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: makie 0.22.10 | Julia 1.11.9
# Quality: 89/100 | Created: 2026-05-28
using CairoMakie
using Colors
using Random
Random.seed!(42)
# Theme tokens
const THEME = get(ENV, "ANYPLOT_THEME", "light")
const PAGE_BG = THEME == "light" ? colorant"#FAF8F1" : colorant"#1A1A17"
const ELEVATED_BG = THEME == "light" ? colorant"#FFFDF6" : colorant"#242420"
const INK = THEME == "light" ? colorant"#1A1A17" : colorant"#F0EFE8"
const INK_SOFT = THEME == "light" ? colorant"#4A4A44" : colorant"#B8B7B0"
const INK_MUTED = THEME == "light" ? colorant"#6B6A63" : colorant"#A8A79F"
const IMPRINT_PALETTE = [
colorant"#009E73", # 1 — brand green (Bias²)
colorant"#C475FD", # 2 — lavender (Variance)
colorant"#4467A3", # 3 — blue (Total Error)
colorant"#BD8233", # 4 — ochre (Irreducible Error)
]
# Data — theoretical bias-variance tradeoff curves
complexity = range(0.1, 10.0, length=80)
bias_sq = 1.0 ./ (1.0 .+ complexity) .+ 0.02
variance = (complexity .^ 1.4) ./ 22.0
irreducible_err = fill(0.15, length(complexity))
total_err = bias_sq .+ variance .+ irreducible_err
# Find optimal complexity (minimum total error)
opt_idx = argmin(total_err)
opt_complexity = complexity[opt_idx]
opt_total = total_err[opt_idx]
# Title
title_str = "curve-bias-variance-tradeoff · julia · makie · anyplot.ai"
n = length(title_str)
default_titlesize = 20.0
titlesize = n > 67 ? max(14.0, round(default_titlesize * 67 / n)) : default_titlesize
# Plot
fig = Figure(
size = (1600, 900),
fontsize = 14,
backgroundcolor = PAGE_BG,
)
ax = Axis(
fig[1, 1];
title = title_str,
titlesize = titlesize,
titlecolor = INK,
xlabel = "Model Complexity",
ylabel = "Prediction Error",
xlabelsize = 14,
ylabelsize = 14,
xlabelcolor = INK,
ylabelcolor = INK,
xticklabelsize = 12,
yticklabelsize = 12,
xticklabelcolor = INK_SOFT,
yticklabelcolor = INK_SOFT,
xtickcolor = INK_SOFT,
ytickcolor = INK_SOFT,
backgroundcolor = PAGE_BG,
topspinevisible = false,
rightspinevisible = false,
leftspinecolor = INK_SOFT,
bottomspinecolor = INK_SOFT,
xgridvisible = false,
ygridcolor = RGBAf(INK.r, INK.g, INK.b, 0.12),
yminorgridvisible = false,
xminorgridvisible = false,
)
# Shaded zones (underfitting / overfitting)
vspan!(ax, complexity[1], opt_complexity;
color = RGBAf(IMPRINT_PALETTE[1].r, IMPRINT_PALETTE[1].g, IMPRINT_PALETTE[1].b, 0.07))
vspan!(ax, opt_complexity, complexity[end];
color = RGBAf(IMPRINT_PALETTE[2].r, IMPRINT_PALETTE[2].g, IMPRINT_PALETTE[2].b, 0.07))
# Curves
lines!(ax, complexity, bias_sq;
color = IMPRINT_PALETTE[1], linewidth = 2.5, linestyle = :solid, label = "Bias²")
lines!(ax, complexity, variance;
color = IMPRINT_PALETTE[2], linewidth = 2.5, linestyle = :dash, label = "Variance")
lines!(ax, complexity, total_err;
color = IMPRINT_PALETTE[3], linewidth = 3.0, linestyle = :dashdot, label = "Total Error")
lines!(ax, complexity, irreducible_err;
color = IMPRINT_PALETTE[4], linewidth = 2.0, linestyle = :dot, label = "Irreducible Error")
# Optimal point vertical line
vlines!(ax, [opt_complexity]; color = INK_SOFT, linewidth = 1.5, linestyle = :dash)
# Optimal point marker on total error curve
scatter!(ax, [opt_complexity], [opt_total];
color = IMPRINT_PALETTE[3], markersize = 14, strokewidth = 1.5,
strokecolor = PAGE_BG)
# Zone labels
text!(ax, complexity[4], maximum(total_err) * 0.78;
text = "Underfitting\n(High Bias)",
color = INK_MUTED, fontsize = 13, align = (:left, :center))
text!(ax, opt_complexity + 0.3, maximum(total_err) * 0.78;
text = "Overfitting\n(High Variance)",
color = INK_MUTED, fontsize = 13, align = (:left, :center))
# Optimal label
text!(ax, opt_complexity, opt_total + 0.03;
text = "Optimal",
color = INK, fontsize = 11, align = (:center, :bottom), font = :bold)
# Direct curve labels (spec requirement: annotate each curve directly on the plot)
lbl_bias_x = 2.0
lbl_bias_y = 1.0 / (1.0 + lbl_bias_x) + 0.02
lbl_var_x = 8.5
lbl_var_y = (lbl_var_x ^ 1.4) / 22.0
lbl_tot_x = 7.5
lbl_tot_y = (1.0 / (1.0 + lbl_tot_x) + 0.02) + ((lbl_tot_x ^ 1.4) / 22.0) + 0.15
lbl_irr_x = 5.0
text!(ax, lbl_bias_x, lbl_bias_y + 0.03;
text = "Bias²",
color = IMPRINT_PALETTE[1], fontsize = 12, align = (:center, :bottom), font = :bold)
text!(ax, lbl_var_x, lbl_var_y + 0.03;
text = "Variance",
color = IMPRINT_PALETTE[2], fontsize = 12, align = (:center, :bottom), font = :bold)
text!(ax, lbl_tot_x, lbl_tot_y + 0.03;
text = "Total Error",
color = IMPRINT_PALETTE[3], fontsize = 12, align = (:center, :bottom), font = :bold)
text!(ax, lbl_irr_x, 0.15 + 0.03;
text = "Irreducible Error",
color = IMPRINT_PALETTE[4], fontsize = 12, align = (:center, :bottom), font = :bold)
# Formula annotation
text!(ax, complexity[end], irreducible_err[1] + 0.01;
text = "Total Error = Bias² + Variance + Irreducible Error",
color = INK_MUTED, fontsize = 12, align = (:right, :bottom))
# Legend
axislegend(ax;
position = :rc,
backgroundcolor = ELEVATED_BG,
framecolor = INK_SOFT,
labelcolor = INK,
titlecolor = INK,
framewidth = 0.8,
labelsize = 12,
)
# Save
save("plot-$(THEME).png", fig; px_per_unit = 2)
Part of Bias-Variance Tradeoff Curve on anyplot.ai.