Andrews Curves for Multivariate Data — Bokeh

Andrews curves visualization transforms multivariate observations into smooth Fourier series curves. Each data point is represented as a continuous function where variable values become coefficients in a Fourier expansion, producing distinctive wave patterns. This technique enables visual comparison of multivariate patterns, cluster identification, and outlier detection—observations with similar values across variables produce similar curves, while outliers appear as distinctly different patterns.

Andrews Curves for Multivariate Data rendered with Bokeh

Python source (Bokeh)

""" anyplot.ai
andrews-curves: Andrews Curves for Multivariate Data
Library: bokeh 3.9.0 | Python 3.13.13
Quality: 92/100 | Updated: 2026-05-15
"""

import os
import time
from pathlib import Path

import numpy as np
from bokeh.io import output_file, save
from bokeh.models import ColumnDataSource, HoverTool, Legend
from bokeh.plotting import figure
from selenium import webdriver
from selenium.webdriver.chrome.options import Options
from sklearn.datasets import load_iris
from sklearn.preprocessing import StandardScaler


# 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"

IMPRINT = ["#009E73", "#C475FD", "#4467A3"]

# Data
iris = load_iris()
X = iris.data
y = iris.target
species_names = ["Setosa", "Versicolor", "Virginica"]

# Normalize variables to similar scales
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# Generate t values from -π to π
t_values = np.linspace(-np.pi, np.pi, 200)

# Create figure
p = figure(
    width=4800,
    height=2700,
    title="andrews-curves · bokeh · anyplot.ai",
    x_axis_label="t (radians)",
    y_axis_label="f(t)",
    background_fill_color=PAGE_BG,
    border_fill_color=PAGE_BG,
)

# Style text
p.title.text_font_size = "28pt"
p.title.text_color = INK
p.xaxis.axis_label_text_font_size = "22pt"
p.xaxis.axis_label_text_color = INK
p.yaxis.axis_label_text_font_size = "22pt"
p.yaxis.axis_label_text_color = INK
p.xaxis.major_label_text_font_size = "18pt"
p.xaxis.major_label_text_color = INK_SOFT
p.yaxis.major_label_text_font_size = "18pt"
p.yaxis.major_label_text_color = INK_SOFT

# Style axes and grid
p.outline_line_color = INK_SOFT
p.xaxis.axis_line_color = INK_SOFT
p.yaxis.axis_line_color = INK_SOFT
p.xaxis.major_tick_line_color = INK_SOFT
p.yaxis.major_tick_line_color = INK_SOFT
p.xgrid.grid_line_color = INK
p.ygrid.grid_line_color = INK
p.xgrid.grid_line_alpha = 0.10
p.ygrid.grid_line_alpha = 0.10

# Store legend items
legend_items = []

# Plot curves for each species using vectorized Andrews curves
for species_idx in range(3):
    species_mask = y == species_idx
    X_species = X_scaled[species_mask]

    first_line = None

    for coeffs in X_species:
        # Vectorized Andrews curve: f(t) = x1/sqrt(2) + x2*sin(t) + x3*cos(t) + x4*sin(2t) + ...
        n = len(coeffs)
        curve_values = coeffs[0] / np.sqrt(2)
        for i in range(1, n):
            if i % 2 == 1:
                curve_values += coeffs[i] * np.sin((i // 2 + 1) * t_values)
            else:
                curve_values += coeffs[i] * np.cos((i // 2) * t_values)

        source = ColumnDataSource(
            data={"x": t_values, "y": curve_values, "species": [species_names[species_idx]] * len(t_values)}
        )

        line = p.line(x="x", y="y", source=source, line_color=IMPRINT[species_idx], line_alpha=0.4, line_width=3)

        if first_line is None:
            first_line = line

    legend_items.append((species_names[species_idx], [first_line]))

# Add hover tool
hover = HoverTool(tooltips=[("t", "@x{0.00}"), ("f(t)", "@y{0.00}"), ("Species", "@species")])
p.add_tools(hover)

# Create and add legend
legend = Legend(items=legend_items, location="top_right")
legend.label_text_font_size = "18pt"
legend.label_text_color = INK_SOFT
legend.background_fill_color = ELEVATED_BG
legend.background_fill_alpha = 0.95
legend.border_line_color = INK_SOFT
p.add_layout(legend, "right")

# Save HTML
output_file(f"plot-{THEME}.html")
save(p)

# Screenshot with headless Chrome via Selenium
W, H = 4800, 2700
opts = Options()
for arg in (
    "--headless=new",
    "--no-sandbox",
    "--disable-dev-shm-usage",
    "--disable-gpu",
    f"--window-size={W},{H}",
    "--hide-scrollbars",
):
    opts.add_argument(arg)
driver = webdriver.Chrome(options=opts)
driver.set_window_size(W, H)
driver.get(f"file://{Path(f'plot-{THEME}.html').resolve()}")
time.sleep(3)
driver.save_screenshot(f"plot-{THEME}.png")
driver.quit()

Part of Andrews Curves for Multivariate Data on anyplot.ai.

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