Vandermonde Polynomial Matrix Builder
Constructs the n × (d + 1) Vandermonde design matrix X = [1, x, x^2, ..., x^d] for arbitrary polynomial degrees.
Statistics & Probability / Polynomial Regression & Overfitting
FREE TO EXPLORELABORATORY 37 · STATISTICS & PROBABILITY
Polynomial Regression, Model Overfitting, and Bias-Variance Tradeoff
Investigate polynomial curve fitting, evaluate Vandermonde design matrices, analyze the Bias-Variance tradeoff, quantify training versus test error across polynomial degrees, and verify parameter scaling. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
FITTED QUADRATIC COMPONENT Ŷ_QUAD—
SAMPLING TELEMETRY
Ready to sample with Regressor x² = 1.0.
Adjust Regressor x², then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
High-degree global polynomials exhibit severe boundary oscillations (Runge's phenomenon) that invalidate extrapolation. The empirical regression of fitted quadratic components against squared regressors confirms parameter estimation with slope m = 0.2000 and R² = 1.000000.
Statistical tools & sampling modules.
Non-Linear Synthetic Data Generator
Generates continuous observations Y = f(x) + ε with controllable Gaussian noise and true non-linear curvature.
Train-Test Split Partition Engine
Splits empirical observations into independent training (70%) and testing (30%) subsets to assess generalization.
Mean Squared Error Evaluator
Computes training error MSE_train and out-of-sample test error MSE_test across polynomial degrees d ∈ {1, ..., 10}.
Bias-Variance Decomposition Unit
Decomposes expected generalization error into Bias² + Variance + Irreducible Noise across Monte Carlo datasets.
Curvature Parameter Regression Unit
Performs linear regression of quadratic response versus x² to confirm the theoretical curvature slope m = 0.200.
FOLLOW THE METHOD
A reason behind every step.
Investigate polynomial curve fitting, evaluate Vandermonde design matrices, analyze the Bias-Variance tradeoff, quantify training versus test error across polynomial degrees, and verify parameter scaling. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 units².
01
Configure Non-Linear Target Function and Noise
Set true underlying model to f(x) = 0.50 x + 0.20 x^2 with Gaussian noise ε ~ N(0, 0.16) on the polynomial simulation panel.
02
Generate Training and Test Partitions
Draw 1,000 observations uniformly on x ∈ [0, 5]. Randomly partition into 700 training observations and 300 out-of-sample test observations.
03
Fit Models Across Polynomial Degrees d = 1 to 10
Construct Vandermonde matrices and fit polynomial models for linear (d = 1), quadratic (d = 2), and high-order (d = 9) configurations.
04
Demonstrate the Overfitting Phenomenon
Observe that training error MSE_train monotonically decreases toward zero as d increases, while test error MSE_test explodes for d > 3 due to Runge-like oscillations.
05
Step Quadratic Regressor Across Standardized Values
Increment the squared regressor coordinate x^2 across standardized settings x^2 ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the fitted quadratic contribution.
06
Conduct Linear Regression on Quadratic Curvature
Fit a linear regression line y = mx to coordinate pairs (x^2, ŷ_quad). Confirm that empirical slope m = 0.2000 validates curvature parameter estimation.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Polynomial Regression & Overfitting mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.2000 · x
| # | Squared Regressor Coordinate x² (units²) | Fitted Quadratic Component ŷ_quad (Component ŷ_quad) | Squared Regressor Coordinate x² | Fitted Quadratic Component ŷ_quad | Remove |
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