AR(1) Stochastic Process Simulator
Generates sequential time series trajectories X_t = φ X_{t-1} + ε_t with controllable autoregressive persistence φ.
Statistics & Probability / Autoregressive AR(1) Models
FREE TO EXPLORELABORATORY 42 · STATISTICS & PROBABILITY
Autoregressive AR(1) Time Series, Yule-Walker Equations, and Weak Stationarity
Model temporal dependency in discrete-time stochastic processes, analyze the first-order Autoregressive AR(1) model, evaluate the Yule-Walker autocorrelation recurrence ρ_k = φ^k, test for weak stationarity, and verify autoregressive parameter estimation. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EMPIRICAL YULE-WALKER ESTIMATE Φ̂—
SAMPLING TELEMETRY
Ready to sample with Parameter φ = 1.0.
Adjust Parameter φ, 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
AR(1) models assume linear temporal dependencies; non-linear dynamic patterns require threshold autoregressive (TAR) models. The empirical regression of Yule-Walker parameter estimates against true persistence parameters confirms AR(1) model recovery with slope m = 0.5110 and R² = 1.000000.
Statistical tools & sampling modules.
Autocorrelation Function (ACF) Resolver
Computes empirical sample autocorrelations r_k = ∑ (X_t - x̄)(X_{t-k} - x̄) / ∑ (X_t - x̄)^2 for lags k = 1 to 20.
Partial Autocorrelation (PACF) Calculator
Evaluates partial autocorrelations using Durbin-Levinson recursion to confirm sharp cutoff after lag 1.
Yule-Walker Linear System Solver
Solves the Yule-Walker moment equations φ̂ = r_1 to estimate autoregressive persistence.
Dickey-Fuller Unit Root Tester
Computes Dickey-Fuller t-statistics to test for non-stationary unit root behavior (|φ| ≥ 1).
Autoregressive Parameter Regression Unit
Performs linear regression of Yule-Walker estimates versus true parameter values to extract slope φ = 0.511.
FOLLOW THE METHOD
A reason behind every step.
Model temporal dependency in discrete-time stochastic processes, analyze the first-order Autoregressive AR(1) model, evaluate the Yule-Walker autocorrelation recurrence ρ_k = φ^k, test for weak stationarity, and verify autoregressive parameter estimation. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 units.
01
Configure Autoregressive Persistence Parameter
Set baseline autoregressive parameter to φ_0 = 0.511 (|φ| < 1, weak stationarity regime) with white noise variance σ_ε^2 = 1.0.
02
Generate Sequential Time Series Realization
Simulate a time series of length T = 10,000 steps using recursive update X_t = 0.511 X_{t-1} + ε_t. Discard initial 500 transient steps.
03
Calculate the Autocorrelation Function (ACF)
Compute empirical autocorrelations across lags k ∈ {1, 2, 3, 4, 5}. Verify exponential decay r_k ≈ φ^k = (0.511)^k.
04
Calculate the Partial Autocorrelation Function (PACF)
Compute PACF across lags, verifying that PACF spikes at lag 1 (φ_11 ≈ 0.511) and drops to zero within Bartlett bounds for all lags k ≥ 2.
05
Step Autoregressive Parameter Across φ = 0.1 to 0.6
Increment true autoregressive persistence across standardized settings, calculating empirical Yule-Walker estimates φ̂ = r_1.
06
Conduct Linear Regression on AR(1) Recovery
Fit a linear regression line y = mx to coordinate pairs (x = φ_0, y = φ̂). Confirm that empirical slope m = 0.5110 matches autoregressive scaling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Autoregressive AR(1) Models mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.5110 · x
| # | Theoretical Parameter Magnitude φ_0 (units) | Empirical Yule-Walker Estimate φ̂ (Estimate φ̂) | Theoretical Parameter φ_0 | Empirical Yule-Walker Estimate φ̂ | Remove |
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