Machine Learning using R How to Implement Multidimentional Scaling

Readme.txt

This R script is a step by step demonstration showing how to implement MDS (both metric and non-metric MDS) using R.  The R functions used are: cmdscale (Base R), mds (smacof) and isoMDS(MASS).   
Click here to view a step by step demo:  https://youtu.be/SkVAzb1IuOU?si=6tTkUsvgu_5vnO4I

R Script

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 #                                             Machine Learning using R   How to Implement Multidimsional Scaling                                                        #
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 ####     Multidimensional Scaling (MDS)
 ##       visualizes similarities/dissimilarities between objects 
 ##       by placing them in low-dimensional space (usually 2D/3D) 
 ##       where distances reflect original relationships
 
 ####    Input: Pairwise dissimilarity matrix (interval, ordinal, or nominal data) 
 ####    Output: Point coordinates + goodness-of-fit measures
 ####    Two main types
 ##             Metric MDS (cmdscale, smacof interval)
 ##                Preserves actual distances; assumes Euclidean structure
 ##             Non-metric/Ordinal MDS (isoMDS): 
 ##                Preserves rank order only; more flexible for rank/perceptual data
 
 #####   Stress value measures fit quality: <0.05=Excellent, <0.10=Good, <0.20=Fair, >0.20=Poor
 ####    cmdscale eigenvalues indicate variance explained per dimension (large positive = good fit)
 
 ####    Shepard diagram shows monotonic regression (observed vs. fitted distances) for ordinal MDS validation
 
# Load required libraries
library(stats)  # for cmdscale (base R MDS)
library(smacof) # for SMACOF MDS

# Set seed for reproducibility
set.seed(1192026)

## PART 1: Interval Data for MDS

# Generate synthetic interval dissimilarity data for 10 objects
n <- 10
diss_matrix <- matrix(runif(n*(n-1)/2, 1, 10), nrow = n, ncol = n)
diss_matrix <- as.dist(diss_matrix)
diag(diss_matrix) <- 0

# Base R MDS using cmdscale (metric/interval MDS)
mds_base <- cmdscale(diss_matrix, k = 2, eig = TRUE)

# SMACOF interval MDS
mds_smacof <- mds(diss_matrix, ndim = 2, type = "interval")

# Comparison
par(mfrow = c(1, 2))
plot(mds_base$points, main = "Base R cmdscale (Interval)", 
     xlab = "Dim 1", ylab = "Dim 2", pch = 16)
plot(mds_smacof$conf, main = "SMACOF (Interval)", 
     xlab = "Dim 1", ylab = "Dim 2", pch = 16)

# Stress values comparison
cat("Base R eigenvalues:", round(mds_base$eig[1:2], 3), "\n")
cat("SMACOF stress:", round(mds_smacof$stress, 3), "\n")

## PART 2: Ordinal Data MDS
# Load required library
library(MASS)

# Set seed for reproducibility
set.seed(123)

## Generate Ordinal Dissimilarity Data for MDS
n_objects <- 12  # Number of objects to compare
max_rank <- 7    # Ordinal scale: 1-7 (like Likert ratings)

# Create symmetric dissimilarity matrix with ordinal ranks (1-7)
# Upper triangle filled with random ordinal values
ordinal_matrix <- matrix(0, n_objects, n_objects)
upper_tri <- upper.tri(ordinal_matrix)

# Fill upper triangle with ordinal ranks (1-7)
ordinal_matrix[upper_tri] <- sample(1:max_rank, sum(upper_tri), replace = TRUE)
ordinal_matrix <- ordinal_matrix + t(ordinal_matrix)  # Make symmetric
diag(ordinal_matrix) <- 0  # Zero diagonal

# Convert to dist object
ordinal_diss <- as.dist(ordinal_matrix)

# Display first few dissimilarities
cat("Sample ordinal dissimilarities (ranks 1-7):\n")
print(round(ordinal_diss[1:6], 1))

## Perform Ordinal MDS using isoMDS
ordinal_mds <- isoMDS(ordinal_diss, k = 2, maxit = 2000)

## Results and Visualization
par(mfrow = c(1, 2))

# Plot MDS configuration
par(mfrow = c(1, 2))

# 1. MDS Configuration Plot
plot(ordinal_mds$points, 
     main = "Ordinal MDS Configuration (isoMDS)",
     xlab = "Dimension 1", ylab = "Dimension 2",
     pch = 16, col = "darkred", cex = 1.3)

# 2. Shepard Diagram (CORRECTED)
shepard_plot <- Shepard(ordinal_diss, ordinal_mds$points)
plot(shepard_plot, pch = 16, col = "blue",
     main = "Shepard Diagram",
     xlab = "Original Ordinal Distances", 
     ylab = "Fitted Distances")
abline(0, 1, col = "red", lwd = 2)
lines(shepard_plot$x, shepard_plot$yf, col = "darkgreen", lwd = 2)


# Stress results
cat("\n=== ORDINAL MDS RESULTS ===\n")
cat("Stress (ordinal):", round(ordinal_mds$stress, 4), "\n")
cat("Stress %:", round(ordinal_mds$stress * 100, 2), "%\n")

VideoDemo

Click here to view a step by step demo:  https://youtu.be/SkVAzb1IuOU?si=6tTkUsvgu_5vnO4I