Unit 5 — Matrices, Data Frames & Exploratory Data Analysis

Topics Covered

1. Matrices

M <- matrix(1:6, nrow = 2, ncol = 3)        # fills by column
M <- matrix(1:6, nrow = 2, byrow = TRUE)    # fill by row instead
dim(M)        # 2 3
M[1, 2]       # element row 1, col 2
M[ , 2]       # entire 2nd column
M[1, ]        # entire 1st row

1.1 Adding / removing rows and columns

A <- matrix(1:4, 2, 2)
A <- rbind(A, c(5, 6))   # add a row
A <- cbind(A, c(7, 8, 9)) # add a column
A <- A[-1, ]             # remove the 1st row

1.2 Matrix operations

B <- matrix(c(1,2,3,4), 2, 2)
t(B)          # transpose
B %*% B       # matrix multiplication
solve(B)      # inverse (B must be square, non-singular)
det(B)        # determinant
B + B; 3 * B  # element-wise add / scalar multiply
rowSums(B); colMeans(B)

M <- matrix(c(0,12,13,8,20, 12,0,15,28,88, 13,15,0,6,9,
              8,28,6,0,33, 20,88,9,33,0), nrow = 5, byrow = TRUE)
rownames(M) <- colnames(M) <- paste0("C", 1:5)
# shortest non-zero distance:
min(M[M > 0])                 # 6
which(M == 6, arr.ind = TRUE) # rows/cols of the pair (C3, C4)

# prices in 2 shops (rows = items, cols = shops)
price <- matrix(c(1.5,1, 2,2.5, 5,4.5, 16,17), nrow = 4, byrow = TRUE)
# demand (rows = persons, cols = items)
demand <- matrix(c(6,5,3,1, 3,6,3,2, 3,4,3,1), nrow = 3, byrow = TRUE)
cost <- demand %*% price       # 3x2: each person's total in each shop
cost
apply(cost, 1, which.min)      # cheapest shop for each person

2. Data Frames

df <- data.frame(
  name  = c("Asha","Ravi","Sita"),
  marks = c(78, 65, 88),
  pass  = c(TRUE, TRUE, TRUE)
)
str(df)        # structure of the data frame
nrow(df); ncol(df); names(df)

2.1 Accessing, adding and removing

df$marks            # a column by name
df[ , "name"]       # same column by index
df[df$marks > 70, ] # rows where marks > 70 (filtering)
df$grade <- ifelse(df$marks >= 75, "A", "B")  # add a column
df <- df[ , -3]      # remove the 3rd column
df <- df[-2, ]       # remove the 2nd row

2.2 Special functions on data frames

head(df); tail(df)         # first / last rows
summary(df)                # per-column summary
apply(df[ ,c("marks")], 2, mean)  # apply a function over columns
aggregate(marks ~ grade, data = df, FUN = mean)  # group means

2.3 Merging data frames

students <- data.frame(id = 1:3, name = c("A","B","C"))
scores   <- data.frame(id = c(1,2,3), marks = c(60, 75, 90))
merged   <- merge(students, scores, by = "id")  # join on id

marks <- data.frame(
  section = c("A","A","A","B","B","B"),
  student = c(1,2,3,1,2,3),
  M1 = c(46,34,56,43,67,76),
  M2 = c(54,55,66,44,76,68),
  M3 = c(45,55,64,45,78,37)
)
marks$total <- marks$M1 + marks$M2 + marks$M3   # row totals
marks                                            # display
aggregate(total ~ section, marks, max)           # highest total per section

marks$M4 <- c(50, 60, 70, 55, 65, 40)   # new column filled with marks
marks$total <- marks$M1 + marks$M2 + marks$M3 + marks$M4  # recompute
colMeans(marks[ , c("M1","M2","M3","M4")])  # subject averages

3. Factors and Tables

gender <- factor(c("M","F","F","M","M","F"))
levels(gender)        # "F" "M"
table(gender)         # F:3  M:3
# two-way table
region <- factor(c("N","S","N","S","N","S"))
table(gender, region) # cross-tabulation
prop.table(table(gender))  # proportions

4. Exploratory Data Analysis (EDA)

4.1 Central tendency and variability

x <- c(12, 15, 14, 10, 18, 20, 13, 16)
mean(x)              # arithmetic mean
median(x)            # median
range(x)             # min and max -> 10 20
diff(range(x))       # numeric range -> 10
var(x)               # sample variance
sd(x)                # sample standard deviation
quantile(x)          # quartiles
# mode (R has no built-in) :
mode_fn <- function(v){ u <- unique(v); u[which.max(tabulate(match(v,u)))] }

4.2 The summary() function

summary(x)
#   Min. 1st Qu. Median  Mean 3rd Qu.  Max.
#  10.00  12.75  14.50 14.75  16.50  20.00

summary() on a data frame reports these five-number-summary statistics (plus the mean) for every numeric column, and counts for factors — the quickest one-line overview of a dataset.

4.3 Handling missing values and outliers

# missing values
y <- c(3, NA, 7, 9, NA, 12)
mean(y, na.rm = TRUE)        # ignore NAs
y[is.na(y)] <- mean(y, na.rm = TRUE)  # impute with the mean

# outliers via the IQR rule
Q <- quantile(x, c(0.25, 0.75)); IQR <- Q[2] - Q[1]
low <- Q[1] - 1.5*IQR;  high <- Q[2] + 1.5*IQR
x[x < low | x > high]        # values flagged as outliers

4.4 Normalization

# min-max normalization
norm01 <- function(v) (v - min(v)) / (max(v) - min(v))
norm01(x)
# z-score standardization
scale(x)   # built-in: subtracts mean, divides by sd

data <- c(46, 54, 45, 34, 55, 64, 78, 90, 41, 66)
summary(data)          # five-number summary + mean
sd(data)               # spread
boxplot(data)          # visual check for outliers

v <- c(20, NA, 35, 50, NA, 80)
v[is.na(v)] <- median(v, na.rm = TRUE)   # impute with median (robust)
v_scaled <- (v - min(v)) / (max(v) - min(v))  # min-max to [0,1]
round(v_scaled, 3)

5. Data Visualization in R

5.1 Basic visualizations (base R)

x <- c(5, 8, 12, 6, 9)
barplot(x, names.arg = c("A","B","C","D","E"), main = "Bar Chart")
hist(rnorm(100), main = "Histogram", col = "skyblue")
boxplot(x, main = "Box Plot")
plot(1:5, x, type = "l", main = "Line Chart")     # line
plot(1:5, x, main = "Scatter Plot")               # scatter (points)
pie(x, labels = c("A","B","C","D","E"))           # pie

5.2 Advanced visualizations (ggplot2)

library(ggplot2)
df <- data.frame(x = 1:5, y = c(5,8,12,6,9))
ggplot(df, aes(x = x, y = y)) +
  geom_line(colour = "steelblue") +
  geom_point(size = 3) +
  labs(title = "ggplot2 line + points", x = "Index", y = "Value") +
  theme_minimal()

5.3 Creating 3-D plots

# scatterplot3d package
library(scatterplot3d)
scatterplot3d(x = mtcars$wt, y = mtcars$hp, z = mtcars$mpg,
              main = "3D Scatter", color = "darkgreen")
# base persp() for a surface
persp(z = volcano, theta = 30, phi = 25, col = "lightblue")

marks <- c(46,54,45,34,55,64,78,90,41,66,72,58,49,61,80)
hist(marks, breaks = 6, col = "lightblue",
     main = "Distribution of Marks", xlab = "Marks")

study <- c(2,4,6,8,10);  score <- c(15,23,28,36,43)
plot(study, score, pch = 19, main = "Study hours vs Score")
abline(lm(score ~ study), col = "red")   # least-squares regression line

Key Take-aways from Unit 5

• Matrices are 2-D, single-mode; use t(), %*%, solve(), det().
• Data frames hold mixed-type columns; access with $, filter with logical conditions, combine with merge().
• Factors store categories; table() builds frequency / contingency tables.
• EDA = centre + spread + shape + missing + outliers; summary() gives a one-line overview.
• Handle NAs with na.rm / imputation; flag outliers with the 1.5×IQR rule; normalize by min-max or z-score (scale()).
• Visualise with base graphics or ggplot2; 3-D via scatterplot3d / persp(); fit lines with lm() + abline().