A scatter plot displaying the relationship between two numeric variables with a fitted polynomial regression curve (degree 2-4). This visualization extends beyond linear regression to capture non-linear relationships in data, making it ideal for modeling curved trends, parabolic patterns, and complex data relationships where a straight line would not adequately represent the underlying pattern.
""" anyplot.ai
scatter-regression-polynomial: Scatter Plot with Polynomial Regression
Library: letsplot 4.9.0 | Python 3.13.13
Quality: 93/100 | Updated: 2026-05-07
"""
import os
import numpy as np
import pandas as pd
from lets_plot import *
LetsPlot.setup_html()
# 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"
RULE = "rgba(26,26,23,0.10)" if THEME == "light" else "rgba(240,239,232,0.10)"
BRAND = "#009E73" # Okabe-Ito position 1 - first categorical series
ACCENT = "#C475FD" # Okabe-Ito position 2 - for regression line
# Data - Simulating diminishing returns pattern (economics example)
np.random.seed(42)
n_points = 80
# Advertising spend (in thousands)
x = np.linspace(5, 50, n_points)
# Sales revenue with diminishing returns (quadratic relationship)
# y = -0.05x² + 4x + 20 + noise
y = -0.05 * x**2 + 4 * x + 20 + np.random.normal(0, 5, n_points)
# Fit polynomial regression (degree 2 - quadratic) using numpy
poly_degree = 2
coefficients = np.polyfit(x, y, poly_degree)
poly_func = np.poly1d(coefficients)
y_pred = poly_func(x)
# Calculate R²
ss_res = np.sum((y - y_pred) ** 2)
ss_tot = np.sum((y - np.mean(y)) ** 2)
r2 = 1 - (ss_res / ss_tot)
# Generate smooth curve for regression line
x_smooth = np.linspace(x.min(), x.max(), 200)
y_smooth = poly_func(x_smooth)
# Generate confidence band (approximate using residual standard error)
residuals = y - y_pred
std_error = np.std(residuals)
y_upper = y_smooth + 1.96 * std_error
y_lower = y_smooth - 1.96 * std_error
# Create dataframes
df_points = pd.DataFrame({"x": x, "y": y})
df_curve = pd.DataFrame({"x": x_smooth, "y": y_smooth, "y_upper": y_upper, "y_lower": y_lower})
# Get polynomial equation
a, b, c = coefficients
equation = f"y = {a:.3f}x² + {b:.3f}x + {c:.2f}"
# Create plot
plot = (
ggplot()
# Confidence band with no border
+ geom_ribbon(aes(x="x", ymin="y_lower", ymax="y_upper"), data=df_curve, fill=BRAND, alpha=0.15, color=None)
# Scatter points
+ geom_point(aes(x="x", y="y"), data=df_points, color=BRAND, size=5, alpha=0.65)
# Polynomial regression line
+ geom_line(aes(x="x", y="y"), data=df_curve, color=ACCENT, size=2.5)
# Labels and title
+ labs(
x="Advertising Spend (thousands $)",
y="Sales Revenue (thousands $)",
title="scatter-regression-polynomial · letsplot · anyplot.ai",
)
# Annotations for R² and equation
+ geom_text(
aes(x="x", y="y", label="label"),
data=pd.DataFrame({"x": [x.max() - 5], "y": [y.max() + 5], "label": [f"R² = {r2:.3f}"]}),
size=18,
color=INK,
hjust=1,
)
+ geom_text(
aes(x="x", y="y", label="label"),
data=pd.DataFrame({"x": [x.max() - 5], "y": [y.max() - 2], "label": [equation]}),
size=14,
color=INK_SOFT,
hjust=1,
)
# Theme
+ theme_minimal()
+ theme(
plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
panel_background=element_rect(fill=PAGE_BG),
panel_grid_major=element_line(color=RULE, size=0.3),
panel_grid_minor=element_blank(),
plot_title=element_text(size=24, face="bold", color=INK),
axis_title=element_text(size=20, color=INK),
axis_text=element_text(size=16, color=INK_SOFT),
axis_line=element_line(color=INK_SOFT),
)
+ ggsize(1600, 900)
)
# Save as PNG (scale 3x for 4800x2700)
ggsave(plot, f"plot-{THEME}.png", path=".", scale=3)
# Save as HTML for interactive viewing
ggsave(plot, f"plot-{THEME}.html", path=".")
Part of Scatter Plot with Polynomial Regression on anyplot.ai.