PCA for materials data · Part 4 of four · October 9, 2026

From five measurements to a PCA picture

We can now connect the pictures to our five measurement columns. We will find combinations of those columns and check how much variation a simpler picture retains.

First, we examine projection

The following activity uses a separate two-coordinate geometric example. It is not a plot of two columns from the materials spreadsheet. Its purpose is to show what projection means before we use five features.

The blue observations stay fixed. Each gold point is the nearest position on the chosen line to a blue point. That nearest position is its orthogonal projection. Turn the line and compare the projected variance. Try to find the largest value before revealing PC1.

The optional 3D example contains 1,155 grid points, or 1,200 random points using seed 714.

Where do eigenvectors enter?

For centered data, the covariance matrix's unit eigenvectors give principal directions. The direction with the largest eigenvalue gives the greatest variance of projected observations. We call it the first principal direction. The next direction is perpendicular to it and captures the greatest variance available under that restriction.

When we project each specimen onto these directions, we obtain its component scores. A direction specifies the combination of features; a score is the value of that combination for one specimen. These are different things.

In five dimensions, each direction has five coefficients. All five measurements participate in calculating the new coordinates. PC1 and PC2 are not two selected spreadsheet columns.

We calculate one step at a time

Download the CSV and save it beside the notebook or Python script. Download the complete Python example. We use Python and NumPy. Run each section before moving to the next. In a notebook, put each section in its own cell.

1. Read and inspect the data

import numpy as np

# We read the five measurements, leaving out the specimen label.
data = np.loadtxt("materials-pca-synthetic.csv",
                  delimiter=",", skiprows=1, usecols=(1, 2, 3, 4, 5))
print(data.shape)
print(data[:3])
assert data.shape == (300, 5)
assert np.isfinite(data).all()

We should see 300 rows and five columns. The first three rows should match the spreadsheet. We check for missing or nonfinite numbers before calculating.

2. Center and scale each measurement column

# We calculate each mean and sample standard deviation across specimens.
means = data.mean(axis=0)
standard_deviations = data.std(axis=0, ddof=1)
assert (standard_deviations > 0).all()
scaled = (data - means) / standard_deviations
print(scaled.mean(axis=0))
print(scaled.var(axis=0, ddof=1))

The means should be close to zero, allowing for rounding in computer arithmetic. The variances should be one. Here axis=0 means calculate down each column; ddof=1 uses the sample convention with divisor n−1.

3. Find the directions and their variances

# We form the covariance matrix of the five standardized features.
covariance = scaled.T @ scaled / (len(scaled) - 1)
values, directions = np.linalg.eigh(covariance)

# We place the largest eigenvalue and its direction first.
order = np.argsort(values)[::-1]
values = values[order]
directions = directions[:, order]

# We choose a sign that makes the yield-strength coefficient positive.
if directions[1, 0] < 0:
    directions[:, 0] *= -1

fractions = values / values.sum()
scores = scaled @ directions
print(100 * fractions)
print(directions[:, 0])

T transposes rows and columns; @ performs matrix multiplication. eigh is NumPy's solver for real symmetric matrices. Each column of directions is one unit eigenvector. The sign of an eigenvector can be reversed without changing the PCA result; our sign choice makes this example easier to discuss.

What does our first combination contain?

These are the calculated PC1 weights for the standardized features. We have reached this table only after explaining what the calculation does.

Positive weights on both strengths and a negative weight on elongation mean that this component describes a tendency toward higher strengths and lower elongation. Grain size and conductivity also contribute. We should interpret the weights together and remember that they were calculated after standardization.

S003 has PC1 score 2.835, S005 has −0.195 and S006 has 2.413. These are dimensionless calculated values, not strengths or grain sizes. One score alone does not fully describe a specimen. A PC1-versus-PC2 plot uses the first two scores for every specimen.

How much does a simpler picture retain?

PC1 accounts for 83.3% of total standardized variance. The first two components together account for 97.6%. These percentages describe variance retained, not the percentage of physical knowledge retained.

# We reconstruct all five standardized measurements using two components.
kept = 2
reconstructed_scaled = scores[:, :kept] @ directions[:, :kept].T
reconstructed = reconstructed_scaled * standard_deviations + means
print("Variance retained (%):", 100 * fractions[:kept].sum())
print("RMS error in standardized values:",
      np.sqrt(np.mean((scaled - reconstructed_scaled)**2)))

# We check that using all five components recovers the input.
full = scores @ directions.T
assert np.allclose(full, scaled)

Reconstruction gives an approximation to the five original measurements, rather than selecting two of them. Using every component recovers the centered and scaled data up to numerical rounding. Using fewer components leaves some differences out.

What can we conclude?

For this synthetic dataset and scaling choice, two combinations retain most of the measured variance. They provide a useful two-dimensional representation of five features. We must still inspect reconstruction errors in each feature's original units and decide whether the omitted differences matter for our purpose.

PCA does not establish causation. A low-variance component is not automatically noise. Important materials differences may occur in a component with relatively little variance. The simpler picture helps us examine the measurements; it does not replace scientific judgment.

For further reading, see the Cross Validated discussion and projection animations. We can also explore PCA reconstruction of small pixel images.