systems.builtin.stochastic.discrete.DiscreteWhiteNoise

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

Pure white noise process - memoryless random sequence.

The fundamental building block of all stochastic processes, representing pure randomness with no memory or structure. White noise is the innovation sequence underlying ARMA models and the limiting case of AR(1) with φ = 0.

Difference Equation

X[k] = σ·w[k]

where w[k] ~ N(0,1) are iid standard normal random variables.

Alternative View: X[k+1] = 0·X[k] + σ·w[k]

This shows white noise as AR(0) - no dependence on past.

Key Features

Memorylessness: Zero autocorrelation: Cov[X[k], X[j]] = 0 for k ≠ j

Each observation independent of all others.

Unpredictability: Best prediction: E[X[k+1] | past] = 0 Prediction error: MSE = σ²

Cannot be improved by using past information.

Stationarity: Strictly stationary - distribution invariant to time shifts.

Flat Spectrum: Power spectral density: S(f) = σ² (constant across frequencies)

Hence “white” - equal power at all frequencies.

Mathematical Properties

Moments: Mean: E[X[k]] = 0 for all k Variance: Var[X[k]] = σ² for all k Higher moments: Standard Gaussian if w ~ N(0,1)

Autocorrelation: γ(h) = σ²·δ_h where δ_h is Kronecker delta: γ(0) = σ² γ(h) = 0 for h ≠ 0

Power Spectrum: S(f) = σ² for all f ∈ [-f_s/2, f_s/2]

where f_s = 1/Δt is sampling frequency.

Physical Interpretation

Standard Deviation σ: - Units: [state] - Scale of random fluctuations - Examples: * Measurement noise: instrument precision * Thermal noise: √(4·k_B·T·R·Δf) * Financial: daily return volatility

No Parameters Besides σ: White noise is completely characterized by variance σ². No memory, no time constants, no structure.

State Space

State: x ∈ ℝ (unbounded) - No persistence between samples - Each value independent - Gaussian: X[k] ~ N(0, σ²)

Control: None (autonomous) - Pure random process - Cannot be controlled or predicted

Parameters

Name	Type	Description	Default
sigma	float	Standard deviation of white noise - Must be positive - σ = 1: Standard white noise - Variance: σ²	1.0
dt	float	Sampling period [time units] - Required for discrete system - Sets time scale	1.0

Stochastic Properties

• Type: Pure random (no deterministic component)
• Innovation: w[k] ~ N(0,1) iid
• Memory: None (memoryless)
• Stationary: Yes (strictly stationary)
• Ergodic: Yes
• Predictable: No

Applications

1. Signal Processing: - Noise modeling in measurements - Filter testing and validation - System identification (input signal) - Detection theory (null hypothesis)

2. Communications: - AWGN channel model - Error analysis - Channel capacity calculations

3. Time Series: - Innovation sequence for ARMA models - Residual diagnostics (should be white) - Benchmark for forecasting performance

4. Control Systems: - Process noise in Kalman filter - Disturbance modeling - Robustness analysis

5. Finance: - Efficient market hypothesis (returns should be white) - Monte Carlo simulation - Risk modeling baseline

6. Testing: - Null hypothesis for independence tests - Benchmark for signal detection - Residual analysis

Numerical Simulation

Direct Generation: X[k] = σ·randn()

Simple, exact, no approximation.

Quality Checks: - Mean ≈ 0 - Variance ≈ σ² - Autocorrelation ≈ 0 for h > 0 - Within confidence bands: ±1.96/√N

Statistical Analysis

Tests for White Noise: - Ljung-Box Q-test (joint autocorrelation) - Durbin-Watson (first-order autocorrelation) - Runs test (randomness) - Portmanteau test

Diagnostics: - ACF plot: Only lag 0 significant - Periodogram: Approximately flat - Q-Q plot: Linear (if Gaussian)

Comparison with Other Processes

vs. Random Walk: - RW: Cumulative sum of white noise - WN: Memoryless, stationary

vs. AR(1): - AR(1): X[k] = φ·X[k-1] + w[k] - WN: AR(1) with φ = 0

vs. Brownian Motion: - BM: Continuous-time integral of white noise - WN: Discrete-time, no integration

Limitations

• Idealization (infinite bandwidth)
• Real noise often band-limited
• May have weak temporal dependence
• Gaussian assumption may not hold

See Also

DiscreteAR1 : White noise is AR(1) with φ=0 DiscreteRandomWalk : Cumulative sum of white noise BrownianMotion : Continuous-time analog

Methods

Name	Description
define_system	Define white noise process.
get_autocorrelation	Get theoretical autocorrelation at given lag.
get_standard_deviation	Get theoretical standard deviation σ.
get_variance	Get theoretical variance σ².

define_system

systems.builtin.stochastic.discrete.DiscreteWhiteNoise.define_system(
    sigma=1.0,
    dt=1.0,
)

Define white noise process.

Sets up pure random sequence: X[k] = σ·w[k]

where w[k] ~ N(0,1) are iid.

Parameters

Name	Type	Description	Default
sigma	float	Standard deviation (must be positive) - σ = 1: Standard white noise - σ² = variance - Typical: 0.1 to 10.0 depending on application	1.0
dt	float	Sampling period [time units] - Required for discrete system - Sets time scale - Does not affect statistics (memoryless)	1.0

Raises

Name	Type	Description
	ValueError	If sigma ≤ 0

Notes

Complete Characterization: White noise is completely specified by variance σ². No other parameters needed - no memory, no time constants.

Independence: Each sample is independent: P(X[k] | X[k-1], …) = P(X[k])

This is the strongest form of memorylessness.

Gaussian White Noise: If w[k] ~ N(0,1), then X[k] ~ N(0, σ²). Uncorrelated + Gaussian ⟹ Independent

Non-Gaussian Extensions: Can use other distributions: - Uniform: w[k] ~ U(-√3, √3) (variance 1) - Laplace: Heavier tails - Student-t: Fat tails

Sampling Period dt: Unlike AR(1) or OU, dt doesn’t affect white noise statistics. Samples at any interval are independent with same variance.

Power Spectral Density: S(f) = σ² for |f| ≤ f_Nyquist = 1/(2·dt)

Flat spectrum - equal power at all frequencies.

Use Cases: - Testing: Null hypothesis for independence - Simulation: Generate innovations for ARMA - Benchmarking: Compare signal detection performance - Modeling: Measurement noise, residuals

Relationship to AR(1): White noise is AR(1) with φ = 0: X[k+1] = 0·X[k] + σ·w[k]

Wold Decomposition: Any stationary process can be written as: X[k] = μ + Σ ψ_j·w[k-j] where w[k] is white noise.

get_autocorrelation

systems.builtin.stochastic.discrete.DiscreteWhiteNoise.get_autocorrelation(lag)

Get theoretical autocorrelation at given lag.

For white noise: ρ(0) = 1 ρ(h) = 0 for h ≠ 0

Parameters

Name	Type	Description	Default
lag	int	Lag h (non-negative)	required

Returns

Name	Type	Description
	float	Autocorrelation: 1 if lag=0, else 0

Examples

>>> wn = DiscreteWhiteNoise(sigma=1.0, dt=1.0)
>>> print(f"ρ(0) = {wn.get_autocorrelation(0)}")  # 1.0
>>> print(f"ρ(1) = {wn.get_autocorrelation(1)}")  # 0.0
>>> print(f"ρ(10) = {wn.get_autocorrelation(10)}")  # 0.0

get_standard_deviation

systems.builtin.stochastic.discrete.DiscreteWhiteNoise.get_standard_deviation()

Get theoretical standard deviation σ.

Returns

Name	Type	Description
	float	Standard deviation

Examples

>>> wn = DiscreteWhiteNoise(sigma=1.5, dt=1.0)
>>> std = wn.get_standard_deviation()
>>> print(f"Std: {std:.3f}")  # 1.5

get_variance

systems.builtin.stochastic.discrete.DiscreteWhiteNoise.get_variance()

Get theoretical variance σ².

For white noise, variance is simply σ².

Returns

Name	Type	Description
	float	Variance

Notes

Unlike AR(1) or OU, white noise variance doesn’t depend on any time constants - it’s just σ².