A lag plot is a scatter plot of a time series against a lagged version of itself, plotting y(t) on the x-axis versus y(t+k) on the y-axis for a given lag order k. If the data is purely random, points scatter uniformly with no visible structure; if autocorrelation is present, distinctive patterns emerge — linear clusters for autoregressive processes, elliptical shapes for seasonal data. This provides a quick visual diagnostic for time series dependence, complementing numerical tools like ACF/PACF.
""" anyplot.ai
scatter-lag: Lag Plot for Time Series Autocorrelation Diagnosis
Library: pygal 3.1.3 | Python 3.13.14
Quality: 89/100 | Updated: 2026-06-24
"""
import os
import sys
import numpy as np
# Avoid name collision: pygal.py filename shadows the pygal package
_cwd = sys.path[0] if sys.path[0] else "."
if _cwd in sys.path:
sys.path.remove(_cwd)
import pygal
from pygal.style import Style
# Theme tokens — Imprint palette, theme-adaptive chrome
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
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 for temporal quartiles + structural reference layers
PALETTE = (
"#009E73", # Q1 — brand green (ALWAYS first)
"#C475FD", # Q2 — lavender
"#4467A3", # Q3 — blue
"#BD8233", # Q4 — ochre
INK_MUTED, # +1σ envelope (structural, muted)
INK_MUTED, # −1σ envelope (structural, muted)
INK_SOFT, # y = x reference diagonal
)
# Data — synthetic AR(1) hourly temperature process, moderate positive autocorrelation
np.random.seed(42)
n = 400
phi = 0.78
noise = np.random.normal(0, 1.0, n)
temperature = np.zeros(n)
temperature[0] = 20.0
for i in range(1, n):
temperature[i] = 20.0 + phi * (temperature[i - 1] - 20.0) + noise[i]
lag = 1
y_t = temperature[:-lag]
y_t_lag = temperature[lag:]
# Temporal quartile masks — color by time to reveal temporal structure
time_idx = np.arange(len(y_t))
q_bounds = np.percentile(time_idx, [25, 50, 75])
early = [
{"value": (float(y_t[i]), float(y_t_lag[i])), "label": f"Hour {i + 1}"}
for i in range(len(y_t))
if time_idx[i] < q_bounds[0]
]
mid_early = [
{"value": (float(y_t[i]), float(y_t_lag[i])), "label": f"Hour {i + 1}"}
for i in range(len(y_t))
if q_bounds[0] <= time_idx[i] < q_bounds[1]
]
mid_late = [
{"value": (float(y_t[i]), float(y_t_lag[i])), "label": f"Hour {i + 1}"}
for i in range(len(y_t))
if q_bounds[1] <= time_idx[i] < q_bounds[2]
]
late = [
{"value": (float(y_t[i]), float(y_t_lag[i])), "label": f"Hour {i + 1}"}
for i in range(len(y_t))
if time_idx[i] >= q_bounds[2]
]
# Correlation coefficient for title annotation
r = np.corrcoef(y_t, y_t_lag)[0, 1]
# Reference geometry — diagonal y = x and ±1σ spread envelope
data_min = float(min(y_t.min(), y_t_lag.min()))
data_max = float(max(y_t.max(), y_t_lag.max()))
pad = (data_max - data_min) * 0.05
ref_start = data_min - pad
ref_end = data_max + pad
ref_line = [(ref_start, ref_start), (ref_end, ref_end)]
sigma = float(np.std(y_t_lag - y_t))
upper_env = [(ref_start, ref_start + sigma), (ref_end, ref_end + sigma)]
lower_env = [(ref_start, ref_start - sigma), (ref_end, ref_end - sigma)]
# Title — length-aware font scaling (baseline 66px for ~67 chars)
title = f"Lag Plot (k={lag}, r={r:.2f}) · scatter-lag · python · pygal · anyplot.ai"
title_len = len(title)
title_font_size = round(66 * 67 / title_len) if title_len > 67 else 66
title_font_size = max(title_font_size, 44)
font = "DejaVu Sans, Helvetica, Arial, sans-serif"
custom_style = Style(
background=PAGE_BG,
plot_background=PAGE_BG,
foreground=INK,
foreground_strong=INK,
foreground_subtle=INK_MUTED,
colors=PALETTE,
font_family=font,
title_font_family=font,
title_font_size=title_font_size,
label_font_size=56,
major_label_font_size=44,
legend_font_size=44,
legend_font_family=font,
value_font_size=36,
tooltip_font_size=32,
tooltip_font_family=font,
opacity=0.45,
opacity_hover=0.95,
stroke_width=2.5,
)
# Chart — canvas at 3200×1800 (landscape 16:9, hard contract)
chart = pygal.XY(
width=3200,
height=1800,
style=custom_style,
title=title,
x_title="y(t) — Temperature (°C)",
y_title=f"y(t+{lag}) — Temp. (°C)",
show_legend=True,
legend_at_bottom=True,
legend_at_bottom_columns=4,
legend_box_size=28,
stroke=False,
dots_size=8,
show_x_guides=True,
show_y_guides=True,
x_value_formatter=lambda x: f"{x:.1f}",
value_formatter=lambda y: f"{y:.1f}",
margin_bottom=65,
margin_left=80,
margin_right=30,
margin_top=40,
range=(ref_start, ref_end),
xrange=(ref_start, ref_end),
x_labels_major_count=8,
y_labels_major_count=8,
print_values=False,
print_zeroes=False,
truncate_legend=40,
)
# Temporal quartile scatter series — size encodes temporal recency (7→10)
chart.add("Hours 1–100", early, stroke=False, dots_size=7)
chart.add("Hours 101–200", mid_early, stroke=False, dots_size=8)
chart.add("Hours 201–300", mid_late, stroke=False, dots_size=9)
chart.add("Hours 301–399", late, stroke=False, dots_size=10)
# ±1σ spread envelope (structural layer — no legend entry)
env_style = {"width": 3, "dasharray": "6, 8", "linecap": "round"}
chart.add(None, upper_env, stroke=True, show_dots=False, stroke_style=env_style)
chart.add(None, lower_env, stroke=True, show_dots=False, stroke_style=env_style)
# Diagonal reference line y = x (strong autocorrelation aligns points to this)
chart.add(
"y = x (±1σ)",
ref_line,
stroke=True,
show_dots=False,
stroke_style={"width": 6, "dasharray": "24, 12", "linecap": "round"},
)
# Save — PNG + interactive HTML (pygal is interactive)
chart.render_to_png(f"plot-{THEME}.png")
with open(f"plot-{THEME}.html", "wb") as f:
f.write(chart.render())
Part of Lag Plot for Time Series Autocorrelation Diagnosis on anyplot.ai.