334.30 151580.48
## 1503 1504 1505 1506 1507 1508 1509
## 249354.45 237142.40 241997.84 202171.79 242153.67 197944.20 178337.71
## 1510 1511 1512 1513 1514 1515 1516
## 158857.88 160057.37 167772.78 147350.52 NA 208644.13 154962.53
## 1517 1518 1519 1520 1521 1522 1523
## 147585.66 138553.70 172617.55 139872.50 148298.24 162768.96 124001.95
## 1524 1525 1526 1527 1528 1529 1530
## 124447.79 128348.65 126359.28 118391.00 132186.45 152306.19 169068.26
## 1531 1532 1533 1534 1535 1536 1537
## 144142.67 NA 141776.20 112785.01 146158.81 114721.19 88742.52
## 1538 1539 1540 1541 1542 1543 1544
## 133231.72 153475.78 NA 139829.89 145340.32 179563.57 93080.36
## 1545 1546 1547 1548 1549 1550 1551
## 108318.33 125852.80 136366.25 135254.42 127337.53 156469.03 122125.55
## 1552 1553 1554 1555 1556 1557 1558
## 155119.35 NA 121873.13 187376.49 133164.86 NA 90322.29
## 1559 1560 1561 1562 1563 1564 1565
## NA 158978.83 NA 120906.22 116411.64 165277.42 156839.89
## 1566 1567 1568 1569 1570 1571 1572
## 221226.82 96081.74 235798.96 144224.41 111735.63 131248.09 144789.17
## 1573 1574 1575 1576 1577 1578 1579
## 184075.03 110828.54 188056.52 231677.55 185964.30 159340.25 136719.32
## 1580 1581 1582 1583 1584 1585 1586
## 195836.41 139711.31 128659.93 299148.02 245175.30 139472.52 76591.07
## 1587 1588 1589 1590 1591 1592 1593
## 94635.71 147657.28 104062.72 125441.23 NA 134116.81 131211.90
## 1594 1595 1596 1597 1598 1599 1600
## NA NA 232188.53 182590.67 179612.22 147680.58 165805.85
## 1601 1602 1603 1604 1605 1606 1607
## 86608.29 99440.43 114073.01 281455.50 188517.89 168184.56 176155.65
## 1608 1609 1610 1611 1612 1613 1614
## 211828.62 185280.45 176401.07 148178.06 175246.53 165974.62 124859.48
## 1615 1616 1617 1618 1619 1620 1621
## NA NA 89788.53 124589.66 129105.03 170898.82 138412.76
## 1622 1623 1624 1625 1626 1627 1628
## 153673.13 203597.69 195252.53 112908.77 169587.89 184654.19 242328.82
## 1629 1630 1631 1632 1633 1634 1635
## 174794.24 333289.16 239562.58 250168.47 170987.52 179880.02 175807.73
## 1636 1637 1638 1639 1640 1641 1642
## 163067.51 206577.73 207776.95 195479.12 251835.45 188428.23 232813.00
## 1643 1644 1645 1646 1647 1648 1649
## 239841.92 233725.08 205420.64 161766.51 160043.63 150606.17 134893.90
## 1650 1651 1652 1653 1654 1655 1656
## 112017.46 120023.82 92483.97 98605.60 162415.91 137012.08 149451.94
## 1657 1658 1659 1660 1661 1662 1663
## 152585.07 161400.08 128854.83 144869.04 448187.10 361942.91 351744.26
## 1664 1665 1666 1667 1668 1669 1670
## 492289.22 336008.06 341445.93 346323.76 310479.56 279771.72 342613.15
## 1671 1672 1673 1674 1675 1676 1677
## 290610.02 484103.04 310913.36 274214.47 209841.73 204222.26 209715.79
## 1678 1679 1680 1681 1682 1683 1684
## 461259.05 431746.74 312120.62 227904.77 319963.74 196887.76 179171.52
## 1685 1686 1687 1688 1689 1690 1691
## 177516.98 174544.73 166348.87 183624.61 187972.19 178098.34 169935.43
## 1692 1693 1694 1695 1696 1697 1698
## 236386.15 164908.21 178787.03 170330.09 283199.34 169742.02 379943.10
## 1699 1700 1701 1702 1703 1704 1705
## 410049.12 262999.28 267826.42 255002.46 272244.42 261592.82 221662.66
## 1706 1707 1708 1709 1710 1711 1712
## 406036.40 210713.58 202813.77 249903.55 212018.41 250357.21 271381.95
## 1713 1714 1715 1716 1717 1718 1719
## 268396.17 239605.90 205443.43 174539.52 163699.52 NA 200631.63
## 1720 1721 1722 1723 1724 1725 1726
## 227535.83 154538.90 NA 153733.43 197635.08 219291.77 197459.97
## 1727 1728 1729 1730 1731 1732 1733
## 176958.63 175932.54 163982.79 167933.54 132814.62 123262.06 113290.59
## 1734 1735 1736 1737 1738 1739 1740
## 130449.71 130343.34 124009.12 364049.65 221899.51 330726.10 296565.91
## 1741 1742 1743 1744 1745 1746 1747
## 186078.05 175334.30 173016.35 268736.65 254427.10 208590.13 204357.14
## 1748 1749 1750 1751 1752 1753 1754
## 232930.74 170400.37 152675.39 235246.41 122670.13 163312.16 222039.60
## 1755 1756 1757 1758 1759 1760 1761
## 168612.82 129609.58 122560.30 164952.04 163737.79 163944.14 160734.11
## 1762 1763 1764 1765 1766 1767 1768
## 199545.74 177058.46 125659.02 152666.22 172437.74 181295.21 146874.62
## 1769 1770 1771 1772 1773 1774 1775
## 159321.72 137401.65 145976.19 149186.65 147407.49 148850.27 131973.59
## 1776 1777 1778 1779 1780 1781 1782
## 118492.84 116261.57 133726.63 126166.83 162720.12 148746.59 107858.03
## 1783 1784 1785 1786 1787 1788 1789
## 129185.26 103741.88 116481.13 195790.62 144644.95 NA 108185.33
## 1790 1791 1792 1793 1794 1795 1796
## 86739.61 203475.96 146590.71 135499.35 141981.04 139048.26 127317.08
## 1797 1798 1799 1800 1801 1802 1803
## 118920.19 133259.24 114162.46 136222.00 132799.83 148886.12 146137.24
## 1804 1805 1806 1807 1808 1809 1810
## 140442.22 135578.22 120204.49 123663.42 116280.79 NA 155966.62
## 1811 1812 1813 1814 1815 1816 1817
## NA NA 116489.40 109814.83 70448.83 97775.68 128435.33
## 1818 1819 1820 1821 1822 1823 1824
## 146176.79 123484.12 NA 128671.48 141911.01 NA 137967.84
## 1825 1826 1827 1828 1829 1830 1831
## 144353.50 112170.35 114095.03 143787.39 134139.06 139421.43 142282.17
## 1832 1833 1834 1835 1836 1837 1838
## NA 142860.17 133506.34 NA 148604.36 NA 133358.62
## 1839 1840 1841 1842 1843 1844 1845
## 110662.84 NA 136224.59 83240.91 137272.53 139737.61 150011.80
## 1846 1847 1848 1849 1850 1851 1852
## 153531.56 197345.99 NA 117891.37 120104.98 147090.95 137914.78
## 1853 1854 1855 1856 1857 1858 1859
## 128190.69 151726.29 152011.22 217251.76 189422.12 151819.69 124937.89
## 1860 1861 1862 1863 1864 1865 1866
## 154776.45 125685.33 297486.12 297486.12 297486.12 273961.86 285566.29
## 1867 1868 1869 1870 1871 1872 1873
## 237064.08 232408.35 190170.08 205081.31 244503.98 174013.00 229375.36
## 1874 1875 1876 1877 1878 1879 1880
## 138309.50 200663.11 210657.59 197269.78 213650.42 122630.09 137870.87
## 1881 1882 1883 1884 1885 1886 1887
## 242585.84 242872.00 198832.19 222120.68 207508.23 238383.53 192818.86
## 1888 1889 1890 1891 1892 1893 1894
## 238976.13 168546.90 129825.04 139176.50 107772.57 139365.00 NA
## 1895 1896 1897 1898 1899 1900 1901
## 141386.27 122780.03 127864.56 139772.63 159646.08 155117.86 204018.17
## 1902 1903 1904 1905 1906 1907 1908
## 149997.70 211319.38 149654.33 237127.48 174201.51 237436.13 141428.31
## 1909 1910 1911 1912 1913 1914 1915
## 140138.98 141124.73 200679.10 279910.58 159942.41 70343.24 292814.37
## 1916 1917 1918 1919 1920 1921 1922
## NA 241963.59 135773.44 162995.75 166063.92 367288.83 256577.35
## 1923 1924 1925 1926 1927 1928 1929
## 228739.54 233148.09 220731.20 355272.07 133741.11 163266.82 129129.27
## 1930 1931 1932 1933 1934 1935 1936
## 139582.63 135351.18 139807.46 179388.24 178427.26 176535.49 182104.76
## 1937 1938 1939 1940 1941 1942 1943
## 184648.73 172068.85 267811.44 196019.92 178340.66 190133.76 196610.79
## 1944 1945 1946 1947 1948 1949 1950
## 335797.95 366350.75 NA 281676.92 196472.49 201429.34 166305.20
## 1951 1952 1953 1954 1955 1956 1957
## 246160.75 231305.18 167458.92 194036.23 157346.18 315357.26 172537.91
## 1958 1959 1960 1961 1962 1963 1964
## 248579.49 139331.75 123679.87 121701.06 90220.91 105043.86 109474.99
## 1965 1966 1967 1968 1969 1970 1971
## 158509.17 144766.35 277859.07 451551.91 385456.19 450864.43 423469.13
## 1972 1973 1974 1975 1976 1977 1978
## 351127.79 279044.12 328988.00 499425.85 290505.98 355026.69 436372.71
## 1979 1980 1981 1982 1983 1984 1985
## 341839.03 199000.10 317840.62 210831.55 201134.51 171114.97 208466.38
## 1986 1987 1988 1989 1990 1991 1992
## 192808.46 169344.02 178954.09 188250.36 214336.12 210677.44 204158.98
## 1993 1994 1995 1996 1997 1998 1999
## 176813.05 234083.26 180645.45 248261.36 300151.43 367040.41 271814.52
## 2000 2001 2002 2003 2004 2005 2006
## 303488.00 304639.33 232785.91 254027.47 254729.31 217018.73 209891.72
## 2007 2008 2009 2010 2011 2012 2013
## 280539.41 197388.85 189427.22 186500.91 NA 166268.83 177561.30
## 2014 2015 2016 2017 2018 2019 2020
## 192841.32 187229.56 191864.43 194762.40 130956.28 121968.35 108533.63
## 2021 2022 2023 2024 2025 2026 2027
## 113367.77 204284.15 150506.47 282261.47 329405.86 192798.54 150674.04
## 2028 2029 2030 2031 2032 2033 2034
## 157090.61 177355.07 274345.25 202777.86 230773.68 250111.76 160273.28
## 2035 2036 2037 2038 2039 2040 2041
## 221335.93 186260.60 185883.31 241796.66 199887.02 305338.78 236499.71
## 2042 2043 2044 2045 2046 2047 2048
## 230175.38 179172.37 210900.91 190163.85 150897.41 148691.80 135985.84
## 2049 2050 2051 2052 2053 2054 2055
## 142161.37 199281.40 116431.16 148875.82 145872.99 98941.48 164735.66
## 2056 2057 2058 2059 2060 2061 2062
## 137303.63 131701.91 199046.33 137787.84 154066.01 181420.54 138162.37
## 2063 2064 2065 2066 2067 2068 2069
## 114259.41 148306.19 120539.76 176909.20 146017.63 146316.74 81078.97
## 2070 2071 2072 2073 2074 2075 2076
## 113471.26 96142.97 150035.87 144549.37 164078.31 135019.91 125184.75
## 2077 2078 2079 2080 2081 2082 2083
## 150027.91 127458.11 132902.84 131236.35 123659.86 NA 145966.05
## 2084 2085 2086 2087 2088 2089 2090
## 121967.54 134165.10 86639.73 122535.83 107988.74 87315.76 131362.53
## 2091 2092 2093 2094 2095 2096 2097
## NA 141177.55 132316.18 NA 143595.44 80902.53 NA
## 2098 2099 2100 2101 2102 2103 2104
## 155766.87 70120.21 NA 128141.42 128790.98 100932.66 125815.06
## 2105 2106 2107 2108 2109 2110 2111
## NA 92381.47 216928.16 109425.67 106493.46 133324.04 146243.98
## 2112 2113 2114 2115 2116 2117 2118
## 147494.35 121529.38 122169.22 148116.44 127021.26 141069.73 122135.37
## 2119 2120 2121 2122 2123 2124 2125
## 120187.49 124528.70 NA 115283.68 79558.11 175789.95 140530.24
## 2126 2127 2128 2129 2130 2131 2132
## 148202.99 NA 131785.14 106288.55 136497.03 182625.48 120500.14
## 2133 2134 2135 2136 2137 2138 2139
## 123853.57 125200.03 103080.53 NA 121186.17 136813.91 138153.19
## 2140 2141 2142 2143 2144 2145 2146
## 137626.18 146029.15 139820.81 142652.02 127458.55 135317.56 140571.87
## 2147 2148 2149 2150 2151 2152 2153
## 186076.21 132281.67 141190.92 230221.25 105637.05 NA 161407.10
## 2154 2155 2156 2157 2158 2159 2160
## NA 118077.96 265788.88 220798.31 218768.59 199775.05 166434.90
## 2161 2162 2163 2164 2165 2166 2167
## 258288.86 328063.14 310674.06 225478.45 204518.66 142989.11 203236.44
## 2168 2169 2170 2171 2172 2173 2174
## 206486.30 189289.55 215691.85 150695.84 152214.52 191043.03 246862.45
## 2175 2176 2177 2178 2179 2180 2181
## 270990.43 282572.33 254113.47 197314.94 142191.75 214489.19 191562.20
## 2182 2183 2184 2185 2186 2187 2188
## 223904.67 190898.92 128265.48 130460.51 136825.08 151772.23 153187.55
## 2189 2190 2191 2192 2193 2194 2195
## 234799.38 NA NA NA NA NA 113170.68
## 2196 2197 2198 2199 2200 2201 2202
## 113075.05 120843.99 153565.42 168418.51 154123.08 151597.93 190635.21
## 2203 2204 2205 2206 2207 2208 2209
## 181572.02 215489.17 148920.81 149679.60 208984.83 246946.78 239732.35
## 2210 2211 2212 2213 2214 2215 2216
## 140090.72 114696.50 111455.51 NA 142806.87 104742.50 168428.24
## 2217 2218 2219 2220 2221 2222 2223
## 70404.85 100476.31 88972.15 100819.36 292814.37 281455.50 253496.06
## 2224 2225 2226 2227 2228 2229 2230
## 194112.12 134153.84 194013.83 210341.38 235059.33 210402.83 174373.22
## 2231 2232 2233 2234 2235 2236 2237
## 216702.62 215902.59 180032.98 205229.89 231895.98 266760.32 338038.12
## 2238 2239 2240 2241 2242 2243 2244
## 190470.00 NA 155844.95 175903.02 134549.44 134175.35 105453.08
## 2245 2246 2247 2248 2249 2250 2251
## 115668.67 141669.09 NA 124141.08 123268.84 128544.64 130518.73
## 2252 2253 2254 2255 2256 2257 2258
## 169410.18 163819.31 181357.44 191517.02 174678.61 228799.36 172171.34
## 2259 2260 2261 2262 2263 2264 2265
## 184714.93 164590.98 181474.03 194332.61 329136.93 524597.24 176885.85
## 2266 2267 2268 2269 2270 2271 2272
## 262437.45 333589.49 356574.01 161943.67 175991.93 206775.40 202721.26
## 2273 2274 2275 2276 2277 2278 2279
## 166581.97 170253.91 173822.11 173136.14 198642.06 148827.02 120981.01
## 2280 2281 2282 2283 2284 2285 2286
## 128113.08 167064.28 169496.45 97590.59 109367.53 139688.97 129152.90
## 2287 2288 2289 2290 2291 2292 2293
## 330274.32 306860.93 401759.30 426935.71 315993.87 402414.73 498650.35
## 2294 2295 2296 2297 2298 2299 2300
## 437119.60 449560.43 266852.04 305668.30 320701.65 453595.86 333941.84
## 2301 2302 2303 2304 2305 2306 2307
## 271068.74 252091.97 264357.50 267138.92 200249.67 200249.67 199059.88
## 2308 2309 2310 2311 2312 2313 2314
## 226903.92 258524.83 212219.03 199638.15 172986.36 180368.24 173015.30
## 2315 2316 2317 2318 2319 2320 2321
## 178082.59 187294.40 181325.65 177929.62 176099.95 173456.14 203687.48
## 2322 2323 2324 2325 2326 2327 2328
## 170533.94 166940.92 166348.87 206152.39 166235.11 212040.89 220567.12
## 2329 2330 2331 2332 2333 2334 2335
## 181906.98 185685.41 424227.81 407879.00 331488.24 300024.39 262534.41
## 2336 2337 2338 2339 2340 2341 2342
## 299758.29 188950.79 242429.37 248657.96 334238.08 217579.91 238292.55
## 2343 2344 2345 2346 2347 2348 2349
## 241317.58 252845.83 244956.33 206818.76 205110.52 223712.50 180058.51
## 2350 2351 2352 2353 2354 2355 2356
## 248619.29 252410.04 282919.98 289768.11 NA NA 150641.44
## 2357 2358 2359 2360 2361 2362 2363
## 190538.99 201577.46 146198.84 123336.27 140002.20 257761.46 137151.68
## 2364 2365 2366 2367 2368 2369 2370
## 146697.94 209169.58 176277.15 240331.82 206145.43 237307.24 175168.60
## 2371 2372 2373 2374 2375 2376 2377
## 177406.09 194801.46 235020.02 274700.98 246981.78 286493.25 322605.72
## 2378 2379 2380 2381 2382 2383 2384
## 157192.91 224295.87 154171.64 168797.96 189004.27 207604.30 216449.79
## 2385 2386 2387 2388 2389 2390 2391
## 164275.47 143264.35 133692.04 108966.28 133718.45 147772.08 145319.30
## 2392 2393 2394 2395 2396 2397 2398
## 121578.77 158225.67 148760.75 202932.22 147262.68 185885.92 130039.64
## 2399 2400 2401 2402 2403 2404 2405
## NA NA 114377.03 139721.41 135748.89 155140.24 164906.99
## 2406 2407 2408 2409 2410 2411 2412
## 127887.15 134382.24 147886.38 140045.35 171538.38 118065.61 152600.28
## 2413 2414 2415 2416 2417 2418 2419
## 147368.03 131959.21 139046.91 133447.82 135000.85 131653.62 134815.69
## 2420 2421 2422 2423 2424 2425 2426
## 126040.94 161631.40 110067.28 NA 153131.46 204040.72 130986.89
## 2427 2428 2429 2430 2431 2432 2433
## NA 152742.95 121251.85 129860.43 113082.23 135545.15 143914.45
## 2434 2435 2436 2437 2438 2439 2440
## 149157.06 150811.14 114480.37 116796.57 129439.41 123593.56 118675.99
## 2441 2442 2443 2444 2445 2446 2447
## 105233.74 114013.86 124023.24 123505.40 92113.18 151244.28 144418.42
## 2448 2449 2450 2451 2452 2453 2454
## 142143.91 124697.48 146110.52 130100.41 176615.26 86657.96 134166.21
## 2455 2456 2457 2458 2459 2460 2461
## 102980.69 137309.54 121777.31 123991.87 118092.39 143830.12 125468.76
## 2462 2463 2464 2465 2466 2467 2468
## 136892.06 125210.45 155745.89 134057.26 124468.99 125396.49 87767.90
## 2469 2470 2471 2472 2473 2474 2475
## 95072.75 185081.58 186420.04 185916.55 128867.41 111236.97 200902.89
## 2476 2477 2478 2479 2480 2481 2482
## 118881.57 128623.63 162700.17 114145.84 144914.45 125350.60 121863.83
## 2483 2484 2485 2486 2487 2488 2489
## 121792.20 129974.12 132540.62 147705.75 189226.74 137366.52 140730.02
## 2490 2491 2492 2493 2494 2495 2496
## 147236.74 101148.08 198576.84 131560.67 153354.20 93263.42 246737.00
## 2497 2498 2499 2500 2501 2502 2503
## 141274.98 118804.98 83053.55 127438.71 134021.28 132060.37 117056.67
## 2504 2505 2506 2507 2508 2509 2510
## 191979.77 208581.03 281959.06 305330.86 270298.49 223335.64 213981.57
## 2511 2512 2513 2514 2515 2516 2517
## 185128.77 209512.29 209598.95 221020.65 147205.29 167445.43 139279.09
## 2518 2519 2520 2521 2522 2523 2524
## 143257.14 226085.15 205169.71 197776.15 223720.30 135402.98 153239.85
## 2525 2526 2527 2528 2529 2530 2531
## 148474.48 145808.11 122104.43 129537.43 137673.80 131277.60 236261.86
## 2532 2533 2534 2535 2536 2537 2538
## 227461.77 201374.00 223944.47 266486.71 230927.70 192822.78 172897.05
## 2539 2540 2541 2542 2543 2544 2545
## 178696.34 184744.58 179111.75 184367.58 121985.73 134638.93 132475.37
## 2546 2547 2548 2549 2550 2551 2552
## 153222.74 126389.62 165887.06 148020.72 326720.23 139176.50 112765.52
## 2553 2554 2555 2556 2557 2558 2559
## NA NA 110154.35 104103.83 104532.26 NA 166938.78
## 2560 2561 2562 2563 2564 2565 2566
## 152552.63 148416.13 156481.27 171160.74 192477.15 165966.39 185990.77
## 2567 2568 2569 2570 2571 2572 2573
## 156254.52 220148.08 216710.61 139681.30 216128.05 150575.46 265422.40
## 2574 2575 2576 2577 2578 2579 2580
## 227732.79 128194.68 NA NA 117344.20 89036.31 NA
## 2581 2582 2583 2584 2585 2586 2587
## 123455.35 115573.20 258101.40 159340.18 197596.56 168595.99 174477.35
## 2588 2589 2590 2591 2592 2593 2594
## 133868.38 144470.81 238404.53 222809.19 241553.03 231023.58 180634.05
## 2595 2596 2597 2598 2599 2600 2601
## 185199.32 347820.20 226979.91 262032.74 305769.23 202659.28 141899.57
## 2602 2603 2604 2605 2606 2607 2608
## 83938.60 103363.35 NA 84118.08 146840.80 172641.17 193829.81
## 2609 2610 2611 2612 2613 2614 2615
## 171286.78 NA 142111.97 150707.84 130361.68 127686.75 162751.36
## 2616 2617 2618 2619 2620 2621 2622
## 137004.65 189345.36 171004.98 204201.85 185040.71 181667.99 200587.40
## 2623 2624 2625 2626 2627 2628 2629
## 230294.51 305719.08 334809.73 171774.87 189184.67 403836.46 443412.45
## 2630 2631 2632 2633 2634 2635 2636
## 329038.34 437237.53 400300.49 287576.68 356841.24 162695.30 198811.78
## 2637 2638 2639 2640 2641 2642 2643
## 161752.81 251376.74 178885.73 154161.44 110557.49 181457.97 105582.25
## 2644 2645 2646 2647 2648 2649 2650
## 139845.61 102769.33 91974.04 98555.91 134427.01 143232.33 138250.23
## 2651 2652 2653 2654 2655 2656 2657
## 158605.28 401916.66 283107.84 298558.03 373045.54 303196.61 289545.23
## 2658 2659 2660 2661 2662 2663 2664
## 269691.20 312077.05 350334.98 320983.78 320298.12 291287.40 256412.36
## 2665 2666 2667 2668 2669 2670 2671
## 324934.47 284522.49 174091.51 175243.48 179125.76 258080.93 178018.10
## 2672 2673 2674 2675 2676 2677 2678
## 181070.35 201233.30 203500.70 162533.58 183995.42 205214.59 247568.61
## 2679 2680 2681 2682 2683 2684 2685
## 284232.96 287745.45 399672.16 351419.05 499000.55 325785.91 378857.79
## 2686 2687 2688 2689 2690 2691 2692
## 252233.64 254431.96 234972.11 194890.50 336016.93 212048.76 NA
## 2693 2694 2695 2696 2697 2698 2699
## 181839.41 NA 194655.02 179630.22 207777.55 195848.75 176716.09
## 2700 2701 2702 2703 2704 2705 2706
## 171791.26 160588.96 121306.23 145328.88 141154.89 125758.14 121170.17
## 2707 2708 2709 2710 2711 2712 2713
## 134491.26 131628.60 NA 126997.41 280028.97 381649.92 173303.64
## 2714 2715 2716 2717 2718 2719 2720
## 154073.23 176326.98 151580.48 176372.74 243920.61 168560.69 168422.29
## 2721 2722 2723 2724 2725 2726 2727
## 142081.45 164732.65 141188.96 129038.57 138202.55 137828.73 162365.44
## 2728 2729 2730 2731 2732 2733 2734
## 167896.45 162797.02 145497.92 125060.39 130516.25 166388.31 162842.78
## 2735 2736 2737 2738 2739 2740 2741
## 137888.65 149395.70 123720.70 139548.71 163953.10 146366.27 147057.77
## 2742 2743 2744 2745 2746 2747 2748
## 158915.37 152563.25 141759.35 140831.31 133692.50 149423.32 130570.97
## 2749 2750 2751 2752 2753 2754 2755
## 125198.00 135663.39 119426.41 200706.90 144739.49 278176.90 132416.20
## 2756 2757 2758 2759 2760 2761 2762
## 97539.01 89056.63 97069.75 159624.40 146246.25 143723.58 140498.97
## 2763 2764 2765 2766 2767 2768 2769
## 182067.32 138614.00 238391.91 137985.50 90982.14 NA 130773.04
## 2770 2771 2772 2773 2774 2775 2776
## 138081.36 120716.27 NA 151658.02 143697.00 140638.96 132770.53
## 2777 2778 2779 2780 2781 2782 2783
## 152869.52 106112.19 128341.71 102808.01 113471.78 102903.34 118579.68
## 2784 2785 2786 2787 2788 2789 2790
## 121852.26 139710.17 84570.30 114309.97 100004.91 170589.20 NA
## 2791 2792 2793 2794 2795 2796 2797
## 114764.39 NA 158898.81 114931.24 123848.16 112489.28 208457.95
## 2798 2799 2800 2801 2802 2803 2804
## 134792.43 115566.91 NA 112450.19 130468.80 150124.51 155581.48
## 2805 2806 2807 2808 2809 2810 2811
## 97817.35 102147.08 129623.83 142489.79 136363.42 135474.86 162308.74
## 2812 2813 2814 2815 2816 2817 2818
## 141631.35 185019.72 173009.49 99422.18 242037.04 173269.14 129013.03
## 2819 2820 2821 2822 2823 2824 2825
## 171481.92 151422.50 115580.89 217461.82 259155.17 210275.36 152532.00
## 2826 2827 2828 2829 2830 2831 2832
## 120403.18 128279.86 224955.06 187671.24 246725.19 191829.51 225333.45
## 2833 2834 2835 2836 2837 2838 2839
## 249499.92 213410.84 223051.52 186727.38 193625.47 158419.05 186025.46
## 2840 2841 2842 2843 2844 2845 2846
## 190866.73 200655.45 215958.52 153926.71 174132.10 131032.21 209286.97
## 2847 2848 2849 2850 2851 2852 2853
## 205036.69 181153.72 214235.70 272590.79 228130.85 245763.14 228046.42
## 2854 2855 2856 2857 2858 2859 2860
## 139472.52 189311.06 201766.39 184370.28 178965.88 139310.61 NA
## 2861 2862 2863 2864 2865 2866 2867
## 113915.96 170898.31 NA 208458.45 139176.50 131712.33 101057.51
## 2868 2869 2870 2871 2872 2873 2874
## 106557.80 128579.04 140496.16 NA 64012.19 105732.48 126515.21
## 2875 2876 2877 2878 2879 2880 2881
## 122386.39 152988.92 142512.16 182967.11 135202.71 113830.59 177126.51
## 2882 2883 2884 2885 2886 2887 2888
## 193544.36 198924.77 189997.51 189757.50 253853.30 105653.33 132085.86
## 2889 2890 2891 2892 2893 2894 2895
## NA 96530.16 136327.19 NA NA NA 285325.63
## 2896 2897 2898 2899 2900 2901 2902
## 281455.50 217382.51 155884.07 188434.17 150538.70 227434.40 179901.16
## 2903 2904 2905 2906 2907 2908 2909
## 328199.76 308384.17 121966.59 187361.93 116467.58 126574.11 163472.84
## 2910 2911 2912 2913 2914 2915 2916
## NA 84222.27 153157.18 83716.89 NA NA 84976.34
## 2917 2918 2919
## 162989.63 NA 226993.12prediction2.df <- data.frame(Id = test_set$Id, SalePrice = prediction)
prediction2.df <- prediction2.df %>% mutate(SalePrice = replace_na(SalePrice,163000))
write.csv(prediction2.df , file = "prediction_random_forest.csv", row.names = FALSE)