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: makie 0.22.10 | Julia 1.11.9
# Quality: 86/100 | Created: 2026-06-10
using CairoMakie
using Colors
using Random
using Statistics
Random.seed!(42)
# Theme tokens
const THEME = get(ENV, "ANYPLOT_THEME", "light")
const PAGE_BG = THEME == "light" ? colorant"#FAF8F1" : colorant"#1A1A17"
const INK = THEME == "light" ? colorant"#1A1A17" : colorant"#F0EFE8"
const INK_SOFT = THEME == "light" ? colorant"#4A4A44" : colorant"#B8B7B0"
const IMPRINT_PALETTE = [
colorant"#009E73", colorant"#C475FD", colorant"#4467A3", colorant"#BD8233",
colorant"#AE3030", colorant"#2ABCCD", colorant"#954477", colorant"#99B314",
]
# Data: AR(2) time series — monthly economic indicator
# x[t] = 0.7*x[t-1] + 0.2*x[t-2] + noise (φ₁=0.7, φ₂=0.2)
# PACF cuts off at lag 2; ACF decays as a mixture of exponentials
n_obs = 300
series = zeros(n_obs)
series[1] = randn()
series[2] = 0.7 * series[1] + randn()
for t in 3:n_obs
series[t] = 0.7 * series[t-1] + 0.2 * series[t-2] + randn()
end
# ACF (lags 0 through max_lag)
max_lag = 35
s_mean = mean(series)
s_var = sum((v - s_mean)^2 for v in series) / n_obs
acf_vals = [sum((series[i] - s_mean) * (series[i-k] - s_mean) for i in (k+1):n_obs) / (n_obs * s_var)
for k in 0:max_lag]
# PACF via Levinson-Durbin recursion
phi = zeros(max_lag, max_lag)
phi[1, 1] = acf_vals[2]
pacf_vals = zeros(max_lag)
pacf_vals[1] = phi[1, 1]
for k in 2:max_lag
numer = acf_vals[k + 1]
denom = 1.0
for j in 1:(k - 1)
numer -= phi[k-1, j] * acf_vals[k - j + 1]
denom -= phi[k-1, j] * acf_vals[j + 1]
end
phi[k, k] = numer / denom
for j in 1:(k - 1)
phi[k, j] = phi[k-1, j] - phi[k, k] * phi[k-1, k-j]
end
pacf_vals[k] = phi[k, k]
end
conf_bound = 1.96 / sqrt(n_obs)
acf_lags = Float64.(0:max_lag)
# Build linesegments! data for ACF: interleaved [x_base, x_tip, ...] pairs per segment
acf_seg_x = vcat([[lag, lag] for lag in acf_lags]...)
acf_seg_y = vcat([[0.0, v] for v in acf_vals]...)
# PACF: split lags into significant (|r| > conf_bound) vs. within-bound for visual hierarchy
pacf_sig_idx = findall(v -> abs(v) > conf_bound, pacf_vals)
pacf_ns_idx = findall(v -> abs(v) <= conf_bound, pacf_vals)
# Muted variant for non-significant PACF lags (same hue, lower opacity)
GREEN_FULL = IMPRINT_PALETTE[1]
GREEN_MUTED = RGBAf(GREEN_FULL.r, GREEN_FULL.g, GREEN_FULL.b, 0.38f0)
function make_segs(idx, vals)
xs = vcat([[Float64(i), Float64(i)] for i in idx]...)
ys = vcat([[0.0, vals[i]] for i in idx]...)
xs, ys
end
# Figure
fig = Figure(
size = (1600, 900),
fontsize = 14,
backgroundcolor = PAGE_BG,
)
# ACF axis (top)
ax_acf = Axis(
fig[1, 1];
title = "AR(2) Process · acf-pacf · julia · makie · anyplot.ai",
titlesize = 20,
titlecolor = INK,
ylabel = "ACF",
ylabelsize = 14,
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,
ygridvisible = true,
ygridcolor = RGBAf(INK.r, INK.g, INK.b, 0.12f0),
)
# PACF axis (bottom)
ax_pacf = Axis(
fig[2, 1];
xlabel = "Lag",
ylabel = "PACF",
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,
ygridvisible = true,
ygridcolor = RGBAf(INK.r, INK.g, INK.b, 0.12f0),
)
linkxaxes!(ax_acf, ax_pacf)
ax_acf.xticklabelsvisible = false
ax_acf.xlabelvisible = false
# ACF stems, tips, and reference lines
linesegments!(ax_acf, acf_seg_x, acf_seg_y;
color = GREEN_FULL, linewidth = 2.0)
scatter!(ax_acf, acf_lags, acf_vals;
color = GREEN_FULL, markersize = 8, strokewidth = 0)
hlines!(ax_acf, [0.0]; color = INK_SOFT, linewidth = 1.0)
hlines!(ax_acf, [conf_bound, -conf_bound];
color = INK_SOFT, linestyle = :dash, linewidth = 1.5)
# PACF: within-bound lags (muted) drawn first, then significant lags on top
if !isempty(pacf_ns_idx)
ns_sx, ns_sy = make_segs(pacf_ns_idx, pacf_vals)
linesegments!(ax_pacf, ns_sx, ns_sy; color = GREEN_MUTED, linewidth = 1.5)
scatter!(ax_pacf, Float64.(pacf_ns_idx), pacf_vals[pacf_ns_idx];
color = GREEN_MUTED, markersize = 7, strokewidth = 0)
end
if !isempty(pacf_sig_idx)
sig_sx, sig_sy = make_segs(pacf_sig_idx, pacf_vals)
linesegments!(ax_pacf, sig_sx, sig_sy; color = GREEN_FULL, linewidth = 2.5)
scatter!(ax_pacf, Float64.(pacf_sig_idx), pacf_vals[pacf_sig_idx];
color = GREEN_FULL, markersize = 9, strokewidth = 0)
end
hlines!(ax_pacf, [0.0]; color = INK_SOFT, linewidth = 1.0)
hlines!(ax_pacf, [conf_bound, -conf_bound];
color = INK_SOFT, linestyle = :dash, linewidth = 1.5)
rowgap!(fig.layout, 1, 20)
save("plot-$(THEME).png", fig; px_per_unit = 2)
Runnable source as JSON, for any HTTP client: https://api.anyplot.ai/specs/acf-pacf/makie/code. Any spec id and library id listed in llms-full.txt fit the same URL shape; every URL below is complete and callable.
{
"spec_id": "acf-pacf",
"language": "julia",
"library": "makie",
"page": "https://anyplot.ai/acf-pacf/julia/makie",
"hub": "https://anyplot.ai/acf-pacf",
"code_json": "https://api.anyplot.ai/specs/acf-pacf/makie/code",
"spec_json": "https://api.anyplot.ai/specs/acf-pacf",
"render_light_png": "https://storage.googleapis.com/anyplot-images/plots/acf-pacf/julia/makie/plot-light.png",
"render_dark_png": "https://storage.googleapis.com/anyplot-images/plots/acf-pacf/julia/makie/plot-dark.png",
"quality_score": 86.0,
"license": "MIT",
"guide": "https://anyplot.ai/llms.txt"
}Part of Autocorrelation and Partial Autocorrelation (ACF/PACF) Plot on anyplot.ai.