R Arrays

An array extends the matrix concept to three or more dimensions. A matrix has rows and columns (2D). An array adds a third dimension — imagine stacking multiple matrices on top of each other, like pages in a book. All elements in an array must be the same data type.

The Dimension Concept

1D → Vector:    [1, 2, 3, 4, 5]
                 single line of values

2D → Matrix:    rows × columns
                 like a table

3D → Array:     rows × columns × layers
                 like a stack of tables

Creating a 3D Array

arr <- array(1:24, dim = c(2, 3, 4))
# dim = c(rows, columns, layers)
# 2 rows × 3 columns × 4 layers = 24 elements

Layer 1:          Layer 2:          Layer 3:         Layer 4:
[,1][,2][,3]     [,1][,2][,3]     [,1][,2][,3]     [,1][,2][,3]
  1    3    5       7    9   11      13   15   17      19   21   23
  2    4    6       8   10   12      14   16   18      20   22   24

Naming Dimensions

# Sales data: 2 products, 3 regions, 2 years
sales_arr <- array(
  data     = c(100,200,150,250,300,180,120,220,160,270,310,195),
  dim      = c(2, 3, 2),
  dimnames = list(
    c("ProductA", "ProductB"),        # rows
    c("North", "South", "East"),      # columns
    c("2023", "2024")                 # layers
  )
)

Accessing Array Elements

arr[1, 2, 3]     # row 1, col 2, layer 3
arr[, , 1]       # all of layer 1 (returns a matrix)
arr[1, , ]       # row 1 across all columns and layers
arr["ProductA", "North", "2024"]  # named access

Array Arithmetic

a1 <- array(1:8, dim = c(2, 2, 2))
a2 <- array(9:16, dim = c(2, 2, 2))

a1 + a2    # element-wise addition across all layers
a1 * 2     # multiply every element by 2

Applying Functions Across Dimensions

# apply(array, margin, function)
# margin: 1=rows, 2=columns, 3=layers, c(1,2)=each cell across layers

apply(sales_arr, 3, sum)   # total sales per year
# 2023   2024
# 1180   1277

apply(sales_arr, 1, mean)  # mean per product
apply(sales_arr, 2, sum)   # total per region

Practical: RGB Image Representation

A digital image is stored as a 3D array: height × width × 3 color channels (Red, Green, Blue).

# Tiny 3×3 pixel image with RGB channels
image_data <- array(
  runif(27, 0, 255),   # random pixel values
  dim = c(3, 3, 3)     # 3 rows, 3 cols, 3 channels
)

# Access the Red channel of all pixels
image_data[, , 1]

# Get one pixel's RGB values (row 2, col 2)
image_data[2, 2, ]

Key Properties

dim(arr)        # dimensions: rows, cols, layers
length(arr)     # total number of elements
nrow(arr)       # number of rows
ncol(arr)       # number of columns
dimnames(arr)   # names of each dimension

When to Use Arrays

Use an Array For:                    Alternative:
──────────────────────────────────   ──────────────────────────────
Time-series data per region/product  Data frames with grouping columns
Image or video data                  Lists of matrices
3D physics or simulation data        Specialized packages (raster, etc.)
Multi-way contingency tables         table() function output

Arrays are less common in everyday data analysis than vectors, matrices, or data frames, but they are essential for spatial data, image analysis, and any problem with three or more dimensions of structure. Understanding arrays also helps you interpret multi-dimensional outputs from statistical functions.