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: plotnine 0.15.5 | Python 3.13.13
Quality: 91/100 | Updated: 2026-06-07
"""
import os
import sys
import numpy as np
import pandas as pd
# Work around naming conflict between plotnine.py script and plotnine package
script_dir = os.path.dirname(os.path.abspath(__file__))
if script_dir in sys.path:
sys.path.remove(script_dir)
if "" in sys.path:
sys.path.remove("")
if "." in sys.path:
sys.path.remove(".")
from plotnine import (
aes,
annotate,
element_blank,
element_line,
element_rect,
element_text,
geom_line,
geom_point,
geom_ribbon,
geom_segment,
ggplot,
guide_legend,
guides,
labs,
scale_color_manual,
scale_fill_manual,
scale_shape_manual,
scale_x_log10,
scale_y_continuous,
theme,
theme_minimal,
)
from scipy import stats
# Theme-adaptive chrome tokens (Imprint palette)
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 categorical palette — canonical hybrid-v3 order
IMPRINT_PALETTE = ["#009E73", "#C475FD", "#4467A3", "#BD8233", "#AE3030", "#2ABCCD", "#954477", "#99B314"]
FIT_COLOR = IMPRINT_PALETTE[2] # blue — Weibull regression line & band
CHAR_COLOR = IMPRINT_PALETTE[4] # matte red — characteristic life focal point
# Data — turbine blade fatigue-life: 25 failures + 7 censored
np.random.seed(42)
n_failures = 25
n_censored = 7
failure_times = np.sort(stats.weibull_min.rvs(c=2.8, scale=5000, size=n_failures))
censored_times = np.sort(stats.uniform.rvs(loc=1000, scale=5000, size=n_censored))
all_times = np.concatenate([failure_times, censored_times])
is_failure = np.array([True] * n_failures + [False] * n_censored)
sort_idx = np.argsort(all_times)
all_times = all_times[sort_idx]
is_failure = is_failure[sort_idx]
failure_order = np.cumsum(is_failure)
median_rank = np.where(is_failure, (failure_order - 0.3) / (n_failures + 0.4), np.nan)
weibull_y = np.where(is_failure, np.log(-np.log(1 - median_rank)), np.nan)
df = pd.DataFrame({"time": all_times, "weibull_y": weibull_y, "status": np.where(is_failure, "Failure", "Censored")})
# Weibull regression on ln(time) vs ln(-ln(1-F))
failures_df = df[df["status"] == "Failure"].dropna()
log_times = np.log(failures_df["time"].values)
slope, intercept, r_value, _, _ = stats.linregress(log_times, failures_df["weibull_y"].values)
beta = slope
eta = np.exp(-intercept / beta)
# Fitted line and 95% prediction band (in raw time units for scale_x_log10)
log_time_range = np.linspace(np.log(all_times.min() * 0.7), np.log(all_times.max() * 1.3), 200)
fitted_y = beta * log_time_range + intercept
n_fit = len(failures_df)
x_mean = log_times.mean()
x_ss = ((log_times - x_mean) ** 2).sum()
residuals = failures_df["weibull_y"].values - (beta * log_times + intercept)
mse = (residuals**2).sum() / (n_fit - 2)
se_pred = np.sqrt(mse * (1 + 1 / n_fit + (log_time_range - x_mean) ** 2 / x_ss))
t_crit = stats.t.ppf(0.975, n_fit - 2)
fit_df = pd.DataFrame(
{
"time": np.exp(log_time_range),
"weibull_y": fitted_y,
"ymin": fitted_y - t_crit * se_pred,
"ymax": fitted_y + t_crit * se_pred,
}
)
# Y-axis: Weibull probability scale (linearised CDF)
prob_levels = np.array([0.01, 0.05, 0.10, 0.20, 0.40, 0.632, 0.80, 0.90, 0.99])
weibull_ticks = np.log(-np.log(1 - prob_levels))
prob_labels = [f"{p * 100:.1f}%".replace(".0%", "%") for p in prob_levels]
# X-axis breaks (raw time, scale_x_log10 handles the transform)
x_tick_values = [1000, 2000, 3000, 5000, 7000, 10000]
x_labels = [f"{v:,}" for v in x_tick_values]
# Reference values
ref_y = np.log(-np.log(1 - 0.632)) # ≈ 0: characteristic life on Weibull y-scale
# Scatter dataframes (raw time units)
failures_plot = failures_df[["time", "weibull_y"]].assign(Status="Failure")
censored_plot = df[df["status"] == "Censored"][["time"]].assign(weibull_y=weibull_ticks[0] + 0.18, Status="Censored")
scatter_df = pd.concat([failures_plot, censored_plot], ignore_index=True)
highlight_df = pd.DataFrame({"time": [eta], "weibull_y": [ref_y]})
# Annotation positions in data space (time units for x; Weibull-y for y)
# Shift further right to avoid crowding with high-probability scatter points
annot_time = eta * 1.55
annot_y = ref_y + 0.42
plot = (
ggplot()
# 95% prediction band
+ geom_ribbon(fit_df, aes(x="time", ymin="ymin", ymax="ymax"), fill=FIT_COLOR, alpha=0.12)
# Weibull regression line
+ geom_line(fit_df, aes(x="time", y="weibull_y"), color=FIT_COLOR, size=0.9, alpha=0.85)
# 63.2% horizontal reference (dashed guide)
+ geom_segment(
aes(x=x_tick_values[0] * 0.75, xend=eta, y=ref_y, yend=ref_y),
linetype="dashed",
color=INK_MUTED,
size=0.35,
alpha=0.65,
)
# Vertical drop guide to x-axis
+ geom_segment(
aes(x=eta, xend=eta, y=weibull_ticks[0] - 0.1, yend=ref_y),
linetype="dashed",
color=INK_MUTED,
size=0.35,
alpha=0.65,
)
# Failure and censored data points
+ geom_point(
scatter_df, aes(x="time", y="weibull_y", color="Status", shape="Status"), size=2.8, alpha=0.82, stroke=0.3
)
# Characteristic life diamond marker
+ geom_point(
highlight_df,
aes(x="time", y="weibull_y"),
color=CHAR_COLOR,
fill=CHAR_COLOR,
size=4.5,
shape="D",
alpha=0.95,
show_legend=False,
)
# Eta annotation label
+ annotate(
"text",
x=annot_time,
y=annot_y,
label=f"η = {eta:,.0f} hrs (63.2%)",
size=3.5,
ha="left",
color=CHAR_COLOR,
fontweight="bold",
fontstyle="italic",
)
# Thin connector from annotation to diamond
+ geom_segment(
aes(x=annot_time * 0.98, xend=eta * 1.02, y=annot_y - 0.10, yend=ref_y + 0.05),
color=CHAR_COLOR,
size=0.28,
alpha=0.50,
)
# Weibull parameters summary (upper-left)
+ annotate(
"text",
x=x_tick_values[0] * 1.06,
y=weibull_ticks[-2] + 0.22,
label=f"β = {beta:.2f} · η = {eta:,.0f} hrs · R² = {r_value**2:.3f}",
size=3.5,
ha="left",
va="top",
color=FIT_COLOR,
fontweight="bold",
)
# Scales — Imprint palette: failures=green (first), censored=lavender (second)
+ scale_color_manual(
values={"Failure": IMPRINT_PALETTE[0], "Censored": IMPRINT_PALETTE[1]},
name="Observation",
limits=["Failure", "Censored"],
)
+ scale_shape_manual(values={"Failure": "o", "Censored": "^"}, name="Observation", limits=["Failure", "Censored"])
+ scale_fill_manual(
values={"Failure": IMPRINT_PALETTE[0], "Censored": IMPRINT_PALETTE[1]},
name="Observation",
limits=["Failure", "Censored"],
)
# scale_x_log10: idiomatic plotnine/ggplot2 log axis (no manual ln transform needed)
+ scale_x_log10(breaks=x_tick_values, labels=x_labels)
+ scale_y_continuous(breaks=weibull_ticks.tolist(), labels=prob_labels)
+ guides(color=guide_legend(override_aes={"size": 3.5, "alpha": 1}))
+ labs(
x="Time to Failure (hours)",
y="Cumulative Failure Probability",
title="probability-weibull · plotnine · anyplot.ai",
caption="Turbine blade fatigue-life — 25 failures, 7 censored observations",
)
+ theme_minimal(base_family="sans-serif")
+ theme(
figure_size=(8, 4.5),
plot_title=element_text(size=12, weight="bold", color=INK, margin={"b": 4}),
plot_caption=element_text(size=7.5, color=INK_MUTED, fontstyle="italic", ha="center"),
axis_title_x=element_text(size=10, color=INK, margin={"t": 8}),
axis_title_y=element_text(size=10, color=INK, margin={"r": 8}),
axis_text_x=element_text(size=8, color=INK_SOFT),
axis_text_y=element_text(size=8, color=INK_SOFT),
panel_grid_major_x=element_line(color=INK, size=0.2, alpha=0.12),
panel_grid_major_y=element_line(color=INK, size=0.2, alpha=0.12),
panel_grid_minor=element_blank(),
panel_background=element_rect(fill=PAGE_BG, color="none"),
plot_background=element_rect(fill=PAGE_BG, color="none"),
legend_title=element_text(size=9, weight="bold", color=INK),
legend_text=element_text(size=8, color=INK_SOFT),
legend_position="right",
legend_background=element_rect(fill=ELEVATED_BG, color=INK_SOFT, size=0.3),
legend_key=element_rect(fill="none", color="none"),
axis_ticks=element_line(color=INK_SOFT, size=0.2),
plot_margin=0.04,
)
)
plot.save(f"plot-{THEME}.png", dpi=400, width=8, height=4.5, units="in", verbose=False)
Part of Weibull Probability Plot for Reliability Analysis on anyplot.ai.