1 Purpose

This report prepares the Ames housing data for regression modeling without fitting a final predictive model. The emphasis is on disciplined data preparation: profiling, cleaning, missingness assessment, validation checks, outlier handling for regression, transformation planning, scaling, normality checks, and feature engineering. A particular focus is placed on the 30 strongest numeric features correlated with SalePrice, along with binning neighborhoods into a regression-ready grouping.

2 Why this matters

Regression quality depends heavily on the quality and structure of the data going into the model. If data are not profiled, cleaned, validated, and transformed appropriately, coefficient estimates and predictions can become unstable, biased, or misleading. This report therefore treats data preparation as a first-class modeling step.

3 Packages

# Install these if needed:
# install.packages(c(
#   "readxl","dplyr","ggplot2","tidyr","stringr","purrr","skimr",
#   "DataExplorer","naniar","corrplot","moments","forcats","scales","knitr"
# ))

library(readxl)
library(dplyr)
library(ggplot2)
library(tidyr)
library(stringr)
library(purrr)
library(skimr)
library(DataExplorer)
library(naniar)
library(corrplot)
library(moments)
library(forcats)
library(scales)
library(knitr)

4 Load the data

This section imports the Ames data and standardizes column names so that later code is easier to write and less error-prone.

# Update the path only if needed
ames <- read_excel("ames_housing_data.xlsx")

# Standardize names for cleaner coding
names(ames) <- make.names(names(ames))

# Basic structure checks
dim(ames)

## [1] 2930   82

names(ames)[1:15]

##  [1] "SID"          "PID"          "SubClass"     "Zoning"       "LotFrontage" 
##  [6] "LotArea"      "Street"       "Alley"        "LotShape"     "LandContour" 
## [11] "Utilities"    "LotConfig"    "LandSlope"    "Neighborhood" "Condition1"

head(ames, 3)

## # A tibble: 3 × 82
##     SID       PID SubClass Zoning LotFrontage LotArea Street Alley LotShape
##   <dbl>     <dbl>    <dbl> <chr>        <dbl>   <dbl> <chr>  <chr> <chr>   
## 1     1 526301100       20 RL             141   31770 Pave   NA    IR1     
## 2     2 526350040       20 RH              80   11622 Pave   NA    Reg     
## 3     3 526351010       20 RL              81   14267 Pave   NA    IR1     
## # ℹ 73 more variables: LandContour <chr>, Utilities <chr>, LotConfig <chr>,
## #   LandSlope <chr>, Neighborhood <chr>, Condition1 <chr>, Condition2 <chr>,
## #   BldgType <chr>, HouseStyle <chr>, OverallQual <dbl>, OverallCond <dbl>,
## #   YearBuilt <dbl>, YearRemodel <dbl>, RoofStyle <chr>, RoofMat <chr>,
## #   Exterior1 <chr>, Exterior2 <chr>, MasVnrType <chr>, MasVnrArea <dbl>,
## #   ExterQual <chr>, ExterCond <chr>, Foundation <chr>, BsmtQual <chr>,
## #   BsmtCond <chr>, BsmtExposure <chr>, BsmtFinType1 <chr>, BsmtFinSF1 <dbl>, …

4.1 Insights

cat(
  paste0(
    "- The raw dataset contains **", nrow(ames), "** rows and **", ncol(ames), "** columns.\n",
    "- Standardizing the column names early helps avoid later coding problems due to spaces or punctuation.\n",
    "- The data include a mix of numeric, ordinal, and nominal fields, so regression preparation must treat variables differently depending on type.\n"
  )
)

The raw dataset contains 2930 rows and 82 columns.
Standardizing the column names early helps avoid later coding problems due to spaces or punctuation.
The data include a mix of numeric, ordinal, and nominal fields, so regression preparation must treat variables differently depending on type.

5 Data profiling

Profiling gives a broad overview of structure, variable types, ranges, and missingness patterns before any filtering or feature engineering.

# Compact profiling summary
skim(ames)

Data summary
Name	ames
Number of rows	2930
Number of columns	82
_______________________
Column type frequency:
character	43
numeric	39
________________________
Group variables	None

Variable type: character

skim_variable	n_missing	complete_rate	min	max	n_unique
Zoning	0	1.00	2	7	7
Street	0	1.00	4	4	2
Alley	0	1.00	2	4	3
LotShape	0	1.00	3	3	4
LandContour	0	1.00	3	3	4
Utilities	0	1.00	6	6	3
LotConfig	0	1.00	3	7	5
LandSlope	0	1.00	3	3	3
Neighborhood	0	1.00	5	7	28
Condition1	0	1.00	4	6	9
Condition2	0	1.00	4	6	8
BldgType	0	1.00	4	6	5
HouseStyle	0	1.00	4	6	8
RoofStyle	0	1.00	3	7	6
RoofMat	0	1.00	4	7	8
Exterior1	0	1.00	5	7	16
Exterior2	0	1.00	5	7	17
MasVnrType	23	0.99	4	7	5
ExterQual	0	1.00	2	2	4
ExterCond	0	1.00	2	2	5
Foundation	0	1.00	4	6	6
BsmtQual	1	1.00	2	2	6
BsmtCond	1	1.00	2	2	6
BsmtExposure	4	1.00	2	2	5
BsmtFinType1	1	1.00	2	3	7
BsmtFinType2	2	1.00	2	3	7
Heating	0	1.00	4	5	6
HeatingQC	0	1.00	2	2	5
CentralAir	0	1.00	1	1	2
Electrical	1	1.00	3	5	5
KitchenQual	0	1.00	2	2	5
Functional	0	1.00	3	4	8
FireplaceQu	0	1.00	2	2	6
GarageType	0	1.00	2	7	7
GarageFinish	2	1.00	2	3	4
GarageQual	1	1.00	2	2	6
GarageCond	1	1.00	2	2	6
PavedDrive	0	1.00	1	1	3
PoolQC	0	1.00	2	2	5
Fence	0	1.00	2	5	5
MiscFeature	0	1.00	2	4	6
SaleType	0	1.00	2	5	10
SaleCondition	0	1.00	6	7	6

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
SID	0	1.00	1465.50	845.96	1	733.25	1465.5	2197.75	2930	▇▇▇▇▇
PID	0	1.00	714464496.99	188730844.65	526301100	528477022.50	535453620.0	907181097.50	1007100110	▇▁▁▆▂
SubClass	0	1.00	57.39	42.64	20	20.00	50.0	70.00	190	▇▅▂▁▁
LotFrontage	490	0.83	69.22	23.37	21	58.00	68.0	80.00	313	▇▃▁▁▁
LotArea	0	1.00	10147.92	7880.02	1300	7440.25	9436.5	11555.25	215245	▇▁▁▁▁
OverallQual	0	1.00	6.09	1.41	1	5.00	6.0	7.00	10	▁▂▇▅▁
OverallCond	0	1.00	5.56	1.11	1	5.00	5.0	6.00	9	▁▁▇▅▁
YearBuilt	0	1.00	1971.36	30.25	1872	1954.00	1973.0	2001.00	2010	▁▂▃▆▇
YearRemodel	0	1.00	1984.27	20.86	1950	1965.00	1993.0	2004.00	2010	▅▂▂▃▇
MasVnrArea	23	0.99	101.90	179.11	0	0.00	0.0	164.00	1600	▇▁▁▁▁
BsmtFinSF1	1	1.00	442.63	455.59	0	0.00	370.0	734.00	5644	▇▁▁▁▁
BsmtFinSF2	1	1.00	49.72	169.17	0	0.00	0.0	0.00	1526	▇▁▁▁▁
BsmtUnfSF	1	1.00	559.26	439.49	0	219.00	466.0	802.00	2336	▇▅▂▁▁
TotalBsmtSF	1	1.00	1051.61	440.62	0	793.00	990.0	1302.00	6110	▇▃▁▁▁
FirstFlrSF	0	1.00	1159.56	391.89	334	876.25	1084.0	1384.00	5095	▇▃▁▁▁
SecondFlrSF	0	1.00	335.46	428.40	0	0.00	0.0	703.75	2065	▇▃▂▁▁
LowQualFinSF	0	1.00	4.68	46.31	0	0.00	0.0	0.00	1064	▇▁▁▁▁
GrLivArea	0	1.00	1499.69	505.51	334	1126.00	1442.0	1742.75	5642	▇▇▁▁▁
BsmtFullBath	2	1.00	0.43	0.52	0	0.00	0.0	1.00	3	▇▆▁▁▁
BsmtHalfBath	2	1.00	0.06	0.25	0	0.00	0.0	0.00	2	▇▁▁▁▁
FullBath	0	1.00	1.57	0.55	0	1.00	2.0	2.00	4	▁▇▇▁▁
HalfBath	0	1.00	0.38	0.50	0	0.00	0.0	1.00	2	▇▁▅▁▁
BedroomAbvGr	0	1.00	2.85	0.83	0	2.00	3.0	3.00	8	▁▇▂▁▁
KitchenAbvGr	0	1.00	1.04	0.21	0	1.00	1.0	1.00	3	▁▇▁▁▁
TotRmsAbvGrd	0	1.00	6.44	1.57	2	5.00	6.0	7.00	15	▁▇▂▁▁
Fireplaces	0	1.00	0.60	0.65	0	0.00	1.0	1.00	4	▇▇▁▁▁
GarageYrBlt	159	0.95	1978.13	25.53	1895	1960.00	1979.0	2002.00	2207	▂▇▁▁▁
GarageCars	1	1.00	1.77	0.76	0	1.00	2.0	2.00	5	▅▇▂▁▁
GarageArea	1	1.00	472.82	215.05	0	320.00	480.0	576.00	1488	▃▇▃▁▁
WoodDeckSF	0	1.00	93.75	126.36	0	0.00	0.0	168.00	1424	▇▁▁▁▁
OpenPorchSF	0	1.00	47.53	67.48	0	0.00	27.0	70.00	742	▇▁▁▁▁
EnclosedPorch	0	1.00	23.01	64.14	0	0.00	0.0	0.00	1012	▇▁▁▁▁
ThreeSsnPorch	0	1.00	2.59	25.14	0	0.00	0.0	0.00	508	▇▁▁▁▁
ScreenPorch	0	1.00	16.00	56.09	0	0.00	0.0	0.00	576	▇▁▁▁▁
PoolArea	0	1.00	2.24	35.60	0	0.00	0.0	0.00	800	▇▁▁▁▁
MiscVal	0	1.00	50.64	566.34	0	0.00	0.0	0.00	17000	▇▁▁▁▁
MoSold	0	1.00	6.22	2.71	1	4.00	6.0	8.00	12	▅▆▇▃▃
YrSold	0	1.00	2007.79	1.32	2006	2007.00	2008.0	2009.00	2010	▇▇▇▇▃
SalePrice	0	1.00	180796.06	79886.69	12789	129500.00	160000.0	213500.00	755000	▇▇▁▁▁

# Global profiling visuals
plot_intro(ames)

plot_missing(ames)

5.1 Insights

num_cols <- sum(sapply(ames, is.numeric))
cat_cols <- ncol(ames) - num_cols
cat(
  paste0(
    "- The dataset contains **", num_cols, "** numeric columns and **", cat_cols, "** non-numeric columns.\n",
    "- Profiling is important because regression preparation depends on identifying which variables are continuous, discrete, ordinal, or nominal.\n",
    "- Missingness is not random in this dataset for some housing attributes, so it should be studied before imputation or row removal.\n"
  )
)

The dataset contains 39 numeric columns and 43 non-numeric columns.
Profiling is important because regression preparation depends on identifying which variables are continuous, discrete, ordinal, or nominal.
Missingness is not random in this dataset for some housing attributes, so it should be studied before imputation or row removal.

6 Initial feature engineering for survey purposes

Before filtering, create a few derived variables that are useful for data understanding and later regression preparation.

ames <- ames %>%
  mutate(
    TotalFloorSF = FirstFlrSF + SecondFlrSF,
    HouseAge = YrSold - YearBuilt,
    RemodAge = YrSold - YearRemodel,
    TotalBath = FullBath + HalfBath * 0.5 + BsmtFullBath + BsmtHalfBath * 0.5,
    QualityIndex = OverallQual * OverallCond,
    GarageAge = ifelse(is.na(GarageYrBlt), NA, YrSold - GarageYrBlt),
    TotalPorchSF = OpenPorchSF + EnclosedPorch  + ScreenPorch,
    TotalOutdoorSF = WoodDeckSF + OpenPorchSF + EnclosedPorch + ScreenPorch,
    logSalePrice = log(SalePrice)
  )

summary(select(ames, SalePrice, logSalePrice, TotalFloorSF, HouseAge, RemodAge, TotalBath, QualityIndex))

##    SalePrice       logSalePrice     TotalFloorSF     HouseAge     
##  Min.   : 12789   Min.   : 9.456   Min.   : 334   Min.   : -1.00  
##  1st Qu.:129500   1st Qu.:11.771   1st Qu.:1120   1st Qu.:  7.00  
##  Median :160000   Median :11.983   Median :1440   Median : 34.00  
##  Mean   :180796   Mean   :12.021   Mean   :1495   Mean   : 36.43  
##  3rd Qu.:213500   3rd Qu.:12.271   3rd Qu.:1740   3rd Qu.: 54.00  
##  Max.   :755000   Max.   :13.534   Max.   :5642   Max.   :136.00  
##                                                                   
##     RemodAge       TotalBath      QualityIndex  
##  Min.   :-2.00   Min.   :1.000   Min.   : 1.00  
##  1st Qu.: 4.00   1st Qu.:1.500   1st Qu.:30.00  
##  Median :15.00   Median :2.000   Median :35.00  
##  Mean   :23.52   Mean   :2.218   Mean   :33.76  
##  3rd Qu.:42.75   3rd Qu.:2.500   3rd Qu.:40.00  
##  Max.   :60.00   Max.   :7.000   Max.   :90.00  
##                  NA's   :2

6.1 Insights

cat(
  "- Derived variables often capture housing value more directly than raw variables alone.\n",
  "- `TotalFloorSF`, `HouseAge`, `TotalBath`, and `QualityIndex` are especially useful because they summarize physical size, age, utility, and construction quality.\n",
  "- `logSalePrice` is created now because home prices are commonly right-skewed, and log-transformation is often useful in regression preparation.\n"
)

Derived variables often capture housing value more directly than raw variables alone.
TotalFloorSF, HouseAge, TotalBath, and QualityIndex are especially useful because they summarize physical size, age, utility, and construction quality.
logSalePrice is created now because home prices are commonly right-skewed, and log-transformation is often useful in regression preparation.

7 Define the regression sample

The goal here is not to keep every sale, but to define a cleaner sample for modeling typical residential transactions.

n0 <- nrow(ames)

sample_df <- ames

# Keep single-family homes only
drop1 <- sample_df %>% filter(BldgType != "1Fam")
sample_df <- sample_df %>% filter(BldgType == "1Fam")
after1 <- nrow(sample_df)

# Remove very large homes as recommended by the Ames documentation
drop2 <- sample_df %>% filter(GrLivArea > 4000)
sample_df <- sample_df %>% filter(GrLivArea <= 4000)
after2 <- nrow(sample_df)

# Remove atypical sale conditions
drop3 <- sample_df %>% filter(SaleCondition %in% c("Partial", "Family", "Abnorml", "Alloca", "AdjLand"))
sample_df <- sample_df %>% filter(!SaleCondition %in% c("Partial", "Family", "Abnorml", "Alloca", "AdjLand"))
after3 <- nrow(sample_df)

# Remove missing or nonpositive sale price
drop4 <- sample_df %>% filter(is.na(SalePrice) | SalePrice <= 0)
sample_df <- sample_df %>% filter(!is.na(SalePrice), SalePrice > 0)
after4 <- nrow(sample_df)

waterfall <- tibble(
  Step = c(
    "Original data",
    "Keep only BldgType = 1Fam",
    "Remove GrLivArea > 4000",
    "Remove atypical sale conditions",
    "Remove missing/nonpositive SalePrice",
    "Final sample"
  ),
  Count = c(n0, after1, after2, after3, after4, nrow(sample_df))
)

kable(waterfall)

Step	Count
Original data	2930
Keep only BldgType = 1Fam	2425
Remove GrLivArea > 4000	2420
Remove atypical sale conditions	2001
Remove missing/nonpositive SalePrice	2001
Final sample	2001

ggplot(waterfall, aes(x = reorder(Step, -seq_along(Step)), y = Count)) +
  geom_col(fill = "steelblue") +
  geom_text(aes(label = comma(Count)), vjust = -0.3, size = 3.5) +
  labs(
    title = "Sample Definition Waterfall",
    x = NULL,
    y = "Count"
  ) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 25, hjust = 1))

7.1 Insights

cat(
  paste0(
    "- The filtered regression sample contains **", nrow(sample_df), "** observations.\n",
    "- These drop conditions improve comparability by focusing on typical single-family residential sales.\n",
    "- Outlier removal at this stage is not arbitrary; it is tied to the practical modeling objective of valuing typical homes rather than every possible transaction type.\n"
  )
)

The filtered regression sample contains 2001 observations.
These drop conditions improve comparability by focusing on typical single-family residential sales.
Outlier removal at this stage is not arbitrary; it is tied to the practical modeling objective of valuing typical homes rather than every possible transaction type.

8 Data missingness

Missingness should be examined before imputation decisions are made. Some missing values in Ames mean “not present” rather than “unknown.”

missing_tbl <- tibble(
  Variable = names(sample_df),
  MissingN = sapply(sample_df, function(x) sum(is.na(x))),
  MissingPct = sapply(sample_df, function(x) mean(is.na(x)) * 100)
) %>%
  arrange(desc(MissingPct))

kable(head(missing_tbl, 20), digits = 2)

Variable	MissingN	MissingPct
LotFrontage	386	19.29
GarageYrBlt	74	3.70
GarageAge	74	3.70
MasVnrType	11	0.55
MasVnrArea	11	0.55
BsmtExposure	2	0.10
BsmtFinType2	1	0.05
Electrical	1	0.05
BsmtFullBath	1	0.05
BsmtHalfBath	1	0.05
TotalBath	1	0.05
SID	0	0.00
PID	0	0.00
SubClass	0	0.00
Zoning	0	0.00
LotArea	0	0.00
Street	0	0.00
Alley	0	0.00
LotShape	0	0.00
LandContour	0	0.00

vis_miss(sample_df, sort_miss = TRUE)

8.1 Missingness handling strategy

# Distinguish likely "structural" NA fields from true missingness
structural_na_vars <- c(
  "Alley", "MasVnrType", "BsmtQual", "BsmtCond", "BsmtExposure",
  "BsmtFinType1", "BsmtFinType2", "FireplaceQu", "GarageType",
  "GarageFinish", "GarageQual", "GarageCond", "PoolQC", "Fence",
  "MiscFeature"
)

# Create a working copy for cleaning
clean_df <- sample_df

# Replace structural missing categories with "None"
for (v in structural_na_vars[structural_na_vars %in% names(clean_df)]) {
  clean_df[[v]] <- as.character(clean_df[[v]])
  clean_df[[v]][is.na(clean_df[[v]])] <- "None"
  clean_df[[v]] <- as.factor(clean_df[[v]])
}

# Replace selected numeric structural missing values with zero
zero_if_absent <- c(
  "MasVnrArea", "BsmtFinSF1", "BsmtFinSF2", "BsmtUnfSF", "TotalBsmtSF",
  "BsmtFullBath", "BsmtHalfBath", "GarageCars", "GarageArea"
)

for (v in zero_if_absent[zero_if_absent %in% names(clean_df)]) {
  clean_df[[v]][is.na(clean_df[[v]])] <- 0
}

# Median imputation for remaining numeric missing values
num_missing_before <- sum(is.na(clean_df[sapply(clean_df, is.numeric)]))

clean_df <- clean_df %>%
  mutate(across(where(is.numeric), ~ ifelse(is.na(.), median(., na.rm = TRUE), .)))

num_missing_after <- sum(is.na(clean_df[sapply(clean_df, is.numeric)]))
c(num_missing_before = num_missing_before, num_missing_after = num_missing_after)

## num_missing_before  num_missing_after 
##                535                  0

8.2 Insights

top_missing <- missing_tbl %>% filter(MissingPct > 0) %>% slice(1:5)
cat("- Not all missingness should be treated the same way.\n")

Not all missingness should be treated the same way.

cat("- In this dataset, several housing attributes are missing because the feature does not exist, which should typically be encoded as `None` or `0` rather than dropped.\n")

In this dataset, several housing attributes are missing because the feature does not exist, which should typically be encoded as None or 0 rather than dropped.

cat("- Remaining numeric missingness is reduced with median imputation in the cleaned working dataset.\n")

Remaining numeric missingness is reduced with median imputation in the cleaned working dataset.

if (nrow(top_missing) > 0) {
  cat("- The most affected variables by missingness in the modeling sample are:\n")
  for (i in seq_len(nrow(top_missing))) {
    cat(paste0("  - `", top_missing$Variable[i], "`: ", round(top_missing$MissingPct[i], 2), "% missing\n"))
  }
}

The most affected variables by missingness in the modeling sample are:
- LotFrontage: 19.29% missing
- GarageYrBlt: 3.7% missing
- GarageAge: 3.7% missing
- MasVnrType: 0.55% missing
- MasVnrArea: 0.55% missing

9 Data validation and logical checks

Validation ensures the cleaned data are still consistent with known housing logic and measurement rules.

rule_checks <- tibble(
  Rule = c(
    "SalePrice > 0",
    "GrLivArea > 0",
    "OverallQual between 1 and 10",
    "OverallCond between 1 and 10",
    "GarageCars >= 0",
    "YearBuilt <= YrSold",
    "HouseAge >= 0",
    "TotalBath >= 0",
    "TotalFloorSF >= GrLivArea not required but should be nonnegative",
    "LotArea > 0"
  ),
  Violations = c(
    sum(clean_df$SalePrice <= 0 | is.na(clean_df$SalePrice)),
    sum(clean_df$GrLivArea <= 0 | is.na(clean_df$GrLivArea)),
    sum(!(clean_df$OverallQual %in% 1:10) | is.na(clean_df$OverallQual)),
    sum(!(clean_df$OverallCond %in% 1:10) | is.na(clean_df$OverallCond)),
    sum(clean_df$GarageCars < 0, na.rm = TRUE),
    sum(clean_df$YearBuilt > clean_df$YrSold, na.rm = TRUE),
    sum(clean_df$HouseAge < 0, na.rm = TRUE),
    sum(clean_df$TotalBath < 0, na.rm = TRUE),
    sum(clean_df$TotalFloorSF < 0, na.rm = TRUE),
    sum(clean_df$LotArea <= 0 | is.na(clean_df$LotArea))
  )
)

kable(rule_checks)

Rule	Violations
SalePrice > 0	0
GrLivArea > 0	0
OverallQual between 1 and 10	0
OverallCond between 1 and 10	0
GarageCars >= 0	0
YearBuilt <= YrSold	0
HouseAge >= 0	0
TotalBath >= 0	0
TotalFloorSF >= GrLivArea not required but should be nonnegative	0
LotArea > 0	0

9.1 Insights

viol_n <- sum(rule_checks$Violations)
cat(
  paste0(
    "- Validation checks are used to confirm that the data still make sense after cleaning.\n",
    "- Total validation violations found across the listed rules: **", viol_n, "**.\n",
    "- These checks are especially important before regression because impossible values can distort coefficients and residual patterns.\n"
  )
)

Validation checks are used to confirm that the data still make sense after cleaning.
Total validation violations found across the listed rules: 0.
These checks are especially important before regression because impossible values can distort coefficients and residual patterns.

10 Top 30 numeric features correlated with SalePrice

Correlation is not a modeling strategy by itself, but it is a useful screening step for identifying variables most strongly associated with the response.

num_data <- clean_df %>% select(where(is.numeric))

if (!"SalePrice" %in% names(num_data)) {
  stop("SalePrice not found among numeric columns.")
}

cor_vals <- sapply(num_data, function(x) cor(x, num_data$SalePrice, use = "pairwise.complete.obs"))

top30_saleprice <- tibble(
  Feature = names(cor_vals),
  Correlation = as.numeric(cor_vals)
) %>%
  filter(Feature != "SalePrice") %>%
  mutate(abs_corr = abs(Correlation)) %>%
  arrange(desc(abs_corr)) %>%
  slice(1:30)

kable(top30_saleprice, digits = 3)

Feature	Correlation	abs_corr
logSalePrice	0.958	0.958
OverallQual	0.806	0.806
TotalFloorSF	0.778	0.778
GrLivArea	0.771	0.771
TotalBath	0.700	0.700
GarageCars	0.666	0.666
TotalBsmtSF	0.651	0.651
GarageArea	0.642	0.642
FirstFlrSF	0.640	0.640
FullBath	0.611	0.611
TotRmsAbvGrd	0.601	0.601
YearBuilt	0.571	0.571
HouseAge	-0.570	0.570
MasVnrArea	0.544	0.544
QualityIndex	0.525	0.525
GarageYrBlt	0.522	0.522
GarageAge	-0.521	0.521
RemodAge	-0.507	0.507
YearRemodel	0.507	0.507
Fireplaces	0.484	0.484
BsmtFinSF1	0.454	0.454
TotalOutdoorSF	0.405	0.405
LotFrontage	0.387	0.387
WoodDeckSF	0.350	0.350
SecondFlrSF	0.350	0.350
HalfBath	0.349	0.349
OpenPorchSF	0.342	0.342
BsmtFullBath	0.294	0.294
LotArea	0.290	0.290
BedroomAbvGr	0.273	0.273

ggplot(top30_saleprice, aes(x = reorder(Feature, abs_corr), y = Correlation)) +
  geom_col(fill = "steelblue") +
  coord_flip() +
  labs(
    title = "Top 30 Numeric Features Correlated with SalePrice",
    x = "Feature",
    y = "Correlation"
  ) +
  theme_minimal()

10.1 Insights

cat("- The top 30 correlation table is a screening tool for regression preparation, not a final variable selection method.\n")

The top 30 correlation table is a screening tool for regression preparation, not a final variable selection method.

cat("- Highly correlated predictors often include size, quality, age, basement, garage, and bath-related features.\n")

Highly correlated predictors often include size, quality, age, basement, garage, and bath-related features.

cat("- Some of these predictors will also be highly correlated with each other, so later regression work should check multicollinearity even after this screening step.\n")

Some of these predictors will also be highly correlated with each other, so later regression work should check multicollinearity even after this screening step.

11 Correlation matrix among the strongest features

This section looks at the correlation structure among the top predictors to identify possible redundancy.

top_corr_vars <- top30_saleprice$Feature[1:min(15, nrow(top30_saleprice))]
corr_subset <- clean_df %>% select(all_of(c("SalePrice", top_corr_vars))) %>% select(where(is.numeric))
corr_matrix <- cor(corr_subset, use = "pairwise.complete.obs")

round(corr_matrix, 3)

##              SalePrice logSalePrice OverallQual TotalFloorSF GrLivArea
## SalePrice        1.000        0.958       0.806        0.778     0.771
## logSalePrice     0.958        1.000       0.828        0.780     0.773
## OverallQual      0.806        0.828       1.000        0.631     0.622
## TotalFloorSF     0.778        0.780       0.631        1.000     0.995
## GrLivArea        0.771        0.773       0.622        0.995     1.000
## TotalBath        0.700        0.731       0.583        0.634     0.627
## GarageCars       0.666        0.690       0.604        0.526     0.519
## TotalBsmtSF      0.651        0.638       0.526        0.397     0.393
## GarageArea       0.642        0.656       0.550        0.490     0.485
## FirstFlrSF       0.640        0.619       0.455        0.529     0.526
## FullBath         0.611        0.646       0.572        0.669     0.668
## TotRmsAbvGrd     0.601        0.618       0.511        0.821     0.829
## YearBuilt        0.571        0.626       0.558        0.301     0.286
## HouseAge        -0.570       -0.625      -0.557       -0.301    -0.286
## MasVnrArea       0.544        0.491       0.429        0.429     0.422
## QualityIndex     0.525        0.566       0.694        0.407     0.403
##              TotalBath GarageCars TotalBsmtSF GarageArea FirstFlrSF FullBath
## SalePrice        0.700      0.666       0.651      0.642      0.640    0.611
## logSalePrice     0.731      0.690       0.638      0.656      0.619    0.646
## OverallQual      0.583      0.604       0.526      0.550      0.455    0.572
## TotalFloorSF     0.634      0.526       0.397      0.490      0.529    0.669
## GrLivArea        0.627      0.519       0.393      0.485      0.526    0.668
## TotalBath        1.000      0.532       0.436      0.486      0.395    0.734
## GarageCars       0.532      1.000       0.440      0.880      0.439    0.515
## TotalBsmtSF      0.436      0.440       1.000      0.455      0.764    0.340
## GarageArea       0.486      0.880       0.455      1.000      0.463    0.450
## FirstFlrSF       0.395      0.439       0.764      0.463      1.000    0.373
## FullBath         0.734      0.515       0.340      0.450      0.373    1.000
## TotRmsAbvGrd     0.516      0.425       0.261      0.366      0.358    0.585
## YearBuilt        0.584      0.536       0.425      0.487      0.316    0.499
## HouseAge        -0.583     -0.536      -0.424     -0.486     -0.315   -0.498
## MasVnrArea       0.353      0.386       0.411      0.386      0.426    0.292
## QualityIndex     0.293      0.307       0.251      0.283      0.236    0.286
##              TotRmsAbvGrd YearBuilt HouseAge MasVnrArea QualityIndex
## SalePrice           0.601     0.571   -0.570      0.544        0.525
## logSalePrice        0.618     0.626   -0.625      0.491        0.566
## OverallQual         0.511     0.558   -0.557      0.429        0.694
## TotalFloorSF        0.821     0.301   -0.301      0.429        0.407
## GrLivArea           0.829     0.286   -0.286      0.422        0.403
## TotalBath           0.516     0.584   -0.583      0.353        0.293
## GarageCars          0.425     0.536   -0.536      0.386        0.307
## TotalBsmtSF         0.261     0.425   -0.424      0.411        0.251
## GarageArea          0.366     0.487   -0.486      0.386        0.283
## FirstFlrSF          0.358     0.316   -0.315      0.426        0.236
## FullBath            0.585     0.499   -0.498      0.292        0.286
## TotRmsAbvGrd        1.000     0.233   -0.233      0.322        0.322
## YearBuilt           0.233     1.000   -0.999      0.314        0.105
## HouseAge           -0.233    -0.999    1.000     -0.315       -0.104
## MasVnrArea          0.322     0.314   -0.315      1.000        0.217
## QualityIndex        0.322     0.105   -0.104      0.217        1.000

corrplot(
  corr_matrix,
  method = "color",
  type = "upper",
  tl.col = "black",
  tl.cex = 0.55,
  number.cex = 0.45
)

11.1 Insights

cat("- Correlation clustering among size-related predictors is expected in housing data.\n")

Correlation clustering among size-related predictors is expected in housing data.

cat("- Strong blocks of correlation suggest that some variables may carry overlapping information.\n")

Strong blocks of correlation suggest that some variables may carry overlapping information.

cat("- This is helpful to know now, because regression modeling later may require choosing among similar predictors or combining them through feature engineering.\n")

This is helpful to know now, because regression modeling later may require choosing among similar predictors or combining them through feature engineering.

12 Data quality summaries for selected high-value variables

This section gives compact summaries for the strongest candidate variables for regression.

quality_numeric <- function(x) {
  tibble(
    n = length(x),
    missing_n = sum(is.na(x)),
    missing_pct = mean(is.na(x)) * 100,
    unique_n = n_distinct(x, na.rm = TRUE),
    min = suppressWarnings(min(x, na.rm = TRUE)),
    q1 = suppressWarnings(quantile(x, 0.25, na.rm = TRUE)),
    median = suppressWarnings(median(x, na.rm = TRUE)),
    mean = suppressWarnings(mean(x, na.rm = TRUE)),
    q3 = suppressWarnings(quantile(x, 0.75, na.rm = TRUE)),
    max = suppressWarnings(max(x, na.rm = TRUE)),
    sd = suppressWarnings(sd(x, na.rm = TRUE)),
    skewness = suppressWarnings(moments::skewness(x, na.rm = TRUE))
  )
}

quality_results <- map_dfr(top30_saleprice$Feature[1:15], function(v) {
  quality_numeric(clean_df[[v]]) %>% mutate(variable = v, .before = 1)
})

kable(quality_results, digits = 2)

variable	n	unique_n	min	q1	median	mean	q3	max	sd	skewness
logSalePrice	2001	721	10.46	11.78	11.99	12.02	12.27	13.35	0.37	0.11
OverallQual	2001	10	1.00	5.00	6.00	5.99	7.00	10.00	1.33	0.17
TotalFloorSF	2001	1083	334.00	1107.00	1442.00	1488.07	1755.00	3820.00	489.19	0.81
GrLivArea	2001	1084	334.00	1111.00	1445.00	1492.91	1760.00	3820.00	491.96	0.84
TotalBath	2001	8	1.00	1.50	2.00	2.15	2.50	4.50	0.79	0.22
GarageCars	2001	6	0.00	1.00	2.00	1.74	2.00	5.00	0.72	-0.07
TotalBsmtSF	2001	859	0.00	801.00	973.00	1030.62	1228.00	3206.00	400.27	0.60
GarageArea	2001	537	0.00	312.00	472.00	467.84	576.00	1488.00	202.10	0.28
FirstFlrSF	2001	898	334.00	882.00	1062.00	1144.38	1344.00	3820.00	360.73	1.24
FullBath	2001	4	0.00	1.00	1.00	1.51	2.00	3.00	0.54	0.25
TotRmsAbvGrd	2001	11	2.00	5.00	6.00	6.43	7.00	12.00	1.42	0.60
YearBuilt	2001	113	1872.00	1950.00	1968.00	1967.50	1996.00	2010.00	29.79	-0.47
HouseAge	2001	125	0.00	11.00	40.00	40.37	58.00	136.00	29.76	0.47
MasVnrArea	2001	359	0.00	0.00	0.00	93.02	143.00	1600.00	170.67	2.81
QualityIndex	2001	34	1.00	30.00	35.00	34.22	40.00	90.00	9.29	0.43

12.1 Insights

cat("- Numeric quality summaries help identify skewed, sparse, or low-variance predictors before modeling.\n")

Numeric quality summaries help identify skewed, sparse, or low-variance predictors before modeling.

cat("- Skewness is especially useful here because strongly non-normal predictors may need transformation or winsorization before regression.\n")

Skewness is especially useful here because strongly non-normal predictors may need transformation or winsorization before regression.

cat("- Variables with extreme maxima relative to quartiles should be reviewed for outliers or long-tailed distributions.\n")

Variables with extreme maxima relative to quartiles should be reviewed for outliers or long-tailed distributions.

13 Outlier handling for regression preparation

Outliers can distort least squares regression. At the preparation stage, the goal is not to erase all extremes, but to identify them and decide whether to trim, cap, or transform.

13.1 IQR-based outlier counts for top features

count_outliers_iqr <- function(x) {
  q1 <- quantile(x, 0.25, na.rm = TRUE)
  q3 <- quantile(x, 0.75, na.rm = TRUE)
  iqr <- q3 - q1
  lo <- q1 - 1.5 * iqr
  hi <- q3 + 1.5 * iqr
  sum(x < lo | x > hi, na.rm = TRUE)
}

outlier_tbl <- tibble(
  Feature = top30_saleprice$Feature[1:15],
  OutlierCount = sapply(clean_df[top30_saleprice$Feature[1:15]], count_outliers_iqr)
) %>%
  arrange(desc(OutlierCount))

kable(outlier_tbl)

Feature	OutlierCount
MasVnrArea	148
TotalBsmtSF	101
QualityIndex	86
FirstFlrSF	41
logSalePrice	32
GrLivArea	32
TotalFloorSF	31
TotRmsAbvGrd	20
GarageArea	18
TotalBath	9
GarageCars	8
YearBuilt	7
HouseAge	4
OverallQual	3
FullBath	0

13.2 Boxplots for top continuous features

top_continuous <- top30_saleprice$Feature[1:9]

plot_df <- clean_df %>%
  select(all_of(top_continuous)) %>%
  pivot_longer(everything(), names_to = "Feature", values_to = "Value")

ggplot(plot_df, aes(x = Feature, y = Value)) +
  geom_boxplot(fill = "lightblue") +
  coord_flip() +
  labs(
    title = "Boxplots of Top Continuous Predictors",
    x = NULL,
    y = "Value"
  ) +
  theme_minimal()

13.3 Winsorization helper (prepared, not yet universally applied)

winsorize_iqr <- function(x) {
  q1 <- quantile(x, 0.25, na.rm = TRUE)
  q3 <- quantile(x, 0.75, na.rm = TRUE)
  iqr <- q3 - q1
  lo <- q1 - 1.5 * iqr
  hi <- q3 + 1.5 * iqr
  pmin(pmax(x, lo), hi)
}

# Example prepared versions for likely long-tailed numeric fields
clean_df <- clean_df %>%
  mutate(
    LotArea_w = winsorize_iqr(LotArea),
    TotalBsmtSF_w = winsorize_iqr(TotalBsmtSF),
    GarageArea_w = winsorize_iqr(GarageArea),
    TotalFloorSF_w = winsorize_iqr(TotalFloorSF)
  )

13.4 Insights

cat("- Outlier handling in regression should be deliberate, not automatic.\n")

Outlier handling in regression should be deliberate, not automatic.

cat("- Some extreme observations are informative and should be transformed rather than removed.\n")

Some extreme observations are informative and should be transformed rather than removed.

cat("- Winsorized versions of long-tailed predictors are created here as optional regression-ready alternatives.\n")

Winsorized versions of long-tailed predictors are created here as optional regression-ready alternatives.

14 Non-normality and transformation planning

Regression does not require every predictor to be normal, but strong skewness and nonlinear scale effects often justify transformations.

14.1 Distribution checks

dist_vars <- c("SalePrice", "logSalePrice", "LotArea", "GrLivArea", "TotalBsmtSF", "GarageArea", "HouseAge", "TotalFloorSF")

dist_df <- clean_df %>%
  select(all_of(dist_vars)) %>%
  pivot_longer(everything(), names_to = "Variable", values_to = "Value")

ggplot(dist_df, aes(x = Value)) +
  geom_histogram(bins = 30, fill = "steelblue", color = "white") +
  facet_wrap(~ Variable, scales = "free", ncol = 2) +
  theme_minimal() +
  labs(title = "Distribution Review for Key Variables")

ggplot(clean_df, aes(sample = SalePrice)) +
  stat_qq() +
  stat_qq_line(color = "blue") +
  labs(title = "Q-Q Plot for SalePrice") +
  theme_minimal()

ggplot(clean_df, aes(sample = logSalePrice)) +
  stat_qq() +
  stat_qq_line(color = "blue") +
  labs(title = "Q-Q Plot for logSalePrice") +
  theme_minimal()

14.2 Transformation creation

clean_df <- clean_df %>%
  mutate(
    logLotArea = log1p(LotArea),
    logGrLivArea = log1p(GrLivArea),
    logTotalBsmtSF = log1p(TotalBsmtSF),
    logGarageArea = log1p(GarageArea),
    sqrtHouseAge = sqrt(pmax(HouseAge, 0)),
    logTotalFloorSF = log1p(TotalFloorSF)
  )

summary(select(clean_df, logLotArea, logGrLivArea, logTotalBsmtSF, logGarageArea, sqrtHouseAge, logTotalFloorSF))

##    logLotArea      logGrLivArea   logTotalBsmtSF  logGarageArea  
##  Min.   : 7.824   Min.   :5.814   Min.   :0.000   Min.   :0.000  
##  1st Qu.: 9.003   1st Qu.:7.014   1st Qu.:6.687   1st Qu.:5.746  
##  Median : 9.185   Median :7.277   Median :6.881   Median :6.159  
##  Mean   : 9.204   Mean   :7.256   Mean   :6.742   Mean   :5.889  
##  3rd Qu.: 9.375   3rd Qu.:7.474   3rd Qu.:7.114   3rd Qu.:6.358  
##  Max.   :12.280   Max.   :8.248   Max.   :8.073   Max.   :7.306  
##   sqrtHouseAge    logTotalFloorSF
##  Min.   : 0.000   Min.   :5.814  
##  1st Qu.: 3.317   1st Qu.:7.010  
##  Median : 6.325   Median :7.274  
##  Mean   : 5.785   Mean   :7.253  
##  3rd Qu.: 7.616   3rd Qu.:7.471  
##  Max.   :11.662   Max.   :8.248

14.3 Insights

cat("- `SalePrice` is usually less normal than `logSalePrice`, so both are reviewed during preparation.\n")

SalePrice is usually less normal than logSalePrice, so both are reviewed during preparation.

cat("- Log transforms are especially useful for right-skewed size-related variables.\n")

Log transforms are especially useful for right-skewed size-related variables.

cat("- The square-root transform for age is a softer transformation that can help when the effect of age is nonlinear but not extremely skewed.\n")

The square-root transform for age is a softer transformation that can help when the effect of age is nonlinear but not extremely skewed.

15 Scaling and normalization

Scaling does not change correlation structure, but it becomes important when regression methods are sensitive to measurement units or penalties.

# Z-score scaling for selected engineered numeric predictors
scale_vars <- c(
  "GrLivArea", "LotArea", "TotalBsmtSF", "GarageArea",
  "TotalFloorSF", "HouseAge", "TotalBath", "QualityIndex",
  "logLotArea", "logGrLivArea", "logTotalBsmtSF", "logGarageArea", "logTotalFloorSF"
)

existing_scale_vars <- scale_vars[scale_vars %in% names(clean_df)]

scaled_matrix <- scale(clean_df[, existing_scale_vars])

scaled_df <- as.data.frame(scaled_matrix)
names(scaled_df) <- paste0(existing_scale_vars, "_z")

clean_df <- bind_cols(clean_df, scaled_df)

summary(select(clean_df, ends_with("_z")))[, 1:6]

##   GrLivArea_z         LotArea_z       TotalBsmtSF_z      GarageArea_z     
##  Min.   :-2.35570   Min.   :-1.0625   Min.   :-2.5748   Min.   :-2.31486  
##  1st Qu.:-0.77630   1st Qu.:-0.3444   1st Qu.:-0.5737   1st Qu.:-0.77109  
##  Median :-0.09739   Median :-0.1373   Median :-0.1440   Median : 0.02058  
##  Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.00000  
##  3rd Qu.: 0.54291   3rd Qu.: 0.1232   3rd Qu.: 0.4931   3rd Qu.: 0.53517  
##  Max.   : 4.73024   Max.   :26.0844   Max.   : 5.4348   Max.   : 5.04773  
##  TotalFloorSF_z       HouseAge_z      
##  Min.   :-2.35916   Min.   :-1.35655  
##  1st Qu.:-0.77899   1st Qu.:-0.98696  
##  Median :-0.09418   Median :-0.01259  
##  Mean   : 0.00000   Mean   : 0.00000  
##  3rd Qu.: 0.54565   3rd Qu.: 0.59219  
##  Max.   : 4.76693   Max.   : 3.21290

15.1 Normalization to 0-1 range (optional alternative)

min_max_norm <- function(x) {
  rng <- range(x, na.rm = TRUE)
  if (rng[1] == rng[2]) return(rep(0, length(x)))
  (x - rng[1]) / (rng[2] - rng[1])
}

norm_vars <- c("SalePrice", "GrLivArea", "LotArea", "TotalFloorSF", "GarageArea")
existing_norm_vars <- norm_vars[norm_vars %in% names(clean_df)]

for (v in existing_norm_vars) {
  clean_df[[paste0(v, "_mm")]] <- min_max_norm(clean_df[[v]])
}

summary(select(clean_df, ends_with("_mm")))

##   SalePrice_mm     GrLivArea_mm      LotArea_mm      TotalFloorSF_mm 
##  Min.   :0.0000   Min.   :0.0000   Min.   :0.00000   Min.   :0.0000  
##  1st Qu.:0.1610   1st Qu.:0.2229   1st Qu.:0.02645   1st Qu.:0.2217  
##  Median :0.2148   Median :0.3187   Median :0.03408   Median :0.3178  
##  Mean   :0.2439   Mean   :0.3324   Mean   :0.03914   Mean   :0.3311  
##  3rd Qu.:0.3005   3rd Qu.:0.4091   3rd Qu.:0.04368   3rd Qu.:0.4076  
##  Max.   :1.0000   Max.   :1.0000   Max.   :1.00000   Max.   :1.0000  
##  GarageArea_mm   
##  Min.   :0.0000  
##  1st Qu.:0.2097  
##  Median :0.3172  
##  Mean   :0.3144  
##  3rd Qu.:0.3871  
##  Max.   :1.0000

15.2 Insights

cat("- Standardization produces mean-centered predictors on a common scale and is especially useful for penalized regression methods.\n")

Standardization produces mean-centered predictors on a common scale and is especially useful for penalized regression methods.

cat("- Min-max normalization is optional and mainly useful for algorithms that expect bounded inputs.\n")

Min-max normalization is optional and mainly useful for algorithms that expect bounded inputs.

cat("- For ordinary least squares, scaling is not mandatory for fit quality, but it still helps with interpretability and later method comparisons.\n")

For ordinary least squares, scaling is not mandatory for fit quality, but it still helps with interpretability and later method comparisons.

16 Neighborhood feature engineering for regression readiness

Neighborhood is a strong housing-price signal, but using many sparse levels can make regression unstable. A practical strategy is to bin neighborhoods using price behavior.

16.1 Neighborhood summary table

nbhd_summary <- clean_df %>%
  group_by(Neighborhood) %>%
  summarise(
    n = n(),
    MeanSalePrice = mean(SalePrice, na.rm = TRUE),
    MedianSalePrice = median(SalePrice, na.rm = TRUE),
    MeanPricePerSF = mean(SalePrice / pmax(GrLivArea, 1), na.rm = TRUE),
    .groups = "drop"
  ) %>%
  arrange(MeanPricePerSF)

kable(nbhd_summary, digits = 2)

Neighborhood	n	MeanSalePrice	MedianSalePrice	MeanPricePerSF
SWISU	34	132983.8	135750	93.10
IDOTRR	62	113263.4	118700	94.38
OldTown	177	128156.1	122000	95.32
BrkSide	96	126740.4	127750	103.94
Edwards	129	132956.2	125000	104.54
NWAmes	113	194384.1	185000	116.73
Gilbert	128	189209.6	184050	117.77
Crawfor	78	199021.4	196500	118.20
NAmes	360	146903.7	142000	119.78
SawyerW	89	190508.2	184900	119.88
Sawyer	121	137326.1	135000	125.23
Blmngtn	1	159895.0	159895	126.30
ClearCr	37	218400.9	225000	128.37
NoRidge	66	319616.0	301750	131.90
CollgCr	213	199779.2	200500	134.44
Mitchel	84	165514.9	156225	135.02
Veenker	17	252491.2	255000	140.69
Timber	49	241995.2	214900	143.03
Somerst	67	248517.9	245000	144.30
StoneBr	13	348963.1	349265	165.07
NridgHt	67	345267.9	326000	166.91

16.2 Visualize neighborhood price-per-square-foot signal

ggplot(nbhd_summary, aes(x = reorder(Neighborhood, MeanPricePerSF), y = MeanPricePerSF)) +
  geom_col(fill = "steelblue") +
  coord_flip() +
  labs(
    title = "Neighborhood Mean Price per Square Foot",
    x = "Neighborhood",
    y = "Mean Price per SF"
  ) +
  theme_minimal()

16.3 Create neighborhood bins

# Create quartile-style bins using mean price per square foot
nbhd_summary <- nbhd_summary %>%
  mutate(
    NbhdBin = ntile(MeanPricePerSF, 4),
    NbhdBin = paste0("grp", NbhdBin)
  )

clean_df <- clean_df %>%
  left_join(nbhd_summary %>% select(Neighborhood, NbhdBin), by = "Neighborhood")

clean_df$NbhdBin <- factor(clean_df$NbhdBin, levels = c("grp1", "grp2", "grp3", "grp4"))

table(clean_df$NbhdBin)

## 
## grp1 grp2 grp3 grp4 
##  611  776  401  213

16.4 Compare sale price by neighborhood bin

ggplot(clean_df, aes(x = NbhdBin, y = SalePrice)) +
  geom_boxplot(fill = "lightblue") +
  labs(
    title = "SalePrice by Neighborhood Bin",
    x = "Neighborhood Bin",
    y = "SalePrice"
  ) +
  theme_minimal()

16.5 Dummy variables for later regression use

clean_df <- clean_df %>%
  mutate(
    NbhdBin_grp1 = ifelse(NbhdBin == "grp1", 1, 0),
    NbhdBin_grp2 = ifelse(NbhdBin == "grp2", 1, 0),
    NbhdBin_grp3 = ifelse(NbhdBin == "grp3", 1, 0)
  )

head(select(clean_df, Neighborhood, NbhdBin, NbhdBin_grp1, NbhdBin_grp2, NbhdBin_grp3))

## # A tibble: 6 × 5
##   Neighborhood NbhdBin NbhdBin_grp1 NbhdBin_grp2 NbhdBin_grp3
##   <chr>        <fct>          <dbl>        <dbl>        <dbl>
## 1 NAmes        grp2               0            1            0
## 2 NAmes        grp2               0            1            0
## 3 NAmes        grp2               0            1            0
## 4 NAmes        grp2               0            1            0
## 5 Gilbert      grp2               0            1            0
## 6 Gilbert      grp2               0            1            0

16.6 Insights

cat("- Neighborhood is an important pricing signal, but using all raw categories can create many regression coefficients with uneven group sizes.\n")

Neighborhood is an important pricing signal, but using all raw categories can create many regression coefficients with uneven group sizes.

cat("- Binning neighborhoods by average price-per-square-foot preserves much of the market signal while reducing dimensionality.\n")

Binning neighborhoods by average price-per-square-foot preserves much of the market signal while reducing dimensionality.

cat("- The grouped neighborhood feature is now ready for later regression modeling without fitting that model here.\n")

The grouped neighborhood feature is now ready for later regression modeling without fitting that model here.

17 Final regression-preparation dataset

This section assembles a curated set of variables that are prepared for later modeling work. No model is fit in this report.

prepared_vars <- c(
  "SalePrice", "logSalePrice",
  "OverallQual", "OverallCond",
  "GrLivArea", "LotArea", "TotalBsmtSF", "GarageArea", "GarageCars",
  "TotalFloorSF", "HouseAge", "RemodAge", "TotalBath", "QualityIndex",
  "logLotArea", "logGrLivArea", "logTotalBsmtSF", "logGarageArea", "logTotalFloorSF",
  "LotArea_w", "TotalBsmtSF_w", "GarageArea_w", "TotalFloorSF_w",
  "Neighborhood", "NbhdBin", "NbhdBin_grp1", "NbhdBin_grp2", "NbhdBin_grp3"
)

prepared_vars <- prepared_vars[prepared_vars %in% names(clean_df)]
prepared_df <- clean_df %>% select(all_of(prepared_vars))

dim(prepared_df)

## [1] 2001   28

head(prepared_df, 5)

## # A tibble: 5 × 28
##   SalePrice logSalePrice OverallQual OverallCond GrLivArea LotArea TotalBsmtSF
##       <dbl>        <dbl>       <dbl>       <dbl>     <dbl>   <dbl>       <dbl>
## 1    215000         12.3           6           5      1656   31770        1080
## 2    105000         11.6           5           6       896   11622         882
## 3    172000         12.1           6           6      1329   14267        1329
## 4    244000         12.4           7           5      2110   11160        2110
## 5    189900         12.2           5           5      1629   13830         928
## # ℹ 21 more variables: GarageArea <dbl>, GarageCars <dbl>, TotalFloorSF <dbl>,
## #   HouseAge <dbl>, RemodAge <dbl>, TotalBath <dbl>, QualityIndex <dbl>,
## #   logLotArea <dbl>, logGrLivArea <dbl>, logTotalBsmtSF <dbl>,
## #   logGarageArea <dbl>, logTotalFloorSF <dbl>, LotArea_w <dbl>,
## #   TotalBsmtSF_w <dbl>, GarageArea_w <dbl>, TotalFloorSF_w <dbl>,
## #   Neighborhood <chr>, NbhdBin <fct>, NbhdBin_grp1 <dbl>, NbhdBin_grp2 <dbl>,
## #   NbhdBin_grp3 <dbl>

# Optional export for later regression modeling
write.csv(prepared_df, "ames_prepared_for_regression.csv", row.names = FALSE)

17.1 Insights

cat(
  paste0(
    "- The final prepared dataset contains **", nrow(prepared_df), "** rows and **", ncol(prepared_df), "** selected columns.\n",
    "- This report intentionally stops after preparation and feature engineering.\n",
    "- The next step, if needed, would be regression modeling using the prepared dataset and a carefully chosen subset of predictors.\n"
  )
)

The final prepared dataset contains 2001 rows and 28 selected columns.
This report intentionally stops after preparation and feature engineering.
The next step, if needed, would be regression modeling using the prepared dataset and a carefully chosen subset of predictors.

18 Conclusion

This report focused on preparing Ames housing data for regression rather than fitting a final model. The process included profiling, sample definition, missingness treatment, rule-based validation, outlier review, transformation planning, scaling, and neighborhood binning. The top 30 correlations with SalePrice were used as a screening device to identify strong candidate predictors, while additional feature engineering created more regression-ready versions of size, age, quality, and neighborhood effects.

A key takeaway is that regression preparation should not be reduced to a single cleaning step. Different issues require different responses: some missing values reflect absent home features, some extreme values should be transformed instead of deleted, and some categorical predictors should be grouped before modeling. By separating these tasks clearly, the resulting dataset is better aligned with the assumptions and practical needs of future regression work.

19 Comparing Correlation Selection of Features with PCA feature selection

PCA is a good screening tool here, PCA does not tell you which variables are best for predicting SalePrice directly. It tells you whether your prepared variables are capturing the main structure and variation among the predictors.

Correlation with SalePrice finds variables individually associated with the target PCA finds combinations of predictors that explain shared variation among the predictors

So PCA may reveal:

redundant variables
hidden feature groups
variables that matter structurally even if their individual correlation with SalePrice is modest

############################################################
# PCA on prepared predictors
############################################################

# Choose numeric predictors only
pca_df <- prepared_df %>%
  select(where(is.numeric)) %>%
  select(-SalePrice, -logSalePrice)   # exclude response variables

pca_df <- na.omit(pca_df)
pca_result <- prcomp(pca_df, center = TRUE, scale. = TRUE)

pca_var <- pca_result$sdev^2
pca_var_exp <- pca_var / sum(pca_var)

plot(pca_var_exp,
     type = "b",
     pch = 19,
     xlab = "Principal Component",
     ylab = "Proportion of Variance Explained",
     main = "Scree Plot")

cumsum(pca_var_exp)

##  [1] 0.4073825 0.5148708 0.6053127 0.6774651 0.7451492 0.8088657 0.8652442
##  [8] 0.8942601 0.9149812 0.9329241 0.9495006 0.9638727 0.9771641 0.9852185
## [15] 0.9911587 0.9951439 0.9970641 0.9979346 0.9985777 0.9991116 0.9996339
## [22] 0.9998360 0.9999903 1.0000000

loadings <- as.data.frame(pca_result$rotation)
round(loadings, 3)

##                    PC1    PC2    PC3    PC4    PC5    PC6    PC7    PC8    PC9
## OverallQual     -0.253  0.041  0.215  0.058 -0.038 -0.088  0.090 -0.161  0.507
## OverallCond      0.062  0.251  0.195  0.145 -0.582 -0.056  0.122  0.190 -0.186
## GrLivArea       -0.271  0.286 -0.080 -0.024  0.124 -0.052 -0.059  0.100 -0.018
## LotArea         -0.109 -0.068 -0.455  0.140 -0.221  0.094  0.176 -0.098 -0.083
## TotalBsmtSF     -0.207 -0.190  0.103  0.430  0.073  0.156 -0.150 -0.023  0.123
## GarageArea      -0.245 -0.175  0.094 -0.274 -0.151  0.196 -0.144 -0.014  0.013
## GarageCars      -0.251 -0.151  0.092 -0.262 -0.118  0.114 -0.120 -0.079  0.017
## TotalFloorSF    -0.273  0.281 -0.077 -0.026  0.123 -0.058 -0.055  0.097 -0.015
## HouseAge         0.190  0.300 -0.126  0.084 -0.192  0.209 -0.215  0.241 -0.065
## RemodAge         0.163  0.012 -0.232  0.024  0.071  0.209 -0.396  0.413  0.394
## TotalBath       -0.241 -0.002  0.032 -0.004  0.140 -0.132  0.104 -0.202 -0.373
## QualityIndex    -0.148  0.230  0.285  0.146 -0.447 -0.105  0.147  0.014  0.294
## logLotArea      -0.176 -0.116 -0.483  0.101 -0.225  0.039  0.124 -0.064  0.003
## logGrLivArea    -0.274  0.272 -0.070 -0.019  0.124 -0.066 -0.072  0.088 -0.031
## logTotalBsmtSF  -0.120 -0.130  0.212  0.481  0.037  0.078 -0.174  0.175 -0.449
## logGarageArea   -0.174 -0.173  0.075 -0.301 -0.253  0.103 -0.248  0.232 -0.238
## logTotalFloorSF -0.275  0.267 -0.067 -0.020  0.123 -0.073 -0.067  0.087 -0.029
## LotArea_w       -0.179 -0.101 -0.430  0.100 -0.203  0.028  0.088 -0.069  0.056
## TotalBsmtSF_w   -0.208 -0.202  0.094  0.418  0.073  0.159 -0.141 -0.024  0.136
## GarageArea_w    -0.247 -0.176  0.096 -0.272 -0.150  0.190 -0.139 -0.013  0.013
## TotalFloorSF_w  -0.274  0.277 -0.072 -0.026  0.131 -0.061 -0.056  0.093 -0.016
## NbhdBin_grp1     0.119  0.349  0.016  0.001 -0.005  0.511 -0.079 -0.328 -0.129
## NbhdBin_grp2     0.052 -0.152 -0.113  0.028 -0.135 -0.637 -0.372  0.101 -0.037
## NbhdBin_grp3    -0.097 -0.167  0.050 -0.061  0.161  0.148  0.579  0.630 -0.001
##                   PC10   PC11   PC12   PC13   PC14   PC15   PC16   PC17   PC18
## OverallQual     -0.288  0.104 -0.307  0.011  0.264  0.078  0.069  0.004  0.000
## OverallCond      0.143  0.051  0.288  0.005 -0.334 -0.055 -0.066 -0.047 -0.012
## GrLivArea        0.045 -0.078  0.030 -0.018 -0.057  0.018 -0.018  0.397  0.089
## LotArea         -0.633 -0.385  0.170 -0.028 -0.049  0.006  0.026  0.018 -0.261
## TotalBsmtSF      0.144 -0.111  0.273  0.179  0.056 -0.040  0.024  0.101 -0.083
## GarageArea       0.078 -0.075  0.114 -0.332 -0.032  0.331  0.079 -0.035 -0.018
## GarageCars      -0.015 -0.038 -0.022 -0.205 -0.008 -0.860 -0.067  0.008  0.006
## TotalFloorSF     0.040 -0.076  0.029 -0.011 -0.070  0.017 -0.020  0.414  0.056
## HouseAge         0.074 -0.120  0.053 -0.192  0.763 -0.036 -0.123 -0.031  0.019
## RemodAge        -0.285  0.445  0.136 -0.092 -0.279 -0.029 -0.049 -0.020 -0.005
## TotalBath       -0.230  0.629  0.432 -0.064  0.269  0.016 -0.031 -0.016  0.006
## QualityIndex    -0.127  0.142 -0.039 -0.016  0.034  0.014  0.053  0.033  0.022
## logLotArea       0.115  0.111 -0.138  0.003 -0.014  0.013  0.026 -0.057  0.757
## logGrLivArea     0.020 -0.070 -0.039  0.051 -0.053 -0.010  0.023 -0.545 -0.032
## logTotalBsmtSF  -0.189  0.070 -0.515 -0.316 -0.111  0.033 -0.047  0.024  0.007
## logGarageArea   -0.164  0.063 -0.179  0.712  0.117  0.076 -0.054  0.070 -0.008
## logTotalFloorSF  0.016 -0.069 -0.040  0.055 -0.062 -0.008  0.022 -0.523 -0.064
## LotArea_w        0.445  0.330 -0.254  0.004  0.034  0.002 -0.030  0.052 -0.566
## TotalBsmtSF_w    0.165 -0.112  0.296  0.202  0.049 -0.045  0.072 -0.103  0.092
## GarageArea_w     0.074 -0.073  0.110 -0.306 -0.027  0.335  0.071 -0.035 -0.006
## TotalFloorSF_w   0.035 -0.071  0.005 -0.003 -0.068  0.008 -0.042  0.237 -0.060
## NbhdBin_grp1    -0.014  0.122 -0.090  0.102 -0.081 -0.065  0.652  0.045 -0.007
## NbhdBin_grp2     0.010 -0.052  0.052 -0.077  0.088 -0.061  0.601  0.042 -0.027
## NbhdBin_grp3     0.026  0.019  0.028 -0.033  0.118 -0.085  0.390  0.007 -0.023
##                   PC19   PC20   PC21   PC22   PC23   PC24
## OverallQual     -0.174  0.080  0.527  0.001  0.009 -0.002
## OverallCond     -0.157  0.063  0.437 -0.004  0.003 -0.003
## GrLivArea        0.096  0.605 -0.039 -0.078  0.012 -0.494
## LotArea          0.041  0.019  0.003  0.007 -0.003  0.000
## TotalBsmtSF     -0.662  0.028 -0.236  0.009  0.010 -0.003
## GarageArea       0.004 -0.015 -0.004  0.033  0.699 -0.001
## GarageCars       0.000  0.004  0.003  0.005 -0.007 -0.001
## TotalFloorSF     0.041 -0.135  0.056  0.584 -0.030  0.504
## HouseAge        -0.008 -0.013  0.041  0.004 -0.003  0.002
## RemodAge        -0.004  0.003  0.011  0.001 -0.004 -0.001
## TotalBath       -0.003  0.000  0.010  0.000  0.002  0.000
## QualityIndex     0.222 -0.092 -0.634 -0.003 -0.005  0.004
## logLotArea      -0.116 -0.062 -0.009 -0.036  0.009 -0.001
## logGrLivArea    -0.032  0.408 -0.095 -0.272  0.021  0.500
## logTotalBsmtSF   0.032  0.001  0.000  0.004 -0.003  0.001
## logGarageArea    0.011  0.003 -0.009 -0.002  0.014 -0.001
## logTotalFloorSF -0.075 -0.315 -0.009  0.407 -0.024 -0.501
## LotArea_w        0.074  0.048  0.020  0.027 -0.006  0.001
## TotalBsmtSF_w    0.645 -0.037  0.222 -0.015 -0.005  0.002
## GarageArea_w    -0.016  0.007  0.001 -0.036 -0.713  0.001
## TotalFloorSF_w  -0.020 -0.566  0.084 -0.639  0.022 -0.010
## NbhdBin_grp1    -0.020 -0.010  0.019 -0.011 -0.004 -0.001
## NbhdBin_grp2    -0.017 -0.004  0.022 -0.012 -0.003  0.000
## NbhdBin_grp3    -0.019 -0.003  0.008 -0.006 -0.001  0.000

top_pc1 <- sort(abs(pca_result$rotation[, 1]), decreasing = TRUE)
top_pc2 <- sort(abs(pca_result$rotation[, 2]), decreasing = TRUE)
top_pc3 <- sort(abs(pca_result$rotation[, 3]), decreasing = TRUE)

head(top_pc1, 10)

## logTotalFloorSF  TotalFloorSF_w    logGrLivArea    TotalFloorSF       GrLivArea 
##       0.2753515       0.2738067       0.2737812       0.2728143       0.2707104 
##     OverallQual      GarageCars    GarageArea_w      GarageArea       TotalBath 
##       0.2531222       0.2510773       0.2469418       0.2445229       0.2407781

head(top_pc2, 10)

##    NbhdBin_grp1        HouseAge       GrLivArea    TotalFloorSF  TotalFloorSF_w 
##       0.3493084       0.3004936       0.2857682       0.2805649       0.2772399 
##    logGrLivArea logTotalFloorSF     OverallCond    QualityIndex   TotalBsmtSF_w 
##       0.2719289       0.2673730       0.2507143       0.2301698       0.2016655

head(top_pc3, 10)

##     logLotArea        LotArea      LotArea_w   QualityIndex       RemodAge 
##      0.4826232      0.4545469      0.4299858      0.2851036      0.2321680 
##    OverallQual logTotalBsmtSF    OverallCond       HouseAge   NbhdBin_grp2 
##      0.2153130      0.2120779      0.1953933      0.1256409      0.1129467

pca_scores <- as.data.frame(pca_result$x)
head(pca_scores)

##          PC1         PC2        PC3        PC4        PC5        PC6        PC7
## 1 -1.8286055 -1.01887502 -4.4608476  0.7135814 -1.3906646 -0.2180562 -0.7614296
## 2  2.1595599 -2.45819030 -0.6257555 -0.6299154 -1.8219632  0.1574683 -1.2567231
## 3  0.7000763 -0.67238656 -1.6125461  2.1145901 -0.8097067 -0.7309959 -0.8641991
## 4 -3.7266429 -0.04073173 -0.2477599  2.1591671  1.1786634 -0.6269818 -2.2496657
## 5 -1.0117876 -0.84249666 -1.6847881 -0.4726617  0.3142807 -1.3751885 -0.3919521
## 6 -0.7888145 -0.11115333  0.1868705 -0.3805976 -0.1814624 -1.7489175 -0.3069621
##          PC8        PC9        PC10        PC11        PC12        PC13
## 1  0.1436286  0.3007578 -0.55189188  0.01720992 -0.57847739 -0.38415446
## 2  0.5654555  0.3929074  0.32668953 -0.12077880 -0.18283853 -0.69894053
## 3  0.6570661  0.9307228  0.32401777  0.37993891 -0.28772980  0.48714609
## 4  0.4012078  0.3410760  0.01260334  0.46306744  1.39948365  0.40371765
## 5 -0.5187309 -1.0197127  0.57595712 -0.02309341 -0.35041140  0.01773298
## 6 -0.3010986 -0.4731845 -0.03589646 -0.02833866 -0.03281889  0.04302094
##          PC14        PC15       PC16        PC17        PC18         PC19
## 1 -0.05693487 -0.03635597 0.13627418 -0.15959511  0.35460959 -0.086891369
## 2 -0.32039767  1.68245966 0.22128263  0.13869436 -0.01994984 -0.025982919
## 3 -0.07937385  0.24479200 0.06834309 -0.09521178 -0.11913305  0.047160479
## 4  0.58821136 -0.15160441 0.26978550  0.09640607  0.02411940 -0.294151123
## 5 -0.43874117 -0.17001512 0.13351956 -0.07880150 -0.15600722  0.004024422
## 6 -0.50400763 -0.20029964 0.19023273 -0.09975696  0.02666305 -0.009482895
##           PC20         PC21        PC22          PC23           PC24
## 1 -0.044990106  0.062240364 -0.02200035 -0.0118288707  0.00031718878
## 2 -0.012471994  0.030976454 -0.01981499 -0.0402253576 -0.00094399065
## 3  0.007112304  0.028136981  0.01528668  0.0152036631  0.00065817054
## 4 -0.048957885 -0.009776146 -0.04322388  0.0080840052  0.00129115202
## 5  0.007312706 -0.065939971  0.01185234 -0.0065133420  0.00007505453
## 6  0.001548306 -0.064137981  0.00164218  0.0003139828 -0.00067596038

pca_scores$SalePrice <- clean_df$SalePrice[as.numeric(rownames(pca_scores))]

pc_saleprice_corr <- sapply(pca_scores %>% select(starts_with("PC")), function(x) {
  cor(x, pca_scores$SalePrice, use = "complete.obs")
})

pc_saleprice_corr

##          PC1          PC2          PC3          PC4          PC5          PC6 
## -0.901773338  0.035176400  0.075964893  0.098840837 -0.007259678 -0.044617026 
##          PC7          PC8          PC9         PC10         PC11         PC12 
##  0.067839931 -0.085764555  0.131801016 -0.004481050  0.008407384  0.079305041 
##         PC13         PC14         PC15         PC16         PC17         PC18 
## -0.015986502 -0.008443691  0.013413584 -0.086041337  0.148539144 -0.003550060 
##         PC19         PC20         PC21         PC22         PC23         PC24 
## -0.068917002  0.031852027  0.011041453  0.033528984 -0.003483513  0.011362201

sort(abs(pc_saleprice_corr), decreasing = TRUE)

##         PC1        PC17         PC9         PC4        PC16         PC8 
## 0.901773338 0.148539144 0.131801016 0.098840837 0.086041337 0.085764555 
##        PC12         PC3        PC19         PC7         PC6         PC2 
## 0.079305041 0.075964893 0.068917002 0.067839931 0.044617026 0.035176400 
##        PC22        PC20        PC13        PC15        PC24        PC21 
## 0.033528984 0.031852027 0.015986502 0.013413584 0.011362201 0.011041453 
##        PC14        PC11         PC5        PC10        PC18        PC23 
## 0.008443691 0.008407384 0.007259678 0.004481050 0.003550060 0.003483513

round(pca_result$rotation[, 1], 3)   # if PC1 is important

##     OverallQual     OverallCond       GrLivArea         LotArea     TotalBsmtSF 
##          -0.253           0.062          -0.271          -0.109          -0.207 
##      GarageArea      GarageCars    TotalFloorSF        HouseAge        RemodAge 
##          -0.245          -0.251          -0.273           0.190           0.163 
##       TotalBath    QualityIndex      logLotArea    logGrLivArea  logTotalBsmtSF 
##          -0.241          -0.148          -0.176          -0.274          -0.120 
##   logGarageArea logTotalFloorSF       LotArea_w   TotalBsmtSF_w    GarageArea_w 
##          -0.174          -0.275          -0.179          -0.208          -0.247 
##  TotalFloorSF_w    NbhdBin_grp1    NbhdBin_grp2    NbhdBin_grp3 
##          -0.274           0.119           0.052          -0.097

round(pca_result$rotation[, 2], 3)   # if PC2 is important

##     OverallQual     OverallCond       GrLivArea         LotArea     TotalBsmtSF 
##           0.041           0.251           0.286          -0.068          -0.190 
##      GarageArea      GarageCars    TotalFloorSF        HouseAge        RemodAge 
##          -0.175          -0.151           0.281           0.300           0.012 
##       TotalBath    QualityIndex      logLotArea    logGrLivArea  logTotalBsmtSF 
##          -0.002           0.230          -0.116           0.272          -0.130 
##   logGarageArea logTotalFloorSF       LotArea_w   TotalBsmtSF_w    GarageArea_w 
##          -0.173           0.267          -0.101          -0.202          -0.176 
##  TotalFloorSF_w    NbhdBin_grp1    NbhdBin_grp2    NbhdBin_grp3 
##           0.277           0.349          -0.152          -0.167

20 Focusing on the most important Components.

By looking at the Scree plot the third bend is on 12th component.

############################################################
# Cumulative variance explained (first 12 PCs)
############################################################


# Variance explained by each component
pca_var <- pca_result$sdev^2

# Proportion of variance explained
pca_var_exp <- pca_var / sum(pca_var)

# Cumulative proportion
pca_cum_var <- cumsum(pca_var_exp)

# Create a table for first 12 components
pca_summary <- data.frame(
  PC = paste0("PC", 1:length(pca_var_exp)),
  Variance_Explained = round(pca_var_exp, 4),
  Cumulative_Variance = round(pca_cum_var, 4)
)

# Show first 12
pca_summary[1:12, ]

##      PC Variance_Explained Cumulative_Variance
## 1   PC1             0.4074              0.4074
## 2   PC2             0.1075              0.5149
## 3   PC3             0.0904              0.6053
## 4   PC4             0.0722              0.6775
## 5   PC5             0.0677              0.7451
## 6   PC6             0.0637              0.8089
## 7   PC7             0.0564              0.8652
## 8   PC8             0.0290              0.8943
## 9   PC9             0.0207              0.9150
## 10 PC10             0.0179              0.9329
## 11 PC11             0.0166              0.9495
## 12 PC12             0.0144              0.9639

pca_summary[1:12, ] %>%
  mutate(
    Variance_Explained = scales::percent(Variance_Explained),
    Cumulative_Variance = scales::percent(Cumulative_Variance)
  )

##      PC Variance_Explained Cumulative_Variance
## 1   PC1             40.74%               40.7%
## 2   PC2             10.75%               51.5%
## 3   PC3              9.04%               60.5%
## 4   PC4              7.22%               67.8%
## 5   PC5              6.77%               74.5%
## 6   PC6              6.37%               80.9%
## 7   PC7              5.64%               86.5%
## 8   PC8              2.90%               89.4%
## 9   PC9              2.07%               91.5%
## 10 PC10              1.79%               93.3%
## 11 PC11              1.66%               95.0%
## 12 PC12              1.44%               96.4%

############################################################
# PCA-based feature importance using first 12 components
############################################################

# Number of components to use
k <- 12

# PCA loadings
loadings <- pca_result$rotation[, 1:k]

# Variance explained by each PC
eigvals <- pca_result$sdev^2
var_exp <- eigvals / sum(eigvals)

# Keep first 12 variance proportions
var_exp_k <- var_exp[1:k]

# Weighted importance:
# absolute loading × variance explained by that PC
feature_importance <- data.frame(
  Feature = rownames(loadings),
  Importance = rowSums(abs(loadings) %*% diag(var_exp_k))
)

# Sort from most important to least important
feature_importance <- feature_importance %>%
  arrange(desc(Importance))

feature_importance

##                         Feature Importance
## GarageArea_w       GarageArea_w  0.1827789
## GarageArea           GarageArea  0.1825329
## TotalBsmtSF_w     TotalBsmtSF_w  0.1804486
## TotalBsmtSF         TotalBsmtSF  0.1801672
## QualityIndex       QualityIndex  0.1782060
## HouseAge               HouseAge  0.1780632
## logGarageArea     logGarageArea  0.1737494
## GarageCars           GarageCars  0.1717215
## OverallQual         OverallQual  0.1709620
## TotalFloorSF       TotalFloorSF  0.1708581
## GrLivArea             GrLivArea  0.1708542
## LotArea_w             LotArea_w  0.1706935
## TotalFloorSF_w   TotalFloorSF_w  0.1705316
## logGrLivArea       logGrLivArea  0.1704284
## logTotalFloorSF logTotalFloorSF  0.1703138
## logLotArea           logLotArea  0.1677544
## RemodAge               RemodAge  0.1655771
## logTotalBsmtSF   logTotalBsmtSF  0.1602580
## TotalBath             TotalBath  0.1596354
## LotArea                 LotArea  0.1587646
## OverallCond         OverallCond  0.1471269
## NbhdBin_grp1       NbhdBin_grp1  0.1407924
## NbhdBin_grp3       NbhdBin_grp3  0.1387341
## NbhdBin_grp2       NbhdBin_grp2  0.1257626

top20_pca_importance <- feature_importance %>%
  slice(1:20)

top20_pca_importance

##                         Feature Importance
## GarageArea_w       GarageArea_w  0.1827789
## GarageArea           GarageArea  0.1825329
## TotalBsmtSF_w     TotalBsmtSF_w  0.1804486
## TotalBsmtSF         TotalBsmtSF  0.1801672
## QualityIndex       QualityIndex  0.1782060
## HouseAge               HouseAge  0.1780632
## logGarageArea     logGarageArea  0.1737494
## GarageCars           GarageCars  0.1717215
## OverallQual         OverallQual  0.1709620
## TotalFloorSF       TotalFloorSF  0.1708581
## GrLivArea             GrLivArea  0.1708542
## LotArea_w             LotArea_w  0.1706935
## TotalFloorSF_w   TotalFloorSF_w  0.1705316
## logGrLivArea       logGrLivArea  0.1704284
## logTotalFloorSF logTotalFloorSF  0.1703138
## logLotArea           logLotArea  0.1677544
## RemodAge               RemodAge  0.1655771
## logTotalBsmtSF   logTotalBsmtSF  0.1602580
## TotalBath             TotalBath  0.1596354
## LotArea                 LotArea  0.1587646

library(ggplot2)

feature_importance %>%
  slice(1:20) %>%
  ggplot(aes(x = reorder(Feature, Importance), y = Importance)) +
  geom_col(fill = "steelblue") +
  coord_flip() +
  labs(
    title = "Top 20 PCA-Based Feature Importances (PC1-PC12)",
    x = "Feature",
    y = "Importance"
  ) +
  theme_minimal()

feature_importance <- data.frame(
  Feature = rownames(loadings),
  Importance = rowSums(sweep(abs(loadings), 2, var_exp_k, FUN = "*"))
) %>%
  mutate(RelativeImportance = Importance / sum(Importance)) %>%
  arrange(desc(RelativeImportance))

feature_importance

##                         Feature Importance RelativeImportance
## GarageArea_w       GarageArea_w  0.1827789         0.04584700
## GarageArea           GarageArea  0.1825329         0.04578528
## TotalBsmtSF_w     TotalBsmtSF_w  0.1804486         0.04526247
## TotalBsmtSF         TotalBsmtSF  0.1801672         0.04519190
## QualityIndex       QualityIndex  0.1782060         0.04469996
## HouseAge               HouseAge  0.1780632         0.04466415
## logGarageArea     logGarageArea  0.1737494         0.04358210
## GarageCars           GarageCars  0.1717215         0.04307344
## OverallQual         OverallQual  0.1709620         0.04288292
## TotalFloorSF       TotalFloorSF  0.1708581         0.04285687
## GrLivArea             GrLivArea  0.1708542         0.04285589
## LotArea_w             LotArea_w  0.1706935         0.04281559
## TotalFloorSF_w   TotalFloorSF_w  0.1705316         0.04277496
## logGrLivArea       logGrLivArea  0.1704284         0.04274907
## logTotalFloorSF logTotalFloorSF  0.1703138         0.04272034
## logLotArea           logLotArea  0.1677544         0.04207835
## RemodAge               RemodAge  0.1655771         0.04153222
## logTotalBsmtSF   logTotalBsmtSF  0.1602580         0.04019800
## TotalBath             TotalBath  0.1596354         0.04004185
## LotArea                 LotArea  0.1587646         0.03982342
## OverallCond         OverallCond  0.1471269         0.03690428
## NbhdBin_grp1       NbhdBin_grp1  0.1407924         0.03531540
## NbhdBin_grp3       NbhdBin_grp3  0.1387341         0.03479911
## NbhdBin_grp2       NbhdBin_grp2  0.1257626         0.03154542

feature_importance %>%
  mutate(RelativeImportance = scales::percent(RelativeImportance))

##                         Feature Importance RelativeImportance
## GarageArea_w       GarageArea_w  0.1827789          4.584700%
## GarageArea           GarageArea  0.1825329          4.578528%
## TotalBsmtSF_w     TotalBsmtSF_w  0.1804486          4.526247%
## TotalBsmtSF         TotalBsmtSF  0.1801672          4.519190%
## QualityIndex       QualityIndex  0.1782060          4.469996%
## HouseAge               HouseAge  0.1780632          4.466415%
## logGarageArea     logGarageArea  0.1737494          4.358210%
## GarageCars           GarageCars  0.1717215          4.307344%
## OverallQual         OverallQual  0.1709620          4.288292%
## TotalFloorSF       TotalFloorSF  0.1708581          4.285687%
## GrLivArea             GrLivArea  0.1708542          4.285589%
## LotArea_w             LotArea_w  0.1706935          4.281559%
## TotalFloorSF_w   TotalFloorSF_w  0.1705316          4.277496%
## logGrLivArea       logGrLivArea  0.1704284          4.274907%
## logTotalFloorSF logTotalFloorSF  0.1703138          4.272034%
## logLotArea           logLotArea  0.1677544          4.207835%
## RemodAge               RemodAge  0.1655771          4.153222%
## logTotalBsmtSF   logTotalBsmtSF  0.1602580          4.019800%
## TotalBath             TotalBath  0.1596354          4.004185%
## LotArea                 LotArea  0.1587646          3.982342%
## OverallCond         OverallCond  0.1471269          3.690428%
## NbhdBin_grp1       NbhdBin_grp1  0.1407924          3.531540%
## NbhdBin_grp3       NbhdBin_grp3  0.1387341          3.479911%
## NbhdBin_grp2       NbhdBin_grp2  0.1257626          3.154542%

20.1 Insights

The cumulative variance explained by the first 12 principal components was examined to assess how much of the total variation in the predictors is captured by a reduced set of components. This helps determine whether the feature space can be effectively summarized or whether important variation is spread across many variables.

PCA-based feature importance was calculated using the absolute variable loadings across the first 12 principal components, weighted by the proportion of variance explained by each component. This provides a measure of how strongly each feature contributes to the main structure of the predictor space. Unlike simple correlation with SalePrice, this approach identifies variables that are influential in explaining the overall variation among predictors, even if their individual association with the response is more modest.

21 PCA versus Correlation

If PCA and top-30 correlation agree:

That means your correlation-based prepared variables are probably capturing the main predictor structure well.

If PCA highlights variables not in the top 30:

That suggests that they may not have a strong individual correlation with SalePrice but they may still be important as part of a broader predictor pattern

For example: one variable may be weak alone but strongly define a component like house size, home quality, or garage/basement capacity. That means the top-30 correlation screen may have missed some structural contributors.

############################################################
# PCA feature importance from first 12 components
############################################################

k <- 12

loadings <- pca_result$rotation[, 1:k]

eigvals <- pca_result$sdev^2
var_exp <- eigvals / sum(eigvals)
var_exp_k <- var_exp[1:k]

pca_importance <- data.frame(
  Feature = rownames(loadings),
  PCA_Importance = rowSums(sweep(abs(loadings), 2, var_exp_k, FUN = "*"))
) %>%
  arrange(desc(PCA_Importance))
############################################################
# PCA feature importance from first 12 components
############################################################

k <- 12

loadings <- pca_result$rotation[, 1:k]

eigvals <- pca_result$sdev^2
var_exp <- eigvals / sum(eigvals)
var_exp_k <- var_exp[1:k]

pca_importance <- data.frame(
  Feature = rownames(loadings),
  PCA_Importance = rowSums(sweep(abs(loadings), 2, var_exp_k, FUN = "*"))
) %>%
  arrange(desc(PCA_Importance))
############################################################
# Correlation table from earlier work
############################################################

corr_importance <- top30_saleprice %>%
  select(Feature, Correlation, abs_corr)
corr_importance <- top30_saleprice %>%
  mutate(abs_corr = abs(Correlation)) %>%
  select(Feature, Correlation, abs_corr)
############################################################
# Merge PCA importance with correlation importance
############################################################

comparison_tbl <- pca_importance %>%
  left_join(corr_importance, by = "Feature") %>%
  mutate(
    In_Top30_Correlation = ifelse(!is.na(abs_corr), "Yes", "No"),
    PCA_Rank = rank(-PCA_Importance, ties.method = "first"),
    Corr_Rank = ifelse(is.na(abs_corr), NA, rank(-abs_corr, ties.method = "first"))
  ) %>%
  arrange(desc(PCA_Importance))

comparison_tbl

##            Feature PCA_Importance Correlation  abs_corr In_Top30_Correlation
## 1     GarageArea_w      0.1827789          NA        NA                   No
## 2       GarageArea      0.1825329   0.6420118 0.6420118                  Yes
## 3    TotalBsmtSF_w      0.1804486          NA        NA                   No
## 4      TotalBsmtSF      0.1801672   0.6512665 0.6512665                  Yes
## 5     QualityIndex      0.1782060   0.5247977 0.5247977                  Yes
## 6         HouseAge      0.1780632  -0.5699966 0.5699966                  Yes
## 7    logGarageArea      0.1737494          NA        NA                   No
## 8       GarageCars      0.1717215   0.6664744 0.6664744                  Yes
## 9      OverallQual      0.1709620   0.8058106 0.8058106                  Yes
## 10    TotalFloorSF      0.1708581   0.7780935 0.7780935                  Yes
## 11       GrLivArea      0.1708542   0.7709897 0.7709897                  Yes
## 12       LotArea_w      0.1706935          NA        NA                   No
## 13  TotalFloorSF_w      0.1705316          NA        NA                   No
## 14    logGrLivArea      0.1704284          NA        NA                   No
## 15 logTotalFloorSF      0.1703138          NA        NA                   No
## 16      logLotArea      0.1677544          NA        NA                   No
## 17        RemodAge      0.1655771  -0.5068690 0.5068690                  Yes
## 18  logTotalBsmtSF      0.1602580          NA        NA                   No
## 19       TotalBath      0.1596354   0.7000760 0.7000760                  Yes
## 20         LotArea      0.1587646   0.2901018 0.2901018                  Yes
## 21     OverallCond      0.1471269          NA        NA                   No
## 22    NbhdBin_grp1      0.1407924          NA        NA                   No
## 23    NbhdBin_grp3      0.1387341          NA        NA                   No
## 24    NbhdBin_grp2      0.1257626          NA        NA                   No
##    PCA_Rank Corr_Rank
## 1         1        NA
## 2         2         7
## 3         3        NA
## 4         4         6
## 5         5         9
## 6         6         8
## 7         7        NA
## 8         8         5
## 9         9         1
## 10       10         2
## 11       11         3
## 12       12        NA
## 13       13        NA
## 14       14        NA
## 15       15        NA
## 16       16        NA
## 17       17        10
## 18       18        NA
## 19       19         4
## 20       20        11
## 21       21        NA
## 22       22        NA
## 23       23        NA
## 24       24        NA

21.1 Insights

A comparison was made between the variables identified through the top-30 correlation screen and those identified as important by PCA using the first 12 principal components. Variables appearing in both lists are especially strong candidates for later regression work, since they are both individually associated with SalePrice and important to the overall structure of the predictor space. Variables highlighted by PCA but not by correlation may represent features that were underemphasized by simple univariate screening, suggesting that the original correlation-based selection may not fully capture all structurally relevant information.

22 Before the Final Decision

top30_pca <- pca_importance %>%
  slice(1:30)

compare_top30 <- top30_pca %>%
  left_join(corr_importance, by = "Feature") %>%
  mutate(In_Correlation_Top30 = ifelse(is.na(abs_corr), "No", "Yes"))

compare_top30

##            Feature PCA_Importance Correlation  abs_corr In_Correlation_Top30
## 1     GarageArea_w      0.1827789          NA        NA                   No
## 2       GarageArea      0.1825329   0.6420118 0.6420118                  Yes
## 3    TotalBsmtSF_w      0.1804486          NA        NA                   No
## 4      TotalBsmtSF      0.1801672   0.6512665 0.6512665                  Yes
## 5     QualityIndex      0.1782060   0.5247977 0.5247977                  Yes
## 6         HouseAge      0.1780632  -0.5699966 0.5699966                  Yes
## 7    logGarageArea      0.1737494          NA        NA                   No
## 8       GarageCars      0.1717215   0.6664744 0.6664744                  Yes
## 9      OverallQual      0.1709620   0.8058106 0.8058106                  Yes
## 10    TotalFloorSF      0.1708581   0.7780935 0.7780935                  Yes
## 11       GrLivArea      0.1708542   0.7709897 0.7709897                  Yes
## 12       LotArea_w      0.1706935          NA        NA                   No
## 13  TotalFloorSF_w      0.1705316          NA        NA                   No
## 14    logGrLivArea      0.1704284          NA        NA                   No
## 15 logTotalFloorSF      0.1703138          NA        NA                   No
## 16      logLotArea      0.1677544          NA        NA                   No
## 17        RemodAge      0.1655771  -0.5068690 0.5068690                  Yes
## 18  logTotalBsmtSF      0.1602580          NA        NA                   No
## 19       TotalBath      0.1596354   0.7000760 0.7000760                  Yes
## 20         LotArea      0.1587646   0.2901018 0.2901018                  Yes
## 21     OverallCond      0.1471269          NA        NA                   No
## 22    NbhdBin_grp1      0.1407924          NA        NA                   No
## 23    NbhdBin_grp3      0.1387341          NA        NA                   No
## 24    NbhdBin_grp2      0.1257626          NA        NA                   No

corr_only <- corr_importance %>%
  filter(!Feature %in% top30_pca$Feature)

corr_only

## # A tibble: 19 × 3
##    Feature        Correlation abs_corr
##    <chr>                <dbl>    <dbl>
##  1 logSalePrice         0.958    0.958
##  2 FirstFlrSF           0.640    0.640
##  3 FullBath             0.611    0.611
##  4 TotRmsAbvGrd         0.601    0.601
##  5 YearBuilt            0.571    0.571
##  6 MasVnrArea           0.544    0.544
##  7 GarageYrBlt          0.522    0.522
##  8 GarageAge           -0.521    0.521
##  9 YearRemodel          0.507    0.507
## 10 Fireplaces           0.484    0.484
## 11 BsmtFinSF1           0.454    0.454
## 12 TotalOutdoorSF       0.405    0.405
## 13 LotFrontage          0.387    0.387
## 14 WoodDeckSF           0.350    0.350
## 15 SecondFlrSF          0.350    0.350
## 16 HalfBath             0.349    0.349
## 17 OpenPorchSF          0.342    0.342
## 18 BsmtFullBath         0.294    0.294
## 19 BedroomAbvGr         0.273    0.273

23 Revisit your Selected features - Final Veification

The NA values in the comparison table do not indicate an error in the PCA. They occur because the PCA was run on an expanded set of engineered predictors, including winsorized, transformed, and neighborhood-bin variables, while the earlier correlation table was based on the original top-30 correlation screen. As a result, some PCA-important variables did not yet have corresponding correlation values in the original comparison table. To make the comparison valid, correlations with SalePrice should be recalculated for the same variables used in the PCA. Even so, the current output already suggests that the original correlation screen identified many strong variables, while PCA additionally highlighted the importance of engineered features such as log-transformed, winsorized, and neighborhood-group variables.

The scree plot was used to evaluate how much variance each principal component explains and to determine whether a relatively small number of components captures most of the information in the predictor space.

############################################################
# Recalculate correlation for all PCA features
############################################################

# Correlation of every PCA variable with SalePrice
corr_all_pca_features <- data.frame(
  Feature = colnames(pca_df),
  Correlation = sapply(pca_df, function(x) {
    cor(x, clean_df$SalePrice[as.numeric(rownames(pca_df))], use = "pairwise.complete.obs")
  })
) %>%
  mutate(abs_corr = abs(Correlation))

corr_all_pca_features############################################################

##                         Feature Correlation  abs_corr
## OverallQual         OverallQual   0.8058106 0.8058106
## OverallCond         OverallCond  -0.1162303 0.1162303
## GrLivArea             GrLivArea   0.7709897 0.7709897
## LotArea                 LotArea   0.2901018 0.2901018
## TotalBsmtSF         TotalBsmtSF   0.6512665 0.6512665
## GarageArea           GarageArea   0.6420118 0.6420118
## GarageCars           GarageCars   0.6664744 0.6664744
## TotalFloorSF       TotalFloorSF   0.7780935 0.7780935
## HouseAge               HouseAge  -0.5699966 0.5699966
## RemodAge               RemodAge  -0.5068690 0.5068690
## TotalBath             TotalBath   0.7000760 0.7000760
## QualityIndex       QualityIndex   0.5247977 0.5247977
## logLotArea           logLotArea   0.4564209 0.4564209
## logGrLivArea       logGrLivArea   0.7452188 0.7452188
## logTotalBsmtSF   logTotalBsmtSF   0.3265755 0.3265755
## logGarageArea     logGarageArea   0.3775568 0.3775568
## logTotalFloorSF logTotalFloorSF   0.7506574 0.7506574
## LotArea_w             LotArea_w   0.4723528 0.4723528
## TotalBsmtSF_w     TotalBsmtSF_w   0.6321503 0.6321503
## GarageArea_w       GarageArea_w   0.6501319 0.6501319
## TotalFloorSF_w   TotalFloorSF_w   0.7714276 0.7714276
## NbhdBin_grp1       NbhdBin_grp1  -0.3576047 0.3576047
## NbhdBin_grp2       NbhdBin_grp2  -0.1793344 0.1793344
## NbhdBin_grp3       NbhdBin_grp3   0.2430058 0.2430058

# Recalculate correlation for all PCA features
############################################################

# Correlation of every PCA variable with SalePrice
corr_all_pca_features <- data.frame(
  Feature = colnames(pca_df),
  Correlation = sapply(pca_df, function(x) {
    cor(x, clean_df$SalePrice[as.numeric(rownames(pca_df))], use = "pairwise.complete.obs")
  })
) %>%
  mutate(abs_corr = abs(Correlation))

corr_all_pca_features

##                         Feature Correlation  abs_corr
## OverallQual         OverallQual   0.8058106 0.8058106
## OverallCond         OverallCond  -0.1162303 0.1162303
## GrLivArea             GrLivArea   0.7709897 0.7709897
## LotArea                 LotArea   0.2901018 0.2901018
## TotalBsmtSF         TotalBsmtSF   0.6512665 0.6512665
## GarageArea           GarageArea   0.6420118 0.6420118
## GarageCars           GarageCars   0.6664744 0.6664744
## TotalFloorSF       TotalFloorSF   0.7780935 0.7780935
## HouseAge               HouseAge  -0.5699966 0.5699966
## RemodAge               RemodAge  -0.5068690 0.5068690
## TotalBath             TotalBath   0.7000760 0.7000760
## QualityIndex       QualityIndex   0.5247977 0.5247977
## logLotArea           logLotArea   0.4564209 0.4564209
## logGrLivArea       logGrLivArea   0.7452188 0.7452188
## logTotalBsmtSF   logTotalBsmtSF   0.3265755 0.3265755
## logGarageArea     logGarageArea   0.3775568 0.3775568
## logTotalFloorSF logTotalFloorSF   0.7506574 0.7506574
## LotArea_w             LotArea_w   0.4723528 0.4723528
## TotalBsmtSF_w     TotalBsmtSF_w   0.6321503 0.6321503
## GarageArea_w       GarageArea_w   0.6501319 0.6501319
## TotalFloorSF_w   TotalFloorSF_w   0.7714276 0.7714276
## NbhdBin_grp1       NbhdBin_grp1  -0.3576047 0.3576047
## NbhdBin_grp2       NbhdBin_grp2  -0.1793344 0.1793344
## NbhdBin_grp3       NbhdBin_grp3   0.2430058 0.2430058

comparison_tbl <- pca_importance %>%
  left_join(corr_all_pca_features, by = "Feature") %>%
  arrange(desc(PCA_Importance))

comparison_tbl

##            Feature PCA_Importance Correlation  abs_corr
## 1     GarageArea_w      0.1827789   0.6501319 0.6501319
## 2       GarageArea      0.1825329   0.6420118 0.6420118
## 3    TotalBsmtSF_w      0.1804486   0.6321503 0.6321503
## 4      TotalBsmtSF      0.1801672   0.6512665 0.6512665
## 5     QualityIndex      0.1782060   0.5247977 0.5247977
## 6         HouseAge      0.1780632  -0.5699966 0.5699966
## 7    logGarageArea      0.1737494   0.3775568 0.3775568
## 8       GarageCars      0.1717215   0.6664744 0.6664744
## 9      OverallQual      0.1709620   0.8058106 0.8058106
## 10    TotalFloorSF      0.1708581   0.7780935 0.7780935
## 11       GrLivArea      0.1708542   0.7709897 0.7709897
## 12       LotArea_w      0.1706935   0.4723528 0.4723528
## 13  TotalFloorSF_w      0.1705316   0.7714276 0.7714276
## 14    logGrLivArea      0.1704284   0.7452188 0.7452188
## 15 logTotalFloorSF      0.1703138   0.7506574 0.7506574
## 16      logLotArea      0.1677544   0.4564209 0.4564209
## 17        RemodAge      0.1655771  -0.5068690 0.5068690
## 18  logTotalBsmtSF      0.1602580   0.3265755 0.3265755
## 19       TotalBath      0.1596354   0.7000760 0.7000760
## 20         LotArea      0.1587646   0.2901018 0.2901018
## 21     OverallCond      0.1471269  -0.1162303 0.1162303
## 22    NbhdBin_grp1      0.1407924  -0.3576047 0.3576047
## 23    NbhdBin_grp3      0.1387341   0.2430058 0.2430058
## 24    NbhdBin_grp2      0.1257626  -0.1793344 0.1793344

############################################################
# Scree plot
############################################################

# Variance explained by each principal component
pca_var <- pca_result$sdev^2

# Proportion of variance explained
pca_var_exp <- pca_var / sum(pca_var)

# Scree plot


scree_df <- data.frame(
  PC = 1:length(pca_var_exp),
  VarianceExplained = pca_var_exp,
  CumulativeVariance = cumsum(pca_var_exp)
)

ggplot(scree_df, aes(x = PC, y = VarianceExplained)) +
  geom_line() +
  geom_point() +
  labs(
    title = "Scree Plot",
    x = "Principal Component",
    y = "Proportion of Variance Explained"
  ) +
  theme_minimal()

ggplot(scree_df, aes(x = PC, y = CumulativeVariance)) +
  geom_line() +
  geom_point() +
  labs(
    title = "Cumulative Scree Plot",
    x = "Principal Component",
    y = "Cumulative Proportion of Variance Explained"
  ) +
  theme_minimal()

Ames Housing: Data Preparation for Regression Modeling

Irene Tsapara