systems.builtin.stochastic.discrete.DiscreteARMA11

systems.builtin.stochastic.discrete.DiscreteARMA11(*args, **kwargs)

ARMA(1,1) process - combines autoregressive and moving average.

The simplest non-trivial ARMA model, combining one AR lag with one MA lag to create a flexible yet parsimonious time series model. This is the workhorse of Box-Jenkins methodology and the foundation of modern forecasting in econometrics.

Difference Equation

Standard ARMA(1,1) form: X[k] = φ·X[k-1] + w[k] + θ·w[k-1]

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

where: - X[k]: Observed series (state) - φ: AR coefficient (persistence, -∞ < φ < ∞) - θ: MA coefficient (smoothing, -∞ < θ < ∞) - w[k]: White noise innovation ~ N(0, σ²) - u[k]: Control/external input

State-Space Form (for simulation): Z[k] = [X[k], w[k-1]]ᵀ (augmented state)

Z[k+1] = [φ  θ]·Z[k] + [1]·w[k+1]
         [0  0]        [1]

Output: X[k] = [1, 0]·Z[k]

Physical Interpretation

AR Component (φ): - Past observation influences current - φ > 0: Positive persistence (typical) - φ < 0: Alternation (rare) - Recursive: Memory of all past (exponentially weighted)

MA Component (θ): - Past innovation influences current - θ > 0: Smoothing (positive autocorrelation at lag 1) - θ < 0: Overshoot (negative autocorrelation) - Finite: Only immediate past innovation matters

Combined Effect: - AR provides long memory (slow decay) - MA adjusts short-term behavior (lag 1) - Together: More flexible than either alone

Why ARMA(1,1) is Popular:

Can approximate: - AR(∞): Via MA(∞) representation - MA(∞): Via AR(∞) representation - Complex ACF: With just 3 parameters

More parsimonious than high-order AR or MA.

Key Features

Stationarity: Requires |φ| < 1 (from AR part).

Invertibility: Requires |θ| < 1 (from MA part).

Parsimony: 3 parameters (φ, θ, σ) provide flexibility of high-order models.

Autocorrelation: - ρ(1): Affected by both φ and θ - ρ(h ≥ 2): Geometric decay at rate φ

Spectral Density: Rational function (ratio of polynomials).

Markov (in Augmented State): State-space form is Markov in (X, w_prev).

Mathematical Properties

Stationarity Condition: |φ| < 1

Invertibility Condition: |θ| < 1

Variance: Var[X] = σ²·(1 + 2θφ + θ²)/(1 - φ²)

Autocorrelation: ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²) ρ(h) = φ·ρ(h-1) for h ≥ 2

Impulse Response: ψ₀ = 1 ψ₁ = φ + θ ψⱼ = φ^(j-1)·(φ + θ) for j ≥ 2

MA(∞) Representation: X[k] = Σψⱼ·w[k-j]

Coefficients decay geometrically: ψⱼ ~ φ^j

AR(∞) Representation: w[k] = Σπⱼ·X[k-j]

Coefficients decay geometrically: πⱼ ~ θ^j (if |θ| < 1)

Physical Interpretation

AR Coefficient φ: - Dimensionless - Fraction of past persisting - Typical: 0.3-0.9 (positive persistence)

MA Coefficient θ: - Dimensionless - Weight on past shock - Typical: -0.5 to +0.5 - θ > 0: Smoothing (positive lag-1 correlation) - θ < 0: Overshoot (negative lag-1 correlation)

Innovation Variance σ²: - Units: [state]² - Unpredictable component each period

Special Cases:

φ = 0.8, θ = 0: Pure AR(1) φ = 0, θ = -0.5: Pure MA(1) φ = 0.8, θ = -0.6: Typical ARMA(1,1)

State Space

Augmented state: Z = [X, w_prev] ∈ ℝ² - X: Observed series - w_prev: Past innovation (latent)

Observable state: X ∈ ℝ - What we actually measure

Control: u ∈ ℝ (optional) - External input

Parameters

Name	Type	Description	Default
phi	float	AR coefficient - Typical: 0.3-0.9 - Must satisfy \|φ\| < 1 for stationarity	`0.7`
theta	float	MA coefficient - Typical: -0.5 to +0.5 - Should satisfy \|θ\| < 1 for invertibility	`0.3`
sigma	float	Innovation standard deviation - Must be positive - Sets scale of unpredictable component	`0.1`
dt	float	Sampling period - Required for discrete system	`1.0`

Stochastic Properties

System Type: LINEAR (ARMA)
Noise Type: ADDITIVE (white noise)
Markov: In augmented state (X, w_prev)
Stationary: If |φ| < 1
Invertible: If |θ| < 1
Gaussian: If w ~ N(0,σ²)

Applications

1. Economic Forecasting: - GDP growth - Inflation - Unemployment - Sales data

2. Financial Time Series: - Asset returns - Interest rate spreads - Volume data

3. Signal Processing: - Digital filter design - Prediction algorithms - Spectral estimation

4. Engineering: - Vibration analysis - Quality control - Process monitoring

5. Environmental: - Temperature series - Rainfall - Pollution levels

Numerical Simulation

State-Space Implementation:

Augment state with past innovation: Z[k+1] = F·Z[k] + G·w[k+1]

where: Z = [X, w_prev]ᵀ F = [φ θ] [0 0] G = [1] [1]

Extract: X[k] = Z[k][0]

Forecasting

One-Step-Ahead: X̂[k+1|k] = φ·X[k] + θ·ε[k]

where ε[k] = X[k] - X̂[k|k-1] is innovation.

h-Step-Ahead: X̂[k+h|k] = φ^h·X[k] + θ·Σφ^(h-1-j)·ε[k-j]

Forecast variance increases with horizon.

Comparison with Other Models

vs. AR(1): - AR(1): Only φ parameter - ARMA(1,1): Adds θ (more flexible) - Same asymptotic ACF (h → ∞) - Different ρ(1)

vs. MA(1): - MA(1): Only θ parameter - ARMA(1,1): Adds φ (long memory)

vs. AR(2): - AR(2): Two AR lags (4 parameters with σ, μ) - ARMA(1,1): One AR, one MA (3 parameters) - Often ARMA(1,1) more parsimonious

vs. ARMA(p,q): - ARMA(1,1): Simplest non-trivial - Higher order: More flexible but more parameters - Start with ARMA(1,1), increase if needed

Limitations

Linear only
Constant parameters
Gaussian innovations (typically)
Stationary (no trends)
Low order (1,1 may not suffice)

Extensions

ARMA(p,q): Higher order
ARIMA: Integrated (for non-stationary)
SARIMA: Seasonal patterns
GARCH: Time-varying variance
Regime-switching ARMA

Examples

Standard ARMA(1,1):

>>> # Typical configuration
>>> arma = DiscreteARMA11(
...     phi=0.7,    # Moderate persistence
...     theta=0.3,  # Positive MA (smoothing)
...     sigma=0.1,
...     dt=1.0
... )
>>>
>>> # Check stationarity and invertibility
>>> print(f"|φ| < 1: {abs(0.7) < 1}")  # Stationary
>>> print(f"|θ| < 1: {abs(0.3) < 1}")  # Invertible

Different configurations:

>>> # High persistence, low MA
>>> high_ar = DiscreteARMA11(phi=0.9, theta=0.1, sigma=0.1)
>>>
>>> # Moderate AR, negative MA (overshoot)
>>> negative_ma = DiscreteARMA11(phi=0.5, theta=-0.3, sigma=0.1)
>>>
>>> # Balanced AR and MA
>>> balanced = DiscreteARMA11(phi=0.6, theta=0.4, sigma=0.1)

Simulation:

>>> # Generate ARMA series
>>> z0 = np.array([0.0, 0.0])  # [X[0], w[-1]]
>>> u_seq = np.zeros((100, 1))
>>>
>>> result = arma.simulate(z0, u_seq, n_steps=100)
>>>
>>> # Extract observed series (first component of augmented state)
>>> X_series = result['states'][:, 0]
>>>
>>> # Compute sample ACF
>>> from statsmodels.tsa.stattools import acf
>>> acf_values = acf(X_series, nlags=20)

References

.. [1] Box, G.E.P. & Jenkins, G.M. (1970). “Time Series Analysis: Forecasting and Control” .. [2] Hamilton, J.D. (1994). “Time Series Analysis” .. [3] Brockwell, P.J. & Davis, R.A. (2016). “Introduction to Time Series and Forecasting” .. [4] Shumway, R.H. & Stoffer, D.S. (2017). “Time Series Analysis and Its Applications”

Methods

Name	Description
define_system	Define ARMA(1,1) process dynamics in state-space form.
get_acf	Get theoretical autocorrelation function.
get_acf_lag1	Get theoretical autocorrelation at lag 1.
get_stationary_variance	Get theoretical stationary variance of X.

define_system

systems.builtin.stochastic.discrete.DiscreteARMA11.define_system(
    phi=0.7,
    theta=0.3,
    sigma=0.1,
    dt=1.0,
)

Define ARMA(1,1) process dynamics in state-space form.

Parameters

Name	Type	Description	Default
phi	float	AR coefficient (persistence) - Typical: 0.3-0.9 - Should satisfy \|φ\| < 1 for stationarity - Similar to AR(1)	`0.7`
theta	float	MA coefficient (smoothing) - Typical: -0.5 to +0.5 - Should satisfy \|θ\| < 1 for invertibility - θ > 0: Smoothing (common) - θ < 0: Overshoot (less common)	`0.3`
sigma	float	Innovation standard deviation - Must be positive - Sets scale of unpredictable shocks	`0.1`
dt	float	Sampling period - Required for discrete system	`1.0`

Raises

Name	Type	Description
	ValueError	If sigma ≤ 0
	UserWarning	If \|φ\| ≥ 1 (non-stationary) If \|θ\| ≥ 1 (non-invertible)

Notes

Stationarity: Requires |φ| < 1 (from AR component). - φ = 1: Unit root (ARIMA needed) - |φ| > 1: Explosive (unstable)

Invertibility: Requires |θ| < 1 (from MA component). - |θ| ≥ 1: Non-unique representation - Multiple ARMA models give same ACF - Estimation problems

Parameter Space: Valid region: |φ| < 1 AND |θ| < 1

State-Space Formulation:

To simulate ARMA, augment state with past innovation: Z = [X, w_prev]ᵀ

Dynamics: Z[k+1] = F·Z[k] + G·w[k+1]

where: F = [φ θ] [0 0]

G = [1]
    [1]

Why Augmented State? Cannot write X[k+1] = f(X[k]) alone because depends on w[k]. Need to track w[k] as auxiliary state.

Variance (Stationary): Var[X] = σ²·(1 + 2θφ + θ²)/(1 - φ²)

Autocorrelation: ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²) ρ(h) = φ·ρ(h-1) for h ≥ 2

Special Cases:

θ = 0: Reduces to AR(1) X[k] = φ·X[k-1] + w[k]

φ = 0: Reduces to MA(1) X[k] = θ·w[k-1] + w[k]

φ = -θ: Simplifies significantly X[k] = φ·(X[k-1] - w[k-1]) + w[k] (Differencing form)

Typical Configurations:

Economic data: - φ = 0.6-0.9 (high persistence) - θ = 0.1-0.5 (moderate smoothing)

Financial returns: - φ = 0.0-0.3 (low persistence) - θ = -0.3-0.0 (slight overshoot or none)

Physical measurements: - φ = 0.5-0.8 (moderate persistence) - θ = 0.2-0.5 (smoothing from measurement)

Examples

>>> # Standard economic series
>>> economic = DiscreteARMA11(
...     phi=0.7,    # High persistence
...     theta=0.3,  # Smoothing
...     sigma=0.1
... )
>>>
>>> # Financial returns (low persistence)
>>> returns = DiscreteARMA11(
...     phi=0.1,     # Weak persistence
...     theta=-0.2,  # Slight overshoot
...     sigma=0.2
... )
>>>
>>> # Physical sensor (measurement smoothing)
>>> sensor = DiscreteARMA11(
...     phi=0.6,
...     theta=0.4,   # Strong smoothing
...     sigma=0.05
... )

get_acf

systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf(max_lag=20)

Get theoretical autocorrelation function.

Parameters

Name	Type	Description	Default
max_lag	int	Maximum lag	`20`

Returns

Name	Type	Description
	np.ndarray	ACF values [ρ(0), ρ(1), …, ρ(max_lag)]

Notes

ρ(0) = 1 ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²) ρ(h) = φ·ρ(h-1) for h ≥ 2

Geometric decay after lag 1.

Examples

>>> arma = DiscreteARMA11(phi=0.7, theta=0.3, sigma=0.1)
>>> acf = arma.get_acf(max_lag=10)
>>> print(f"ACF: {acf}")

get_acf_lag1

systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf_lag1()

Get theoretical autocorrelation at lag 1.

ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²)

Returns

Name	Type	Description
	float	Lag-1 autocorrelation

Notes

This is where ARMA differs from AR(1). For AR(1): ρ(1) = φ For ARMA(1,1): ρ(1) ≠ φ (MA effect)

Examples

>>> arma = DiscreteARMA11(phi=0.7, theta=0.3, sigma=0.1)
>>> rho1 = arma.get_acf_lag1()
>>> print(f"ρ(1) = {rho1:.3f}")
>>> print(f"Compare with φ = 0.7")

get_stationary_variance

systems.builtin.stochastic.discrete.DiscreteARMA11.get_stationary_variance()

Get theoretical stationary variance of X.

Var[X] = σ²·(1 + 2θφ + θ²)/(1 - φ²)

Returns

Name	Type	Description
	float	Stationary variance

Raises

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

Examples

>>> arma = DiscreteARMA11(phi=0.7, theta=0.3, sigma=0.1)
>>> var = arma.get_stationary_variance()
>>> print(f"Stationary variance: {var:.4f}")

# systems.builtin.stochastic.discrete.DiscreteARMA11 { #cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11 } ```python systems.builtin.stochastic.discrete.DiscreteARMA11(*args, **kwargs) ``` ARMA(1,1) process - combines autoregressive and moving average. The simplest non-trivial ARMA model, combining one AR lag with one MA lag to create a flexible yet parsimonious time series model. This is the workhorse of Box-Jenkins methodology and the foundation of modern forecasting in econometrics. ## Difference Equation {.doc-section .doc-section-difference-equation} Standard ARMA(1,1) form: X[k] = φ·X[k-1] + w[k] + θ·w[k-1] With control: X[k] = φ·X[k-1] + u[k] + w[k] + θ·w[k-1] where: - X[k]: Observed series (state) - φ: AR coefficient (persistence, -∞ < φ < ∞) - θ: MA coefficient (smoothing, -∞ < θ < ∞) - w[k]: White noise innovation ~ N(0, σ²) - u[k]: Control/external input **State-Space Form (for simulation):** Z[k] = [X[k], w[k-1]]ᵀ (augmented state) Z[k+1] = [φ θ]·Z[k] + [1]·w[k+1] [0 0] [1] Output: X[k] = [1, 0]·Z[k] ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **AR Component (φ):** - Past observation influences current - φ > 0: Positive persistence (typical) - φ < 0: Alternation (rare) - Recursive: Memory of all past (exponentially weighted) **MA Component (θ):** - Past innovation influences current - θ > 0: Smoothing (positive autocorrelation at lag 1) - θ < 0: Overshoot (negative autocorrelation) - Finite: Only immediate past innovation matters **Combined Effect:** - AR provides long memory (slow decay) - MA adjusts short-term behavior (lag 1) - Together: More flexible than either alone **Why ARMA(1,1) is Popular:** Can approximate: - AR(∞): Via MA(∞) representation - MA(∞): Via AR(∞) representation - Complex ACF: With just 3 parameters More parsimonious than high-order AR or MA. ## Key Features {.doc-section .doc-section-key-features} **Stationarity:** Requires |φ| < 1 (from AR part). **Invertibility:** Requires |θ| < 1 (from MA part). **Parsimony:** 3 parameters (φ, θ, σ) provide flexibility of high-order models. **Autocorrelation:** - ρ(1): Affected by both φ and θ - ρ(h ≥ 2): Geometric decay at rate φ **Spectral Density:** Rational function (ratio of polynomials). **Markov (in Augmented State):** State-space form is Markov in (X, w_prev). ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **Stationarity Condition:** |φ| < 1 **Invertibility Condition:** |θ| < 1 **Variance:** Var[X] = σ²·(1 + 2θφ + θ²)/(1 - φ²) **Autocorrelation:** ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²) ρ(h) = φ·ρ(h-1) for h ≥ 2 **Impulse Response:** ψ₀ = 1 ψ₁ = φ + θ ψⱼ = φ^(j-1)·(φ + θ) for j ≥ 2 **MA(∞) Representation:** X[k] = Σψⱼ·w[k-j] Coefficients decay geometrically: ψⱼ ~ φ^j **AR(∞) Representation:** w[k] = Σπⱼ·X[k-j] Coefficients decay geometrically: πⱼ ~ θ^j (if |θ| < 1) ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **AR Coefficient φ:** - Dimensionless - Fraction of past persisting - Typical: 0.3-0.9 (positive persistence) **MA Coefficient θ:** - Dimensionless - Weight on past shock - Typical: -0.5 to +0.5 - θ > 0: Smoothing (positive lag-1 correlation) - θ < 0: Overshoot (negative lag-1 correlation) **Innovation Variance σ²:** - Units: [state]² - Unpredictable component each period **Special Cases:** φ = 0.8, θ = 0: Pure AR(1) φ = 0, θ = -0.5: Pure MA(1) φ = 0.8, θ = -0.6: Typical ARMA(1,1) ## State Space {.doc-section .doc-section-state-space} Augmented state: Z = [X, w_prev] ∈ ℝ² - X: Observed series - w_prev: Past innovation (latent) Observable state: X ∈ ℝ - What we actually measure Control: u ∈ ℝ (optional) - External input ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|------------------------------------------------------------------------------------------|-----------| | phi | float | AR coefficient - Typical: 0.3-0.9 - Must satisfy \|φ\| < 1 for stationarity | `0.7` | | theta | float | MA coefficient - Typical: -0.5 to +0.5 - Should satisfy \|θ\| < 1 for invertibility | `0.3` | | sigma | float | Innovation standard deviation - Must be positive - Sets scale of unpredictable component | `0.1` | | dt | float | Sampling period - Required for discrete system | `1.0` | ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} - System Type: LINEAR (ARMA) - Noise Type: ADDITIVE (white noise) - Markov: In augmented state (X, w_prev) - Stationary: If |φ| < 1 - Invertible: If |θ| < 1 - Gaussian: If w ~ N(0,σ²) ## Applications {.doc-section .doc-section-applications} **1. Economic Forecasting:** - GDP growth - Inflation - Unemployment - Sales data **2. Financial Time Series:** - Asset returns - Interest rate spreads - Volume data **3. Signal Processing:** - Digital filter design - Prediction algorithms - Spectral estimation **4. Engineering:** - Vibration analysis - Quality control - Process monitoring **5. Environmental:** - Temperature series - Rainfall - Pollution levels ## Numerical Simulation {.doc-section .doc-section-numerical-simulation} **State-Space Implementation:** Augment state with past innovation: Z[k+1] = F·Z[k] + G·w[k+1] where: Z = [X, w_prev]ᵀ F = [φ θ] [0 0] G = [1] [1] Extract: X[k] = Z[k][0] ## Forecasting {.doc-section .doc-section-forecasting} **One-Step-Ahead:** X̂[k+1|k] = φ·X[k] + θ·ε[k] where ε[k] = X[k] - X̂[k|k-1] is innovation. **h-Step-Ahead:** X̂[k+h|k] = φ^h·X[k] + θ·Σφ^(h-1-j)·ε[k-j] Forecast variance increases with horizon. ## Comparison with Other Models {.doc-section .doc-section-comparison-with-other-models} **vs. AR(1):** - AR(1): Only φ parameter - ARMA(1,1): Adds θ (more flexible) - Same asymptotic ACF (h → ∞) - Different ρ(1) **vs. MA(1):** - MA(1): Only θ parameter - ARMA(1,1): Adds φ (long memory) **vs. AR(2):** - AR(2): Two AR lags (4 parameters with σ, μ) - ARMA(1,1): One AR, one MA (3 parameters) - Often ARMA(1,1) more parsimonious **vs. ARMA(p,q):** - ARMA(1,1): Simplest non-trivial - Higher order: More flexible but more parameters - Start with ARMA(1,1), increase if needed ## Limitations {.doc-section .doc-section-limitations} - Linear only - Constant parameters - Gaussian innovations (typically) - Stationary (no trends) - Low order (1,1 may not suffice) ## Extensions {.doc-section .doc-section-extensions} - ARMA(p,q): Higher order - ARIMA: Integrated (for non-stationary) - SARIMA: Seasonal patterns - GARCH: Time-varying variance - Regime-switching ARMA ## Examples {.doc-section .doc-section-examples} Standard ARMA(1,1): ```python >>> # Typical configuration >>> arma = DiscreteARMA11( ... phi=0.7, # Moderate persistence ... theta=0.3, # Positive MA (smoothing) ... sigma=0.1, ... dt=1.0 ... ) >>> >>> # Check stationarity and invertibility >>> print(f"|φ| < 1: {abs(0.7) < 1}") # Stationary >>> print(f"|θ| < 1: {abs(0.3) < 1}") # Invertible ``` Different configurations: ```python >>> # High persistence, low MA >>> high_ar = DiscreteARMA11(phi=0.9, theta=0.1, sigma=0.1) >>> >>> # Moderate AR, negative MA (overshoot) >>> negative_ma = DiscreteARMA11(phi=0.5, theta=-0.3, sigma=0.1) >>> >>> # Balanced AR and MA >>> balanced = DiscreteARMA11(phi=0.6, theta=0.4, sigma=0.1) ``` Simulation: ```python >>> # Generate ARMA series >>> z0 = np.array([0.0, 0.0]) # [X[0], w[-1]] >>> u_seq = np.zeros((100, 1)) >>> >>> result = arma.simulate(z0, u_seq, n_steps=100) >>> >>> # Extract observed series (first component of augmented state) >>> X_series = result['states'][:, 0] >>> >>> # Compute sample ACF >>> from statsmodels.tsa.stattools import acf >>> acf_values = acf(X_series, nlags=20) ``` ## See Also {.doc-section .doc-section-see-also} DiscreteAR1 : Pure autoregressive (θ=0) DiscreteMA1 : Pure moving average (φ=0, future) DiscreteARMA : General ARMA(p,q) (future) ## References {.doc-section .doc-section-references} .. [1] Box, G.E.P. & Jenkins, G.M. (1970). "Time Series Analysis: Forecasting and Control" .. [2] Hamilton, J.D. (1994). "Time Series Analysis" .. [3] Brockwell, P.J. & Davis, R.A. (2016). "Introduction to Time Series and Forecasting" .. [4] Shumway, R.H. & Stoffer, D.S. (2017). "Time Series Analysis and Its Applications" ## Methods | Name | Description | | --- | --- | | [define_system](#cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.define_system) | Define ARMA(1,1) process dynamics in state-space form. | | [get_acf](#cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf) | Get theoretical autocorrelation function. | | [get_acf_lag1](#cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf_lag1) | Get theoretical autocorrelation at lag 1. | | [get_stationary_variance](#cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.get_stationary_variance) | Get theoretical stationary variance of X. | ### define_system { #cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.define_system } ```python systems.builtin.stochastic.discrete.DiscreteARMA11.define_system( phi=0.7, theta=0.3, sigma=0.1, dt=1.0, ) ``` Define ARMA(1,1) process dynamics in state-space form. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|--------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | phi | float | AR coefficient (persistence) - Typical: 0.3-0.9 - Should satisfy \|φ\| < 1 for stationarity - Similar to AR(1) | `0.7` | | theta | float | MA coefficient (smoothing) - Typical: -0.5 to +0.5 - Should satisfy \|θ\| < 1 for invertibility - θ > 0: Smoothing (common) - θ < 0: Overshoot (less common) | `0.3` | | sigma | float | Innovation standard deviation - Must be positive - Sets scale of unpredictable shocks | `0.1` | | dt | float | Sampling period - Required for discrete system | `1.0` | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|-------------|-------------------------------------------------------------| | | ValueError | If sigma ≤ 0 | | | UserWarning | If \|φ\| ≥ 1 (non-stationary) If \|θ\| ≥ 1 (non-invertible) | #### Notes {.doc-section .doc-section-notes} **Stationarity:** Requires |φ| < 1 (from AR component). - φ = 1: Unit root (ARIMA needed) - |φ| > 1: Explosive (unstable) **Invertibility:** Requires |θ| < 1 (from MA component). - |θ| ≥ 1: Non-unique representation - Multiple ARMA models give same ACF - Estimation problems **Parameter Space:** Valid region: |φ| < 1 AND |θ| < 1 **State-Space Formulation:** To simulate ARMA, augment state with past innovation: Z = [X, w_prev]ᵀ Dynamics: Z[k+1] = F·Z[k] + G·w[k+1] where: F = [φ θ] [0 0] G = [1] [1] **Why Augmented State?** Cannot write X[k+1] = f(X[k]) alone because depends on w[k]. Need to track w[k] as auxiliary state. **Variance (Stationary):** Var[X] = σ²·(1 + 2θφ + θ²)/(1 - φ²) **Autocorrelation:** ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²) ρ(h) = φ·ρ(h-1) for h ≥ 2 **Special Cases:** θ = 0: Reduces to AR(1) X[k] = φ·X[k-1] + w[k] φ = 0: Reduces to MA(1) X[k] = θ·w[k-1] + w[k] φ = -θ: Simplifies significantly X[k] = φ·(X[k-1] - w[k-1]) + w[k] (Differencing form) **Typical Configurations:** Economic data: - φ = 0.6-0.9 (high persistence) - θ = 0.1-0.5 (moderate smoothing) Financial returns: - φ = 0.0-0.3 (low persistence) - θ = -0.3-0.0 (slight overshoot or none) Physical measurements: - φ = 0.5-0.8 (moderate persistence) - θ = 0.2-0.5 (smoothing from measurement) #### Examples {.doc-section .doc-section-examples} ```python >>> # Standard economic series >>> economic = DiscreteARMA11( ... phi=0.7, # High persistence ... theta=0.3, # Smoothing ... sigma=0.1 ... ) >>> >>> # Financial returns (low persistence) >>> returns = DiscreteARMA11( ... phi=0.1, # Weak persistence ... theta=-0.2, # Slight overshoot ... sigma=0.2 ... ) >>> >>> # Physical sensor (measurement smoothing) >>> sensor = DiscreteARMA11( ... phi=0.6, ... theta=0.4, # Strong smoothing ... sigma=0.05 ... ) ``` ### get_acf { #cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf } ```python systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf(max_lag=20) ``` Get theoretical autocorrelation function. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |---------|--------|---------------|-----------| | max_lag | int | Maximum lag | `20` | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|------------------------------------------| | | np.ndarray | ACF values [ρ(0), ρ(1), ..., ρ(max_lag)] | #### Notes {.doc-section .doc-section-notes} ρ(0) = 1 ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²) ρ(h) = φ·ρ(h-1) for h ≥ 2 Geometric decay after lag 1. #### Examples {.doc-section .doc-section-examples} ```python >>> arma = DiscreteARMA11(phi=0.7, theta=0.3, sigma=0.1) >>> acf = arma.get_acf(max_lag=10) >>> print(f"ACF: {acf}") ``` ### get_acf_lag1 { #cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf_lag1 } ```python systems.builtin.stochastic.discrete.DiscreteARMA11.get_acf_lag1() ``` Get theoretical autocorrelation at lag 1. ρ(1) = (φ + θ)·(1 + θφ)/(1 + 2θφ + θ²) #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|-----------------------| | | float | Lag-1 autocorrelation | #### Notes {.doc-section .doc-section-notes} This is where ARMA differs from AR(1). For AR(1): ρ(1) = φ For ARMA(1,1): ρ(1) ≠ φ (MA effect) #### Examples {.doc-section .doc-section-examples} ```python >>> arma = DiscreteARMA11(phi=0.7, theta=0.3, sigma=0.1) >>> rho1 = arma.get_acf_lag1() >>> print(f"ρ(1) = {rho1:.3f}") >>> print(f"Compare with φ = 0.7") ``` ### get_stationary_variance { #cdesym.systems.builtin.stochastic.discrete.DiscreteARMA11.get_stationary_variance } ```python systems.builtin.stochastic.discrete.DiscreteARMA11.get_stationary_variance() ``` Get theoretical stationary variance of X. Var[X] = σ²·(1 + 2θφ + θ²)/(1 - φ²) #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------| | | float | Stationary variance | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|------------|-------------------------------| | | ValueError | If \|φ\| ≥ 1 (non-stationary) | #### Examples {.doc-section .doc-section-examples} ```python >>> arma = DiscreteARMA11(phi=0.7, theta=0.3, sigma=0.1) >>> var = arma.get_stationary_variance() >>> print(f"Stationary variance: {var:.4f}") ```