systems.builtin.stochastic.discrete.DiscreteAR1

systems.builtin.stochastic.discrete.DiscreteAR1(*args, **kwargs)

First-order autoregressive process with additive noise.

The AR(1) model is the fundamental discrete-time stochastic process exhibiting memory and mean reversion. It is the discrete analog of the Ornstein-Uhlenbeck process and the building block for ARMA and ARIMA models.

Difference Equation

Standard form: X[k+1] = φ·X[k] + σ·w[k]

With control: X[k+1] = φ·X[k] + u[k] + σ·w[k]

where: - X[k]: State at time k - φ: Autoregressive coefficient (persistence parameter) - u[k]: Control input (optional) - σ: Innovation standard deviation - w[k] ~ N(0,1): Standard normal white noise

Key Features

Persistence: Parameter φ controls memory: - φ near 1: High persistence (long memory) - φ near 0: Low persistence (short memory) - φ = 0: White noise (no memory)

Stationarity: Process stationary if and only if |φ| < 1: - |φ| < 1: Stable, mean-reverting - φ = 1: Unit root (random walk) - |φ| > 1: Explosive

Additive Noise: Innovation w[k] independent of state. Constant variance σ².

Markov Property: Future depends only on present, not past history.

Mathematical Properties

Stationary Distribution (|φ| < 1): Mean: E[X[∞]] = u/(1-φ) Variance: Var[X[∞]] = σ²/(1-φ²) Distribution: N(u/(1-φ), σ²/(1-φ²))

Autocorrelation: ρ(h) = φ^h

Geometric decay with lag h.

Mean Reversion: Half-life: h = ln(0.5)/ln(φ) periods Time constant: τ = -1/ln(φ) periods

Physical Interpretation

Autoregressive Coefficient φ: - Dimensionless (ratio) - Fraction of value persisting to next period - Typical range: 0 to 0.95

Interpretation by Value: - φ = 0.9: High persistence, slow decay - φ = 0.5: Moderate persistence - φ = 0.1: Low persistence, fast decay - φ = 1: Random walk (non-stationary)

Innovation Variance σ²: - Units: [state]² - Shock variance each period - Stationary variance: σ²/(1-φ²)

State Space

State: x ∈ ℝ - Unbounded (can take any real value) - Equilibrium: u/(1-φ) (for |φ| < 1)

Control: u ∈ ℝ (optional) - Shifts equilibrium - External forcing

Parameters

Name	Type	Description	Default
phi	float	Autoregressive coefficient - |φ| < 1: Stationary (typical) - φ = 1: Unit root (special case) - |φ| > 1: Explosive (avoid) - Typical: 0.5 to 0.95	0.9
sigma	float	Innovation standard deviation (must be positive) - Controls noise magnitude - Stationary std: σ/√(1-φ²)	0.1
dt	float	Sampling period (time between observations) - Sets time units - Needed for discrete system	1.0

Stochastic Properties

• Noise Type: ADDITIVE
• Innovation: w[k] ~ N(0,1) iid
• Markov: Memoryless given current state
• Stationary: If |φ| < 1
• Ergodic: If |φ| < 1

Applications

1. Econometrics: - GDP growth (quarterly data) - Inflation rates (monthly data) - Interest rates (daily/weekly) - Unemployment rates

2. Finance: - Asset returns (daily) - Volatility models (log-volatility) - Exchange rates

3. Signal Processing: - Colored noise generation - Digital filtering (one-pole filter) - Prediction algorithms

4. Control Systems: - Disturbance models - State estimation (Kalman filter) - Model predictive control

5. Time Series: - Foundation for ARMA, ARIMA - Benchmark for forecasting

Numerical Simulation

Exact Sampling: X[k+1] = φ·X[k] + u + σ·Z[k]

where Z[k] ~ N(0,1).

This is exact (no discretization needed).

Vectorized: Can use scipy.signal.lfilter for efficient generation.

Statistical Analysis

Parameter Estimation: - OLS: φ̂ = Σ X[k]·X[k+1] / Σ X[k]² - MLE: Same as OLS for Gaussian - Yule-Walker: From autocorrelations

Model Validation: - Unit root test (Dickey-Fuller) - Residual diagnostics (Ljung-Box) - Information criteria (AIC, BIC)

Comparison with Other Models

vs. White Noise: - WN: φ = 0 (no persistence) - AR(1): φ ≠ 0 (memory)

vs. Random Walk: - RW: φ = 1 (unit root) - AR(1): |φ| < 1 (stationary)

vs. Ornstein-Uhlenbeck: - OU: Continuous time - AR(1): Discrete time - Connection: φ = e^(-α·Δt)

Limitations

• Linear dynamics only
• Constant parameters
• Gaussian innovations
• Short memory (one lag)

Extensions: - AR(p): Higher-order lags - ARMA: Add moving average - GARCH: Time-varying variance - TAR: Threshold/regime-switching

See Also

OrnsteinUhlenbeck : Continuous-time analog DiscreteRandomWalk : Unit root case (φ=1) DiscreteWhiteNoise : No persistence (φ=0)

Methods

Name	Description
define_system	Define AR(1) process dynamics.
get_half_life	Get half-life: number of periods to reduce deviation by 50%.
get_stationary_variance	Get theoretical stationary variance σ²/(1-φ²).

define_system

systems.builtin.stochastic.discrete.DiscreteAR1.define_system(
    phi=0.9,
    sigma=0.1,
    dt=1.0,
)

Define AR(1) process dynamics.

Sets up the difference equation: X[k+1] = φ·X[k] + u[k] + σ·w[k]

Parameters

Name	Type	Description	Default
phi	float	Autoregressive coefficient - |φ| < 1: Stationary (mean-reverting) - φ = 1: Unit root (random walk) - |φ| > 1: Explosive (unstable) - Typical: 0.5 to 0.95	0.9
sigma	float	Innovation standard deviation (must be positive) - Controls shock magnitude - Stationary std: σ/√(1-φ²)	0.1
dt	float	Sampling period [time units] - Required for discrete system - Sets time scale	1.0

Raises

Name	Type	Description
	ValueError	If sigma ≤ 0
	UserWarning	If |phi| ≥ 1 (non-stationary)

Notes

Stationarity Condition: Process is stationary if and only if |φ| < 1.

Critical Cases: - φ = 1: Unit root (random walk) * Non-stationary * Variance grows linearly: Var[X[k]] = k·σ² * Requires different statistical treatment

• φ = -1: Perfect negative autocorrelation
  • Alternates between extremes
  • Non-stationary
• |φ| > 1: Explosive
  • Variance grows exponentially
  • Diverges to ±∞

Stationary Properties: For |φ| < 1: - Mean: μ = u/(1-φ) - Variance: γ(0) = σ²/(1-φ²) - Autocorrelation: ρ(h) = φ^h - Half-life: ln(0.5)/ln(φ) periods

Relationship to Continuous Time: If discretizing OU process with parameter α: φ = exp(-α·dt) σ ≈ σ_continuous·√(2α·dt) (approximate)

Parameter Selection: - High persistence (φ ≈ 0.9): Financial returns, macro data - Moderate (φ ≈ 0.5): GDP growth, some commodities - Low (φ ≈ 0.1): Nearly white noise

Innovation Variance: Total variance decomposes: - Stationary variance: σ²/(1-φ²) - Innovation variance: σ² - Ratio: 1/(1-φ²) ≥ 1

get_half_life

systems.builtin.stochastic.discrete.DiscreteAR1.get_half_life()

Get half-life: number of periods to reduce deviation by 50%.

Formula: h = ln(0.5) / ln(φ)

Only meaningful for 0 < φ < 1.

Returns

Name	Type	Description
	float	Half-life [periods]

get_stationary_variance

systems.builtin.stochastic.discrete.DiscreteAR1.get_stationary_variance()

Get theoretical stationary variance σ²/(1-φ²).

Only valid for |φ| < 1.

Returns

Name	Type	Description
	float	Stationary variance

Raises

Name	Type	Description
	ValueError	If |φ| ≥ 1 (non-stationary)