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: ggplot2 3.5.1 | R 4.4.1
#' Quality: 92/100 | Created: 2026-05-28
library(ggplot2)
library(ragg)
set.seed(42)
# --- Theme tokens ---
THEME <- Sys.getenv("ANYPLOT_THEME", "light")
PAGE_BG <- if (THEME == "light") "#FAF8F1" else "#1A1A17"
INK <- if (THEME == "light") "#1A1A17" else "#F0EFE8"
INK_SOFT <- if (THEME == "light") "#4A4A44" else "#B8B7B0"
INK_MUTED <- if (THEME == "light") "#6B6A63" else "#A8A79F"
ELEVATED_BG <- if (THEME == "light") "#FFFDF6" else "#242420"
IMPRINT_PALETTE <- c("#009E73", "#C475FD", "#4467A3", "#BD8233")
# --- Data ---
n <- 100
complexity <- seq(1, 10, length.out = n)
irreducible <- rep(0.15, n)
bias_squared <- 0.70 * exp(-0.5 * (complexity - 1))
variance <- 0.70 * exp(-0.5 * (10 - complexity))
total_error <- bias_squared + variance + irreducible
optimal_idx <- which.min(total_error)
optimal_x <- complexity[optimal_idx]
optimal_y <- total_error[optimal_idx]
bias_ann_y <- bias_squared[which.min(abs(complexity - 1.4))] + 0.04
var_ann_y <- variance[which.min(abs(complexity - 8.5))] + 0.04
tot_ann_y <- total_error[which.min(abs(complexity - 3.0))] + 0.05
# --- Long format ---
components <- c("Bias²", "Variance", "Total Error", "Irreducible Error")
df_long <- data.frame(
complexity = rep(complexity, 4),
error = c(bias_squared, variance, total_error, irreducible),
component = factor(rep(components, each = n), levels = components)
)
curve_colors <- setNames(IMPRINT_PALETTE, components)
curve_linetypes <- c("Bias²" = "solid", "Variance" = "longdash",
"Total Error" = "solid", "Irreducible Error" = "dotted")
curve_lwidths <- c("Bias²" = 1.0, "Variance" = 1.0,
"Total Error" = 1.7, "Irreducible Error" = 0.9)
title_str <- "curve-bias-variance-tradeoff · r · ggplot2 · anyplot.ai"
# --- Plot ---
p <- ggplot(df_long,
aes(x = complexity, y = error,
color = component, linetype = component,
linewidth = component)) +
annotate("rect", xmin = 1, xmax = optimal_x,
ymin = -Inf, ymax = Inf, fill = "#009E73", alpha = 0.05) +
annotate("rect", xmin = optimal_x, xmax = 10,
ymin = -Inf, ymax = Inf, fill = "#AE3030", alpha = 0.05) +
geom_vline(xintercept = optimal_x, color = INK_SOFT,
linetype = "dashed", linewidth = 0.5) +
geom_line() +
scale_color_manual(values = curve_colors) +
scale_linetype_manual(values = curve_linetypes) +
scale_linewidth_manual(values = curve_lwidths) +
annotate("text", x = (1 + optimal_x) / 2, y = 0.86,
label = "Underfitting", color = INK_MUTED, size = 3.0) +
annotate("text", x = (optimal_x + 10) / 2, y = 0.86,
label = "Overfitting", color = INK_MUTED, size = 3.0) +
annotate("text", x = optimal_x + 0.2, y = optimal_y - 0.04,
label = "Optimal", color = INK_SOFT, size = 2.8, hjust = 0) +
annotate("text", x = 1.4, y = bias_ann_y,
label = "Bias²", color = IMPRINT_PALETTE[1],
size = 3.2, hjust = 0, fontface = "bold") +
annotate("text", x = 8.5, y = var_ann_y,
label = "Variance", color = IMPRINT_PALETTE[2],
size = 3.2, hjust = 0.5, fontface = "bold") +
annotate("text", x = 3.0, y = tot_ann_y,
label = "Total Error", color = IMPRINT_PALETTE[3],
size = 3.0, hjust = 0.5, fontface = "bold") +
annotate("text", x = 7.0, y = 0.10,
label = "Irreducible Error", color = IMPRINT_PALETTE[4],
size = 2.8, hjust = 0.5) +
scale_x_continuous(
breaks = c(1, optimal_x, 10),
labels = c("Low", "Optimal", "High"),
expand = c(0.02, 0)
) +
scale_y_continuous(expand = c(0.02, 0)) +
labs(
title = title_str,
subtitle = "Total Error = Bias² + Variance + Irreducible Error",
x = "Model Complexity",
y = "Prediction Error"
) +
theme_minimal(base_size = 8) +
theme(
plot.background = element_rect(fill = PAGE_BG, color = PAGE_BG),
panel.background = element_rect(fill = PAGE_BG, color = NA),
panel.grid.major.y = element_line(color = INK, linewidth = 0.15),
panel.grid.major.x = element_blank(),
panel.grid.minor = element_blank(),
panel.border = element_blank(),
axis.title = element_text(color = INK, size = 10),
axis.text = element_text(color = INK_SOFT, size = 8),
axis.line = element_line(color = INK_SOFT, linewidth = 0.4),
plot.title = element_text(color = INK, size = 12, face = "bold"),
plot.subtitle = element_text(color = INK_MUTED, size = 8),
legend.position = "none",
plot.margin = margin(15, 20, 15, 15)
)
# --- Save ---
ggsave(
filename = sprintf("plot-%s.png", THEME),
plot = p,
device = ragg::agg_png,
width = 8,
height = 4.5,
units = "in",
dpi = 400
)
Part of Bias-Variance Tradeoff Curve on anyplot.ai.