PCA Biplot with Scores and Loading Vectors — lets-plot

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.

PCA Biplot with Scores and Loading Vectors rendered with lets-plot

Python source (lets-plot)

""" anyplot.ai
biplot-pca: PCA Biplot with Scores and Loading Vectors
Library: letsplot 4.9.0 | Python 3.13.13
Quality: 90/100 | Updated: 2026-05-17
"""

import os

import numpy as np
import pandas as pd
from lets_plot import *
from lets_plot.export import ggsave as export_ggsave
from sklearn.datasets import load_iris
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler


LetsPlot.setup_html()

# Theme tokens
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
BRAND = "#009E73"  # Okabe-Ito position 1

# Okabe-Ito palette
IMPRINT = ["#009E73", "#C475FD", "#4467A3"]

# Load Iris dataset
iris = load_iris()
X = iris.data
y = iris.target
feature_names = iris.feature_names
target_names = iris.target_names

# Standardize features
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# Perform PCA
pca = PCA(n_components=2)
scores = pca.fit_transform(X_scaled)
loadings = pca.components_.T  # Variables x Components

# Variance explained
var_explained = pca.explained_variance_ratio_ * 100

# Create dataframe for scores
scores_df = pd.DataFrame({"PC1": scores[:, 0], "PC2": scores[:, 1], "Species": [target_names[i] for i in y]})

# Scale loadings for visibility alongside scores
score_range = max(np.abs(scores).max(), 1)
loading_scale = score_range * 1.5

# Create dataframe for loading arrows
clean_names = ["Sepal Length", "Sepal Width", "Petal Length", "Petal Width"]
loadings_df = pd.DataFrame(
    {
        "x_start": [0] * len(feature_names),
        "y_start": [0] * len(feature_names),
        "x_end": loadings[:, 0] * loading_scale,
        "y_end": loadings[:, 1] * loading_scale,
        "variable": clean_names,
    }
)

# Label positions with smart offset to avoid overlap
label_offsets = []
for i, name in enumerate(clean_names):
    x_end = loadings_df["x_end"].iloc[i]
    y_end = loadings_df["y_end"].iloc[i]
    if name == "Petal Width":
        label_offsets.append((x_end * 1.15, y_end * 1.15 + 0.4))
    elif name == "Petal Length":
        label_offsets.append((x_end * 1.15, y_end * 1.15 - 0.3))
    else:
        label_offsets.append((x_end * 1.15, y_end * 1.15))

loadings_df["label_x"] = [offset[0] for offset in label_offsets]
loadings_df["label_y"] = [offset[1] for offset in label_offsets]

# Unit circle for reference
circle_theta = np.linspace(0, 2 * np.pi, 100)
circle_df = pd.DataFrame({"x": np.cos(circle_theta), "y": np.sin(circle_theta)})

# Build the plot
plot = (
    ggplot()
    + geom_point(
        data=scores_df,
        mapping=aes(x="PC1", y="PC2", color="Species"),
        size=5,
        alpha=0.8,
    )
    + geom_segment(
        data=loadings_df,
        mapping=aes(x="x_start", y="y_start", xend="x_end", yend="y_end"),
        color=INK_SOFT,
        size=1.8,
        arrow=arrow(length=15, type="open"),
    )
    + geom_text(
        data=loadings_df,
        mapping=aes(x="label_x", y="label_y", label="variable"),
        size=14,
        color=INK_SOFT,
    )
    + geom_path(
        data=circle_df,
        mapping=aes(x="x", y="y"),
        color=INK_MUTED,
        size=0.5,
    )
    + geom_hline(yintercept=0, color=INK_MUTED, size=0.5, linetype="dashed")
    + geom_vline(xintercept=0, color=INK_MUTED, size=0.5, linetype="dashed")
    + labs(
        x=f"PC1 ({var_explained[0]:.1f}%)",
        y=f"PC2 ({var_explained[1]:.1f}%)",
        title="biplot-pca · letsplot · anyplot.ai",
        color="Species",
    )
    + scale_color_manual(values=IMPRINT)
    + scale_x_continuous(expand=[0.15, 0.15])
    + scale_y_continuous(expand=[0.15, 0.15])
    + theme_minimal()
    + theme(
        plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
        panel_background=element_rect(fill=PAGE_BG),
        panel_grid_major=element_line(
            color=INK, size=0.3
        ),
        panel_grid_minor=element_blank(),
        axis_title=element_text(size=20, color=INK),
        axis_text=element_text(size=16, color=INK_SOFT),
        axis_line=element_line(color=INK_SOFT, size=0.5),
        plot_title=element_text(size=24, color=INK),
        legend_background=element_rect(
            fill=ELEVATED_BG, color=INK_SOFT
        ),
        legend_text=element_text(size=16, color=INK_SOFT),
        legend_title=element_text(size=18, color=INK),
    )
    + ggsize(1600, 900)
)

# Save PNG (scale 3x for 4800x2700)
export_ggsave(plot, filename=f"plot-{THEME}.png", path=".", scale=3)

# Save HTML for interactivity
export_ggsave(plot, filename=f"plot-{THEME}.html", path=".")

Part of PCA Biplot with Scores and Loading Vectors on anyplot.ai.

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