A Weibull probability plot displays failure or lifetime data on Weibull probability paper (logarithmic x-axis for time/cycles, linearized Weibull CDF on y-axis) with a fitted straight line. It is the standard tool in reliability engineering for estimating Weibull distribution parameters (shape and scale), assessing whether data follow a Weibull distribution, and extrapolating failure probabilities. The slope of the fitted line gives the shape parameter (beta), while the characteristic life (eta) is read at the 63.2% failure probability crossing.
""" anyplot.ai
probability-weibull: Weibull Probability Plot for Reliability Analysis
Library: letsplot 4.10.1 | Python 3.13.13
Quality: 86/100 | Updated: 2026-06-07
"""
# ruff: noqa: F403, F405, E402
"""anyplot.ai
probability-weibull: Weibull Probability Plot for Reliability Analysis
Library: letsplot | Python 3.13
Quality: pending | Created: 2026-06-07
"""
import os
import numpy as np
import pandas as pd
from lets_plot import *
from lets_plot.export import ggsave
from scipy import stats
LetsPlot.setup_html()
# Theme tokens (Imprint palette — see prompts/default-style-guide.md)
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"
# Imprint palette — categorical (theme-independent)
BRAND = "#009E73" # position 1 — Failure observations
COLOR_CENSORED = "#C475FD" # position 2 — Censored observations
COLOR_ETA = "#4467A3" # position 3 — characteristic life marker
# Data — Bearing wear-out fatigue: shape=3.2 (steep wear-out mode), scale=8000 h
np.random.seed(42)
shape_param = 3.2 # beta: wear-out failure mode (high shape → tight failure band)
scale_param = 8000 # eta: characteristic life in hours
n_failures = 25
n_censored = 8
failure_times = np.sort(stats.weibull_min.rvs(shape_param, scale=scale_param, size=n_failures))
censored_times = np.sort(np.random.uniform(3000, 10000, n_censored))
all_times = np.concatenate([failure_times, censored_times])
is_failure = np.concatenate([np.ones(n_failures), np.zeros(n_censored)])
sort_idx = np.argsort(all_times)
all_times = all_times[sort_idx]
is_failure = is_failure[sort_idx]
# Median rank plotting positions (Bernard's approximation): (i - 0.3) / (n + 0.4)
n_total = len(all_times)
failure_rank = np.cumsum(is_failure)
median_ranks = np.clip((failure_rank - 0.3) / (n_total + 0.4), 1e-6, 1 - 1e-6)
# Weibull linearization: y = ln(-ln(1 - F))
weibull_y = np.log(-np.log(1 - median_ranks))
log_times = np.log(all_times)
df_all = pd.DataFrame(
{"log_time": log_times, "weibull_y": weibull_y, "status": np.where(is_failure == 1, "Failure", "Censored")}
)
df_failures = df_all[df_all["status"] == "Failure"].copy()
# Linear fit on Weibull-linearized failure points: beta = slope, eta = exp(-intercept/slope)
slope, intercept, r_value, _, _ = stats.linregress(df_failures["log_time"], df_failures["weibull_y"])
beta_fit = slope
eta_fit = np.exp(-intercept / slope)
# Fitted regression line spanning data range with padding
fit_x = np.linspace(np.log(all_times.min() * 0.6), np.log(all_times.max() * 1.4), 100)
fit_y = slope * fit_x + intercept
df_fit = pd.DataFrame({"log_time": fit_x, "weibull_y": fit_y})
# 63.2% characteristic life reference (where Weibull CDF = 1 - 1/e ≈ 0.632)
ref_y = np.log(-np.log(1 - 0.632)) # ≈ 0.0
log_eta = np.log(eta_fit)
# Y-axis ticks — Weibull probability scale
prob_levels = [0.01, 0.05, 0.10, 0.20, 0.50, 0.632, 0.90, 0.99]
y_ticks = [np.log(-np.log(1 - p)) for p in prob_levels]
y_labels = ["1%", "5%", "10%", "20%", "50%", "63.2%", "90%", "99%"]
# X-axis ticks — hours on log scale
x_vals = [1000, 2000, 4000, 6000, 8000, 12000]
x_ticks = [np.log(v) for v in x_vals]
x_labels = ["1,000", "2,000", "4,000", "6,000", "8,000", "12,000"]
# Crosshair segments emphasising the characteristic life intersection
df_h_seg = pd.DataFrame({"x": [log_eta - 0.4], "xend": [log_eta + 0.4], "y": [ref_y], "yend": [ref_y]})
df_v_seg = pd.DataFrame({"x": [log_eta], "xend": [log_eta], "y": [ref_y - 0.28], "yend": [ref_y + 0.28]})
# Characteristic life label — positioned above the crosshair to avoid data crowding
df_eta_label = pd.DataFrame({"x": [log_eta], "y": [ref_y + 0.55], "label": [f"η = {eta_fit:.0f} hrs\n(63.2%)"]})
# Parameter annotation — upper-left (high-probability region is sparse for wear-out data)
df_annot = pd.DataFrame(
{
"x": [x_ticks[0] + 0.1],
"y": [y_ticks[-2] - 0.1],
"label": [f"β = {beta_fit:.2f}\nη = {eta_fit:.0f} hrs\nR² = {r_value**2:.4f}"],
}
)
TITLE = "probability-weibull · python · letsplot · anyplot.ai"
anyplot_theme = theme(
plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
panel_background=element_rect(fill=PAGE_BG),
panel_border=element_blank(),
panel_grid_major=element_line(color=INK_MUTED, size=0.25),
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),
plot_title=element_text(color=INK, size=16),
legend_background=element_rect(fill=ELEVATED_BG, color=INK_SOFT),
legend_text=element_text(color=INK_SOFT, size=10),
legend_title=element_text(color=INK, size=10),
legend_position=[0.05, 0.17],
)
# Plot
plot = (
ggplot(df_all, aes(x="log_time", y="weibull_y", color="status", shape="status"))
# Fitted regression line
+ geom_line(
data=df_fit, mapping=aes(x="log_time", y="weibull_y"), color=INK, size=1.6, alpha=0.90, inherit_aes=False
)
# 63.2% horizontal reference
+ geom_hline(yintercept=ref_y, linetype="dashed", color=INK_MUTED, size=0.6)
# Vertical reference at eta
+ geom_vline(xintercept=log_eta, linetype="dotted", color=INK_MUTED, size=0.6)
# Crosshair emphasis at characteristic life intersection
+ geom_segment(
data=df_h_seg,
mapping=aes(x="x", y="y", xend="xend", yend="yend"),
color=COLOR_ETA,
size=1.8,
alpha=0.8,
inherit_aes=False,
)
+ geom_segment(
data=df_v_seg,
mapping=aes(x="x", y="y", xend="xend", yend="yend"),
color=COLOR_ETA,
size=1.8,
alpha=0.8,
inherit_aes=False,
)
# Failure / censored data points (color + shape redundancy for accessibility)
+ geom_point(size=5, alpha=0.9, stroke=1.0)
# Diamond marker at characteristic life intersection
+ geom_point(
data=pd.DataFrame({"x": [log_eta], "y": [ref_y]}),
mapping=aes(x="x", y="y"),
color=COLOR_ETA,
fill=COLOR_ETA,
size=8,
shape=18,
alpha=0.95,
inherit_aes=False,
)
# Characteristic life label above intersection to avoid crowding
+ geom_text(
data=df_eta_label,
mapping=aes(x="x", y="y", label="label"),
size=5,
color=COLOR_ETA,
hjust=0.5,
fontface="bold",
inherit_aes=False,
)
# Weibull parameter annotation (upper-left, away from dense data region)
+ geom_text(
data=df_annot,
mapping=aes(x="x", y="y", label="label"),
size=6,
color=INK_SOFT,
hjust=0,
fontface="bold",
inherit_aes=False,
)
+ scale_color_manual(name="Observation", values={"Failure": BRAND, "Censored": COLOR_CENSORED})
+ scale_shape_manual(name="Observation", values={"Failure": 16, "Censored": 1})
+ scale_x_continuous(breaks=x_ticks, labels=x_labels)
+ scale_y_continuous(breaks=y_ticks, labels=y_labels)
+ labs(x="Time to Failure (hours)", y="Cumulative Failure Probability", title=TITLE)
+ ggsize(800, 450)
+ anyplot_theme
)
# Save
ggsave(plot, f"plot-{THEME}.png", path=".", scale=4)
ggsave(plot, f"plot-{THEME}.html", path=".")
Part of Weibull Probability Plot for Reliability Analysis on anyplot.ai.