A recurrence plot is a binary or distance-based matrix visualization that reveals when states in a dynamical system recur over time. Both axes represent time indices, and a point is plotted at position (i, j) when the system state at time i is sufficiently similar to the state at time j (distance below a threshold). The resulting symmetric matrix exposes hidden structure in complex, nonlinear dynamics — diagonal lines indicate determinism, vertical/horizontal lines reveal laminar states, and block structures signal regime changes.
""" anyplot.ai
recurrence-basic: Recurrence Plot for Nonlinear Time Series
Library: altair 6.2.1 | Python 3.13.13
Quality: 90/100 | Updated: 2026-06-10
"""
import importlib
import os
import sys
from PIL import Image
# Drop script directory from sys.path so the `altair` package resolves, not this file
sys.path[:] = [p for p in sys.path if os.path.abspath(p or ".") != os.path.dirname(os.path.abspath(__file__))]
alt = importlib.import_module("altair")
np = importlib.import_module("numpy")
pd = importlib.import_module("pandas")
# Theme tokens — Imprint palette, theme-adaptive chrome
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 sequential colormap for continuous single-polarity distances
SEQ_LOW = "#009E73" # brand green — near-identical states (distance ≈ 0)
SEQ_HIGH = "#4467A3" # blue — close to threshold
# Data — logistic map near type-I intermittency (r = 3.8284)
# 100 steps (reduced from 152) for larger cells; ε=0.17 (raised) for more V/H structure
np.random.seed(42)
n_steps = 100
r = 3.8284
x = np.zeros(n_steps)
x[0] = 0.1
for i in range(1, n_steps):
x[i] = r * x[i - 1] * (1 - x[i - 1])
# Time-delay embedding (dimension=3, delay=1) per Takens' theorem
embedding_dim = 3
delay = 1
n_embedded = n_steps - (embedding_dim - 1) * delay
embedded = np.column_stack([x[d * delay : d * delay + n_embedded] for d in range(embedding_dim)])
# Pairwise Euclidean distance matrix + threshold
diff = embedded[:, np.newaxis, :] - embedded[np.newaxis, :, :]
distance_matrix = np.sqrt(np.sum(diff**2, axis=2))
threshold = 0.17
recurrence_matrix = distance_matrix < threshold
# Sparse encoding: recurrent points only (absent cells render as PAGE_BG background)
rows, cols = np.where(recurrence_matrix)
distances = distance_matrix[rows, cols]
df = pd.DataFrame({"time_i": rows, "time_j": cols, "distance": distances})
# Interactive hover selection for cross-highlighting
hover = alt.selection_point(on="pointerover", fields=["time_i"], nearest=True, empty=False)
# Imprint sequential scale: distance=0 (self-recurrence) → green; near-threshold → blue
color_scale = alt.Scale(domain=[0, threshold], range=[SEQ_LOW, SEQ_HIGH])
# Recurrence heatmap
heatmap = (
alt.Chart(df)
.mark_rect(stroke=None)
.encode(
x=alt.X(
"time_i:O",
title="Time Index (step)",
axis=alt.Axis(
labelFontSize=10,
titleFontSize=12,
titlePadding=10,
values=list(range(0, n_embedded, 20)),
grid=False,
domainColor=INK_SOFT,
tickColor=INK_SOFT,
labelColor=INK_SOFT,
titleColor=INK,
),
scale=alt.Scale(paddingInner=0, paddingOuter=0),
),
y=alt.Y(
"time_j:O",
title="Time Index (step)",
axis=alt.Axis(
labelFontSize=10,
titleFontSize=12,
titlePadding=10,
values=list(range(0, n_embedded, 20)),
grid=False,
domainColor=INK_SOFT,
tickColor=INK_SOFT,
labelColor=INK_SOFT,
titleColor=INK,
),
scale=alt.Scale(paddingInner=0, paddingOuter=0),
),
color=alt.Color(
"distance:Q",
title="Distance",
scale=color_scale,
legend=alt.Legend(
titleFontSize=10,
labelFontSize=10,
orient="right",
direction="vertical",
gradientLength=200,
gradientThickness=12,
titlePadding=6,
offset=8,
labelColor=INK_SOFT,
titleColor=INK,
),
),
opacity=alt.condition(hover, alt.value(1.0), alt.value(0.88)),
tooltip=[
alt.Tooltip("time_i:Q", title="Time i"),
alt.Tooltip("time_j:Q", title="Time j"),
alt.Tooltip("distance:Q", title="Distance", format=".4f"),
],
)
.add_params(hover)
)
# Annotation markers for key structural features
annotations_data = pd.DataFrame(
{
"label": ["Main diagonal (self-recurrence)", "Laminar phase", "Diagonal lines (determinism)"],
"x": [20, 68, 42],
"y": [14, 60, 30],
}
)
annotation_shadow = (
alt.Chart(annotations_data)
.mark_text(fontSize=11, fontWeight="bold", color=PAGE_BG, align="left", dx=9, dy=-5, strokeWidth=3)
.encode(x=alt.X("x:O"), y=alt.Y("y:O"), text="label:N")
)
annotation_marks = (
alt.Chart(annotations_data)
.mark_text(fontSize=11, fontWeight="bold", color="#AE3030", align="left", dx=9, dy=-5)
.encode(x=alt.X("x:O"), y=alt.Y("y:O"), text="label:N")
)
annotation_dots = (
alt.Chart(annotations_data)
.mark_point(size=70, color="#AE3030", filled=True, opacity=0.9)
.encode(x=alt.X("x:O"), y=alt.Y("y:O"))
)
# Chart assembly — square target 2400 × 2400; inner view 450 × 450 + scale 4.0
title_str = "recurrence-basic · python · altair · anyplot.ai"
chart = (
alt.layer(heatmap, annotation_dots, annotation_shadow, annotation_marks)
.properties(
width=450,
height=450,
background=PAGE_BG,
title=alt.Title(
title_str,
subtitle=[
"Logistic map r=3.8284 (type-I intermittency) · ε=0.17 · Takens d=3, τ=1",
"Green = near-identical states · Diagonal lines = determinism · Blocks = laminar phases",
],
fontSize=16,
subtitleFontSize=10,
subtitleColor=INK_MUTED,
anchor="start",
offset=12,
color=INK,
),
padding={"left": 10, "right": 10, "top": 10, "bottom": 10},
)
.configure_axis(grid=False)
.configure_title(color=INK)
.configure_view(strokeWidth=0, fill=PAGE_BG)
.configure_legend(fillColor=ELEVATED_BG, strokeColor=INK_SOFT, labelColor=INK_SOFT, titleColor=INK)
)
# Save PNG then pad to exact 2400 × 2400 (square target)
TW, TH = 2400, 2400
chart.save(f"plot-{THEME}.png", scale_factor=4.0)
_img = Image.open(f"plot-{THEME}.png").convert("RGB")
_w, _h = _img.size
if _w > TW or _h > TH:
raise SystemExit(
f"altair vl-convert produced {_w}×{_h}, exceeds target {TW}×{TH}. "
f"Shrink chart .properties(width=, height=) values and re-render."
)
if _w < TW or _h < TH:
_canvas = Image.new("RGB", (TW, TH), PAGE_BG)
_canvas.paste(_img, ((TW - _w) // 2, (TH - _h) // 2))
_canvas.save(f"plot-{THEME}.png")
# Save HTML (interactive view)
chart.save(f"plot-{THEME}.html")
Part of Recurrence Plot for Nonlinear Time Series on anyplot.ai.