Displays the autocorrelation function (ACF) and partial autocorrelation function (PACF) of a time series as vertical stem/bar plots arranged in two vertically stacked subplots. Each lag is represented by a vertical line from zero to the correlation value, with horizontal dashed lines indicating 95% confidence bounds. These plots are essential for identifying the order of AR and MA components in ARIMA modeling and for diagnosing residual independence.
""" anyplot.ai
acf-pacf: Autocorrelation and Partial Autocorrelation (ACF/PACF) Plot
Library: plotly 6.8.0 | Python 3.13.13
Quality: 91/100 | Updated: 2026-06-10
"""
import os
import numpy as np
import plotly.graph_objects as go
from plotly.subplots import make_subplots
# Theme tokens
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"
GRID = "rgba(26,26,23,0.15)" if THEME == "light" else "rgba(240,239,232,0.15)"
# Imprint palette — position 1 for significant lags
BRAND = "#009E73"
# Confidence band fill/border (INK_MUTED at low opacity)
band_fill = "rgba(107,106,99,0.12)" if THEME == "light" else "rgba(168,167,159,0.12)"
band_border = "rgba(107,106,99,0.40)" if THEME == "light" else "rgba(168,167,159,0.40)"
# Data — monthly retail sales AR(2) process
np.random.seed(42)
n_obs = 200
ar1_coeff, ar2_coeff = 0.7, -0.3
series = np.zeros(n_obs)
noise = np.random.normal(0, 1, n_obs)
for t in range(2, n_obs):
series[t] = ar1_coeff * series[t - 1] + ar2_coeff * series[t - 2] + noise[t]
# Compute ACF
n_lags = 35
series_centered = series - np.mean(series)
variance = np.sum(series_centered**2)
acf_values = np.array(
[np.sum(series_centered[: n_obs - k] * series_centered[k:]) / variance for k in range(n_lags + 1)]
)
# Compute PACF via Durbin-Levinson recursion
pacf_values = np.zeros(n_lags + 1)
pacf_values[0] = 1.0
pacf_values[1] = acf_values[1]
phi = np.zeros((n_lags + 1, n_lags + 1))
phi[1, 1] = acf_values[1]
for k in range(2, n_lags + 1):
num = acf_values[k] - np.sum(phi[k - 1, 1:k] * acf_values[k - 1 : 0 : -1])
den = 1.0 - np.sum(phi[k - 1, 1:k] * acf_values[1:k])
phi[k, k] = num / den if abs(den) > 1e-12 else 0.0
for j in range(1, k):
phi[k, j] = phi[k - 1, j] - phi[k, k] * phi[k - 1, k - j]
pacf_values[k] = phi[k, k]
conf_bound = 1.96 / np.sqrt(n_obs)
lags_acf = np.arange(0, n_lags + 1)
lags_pacf = np.arange(1, n_lags + 1)
pacf_plot = pacf_values[1:]
acf_significant = np.abs(acf_values) > conf_bound
pacf_significant = np.abs(pacf_plot) > conf_bound
# Build stem coords using None separators (one trace per class, not per stem)
acf_sig_x, acf_sig_y, acf_nsig_x, acf_nsig_y = [], [], [], []
for lag, val, sig in zip(lags_acf, acf_values, acf_significant, strict=False):
(acf_sig_x if sig else acf_nsig_x).extend([int(lag), int(lag), None])
(acf_sig_y if sig else acf_nsig_y).extend([0, float(val), None])
pacf_sig_x, pacf_sig_y, pacf_nsig_x, pacf_nsig_y = [], [], [], []
for lag, val, sig in zip(lags_pacf, pacf_plot, pacf_significant, strict=False):
(pacf_sig_x if sig else pacf_nsig_x).extend([int(lag), int(lag), None])
(pacf_sig_y if sig else pacf_nsig_y).extend([0, float(val), None])
# Plot
fig = make_subplots(
rows=2,
cols=1,
shared_xaxes=True,
vertical_spacing=0.10,
subplot_titles=["Autocorrelation (ACF)", "Partial Autocorrelation (PACF)"],
)
hover_tpl = "Lag %{x}<br>Correlation: %{y:.3f}<extra></extra>"
# ACF stems (row 1)
if acf_sig_x:
fig.add_trace(
go.Scatter(
x=acf_sig_x,
y=acf_sig_y,
mode="lines",
line={"color": BRAND, "width": 3},
showlegend=False,
hoverinfo="skip",
),
row=1,
col=1,
)
if acf_nsig_x:
fig.add_trace(
go.Scatter(
x=acf_nsig_x,
y=acf_nsig_y,
mode="lines",
line={"color": INK_MUTED, "width": 2},
showlegend=False,
hoverinfo="skip",
),
row=1,
col=1,
)
if np.any(acf_significant):
fig.add_trace(
go.Scatter(
x=lags_acf[acf_significant],
y=acf_values[acf_significant],
mode="markers",
name="Significant",
marker={"size": 12, "color": BRAND, "line": {"color": PAGE_BG, "width": 2}},
hovertemplate=hover_tpl,
),
row=1,
col=1,
)
if np.any(~acf_significant):
fig.add_trace(
go.Scatter(
x=lags_acf[~acf_significant],
y=acf_values[~acf_significant],
mode="markers",
name="Non-significant",
marker={"size": 9, "color": INK_MUTED, "line": {"color": PAGE_BG, "width": 1.5}},
hovertemplate=hover_tpl,
),
row=1,
col=1,
)
# PACF stems (row 2)
if pacf_sig_x:
fig.add_trace(
go.Scatter(
x=pacf_sig_x,
y=pacf_sig_y,
mode="lines",
line={"color": BRAND, "width": 3},
showlegend=False,
hoverinfo="skip",
),
row=2,
col=1,
)
if pacf_nsig_x:
fig.add_trace(
go.Scatter(
x=pacf_nsig_x,
y=pacf_nsig_y,
mode="lines",
line={"color": INK_MUTED, "width": 2},
showlegend=False,
hoverinfo="skip",
),
row=2,
col=1,
)
if np.any(pacf_significant):
fig.add_trace(
go.Scatter(
x=lags_pacf[pacf_significant],
y=pacf_plot[pacf_significant],
mode="markers",
showlegend=False,
marker={"size": 12, "color": BRAND, "line": {"color": PAGE_BG, "width": 2}},
hovertemplate=hover_tpl,
),
row=2,
col=1,
)
if np.any(~pacf_significant):
fig.add_trace(
go.Scatter(
x=lags_pacf[~pacf_significant],
y=pacf_plot[~pacf_significant],
mode="markers",
showlegend=False,
marker={"size": 9, "color": INK_MUTED, "line": {"color": PAGE_BG, "width": 1.5}},
hovertemplate=hover_tpl,
),
row=2,
col=1,
)
# Confidence bands and zero baselines
for row in [1, 2]:
x_start, x_end = (0, n_lags) if row == 1 else (1, n_lags)
fig.add_trace(
go.Scatter(
x=[x_start, x_end, x_end, x_start],
y=[conf_bound, conf_bound, -conf_bound, -conf_bound],
fill="toself",
fillcolor=band_fill,
line={"color": band_border, "width": 1, "dash": "dash"},
showlegend=(row == 1),
name="95% Confidence",
hoverinfo="skip",
),
row=row,
col=1,
)
fig.add_trace(
go.Scatter(
x=[x_start, x_end],
y=[0, 0],
mode="lines",
line={"color": INK_SOFT, "width": 1.5},
showlegend=False,
hoverinfo="skip",
),
row=row,
col=1,
)
# Title (length-adaptive font size)
title_str = "Monthly Retail Sales · acf-pacf · python · plotly · anyplot.ai"
subtitle = "AR(2) Process (n=200, φ₁=0.7, φ₂=−0.3)"
n = len(title_str)
title_size = round(16 * (67 / n)) if n > 67 else 16
# Layout
fig.update_layout(
autosize=False,
width=800,
height=450,
template="plotly_white",
paper_bgcolor=PAGE_BG,
plot_bgcolor=PAGE_BG,
margin={"l": 90, "r": 50, "t": 130, "b": 100},
title={
"text": f"{title_str}<br><sup style='color:{INK_SOFT};font-size:11px'>{subtitle}</sup>",
"font": {"size": title_size, "color": INK},
"x": 0.5,
"xanchor": "center",
},
legend={
"orientation": "h",
"yanchor": "top",
"y": -0.18,
"xanchor": "center",
"x": 0.5,
"font": {"size": 10, "color": INK_SOFT},
"bgcolor": ELEVATED_BG,
"bordercolor": INK_SOFT,
"borderwidth": 1,
},
hoverlabel={"font_size": 13},
)
# Subplot title font
for annotation in fig.layout.annotations:
annotation.font = {"size": 13, "color": INK_SOFT}
# Y-axes
for row, label in [(1, "ACF (correlation)"), (2, "PACF (correlation)")]:
fig.update_yaxes(
title_text=label,
title_font={"size": 12, "color": INK},
tickfont={"size": 10, "color": INK_SOFT},
showgrid=True,
gridcolor=GRID,
zeroline=False,
showline=True,
linecolor=INK_SOFT,
row=row,
col=1,
)
# X-axes (global then row-specific)
fig.update_xaxes(
tickfont={"size": 10, "color": INK_SOFT},
linecolor=INK_SOFT,
showgrid=False,
showspikes=True,
spikecolor=INK_MUTED,
spikethickness=1,
spikedash="dot",
spikemode="across",
)
fig.update_xaxes(showline=True, title_text="Lag (periods)", title_font={"size": 12, "color": INK}, row=2, col=1)
fig.update_xaxes(showline=False, row=1, col=1)
# Save
fig.write_image(f"plot-{THEME}.png", width=800, height=450, scale=4)
fig.write_html(f"plot-{THEME}.html", include_plotlyjs="cdn")
Part of Autocorrelation and Partial Autocorrelation (ACF/PACF) Plot on anyplot.ai.