Multivariate Matrix Design Engine
Constructs the n × p design matrix X, appends unit intercept column, and computes the Gram matrix X^T X.
Statistics & Probability / Multiple Linear Regression
FREE TO EXPLORELABORATORY 35 · STATISTICS & PROBABILITY
Multiple Linear Regression, Matrix Formulation, and Multicollinearity Diagnostics
Formulate multivariate multiple linear regression using matrix projection notation, compute the OLS vector β̂ = (X^T X)^(-1) X^T Y, diagnose multicollinearity via Variance Inflation Factors (VIF), and verify joint regression parameter scaling. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
PARTIAL FITTED CONTRIBUTION Ŷ_PART—
SAMPLING TELEMETRY
Ready to sample with Regressor x_2 = 1.0.
Adjust Regressor x_2, 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
Multiple regression identifies predictive linear associations, but cannot prove causal relationships without experimental control. The empirical regression of partial fitted contributions against regressor coordinate x_2 confirms multivariate OLS parameter extraction with slope m = 0.2500 and R² = 1.000000.
Statistical tools & sampling modules.
Cholesky Matrix Inverter
Inverts the symmetric positive-definite normal matrix (X^T X) via Cholesky decomposition to evaluate β̂ = (X^T X)^(-1) X^T Y.
Hat Matrix Projection Unit
Computes the orthogonal projection matrix H = X (X^T X)^(-1) X^T and evaluates leverage diagonals h_{ii}.
Variance Inflation Factor (VIF) Meter
Computes auxiliary regressions to calculate VIF_j = 1 / (1 - R_j^2) for diagnosing multicollinearity.
Adjusted R-Squared Optimizer
Computes R^2_adj = 1 - [SSE/(n - p)] / [SST/(n - 1)] to penalize model over-parameterization.
Partial Coefficient Regression Unit
Performs linear regression of partial fitted responses versus x_2 to extract the slope β_2 = 0.250.
FOLLOW THE METHOD
A reason behind every step.
Formulate multivariate multiple linear regression using matrix projection notation, compute the OLS vector β̂ = (X^T X)^(-1) X^T Y, diagnose multicollinearity via Variance Inflation Factors (VIF), and verify joint regression parameter scaling. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 units.
01
Construct Multivariate Design Matrix
Set sample size n = 5,000 with p = 2 regressors: X_1 (independent) and X_2 (uncorrelated). Establish true coefficients β = [0.0, 0.500, 0.250].
02
Synthesize Bivariate Predictor Response Data
Generate response Y = 0.500 X_1 + 0.250 X_2 + ε with ε ~ N(0, 0.25). Compute the Gram matrix X^T X and cross-product X^T Y.
03
Execute Matrix OLS Inversion
Invert X^T X and evaluate the coefficient vector β̂ = (X^T X)^(-1) X^T Y. Confirm convergence to [0.000, 0.500, 0.250].
04
Inject Inter-Predictor Multicollinearity
Increase correlation between X_1 and X_2 to r = 0.95. Observe that coefficient standard errors explode while VIF exceeds 10.0.
05
Step Second Regressor Coordinate Across x_2 = 1 to 6
Increment the x_2 coordinate across standardized settings x_2 ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the partial contribution ŷ_part.
06
Conduct Linear Regression on Partial Slope
Fit a linear regression model y = mx to coordinate pairs (x_2, ŷ_part). Confirm that empirical slope m = 0.2500 recovers the partial coefficient β_2.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Multiple Linear Regression mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.2500 · x
| # | Second Regressor Coordinate x_2 (units) | Partial Fitted Contribution ŷ_part (Contribution ŷ_part) | Regressor Coordinate x_2 | Partial Fitted Contribution ŷ_part | Remove |
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