A theoretical visualization of the bias-variance tradeoff showing how total prediction error decomposes into bias squared, variance, and irreducible noise as a function of model complexity. The plot displays multiple curves: bias squared (decreasing with complexity), variance (increasing with complexity), irreducible error (constant), and total error (U-shaped). This is one of the most fundamental conceptual plots in machine learning for understanding model selection, overfitting, and underfitting.
""" anyplot.ai
curve-bias-variance-tradeoff: Bias-Variance Tradeoff Curve
Library: letsplot 4.10.1 | Python 3.13.13
Quality: 88/100 | Updated: 2026-05-28
"""
import os
import numpy as np
import pandas as pd
from lets_plot import *
LetsPlot.setup_html()
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"
# Data — formula distinct from seaborn sibling (change request)
np.random.seed(42)
complexity = np.linspace(0.5, 10.0, 100)
bias_squared = 3.0 / (1 + 0.7 * complexity)
variance = 0.08 * complexity**1.3
irreducible_error = np.full_like(complexity, 0.25)
total_error = bias_squared + variance + irreducible_error
optimal_idx = int(np.argmin(total_error))
optimal_complexity = float(complexity[optimal_idx])
optimal_error = float(total_error[optimal_idx])
component_order = ["Bias²", "Variance", "Irreducible Error", "Total Error"]
df = pd.DataFrame(
{
"complexity": np.tile(complexity, 4),
"error": np.concatenate([bias_squared, variance, irreducible_error, total_error]),
"component": pd.Categorical(
["Bias²"] * 100 + ["Variance"] * 100 + ["Irreducible Error"] * 100 + ["Total Error"] * 100,
categories=component_order,
),
}
)
# Curve values at x=10 for right-side direct labels
x_lbl = 10.0
bias_at_lbl = float(3.0 / (1 + 0.7 * x_lbl))
var_at_lbl = float(0.08 * x_lbl**1.3)
irred_at_lbl = 0.25
total_at_lbl = bias_at_lbl + var_at_lbl + irred_at_lbl
label_df = pd.DataFrame(
{
"complexity": [x_lbl + 0.3] * 4,
"error": [bias_at_lbl + 0.15, var_at_lbl + 0.05, irred_at_lbl - 0.22, total_at_lbl + 0.04],
"label": ["Bias²", "Variance", "Irred.\nError", "Total\nError"],
"component": pd.Categorical(component_order, categories=component_order),
}
)
# Shaded underfitting / overfitting zones
under_df = pd.DataFrame({"xmin": [0.4], "xmax": [optimal_complexity], "ymin": [-0.15], "ymax": [3.38]})
over_df = pd.DataFrame({"xmin": [optimal_complexity], "xmax": [10.5], "ymin": [-0.15], "ymax": [3.38]})
# Zone labels
zone_df = pd.DataFrame(
{
"x": [optimal_complexity * 0.42, (optimal_complexity + 10.5) * 0.5],
"y": [3.12, 3.12],
"label": ["← Underfitting\n(High Bias)", "Overfitting →\n(High Variance)"],
}
)
# Optimal complexity annotation
opt_df = pd.DataFrame({"x": [optimal_complexity + 0.25], "y": [optimal_error + 0.40], "label": ["Optimal\nComplexity"]})
# Formula annotation
formula_df = pd.DataFrame({"x": [5.25], "y": [2.80], "label": ["Total Error = Bias² + Variance + Irreducible Error"]})
# Imprint palette: Bias²=pos1 (green), Variance=pos2 (purple), Irred.=pos3 (blue), Total=red (semantic error)
color_values = ["#009E73", "#C475FD", "#4467A3", "#AE3030"]
linetype_values = ["dashed", "longdash", "dotted", "solid"]
title_str = "curve-bias-variance-tradeoff · python · letsplot · anyplot.ai"
title_n = len(title_str)
title_size = max(11, round(16 * 67 / title_n)) if title_n > 67 else 16
anyplot_theme = theme(
plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
panel_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
panel_border=element_blank(),
panel_grid_major_x=element_blank(),
panel_grid_major_y=element_line(color=INK_SOFT, size=0.12),
panel_grid_minor=element_blank(),
axis_title=element_text(color=INK, size=12),
axis_text=element_text(color=INK_SOFT, size=10),
axis_line=element_line(color=INK_SOFT, size=0.5),
plot_title=element_text(color=INK, size=title_size),
legend_position="none",
)
plot = (
ggplot(df, aes(x="complexity", y="error", color="component"))
+ geom_rect(
data=under_df,
mapping=aes(xmin="xmin", xmax="xmax", ymin="ymin", ymax="ymax"),
fill="#009E73",
color=PAGE_BG,
alpha=0.06,
inherit_aes=False,
)
+ geom_rect(
data=over_df,
mapping=aes(xmin="xmin", xmax="xmax", ymin="ymin", ymax="ymax"),
fill="#AE3030",
color=PAGE_BG,
alpha=0.06,
inherit_aes=False,
)
+ geom_line(aes(linetype="component"), size=1.0, tooltips=layer_tooltips(["component", "error"]))
+ geom_vline(xintercept=optimal_complexity, linetype="dotdash", color=INK_SOFT, size=0.65)
+ geom_text(
data=zone_df, mapping=aes(x="x", y="y", label="label"), inherit_aes=False, color=INK_SOFT, size=3.3, hjust=0.5
)
+ geom_text(data=opt_df, mapping=aes(x="x", y="y", label="label"), inherit_aes=False, color=INK, size=3.2, hjust=0)
+ geom_text(
data=formula_df,
mapping=aes(x="x", y="y", label="label"),
inherit_aes=False,
color=INK_SOFT,
size=3.3,
hjust=0.5,
)
+ geom_text(
data=label_df,
mapping=aes(x="complexity", y="error", label="label", color="component"),
inherit_aes=False,
size=3.2,
hjust=0,
)
+ scale_color_manual(values=color_values)
+ scale_linetype_manual(values=linetype_values)
+ scale_x_continuous(limits=[0.4, 12.0], breaks=[2, 4, 6, 8, 10])
+ scale_y_continuous(limits=[-0.15, 3.45])
+ labs(x="Model Complexity", y="Prediction Error", title=title_str)
+ ggsize(800, 450)
+ theme_classic()
+ anyplot_theme
)
ggsave(plot, f"plot-{THEME}.png", path=".", scale=4)
ggsave(plot, f"plot-{THEME}.html", path=".")
Part of Bias-Variance Tradeoff Curve on anyplot.ai.