A PCA biplot simultaneously displays both observation scores (as points) and variable loadings (as arrows) in the principal component space. This dual representation is essential for interpreting PCA results, revealing how observations relate to each other and which original variables drive the separation along each principal component. The length and direction of loading arrows indicate variable importance and correlation with the components.
""" anyplot.ai
biplot-pca: PCA Biplot with Scores and Loading Vectors
Library: pygal 3.1.0 | Python 3.13.13
Quality: 92/100 | Updated: 2026-05-17
"""
import math
import os
import sys
from importlib import import_module
from sklearn.datasets import load_iris
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
# Ensure we import the installed pygal package, not the local script
site_packages = [p for p in sys.path if "site-packages" in p or "dist-packages" in p]
if site_packages:
sys.path = site_packages + [p for p in sys.path if "site-packages" not in p and "dist-packages" not in p]
pygal = import_module("pygal")
Style = import_module("pygal.style").Style
# Theme tokens
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
# Okabe-Ito palette: first series always #009E73 (brand), then position 2 and 3
IMPRINT = ("#009E73", "#C475FD", "#4467A3", "#BD8233", "#AE3030", "#2ABCCD", "#954477")
# Load and prepare data
iris = load_iris()
X = iris.data
y = iris.target
target_names = iris.target_names
# Standardize features
X_scaled = StandardScaler().fit_transform(X)
# Perform PCA
pca = PCA(n_components=2)
scores = pca.fit_transform(X_scaled)
loadings = pca.components_.T
variance_explained = pca.explained_variance_ratio_ * 100
# Use correlation biplot scaling
score_range = max(scores[:, 0].max() - scores[:, 0].min(), scores[:, 1].max() - scores[:, 1].min())
loading_scale = score_range * 0.4
# Custom style for theme-adaptive rendering
custom_style = Style(
background=PAGE_BG,
plot_background=PAGE_BG,
foreground=INK,
foreground_strong=INK,
foreground_subtle=INK_MUTED,
colors=IMPRINT,
title_font_size=28,
label_font_size=22,
major_label_font_size=18,
legend_font_size=16,
value_font_size=14,
stroke_width=2,
)
# Create XY chart for biplot
chart = pygal.XY(
width=4800,
height=2700,
style=custom_style,
title="biplot-pca · pygal · anyplot.ai",
x_title=f"PC1 ({variance_explained[0]:.1f}%)",
y_title=f"PC2 ({variance_explained[1]:.1f}%)",
show_legend=True,
legend_at_bottom=True,
dots_size=10,
stroke=False,
show_x_guides=True,
show_y_guides=True,
truncate_legend=0,
explicit_size=True,
)
# Add score points for each species (class)
for i, name in enumerate(target_names):
mask = y == i
points = [(float(scores[j, 0]), float(scores[j, 1])) for j in range(len(y)) if mask[j]]
chart.add(name.capitalize(), points, stroke=False, dots_size=10)
# Full feature names for legend clarity
full_names = ["Sepal Length", "Sepal Width", "Petal Length", "Petal Width"]
# Add each loading vector as a thin line arrow
for i, full_name in enumerate(full_names):
tip_x = float(loadings[i, 0] * loading_scale)
tip_y = float(loadings[i, 1] * loading_scale)
dx, dy = tip_x, tip_y
length = math.sqrt(dx * dx + dy * dy)
ux = dx / length if length > 0 else 0
uy = dy / length if length > 0 else 0
px, py = -uy, ux # Perpendicular vector
head_len = 0.06 * loading_scale
head_wid = 0.03 * loading_scale
hb_x = tip_x - ux * head_len
hb_y = tip_y - uy * head_len
arrow_line = [
(0.0, 0.0),
(tip_x, tip_y),
(hb_x + px * head_wid, hb_y + py * head_wid),
(tip_x, tip_y),
(hb_x - px * head_wid, hb_y - py * head_wid),
(tip_x, tip_y),
]
chart.add(full_name, arrow_line, stroke=True, show_dots=False, fill=False, stroke_style={"width": 2})
# Save outputs
chart.render_to_png(f"plot-{THEME}.png")
with open(f"plot-{THEME}.html", "wb") as f:
f.write(chart.render())
Part of PCA Biplot with Scores and Loading Vectors on anyplot.ai.