Type:	Package
Title:	Polynomial Maximization Method for Non-Gaussian Regression
Version:	0.3.1
Date:	2026-04-06
Description:	Implements the Polynomial Maximization Method ('PMM') for parameter estimation in linear and time series models when error distributions deviate from normality. The 'PMM2' variant achieves lower variance parameter estimates compared to ordinary least squares ('OLS') when errors exhibit significant skewness. The 'PMM3' variant (S=3) targets symmetric platykurtic error distributions, reducing variance when excess kurtosis is negative. Includes automatic method selection ('pmm_dispatch'), linear regression, 'AR'/'MA'/'ARMA'/'ARIMA' models, and bootstrap inference. Methodology described in Zabolotnii, Warsza, and Tkachenko (2018) <doi:10.1007/978-3-319-77179-3_75>, Zabolotnii, Tkachenko, and Warsza (2022) <doi:10.1007/978-3-031-03502-9_37>, and Zabolotnii, Tkachenko, and Warsza (2023) <doi:10.1007/978-3-031-25844-2_21>.
License:	GPL-3
Encoding:	UTF-8
Depends:	R (≥ 3.5.0)
Imports:	methods, stats, graphics, utils
Suggests:	dplyr, ggplot2, gridExtra, testthat (≥ 3.0.0), rmarkdown, knitr, MASS, numDeriv
LazyData:	true
RoxygenNote:	7.3.3
Config/testthat/edition:	3
URL:	https://github.com/SZabolotnii/EstemPMM
BugReports:	https://github.com/SZabolotnii/EstemPMM/issues
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-04-06 18:44:35 UTC; docua
Author:	Serhii Zabolotnii [aut, cre]
Maintainer:	Serhii Zabolotnii <zabolotniua@gmail.com>
Repository:	CRAN
Date/Publication:	2026-04-07 05:20:20 UTC

EstemPMM: Polynomial Maximization Method for Robust Regression and Time Series

Description

The EstemPMM package provides robust methods for estimating parameters of linear models and time series models that are robust to non-Gaussian errors.

Linear Regression Functions

lm_pmm2 - Fit linear models using PMM2

compare_with_ols - Compare PMM2 with OLS

Time Series Functions

ts_pmm2 - General function for fitting time series models using PMM2

ar_pmm2 - Fit AR models

ma_pmm2 - Fit MA models

arma_pmm2 - Fit ARMA models

arima_pmm2 - Fit ARIMA models

compare_ts_methods - Compare PMM2 with classical methods

Statistical Inference

pmm2_inference - Bootstrap inference for linear models

ts_pmm2_inference - Bootstrap inference for time series models

PMM3 (Symmetric Errors)

lm_pmm3 - Fit linear models using PMM3 (S=3, symmetric platykurtic errors)

compute_moments_pmm3 - Compute moments for PMM3

pmm3_variance_factor - PMM3 theoretical variance reduction factor

pmm_gamma6 - Compute sixth-order cumulant coefficient

test_symmetry - Test residual symmetry for PMM2 vs PMM3 choice

PMM3 Time Series Functions

ts_pmm3 - General function for fitting time series models using PMM3

ar_pmm3 - Fit AR models using PMM3

ma_pmm3 - Fit MA models using PMM3

arma_pmm3 - Fit ARMA models using PMM3

arima_pmm3 - Fit ARIMA models using PMM3

Method Selection

pmm_dispatch - Automatic PMM method selection (OLS / PMM2 / PMM3)

Utilities

pmm_skewness - Compute skewness

pmm_kurtosis - Compute kurtosis

compute_moments - Compute moments and cumulants

Author(s)

Maintainer: Serhii Zabolotnii zabolotniua@gmail.com (ORCID)

See Also

Useful links:

• https://github.com/SZabolotnii/EstemPMM

• Report bugs at https://github.com/SZabolotnii/EstemPMM/issues

PMM2 theoretical efficiency factor

Description

PMM2 theoretical efficiency factor

Usage

.g2_factor(gamma3, gamma4)

Arguments

Value

Numeric scalar: g2 factor

PMM3 theoretical efficiency factor

Description

PMM3 theoretical efficiency factor

Usage

.g3_factor(gamma4, gamma6)

Arguments

Value

Numeric scalar: g3 factor

PMM2 fitting algorithm - unified implementation

Description

Core iterative algorithm for PMM2 parameter estimation

Usage

.pmm2_fit(
  b_init,
  X,
  y,
  m2,
  m3,
  m4,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Value

A list with components:

PMM3 Newton-Raphson solver (vectorised)

Description

Solves the PMM3 score equations using Newton-Raphson iteration. The score is Z = X'(\varepsilon(\kappa - \varepsilon^2)) and the Jacobian is J = X' \mathrm{diag}(3\varepsilon^2 - \kappa) X.

Usage

.pmm3_nr_solver(
  B_ols,
  X,
  Y,
  kappa,
  adaptive = FALSE,
  tol = 1e-06,
  max_iter = 100,
  step_max = 5
)

Arguments

Value

List: B (estimates), converged, iter, kappa, near_gaussian

PMM3 Newton-Raphson solver for nonlinear models

Description

Uses a residual function and numerical Jacobian (via numDeriv) instead of a fixed design matrix. This is necessary for ARIMA and other models where the linearized design matrix does not capture full dynamics.

Usage

.pmm3_nr_solver_nonlinear(
  theta_init,
  fn_residuals,
  kappa,
  adaptive = FALSE,
  tol = 1e-06,
  max_iter = 100,
  step_max = 5
)

Arguments

Details

The PMM3 estimating equation is:

\sum_t \varepsilon_t(\kappa - \varepsilon_t^2) \cdot \frac{\partial \varepsilon_t}{\partial \theta} = 0

Value

List: B (estimates), converged, iter, kappa, near_gaussian

Compute residual cumulants up to 6th order

Description

Compute residual cumulants up to 6th order

Usage

.residual_cumulants(residuals)

Arguments

Value

Named list: m2, m3, m4, m6, gamma3, gamma4, gamma6

PMM2 fitting algorithm for time series models

Description

Implementation of PMM2 algorithm for time series models with various approaches depending on the model structure (AR, MA, ARMA, ARIMA)

Usage

.ts_pmm2_fit(
  b_init,
  innovations_init,
  model_info,
  m2,
  m3,
  m4,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Value

A list with components:

Calculate AIC for PMM2fit object

Description

Calculate AIC for PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
AIC(object, ..., k = 2)

Arguments

Value

AIC value

Calculate AIC for PMM3fit object

Description

Calculate AIC for PMM3fit object

Usage

## S4 method for signature 'PMM3fit'
AIC(object, ..., k = 2)

Arguments

Value

AIC value

Calculate AIC for TS3fit object

Description

Calculate AIC for TS3fit object

Usage

## S4 method for signature 'TS3fit'
AIC(object, ..., k = 2)

Arguments

Value

AIC value

S4 class for storing PMM2 ARIMA model results

Description

S4 class for storing PMM2 ARIMA model results

S4 class for PMM3 ARIMA model results

Description

S4 class for PMM3 ARIMA model results

S4 class for storing PMM2 ARMA model results

Description

S4 class for storing PMM2 ARMA model results

S4 class for PMM3 ARMA model results

Description

S4 class for PMM3 ARMA model results

S4 class for storing PMM2 AR model results

Description

S4 class for storing PMM2 AR model results

S4 class for PMM3 AR model results

Description

S4 class for PMM3 AR model results

Base S4 class for storing PMM2 model results

Description

Base S4 class for storing PMM2 model results

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals

m2

numeric second central moment of initial residuals

m3

numeric third central moment of initial residuals

m4

numeric fourth central moment of initial residuals

convergence

logical or integer code indicating whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

WTI Crude Oil Prices

Description

Daily spot prices for West Texas Intermediate (WTI) crude oil in U.S. dollars per barrel.

Usage

DCOILWTICO

Format

A data frame with observations for each trading day:

observation_date

Date of observation in YYYY-MM-DD format

DCOILWTICO

Crude Oil Price: West Texas Intermediate (WTI) in USD per barrel

Source

Federal Reserve Economic Data (FRED), Federal Reserve Bank of St. Louis https://fred.stlouisfed.org/series/DCOILWTICO

Examples

data(DCOILWTICO)
head(DCOILWTICO)
summary(DCOILWTICO$DCOILWTICO)

S4 class for storing PMM2 MA model results

Description

S4 class for storing PMM2 MA model results

S4 class for PMM3 MA model results

Description

S4 class for PMM3 MA model results

S4 class for storing PMM2 regression model results

Description

Class for storing results of linear model estimation using PMM2

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals

m2

numeric second central moment of initial residuals

m3

numeric third central moment of initial residuals

m4

numeric fourth central moment of initial residuals

convergence

logical or integer code indicating whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

Slots

coefficients

Estimated coefficients

residuals

Final residuals

m2

Second central moment

m3

Third central moment

m4

Fourth central moment

convergence

Convergence status

iterations

Number of iterations performed

call

Original call

S4 class for storing PMM3 regression model results

Description

PMM3 (S=3) is designed for symmetric platykurtic error distributions. This class is fully standalone and does NOT inherit from BasePMM2.

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals

m2

numeric second central moment of initial residuals

m4

numeric fourth central moment of initial residuals

m6

numeric sixth central moment of initial residuals

gamma4

numeric excess kurtosis coefficient

gamma6

numeric sixth-order cumulant coefficient

g_coefficient

numeric theoretical variance reduction factor g3

kappa

numeric moment ratio used in NR solver

convergence

logical indicating whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

S4 class for Seasonal ARIMA model results with PMM2

Description

This class stores the results of fitting a Seasonal Autoregressive Integrated Moving Average (SARIMA) model using the PMM2 method. It extends SARMA with differencing operators.

Details

The SARIMAPMM2 class represents fitted SARIMA(p,d,q) x (P,D,Q)_s models:

(1 - \phi_1 B - \dots - \phi_p B^p)(1 - \Phi_1 B^s - \dots - \Phi_P B^{Ps})(1 - B)^d (1 - B^s)^D y_t = (1 + \theta_1 B + \dots + \theta_q B^q)(1 + \Theta_1 B^s + \dots + \Theta_Q B^{Qs}) \epsilon_t.

Where B is the backshift operator.

Slots

coefficients

Numeric vector of estimated parameters

residuals

Numeric vector of residuals/innovations

m2

Second central moment of residuals

m3

Third central moment of residuals

m4

Fourth central moment of residuals

convergence

Logical, whether PMM2 algorithm converged

iterations

Integer, number of iterations performed

call

Original function call

model_type

Character, model type identifier ("sarima")

intercept

Numeric, intercept/mean term

original_series

Numeric vector, original time series data

order

List with model specification: list(ar, sar, ma, sma, d, D, period)

• ar: Non-seasonal AR order (p)

• sar: Seasonal AR order (P)

• ma: Non-seasonal MA order (q)

• sma: Seasonal MA order (Q)

• d: Non-seasonal differencing order

• D: Seasonal differencing order

• period: Seasonal period (s)

See Also

sarima_pmm2 for fitting SARIMA models

S4 class for Seasonal ARMA model results with PMM2

Description

This class stores the results of fitting a Seasonal Autoregressive Moving Average (SARMA) model using the PMM2 method. It combines both seasonal AR and seasonal MA components.

Details

The SARMAPMM2 class represents fitted SARMA models combining:

• AR(p): \phi_1 y_{t-1} + \dots + \phi_p y_{t-p}

• Seasonal AR component: \Phi_1 y_{t-s} + \dots + \Phi_P y_{t-Ps}

• MA(q): \theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q}

• Seasonal MA component: \Theta_1 \epsilon_{t-s} + \dots + \Theta_Q \epsilon_{t-Qs}

Slots

coefficients

Numeric vector of estimated parameters (SAR and SMA coefficients)

residuals

Numeric vector of residuals/innovations

m2

Second central moment of residuals

m3

Third central moment of residuals

m4

Fourth central moment of residuals

convergence

Logical, whether PMM2 algorithm converged

iterations

Integer, number of iterations performed

call

Original function call

model_type

Character, model type identifier ("sarma")

intercept

Numeric, intercept/mean term

original_series

Numeric vector, original time series data

order

List with model specification: list(ar, sar, ma, sma, period)

• ar: Non-seasonal AR order (p)

• sar: Seasonal AR order (P)

• ma: Non-seasonal MA order (q)

• sma: Seasonal MA order (Q)

• period: Seasonal period (s)

See Also

sarma_pmm2 for fitting SARMA models

S4 class for Seasonal AR model results with PMM2

Description

This class stores the results of fitting a Seasonal Autoregressive (SAR) model using the PMM2 method. It extends the TS2fit class with additional slots specific to seasonal models.

Details

The SARPMM2 class represents fitted SAR models of the form

y_t = \phi_1 y_{t-1} + \dots + \phi_p y_{t-p} + \Phi_1 y_{t-s} + \dots + \Phi_P y_{t-Ps} + \epsilon_t.

Where:

• p is the non-seasonal AR order

• P is the seasonal AR order

• s is the seasonal period

Slots

coefficients

Numeric vector of estimated parameters (AR and SAR coefficients)

residuals

Numeric vector of residuals/innovations

m2

Second central moment of residuals

m3

Third central moment of residuals

m4

Fourth central moment of residuals

convergence

Logical, whether PMM2 algorithm converged

iterations

Integer, number of iterations performed

call

Original function call

model_type

Character, model type identifier ("sar")

intercept

Numeric, intercept/mean term

original_series

Numeric vector, original time series data

order

List with model specification: list(ar, sar, period)

• ar: Non-seasonal AR order (p)

• sar: Seasonal AR order (P)

• period: Seasonal period (s)

See Also

sar_pmm2 for fitting SAR models

S4 class for Seasonal MA PMM2 results

Description

This class stores the results of fitting a Seasonal Moving Average (SMA) model using the Polynomial Maximization Method (PMM2).

Details

The SMA(Q)_s model is expressed as

y_t = \mu + \epsilon_t + \Theta_1 \epsilon_{t-s} + \Theta_2 \epsilon_{t-2s} + \dots + \Theta_Q \epsilon_{t-Qs}.

Where:

• Q is the seasonal MA order

• s is the seasonal period

• epsilon_t are innovations

Slots

coefficients

Estimated seasonal MA coefficients (Theta_1, Theta_2, ..., Theta_Q)

innovations

Estimated innovations (residuals epsilon_t)

m2

Second central moment (variance) of innovations

m3

Third central moment (skewness indicator) of innovations

m4

Fourth central moment (kurtosis indicator) of innovations

convergence

Logical indicating whether PMM2 algorithm converged

iterations

Number of iterations required for convergence

call

The function call that created this object

model_type

Character string "sma"

intercept

Model intercept (mean)

original_series

Original time series data

order

List with Q (seasonal MA order) and s (seasonal period)

See Also

sma_pmm2 for fitting SMA models

Base S4 class for storing PMM2 time series model results

Description

Base class for storing results of time series model estimation using PMM2

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals

m2

numeric second central moment of initial residuals

m3

numeric third central moment of initial residuals

m4

numeric fourth central moment of initial residuals

convergence

logical or integer code indicating whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

model_type

character string indicating model type

intercept

numeric value of intercept

original_series

numeric vector of original time series

order

list of order parameters

Slots

coefficients

Estimated coefficients

residuals

Final residuals

m2

Second central moment

m3

Third central moment

m4

Fourth central moment

convergence

Convergence status

iterations

Number of iterations performed

call

Original call

model_type

Model type

intercept

Intercept

original_series

Original time series

order

Model orders

Base S4 class for PMM3 time series model results

Description

Stores results from time series estimation using PMM3 (S=3). Designed for symmetric platykurtic innovations. Does NOT inherit from BasePMM2 or TS2fit.

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals/innovations

m2

numeric second central moment of initial residuals

m4

numeric fourth central moment of initial residuals

m6

numeric sixth central moment of initial residuals

gamma4

numeric excess kurtosis coefficient

gamma6

numeric sixth-order cumulant coefficient

g_coefficient

numeric theoretical variance reduction factor g3

kappa

numeric moment ratio used in NR solver

convergence

logical whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

model_type

character string indicating model type

intercept

numeric intercept value

original_series

numeric vector of original time series

order

list of order parameters

Fit an AR model using PMM2 (wrapper)

Description

Estimates autoregressive model parameters using the Pearson Moment Method (PMM2). PMM2 exploits third and fourth moment information to achieve more accurate parameter estimates than classical maximum likelihood, particularly for non-Gaussian innovations.

Usage

ar_pmm2(
  x,
  order = 1,
  method = "pmm2",
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

PMM2 Variants:

• "unified_global" (default): One-step correction from MLE/CSS estimates. Fast and stable. Recommended for most AR models. Typical improvement: 3-5\

• "unified_iterative": Full Newton-Raphson optimization starting from classical estimates. More accurate but computationally intensive. Best for complex models with strong non-Gaussian features.

• "linearized": Uses first-order Taylor expansion. Not recommended for AR models; designed for MA/SMA where Jacobian computation is complex.

Variant Selection Guidelines:

• For AR(p) models: Use "unified_global" (default)

• If convergence issues occur: Try "unified_iterative"

• For highly skewed/heavy-tailed innovations: Use "unified_iterative"

Computational Characteristics:

• unified_global: ~2x slower than MLE (single correction step)

• unified_iterative: 5-10x slower than MLE (iterative refinement)

• linearized: ~1.5x slower than MLE (approximation-based)

Value

An S4 object of class ARPMM2 containing fitted autoregressive coefficients, residuals, central moment estimates (m2-m4), model order, intercept, original series, and convergence diagnostics.

References

Monte Carlo validation (R=50, n=200): Unified Iterative showed 2.9\ improvement over MLE for AR(1) models. See NEWS.md (Version 0.2.0) for details.

See Also

ma_pmm2, arma_pmm2, arima_pmm2

Examples

# Fit AR(2) model with default variant
x <- arima.sim(n = 200, list(ar = c(0.7, -0.3)))
fit1 <- ar_pmm2(x, order = 2)
coef(fit1)

# Compare variants
fit2 <- ar_pmm2(x, order = 2, pmm2_variant = "unified_iterative")
fit3 <- ar_pmm2(x, order = 2, pmm2_variant = "linearized")

  

Fit an AR model using PMM3

Description

Estimates autoregressive model parameters using the Polynomial Maximization Method of order 3 (PMM3). Designed for symmetric platykurtic innovations.

Usage

ar_pmm3(
  x,
  order = 1,
  max_iter = 100,
  tol = 1e-06,
  adaptive = FALSE,
  step_max = 5,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  verbose = FALSE
)

Arguments

Value

An S4 object of class ARPMM3

See Also

ar_pmm2, ma_pmm3, lm_pmm3

Examples

set.seed(42)
x <- arima.sim(n = 200, list(ar = 0.7),
               innov = runif(200, -sqrt(3), sqrt(3)))
fit <- ar_pmm3(x, order = 1)
coef(fit)

Fit an ARIMA model using PMM2 (wrapper)

Description

Estimates autoregressive integrated moving-average model parameters using PMM2. ARIMA models extend ARMA to non-stationary series via differencing. PMM2 provides robust parameter estimates for the stationary ARMA component after differencing is applied.

Usage

arima_pmm2(
  x,
  order = c(1, 1, 1),
  method = "pmm2",
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

PMM2 Variants:

• "unified_global" (default): One-step correction from MLE/CSS estimates. Fast and reliable for most ARIMA specifications.

• "unified_iterative": Full Newton-Raphson optimization. Recommended for ARIMA models with complex dynamics or when unified_global shows residual non-Gaussianity.

• "linearized": First-order approximation. Not typically recommended for ARIMA unless MA component dominates.

Variant Selection Guidelines:

• For standard ARIMA(p,d,q): Use "unified_global" (default)

• For models with d >= 2 or high orders: Try "unified_iterative"

• If MA component is large relative to AR: Consider "unified_iterative"

ARIMA Estimation Workflow:

1. Apply differencing of order d to achieve stationarity

2. Estimate ARMA(p,q) model on differenced series using PMM2

3. Return coefficients and diagnostics for the integrated model

Differencing Notes: The d parameter determines how many times the series is differenced. d=0 reduces to ARMA, d=1 handles unit root processes, d=2 is rare but useful for some economic series with trend acceleration.

Value

An S4 object of class ARIMAPMM2 containing fitted AR and MA coefficients, residual series, central moments, differencing order, intercept, original series, and convergence diagnostics.

See Also

arma_pmm2, sarima_pmm2, ar_pmm2

Examples

# Fit ARIMA(1,1,1) model to non-stationary series
x <- cumsum(arima.sim(n = 200, list(ar = 0.6, ma = -0.4)))
fit1 <- arima_pmm2(x, order = c(1, 1, 1))
coef(fit1)

# ARIMA(2,1,0) - random walk with AR(2) innovations
x2 <- cumsum(arima.sim(n = 250, list(ar = c(0.7, -0.3))))
fit2 <- arima_pmm2(x2, order = c(2, 1, 0), pmm2_variant = "unified_global")

# ARIMA(0,2,2) - double differencing with MA(2)
x3 <- cumsum(cumsum(arima.sim(n = 300, list(ma = c(0.5, 0.3)))))
fit3 <- arima_pmm2(x3, order = c(0, 2, 2), pmm2_variant = "unified_iterative")

  

Fit an ARIMA model using PMM3

Description

Estimates ARIMA model parameters using PMM3.

Usage

arima_pmm3(
  x,
  order = c(1, 1, 1),
  max_iter = 100,
  tol = 1e-06,
  adaptive = FALSE,
  step_max = 5,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  verbose = FALSE
)

Arguments

Value

An S4 object of class ARIMAPMM3

See Also

arima_pmm2, arma_pmm3, lm_pmm3

Examples

set.seed(42)
x <- cumsum(arima.sim(n = 200, list(ar = 0.6),
            innov = runif(200, -sqrt(3), sqrt(3))))
fit <- arima_pmm3(x, order = c(1, 1, 0))
coef(fit)

Fit an ARMA model using PMM2 (wrapper)

Description

Estimates autoregressive moving-average model parameters using PMM2. ARMA models combine AR and MA components, capturing both direct past value dependencies and innovation structure. PMM2 leverages higher moments to improve parameter estimation accuracy.

Usage

arma_pmm2(
  x,
  order = c(1, 1),
  method = "pmm2",
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

PMM2 Variants:

• "unified_global" (default): One-step correction from MLE/CSS estimates. Balances speed and accuracy for ARMA models. Recommended for most use cases.

• "unified_iterative": Full Newton-Raphson optimization. Best for models with complex dynamics or strong non-Gaussian features.

• "linearized": First-order approximation. Generally not recommended for ARMA; better suited for pure MA/SMA models.

Variant Selection Guidelines:

• For ARMA(p,q) with p,q <= 2: Use "unified_global" (default)

• For higher-order ARMA or ill-conditioned models: Try "unified_iterative"

• For quick exploration: Start with "unified_global"

ARMA Estimation Challenges: ARMA models have more parameters than pure AR or MA models, making estimation more sensitive to initialization and numerical stability. PMM2 benefits from robust moment-based constraints that help regularize the parameter space.

Value

An S4 object of class ARMAPMM2 containing fitted AR and MA coefficients, residuals, central moments, model specification, intercept, original series, and convergence diagnostics.

See Also

ar_pmm2, ma_pmm2, arima_pmm2

Examples

# Fit ARMA(2,1) model
x <- arima.sim(n = 250, list(ar = c(0.7, -0.3), ma = 0.5))
fit1 <- arma_pmm2(x, order = c(2, 1))
coef(fit1)

# Try iterative variant for better accuracy
fit2 <- arma_pmm2(x, order = c(2, 1), pmm2_variant = "unified_iterative")

# Higher-order ARMA
x2 <- arima.sim(n = 300, list(ar = c(0.6, -0.2), ma = c(0.4, 0.3)))
fit3 <- arma_pmm2(x2, order = c(2, 2), pmm2_variant = "unified_global")

  

Fit an ARMA model using PMM3

Description

Estimates ARMA model parameters using PMM3.

Usage

arma_pmm3(
  x,
  order = c(1, 1),
  max_iter = 100,
  tol = 1e-06,
  adaptive = FALSE,
  step_max = 5,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  verbose = FALSE
)

Arguments

Value

An S4 object of class ARMAPMM3

See Also

arma_pmm2, arima_pmm3, lm_pmm3

Examples

set.seed(42)
x <- arima.sim(n = 250, list(ar = 0.7, ma = -0.3),
               innov = runif(250, -sqrt(3), sqrt(3)))
fit <- arma_pmm3(x, order = c(1, 1))
coef(fit)

Auto MPG Dataset

Description

Fuel consumption and vehicle characteristics for 398 automobiles from the 1970s and 1980s. This dataset is used in published PMM research to demonstrate both PMM2 (asymmetric residuals: MPG vs Weight) and PMM3 (symmetric platykurtic residuals: MPG vs Horsepower).

Usage

auto_mpg

Format

A data frame with 398 rows and 9 variables:

mpg

Miles per gallon (fuel efficiency)

cylinders

Number of cylinders (4, 6, or 8)

displacement

Engine displacement (cubic inches)

horsepower

Engine horsepower (6 missing values)

weight

Vehicle weight (pounds)

acceleration

Time to accelerate from 0 to 60 mph (seconds)

model_year

Model year (70-82, i.e., 1970-1982)

origin

Origin (1 = American, 2 = European, 3 = Japanese)

car_name

Car model name

Details

Three regression examples from published PMM papers:

• MPG vs Acceleration (PMM2, linear): residuals have gamma3 = 0.49, g2 = 0.86 (Zabolotnii et al., 2018)

• MPG vs Weight (PMM2, quadratic): residuals have gamma3 = 0.8, g2 = 0.83 (Zabolotnii et al., 2025)

• MPG vs Horsepower (PMM3, quadratic): residuals have gamma3 ~ 0.2, gamma4 = 1.3, g3 = 0.89 (Zabolotnii et al., 2025)

Source

UCI Machine Learning Repository https://archive.ics.uci.edu/dataset/9/auto+mpg

References

Quinlan, J.R. (1993). Combining Instance-Based and Model-Based Learning. In Proceedings on the Tenth International Conference of Machine Learning, 236-243.

Zabolotnii S., Warsza Z.L., Tkachenko O. (2018) Polynomial Estimation of Linear Regression Parameters for the Asymmetric PDF of Errors. Springer AISC, vol 743. doi:10.1007/978-3-319-77179-3_75

Examples

data(auto_mpg)
# PMM2 example: MPG vs Acceleration (asymmetric residuals)
fit_ols <- lm(mpg ~ acceleration, data = auto_mpg)
pmm_skewness(residuals(fit_ols))  # gamma3 ~ 0.5 -> PMM2
pmm_dispatch(residuals(fit_ols))
fit_pmm2 <- lm_pmm2(mpg ~ acceleration, data = auto_mpg, na.action = na.omit)
coef(fit_pmm2)  # compare with coef(fit_ols)

Extract coefficients from PMM2fit object

Description

Extract coefficients from PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
coef(object, ...)

Arguments

Value

Vector of coefficients

Extract coefficients from PMM3fit object

Description

Extract coefficients from PMM3fit object

Usage

## S4 method for signature 'PMM3fit'
coef(object, ...)

Arguments

Value

Vector of coefficients

Extract coefficients from SARPMM2 object

Description

Extract coefficients from SARPMM2 object

Usage

## S4 method for signature 'SARPMM2'
coef(object, ...)

Arguments

Value

Named numeric vector of coefficients

Extract coefficients from SMAPMM2 object

Description

Extract coefficients from SMAPMM2 object

Usage

## S4 method for signature 'SMAPMM2'
coef(object, ...)

Arguments

Value

Named vector of seasonal MA coefficients

Extract coefficients from TS2fit object

Description

Extract coefficients from TS2fit object

Usage

## S4 method for signature 'TS2fit'
coef(object, ...)

Arguments

Value

Named vector of coefficients

Extract coefficients from TS3fit object

Description

Extract coefficients from TS3fit object

Usage

## S4 method for signature 'TS3fit'
coef(object, ...)

Arguments

Value

Named vector of coefficients

Compare AR methods

Description

Compare AR methods

Usage

compare_ar_methods(x, order = 1, include.mean = TRUE, pmm2_args = list())

Arguments

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare ARIMA methods

Description

Compare ARIMA methods

Usage

compare_arima_methods(
  x,
  order = c(1, 1, 1),
  include.mean = TRUE,
  pmm2_args = list()
)

Arguments

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare ARMA methods

Description

Compare ARMA methods

Usage

compare_arma_methods(
  x,
  order = c(1, 1),
  include.mean = TRUE,
  pmm2_args = list()
)

Arguments

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare MA methods

Description

Compare MA methods

Usage

compare_ma_methods(x, order = 1, include.mean = TRUE, pmm2_args = list())

Arguments

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare SAR model estimation methods

Description

Compares different estimation methods (OLS, PMM2, CSS, ML) for SAR models on the same data.

Usage

compare_sar_methods(
  x,
  order = c(1, 1),
  period = 12,
  methods = c("ols", "pmm2", "css"),
  verbose = TRUE
)

Arguments

Value

Data frame with comparison results (invisibly)

Examples

set.seed(42)
y <- arima.sim(n = 120,
  model = list(order = c(1, 0, 0), ar = 0.7,
    seasonal = list(order = c(1, 0, 0), ar = 0.5, period = 12)))
compare_sar_methods(y, order = c(1, 1), period = 12)

Compare PMM2 with classical time series estimation methods

Description

Compare PMM2 with classical time series estimation methods

Usage

compare_ts_methods(
  x,
  order,
  model_type = c("ar", "ma", "arma", "arima"),
  include.mean = TRUE,
  pmm2_args = list()
)

Arguments

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare PMM2 with OLS

Description

Compare PMM2 with OLS

Usage

compare_with_ols(formula, data, pmm2_args = list())

Arguments

Value

List with OLS and PMM2 fit objects

Examples

result <- compare_with_ols(mpg ~ wt, data = mtcars)

Calculate moments and cumulants of error distribution

Description

Calculate moments and cumulants of error distribution

Usage

compute_moments(errors)

Arguments

Value

list with moments, cumulants and theoretical variance reduction coefficient

Compute central moments for PMM3 from residuals

Description

Computes the second, fourth, and sixth central moments, along with standardised cumulant coefficients gamma3, gamma4, gamma6, the theoretical efficiency factor g3, and the moment ratio kappa used in the PMM3 solver.

Usage

compute_moments_pmm3(residuals)

Arguments

Value

A list with components:

Unified PMM2 Estimator for Nonlinear Regression Models

Description

This file contains a unified PMM2 implementation suitable for arbitrary nonlinear regression models (including SARIMAX) for which residuals and their derivatives (Jacobian) can be computed.

Usage

compute_numerical_jacobian(fn_residuals, theta, method = "Richardson")

Arguments

Details

Two approaches are implemented:

1. Iterative: Full iterative procedure (Nonlinear PMM2).

2. One-step (Global): Single-step correction after a classical estimator (e.g., MLE). Compute numerical Jacobian of the residual function

Uses numDeriv::jacobian to compute the derivative matrix.

Value

Jacobian matrix (n x p)

Compute PMM2 weights and components

Description

Compute PMM2 weights and components

Usage

compute_pmm2_components(residuals)

Arguments

Value

List with moments and PMM2 parameters (gamma3, gamma4, weights)

Compute final residuals for time series models

Description

Compute final residuals for time series models

Usage

compute_ts_residuals(coefs, model_info)

Arguments

Value

Vector of residuals

Create design matrix for AR model

Description

Create design matrix for AR model

Usage

create_ar_matrix(x, p)

Arguments

Value

Design matrix with lagged values

Create residual function for PMM2 optimization

Description

Create residual function for PMM2 optimization

Usage

create_residual_function(x, order, model_type, seasonal, include.mean)

Arguments

Value

Function that computes residuals given parameters

Create design matrix for seasonal AR model

Description

Constructs a design matrix for Seasonal Autoregressive (SAR) models, optionally including non-seasonal AR lags and multiplicative cross-terms.

Usage

create_sar_matrix(x, p = 0, P = 1, s = 12, multiplicative = FALSE)

Arguments

Details

For a SAR(P)_s model:

y_t = \Phi_1 y_{t-s} + \dots + \Phi_P y_{t-Ps} + \epsilon_t.

For an additive AR(p)+SAR(P)_s model:

y_t = \phi_1 y_{t-1} + \dots + \phi_p y_{t-p} + \Phi_1 y_{t-s} + \dots + \Phi_P y_{t-Ps} + \epsilon_t.

For a multiplicative AR(p) x SAR(P)_s model (multiplicative = TRUE): Includes cross-terms such as y_{t-1-s}, y_{t-1-2s}, etc.

Value

Design matrix with lagged values. Columns are:

• First p columns: non-seasonal lags (y_{t-1}, \dots, y_{t-p})

• Next P columns: seasonal lags (y_{t-s}, \dots, y_{t-Ps})

• If multiplicative = TRUE: additional p \times P columns for cross-terms

Examples

# Simple SAR(1)_12 model
x <- rnorm(120)
X <- create_sar_matrix(x, p = 0, P = 1, s = 12)

# AR(1) + SAR(1)_12 additive model
X <- create_sar_matrix(x, p = 1, P = 1, s = 12)

# AR(1) x SAR(1)_12 multiplicative model
X <- create_sar_matrix(x, p = 1, P = 1, s = 12, multiplicative = TRUE)

Create design matrix for seasonal ARMA model

Description

Constructs a design matrix for Seasonal ARMA (SARMA) models, combining non-seasonal and seasonal AR and MA components.

Usage

create_sarma_matrix(
  x,
  residuals,
  p = 0,
  P = 0,
  q = 0,
  Q = 0,
  s = 12,
  multiplicative = FALSE
)

Arguments

Details

For a SARMA(p,q) x (P,Q)_s model:

y_t = \phi_1 y_{t-1} + \dots + \phi_p y_{t-p} + \Phi_1 y_{t-s} + \dots + \Phi_P y_{t-Ps} + \theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q} + \Theta_1 \epsilon_{t-s} + \dots + \Theta_Q \epsilon_{t-Qs} + \epsilon_t.

Where:

• p, P are non-seasonal and seasonal AR orders

• q, Q are non-seasonal and seasonal MA orders

• s is the seasonal period

Value

Design matrix with lagged values. Columns are ordered as:

• First p columns: non-seasonal AR lags (y_{t-1}, \dots, y_{t-p})

• Next P columns: seasonal AR lags (y_{t-s}, \dots, y_{t-Ps})

• Next q columns: non-seasonal MA lags (\epsilon_{t-1}, \dots, \epsilon_{t-q})

• Next Q columns: seasonal MA lags (\epsilon_{t-s}, \dots, \epsilon_{t-Qs})

• If multiplicative=TRUE: AR cross-terms then MA cross-terms

Examples

# Simple SARMA(1,0) x (1,0)_12 model (AR+SAR, no MA)
x <- rnorm(120)
residuals <- rnorm(120)
X <- create_sarma_matrix(x, residuals, p = 1, P = 1, q = 0, Q = 0, s = 12)

# Full SARMA(1,1) x (1,1)_12 model
X <- create_sarma_matrix(x, residuals, p = 1, P = 1, q = 1, Q = 1, s = 12)

Create design matrix for time series

Description

Create design matrix for time series

Usage

create_ts_design_matrix(x, model_info, innovations = NULL)

Arguments

Value

List with design matrix, response variable, and other components

Dow Jones Industrial Average Daily Data (July-December 2002)

Description

Daily closing prices and changes of the Dow Jones Industrial Average for the second half of 2002. Used in published PMM2 research to demonstrate AR(1) estimation with asymmetric innovations.

Usage

djia2002

Format

A data frame with 127 rows and 3 variables:

date

Trading date

close

DJIA closing price (USD)

change

Daily change in closing price (first difference; NA for the first observation)

Details

The daily changes exhibit positive skewness, making this dataset suitable for PMM2 estimation. In the original paper, an AR(1) model fitted to the change series yielded PMM2 coefficient a1 = -0.43 versus OLS a1 = -0.49, with g2 = 0.77 (23\

Source

Yahoo Finance via the quantmod R package.

References

Zabolotnii S., Tkachenko O., Warsza Z.L. (2022) Application of the Polynomial Maximization Method for Estimation Parameters of Autoregressive Models with Asymmetric Innovations. Springer AISC, vol 1427. doi:10.1007/978-3-031-03502-9_37

Examples

data(djia2002)
# AR(1) with PMM2
changes <- na.omit(djia2002$change)
pmm_skewness(changes)  # positive skewness -> PMM2
fit <- ar_pmm2(changes, order = 1)
summary(fit)

EstemPMM-style PMM2 estimator for MA models

Description

EstemPMM-style PMM2 estimator for MA models

Usage

estpmm_style_ma(x, q = 1, include.mean = TRUE, max_iter = 50, verbose = FALSE)

Arguments

Value

List with ma_coef, mean, innovations, convergence, method

EstemPMM-style PMM2 estimator for combined MA+SMA models

Description

EstemPMM-style PMM2 estimator for combined MA+SMA models

Usage

estpmm_style_ma_sma(
  x,
  q = 1,
  Q = 1,
  s = 12,
  include.mean = TRUE,
  max_iter = 50,
  verbose = FALSE,
  multiplicative = TRUE
)

Arguments

Value

List with ma_coef, sma_coef, mean, innovations, convergence, method

EstemPMM-style PMM2 estimator for SMA models

Description

EstemPMM-style PMM2 estimator for SMA models

Usage

estpmm_style_sma(
  x,
  Q = 1,
  s = 4,
  include.mean = TRUE,
  max_iter = 50,
  verbose = FALSE
)

Arguments

Value

List with sma_coef, mean, innovations, convergence, method

Extract fitted values from PMM2fit object

Description

Extract fitted values from PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
fitted(object, data = NULL, ...)

Arguments

Value

Vector of fitted values

Extract fitted values from PMM3fit object

Description

Extract fitted values from PMM3fit object

Usage

## S4 method for signature 'PMM3fit'
fitted(object, ...)

Arguments

Value

Vector of fitted values

Extract fitted values from TS2fit object

Description

Extract fitted values from TS2fit object

Usage

## S4 method for signature 'TS2fit'
fitted(object, ...)

Arguments

Value

Vector of fitted values

Extract fitted values from TS3fit object

Description

Extract fitted values from TS3fit object

Usage

## S4 method for signature 'TS3fit'
fitted(object, ...)

Arguments

Value

Vector of fitted values

Helper function for extracting fitted values

Description

Helper function for extracting fitted values

Usage

fitted_values(object, data = NULL)

Arguments

Value

Vector of fitted values

Get fitted values for AR model

Description

Get fitted values for AR model

Usage

get_ar_fitted(object)

Arguments

Value

Vector of fitted values

Get classical (MLE/CSS) estimates for initialization

Description

Get classical (MLE/CSS) estimates for initialization

Usage

get_classical_estimates(x, order, model_type, seasonal, include.mean)

Arguments

Value

Vector of initial parameter estimates

Get initial parameter estimates for time series models

Description

Get initial parameter estimates for time series models

Usage

get_initial_estimates(
  model_params,
  initial = NULL,
  method = "pmm2",
  verbose = FALSE
)

Arguments

Value

List containing:

Calculate SARIMAX Jacobian (Numerical)

Description

Calculate SARIMAX Jacobian (Numerical)

Usage

get_sarimax_jacobian(theta, ...)

Arguments

Value

Jacobian matrix (n x p)

Calculate SARIMAX Residuals

Description

Calculate SARIMAX Residuals

Usage

get_sarimax_residuals(
  theta,
  y,
  xreg = NULL,
  order = c(0, 0, 0),
  seasonal = list(order = c(0, 0, 0), period = NA),
  include.mean = TRUE
)

Arguments

Value

Vector of residuals

Get Yule-Walker estimates for AR(p)

Description

Get Yule-Walker estimates for AR(p)

Usage

get_yw_estimates(x, p)

Arguments

Value

Numeric vector of length p (AR coefficients)

Wrapper for linearized PMM2 approach (MA/SMA models)

Description

Wrapper for linearized PMM2 approach (MA/SMA models)

Usage

linearized_pmm2_wrapper(
  x,
  order,
  model_type,
  seasonal,
  include.mean,
  max_iter,
  tol,
  verbose
)

Arguments

Value

PMM2 estimation results

PMM2: Main function for PMM2 (S=2)

Description

Fits a linear model using the Polynomial Maximization Method (order 2), which is robust to non-Gaussian errors.

Usage

lm_pmm2(
  formula,
  data,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  na.action = na.fail,
  weights = NULL,
  verbose = FALSE
)

Arguments

Details

The PMM2 algorithm works as follows:

1. Fits ordinary least squares (OLS) regression to obtain initial estimates

2. Computes central moments (m2, m3, m4) from OLS residuals

3. Iteratively improves parameter estimates using a gradient-based approach

PMM2 is especially useful when error terms are not Gaussian.

Value

S4 object of class PMM2fit

Examples

set.seed(123)
n <- 80
x <- rnorm(n)
y <- 2 + 3 * x + rt(n, df = 3)
dat <- data.frame(y = y, x = x)

fit <- lm_pmm2(y ~ x, data = dat)
summary(fit, formula = y ~ x, data = dat)

PMM3: Fit linear model using Polynomial Maximization Method (S=3)

Description

Fits a linear model using PMM3, which is designed for symmetric platykurtic error distributions. Uses a cubic stochastic polynomial with Newton-Raphson solver.

Usage

lm_pmm3(
  formula,
  data,
  max_iter = 100,
  tol = 1e-06,
  adaptive = FALSE,
  step_max = 5,
  na.action = na.fail,
  verbose = FALSE
)

Arguments

Details

The PMM3 algorithm works as follows:

1. Fits OLS regression to obtain initial estimates

2. Computes central moments (m2, m4, m6) from OLS residuals

3. Checks symmetry: warns if |gamma3| > 0.3 (PMM2 may be more appropriate)

4. Computes moment ratio kappa = (m6 - 3m4m2) / (m4 - 3*m2^2)

5. Iterates Newton-Raphson with score Z = X'(eps*(kappa - eps^2))

PMM3 achieves lower variance than OLS when errors are symmetric and platykurtic (gamma4 < 0), with theoretical efficiency g3 = 1 - gamma4^2 / (6 + 9*gamma4 + gamma6).

Value

S4 object of class PMM3fit

Examples

set.seed(123)
n <- 100
x <- rnorm(n)
y <- 2 + 3 * x + runif(n, -sqrt(3), sqrt(3))
dat <- data.frame(y = y, x = x)

fit <- lm_pmm3(y ~ x, data = dat)
summary(fit)

Build design matrix for MA

Description

Build design matrix for MA

Usage

ma_build_design(intercept, residuals, x, q)

Arguments

Value

List with X (design matrix) and y (response)

Compute MA innovations (forward recursion)

Description

Compute MA innovations (forward recursion)

Usage

ma_compute_innovations(x, theta, q)

Arguments

Value

Vector of innovations

Fit an MA model using PMM2 (wrapper)

Description

Estimates moving-average model parameters using the Pearson Moment Method (PMM2). For MA models, PMM2 can achieve substantial improvements over MLE, particularly when innovations are non-Gaussian. Monte Carlo experiments showed up to 23\ reduction for MA(1) models.

Usage

ma_pmm2(
  x,
  order = 1,
  method = "pmm2",
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

PMM2 Variants:

• "unified_global" (default): One-step correction from MLE/CSS estimates. Fast and reliable. Typical improvement: 15-23\

• "unified_iterative": Full Newton-Raphson optimization. Slower but can achieve better accuracy for complex MA models.

• "linearized": Uses first-order Taylor expansion around MLE. Recommended for MA/SMA models as it avoids complex Jacobian computation while maintaining accuracy. Fastest option for MA models.

Variant Selection Guidelines:

• For MA(q) models: Use "linearized" (fastest, MA-optimized)

• If you need maximum accuracy: Try "unified_global" (best MSE)

• For exploration: Compare all three variants

Computational Characteristics:

• linearized: ~1.5x slower than MLE (recommended)

• unified_global: ~2x slower than MLE (high accuracy)

• unified_iterative: 5-10x slower than MLE (iterative)

Why MA Models Benefit Most: MA parameter estimation from MLE has known numerical difficulties due to non-identifiability and flat likelihood regions. PMM2 uses moment constraints that better resolve these issues, leading to larger improvements than for AR models.

Value

An S4 object of class MAPMM2 containing moving-average coefficients, residual innovations, central moments, model order, intercept, original series, and convergence diagnostics.

References

Monte Carlo validation (R=50, n=200): Unified Global showed 23.0\ improvement over MLE for MA(1) models - the largest improvement among all model types. See NEWS.md (Version 0.2.0) for full comparison.

See Also

ar_pmm2, arma_pmm2, sma_pmm2

Examples

# Fit MA(1) model with linearized variant (recommended)
x <- arima.sim(n = 200, list(ma = 0.6))
fit1 <- ma_pmm2(x, order = 1, pmm2_variant = "linearized")
coef(fit1)

# Compare with unified_global (best accuracy)
fit2 <- ma_pmm2(x, order = 1, pmm2_variant = "unified_global")

# Higher-order MA
x2 <- arima.sim(n = 300, list(ma = c(0.7, -0.4, 0.2)))
fit3 <- ma_pmm2(x2, order = 3, pmm2_variant = "linearized")

  

Fit an MA model using PMM3

Description

Estimates moving-average model parameters using PMM3.

Usage

ma_pmm3(
  x,
  order = 1,
  max_iter = 100,
  tol = 1e-06,
  adaptive = FALSE,
  step_max = 5,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  verbose = FALSE
)

Arguments

Value

An S4 object of class MAPMM3

See Also

ma_pmm2, ar_pmm3, lm_pmm3

Examples

set.seed(42)
x <- arima.sim(n = 200, list(ma = 0.6),
               innov = runif(200, -sqrt(3), sqrt(3)))
fit <- ma_pmm3(x, order = 1)
coef(fit)

Build design matrix for combined MA+SMA model

Description

Build design matrix for combined MA+SMA model

Usage

ma_sma_build_design(intercept, residuals, x, q, Q, s, multiplicative = TRUE)

Arguments

Value

List with X (design matrix) and y (response)

Compute combined MA+SMA innovations (forward recursion)

Description

Compute combined MA+SMA innovations (forward recursion)

Usage

ma_sma_compute_innovations(
  x,
  theta_ma,
  theta_sma,
  q,
  Q,
  s,
  multiplicative = TRUE
)

Arguments

Value

Vector of innovations

PMM2 solver (EstemPMM formula)

Description

PMM2 solver (EstemPMM formula)

Usage

ma_solve_pmm2(
  b_init,
  X,
  Y,
  m2,
  m3,
  m4,
  max_iter = 50,
  tol = 1e-06,
  verbose = FALSE
)

Arguments

Value

List with coefficients, convergence, iterations

Optimized Direct Nonlinear PMM2 (DEPRECATED)

Description

DEPRECATED: This function has been deprecated and should not be used.

Reason for Deprecation: Monte Carlo experiments (R=50, n=200) showed this direct optimization approach performs catastrophically worse than classical methods:

• AR(1): 17.0x worse MSE than MLE (3.67 vs 0.22)

• MA(1): Failed to converge in most replications

• SARIMA: Severe numerical instability

Recommended Alternatives: Use the Unified PMM2 framework instead, which showed 3-23\

• ar_pmm2 with pmm2_variant = "unified_global" (default)

• arima_pmm2 with pmm2_variant = "unified_iterative"

• ma_pmm2 with pmm2_variant = "linearized"

Usage

optimized_direct_pmm2(
  theta_init,
  y,
  order = c(0, 0, 0),
  seasonal = list(order = c(0, 0, 0), period = NA)
)

Arguments

Details

This function attempted to directly maximize the PMM2 objective function:

J(\theta) = \gamma_3^2 / (2 + \gamma_4)

However, this objective surface is highly non-convex with many local optima and flat regions, making direct optimization unreliable. The Unified PMM2 approach (global/iterative corrections from classical estimates) proved far more stable and accurate.

Technical Issues:

• Objective function has flat regions near optimal values

• Gradient becomes numerically unstable when \gamma_4 \approx -2

• Sensitive to initialization (even from MLE estimates)

• No guarantee of convergence to global optimum

Value

This function will throw an error directing users to unified methods

Optimization result

References

Monte Carlo comparison results documented in NEWS.md (Version 0.2.0)

See Also

ar_pmm2, ma_pmm2, arima_pmm2

Optimized Direct Nonlinear PMM2

Uses analytical gradients to maximize the PMM2 objective function. J(theta) = gamma3^2 / (2 + gamma4)

Plot diagnostic plots for PMM2fit object

Description

Plot diagnostic plots for PMM2fit object

Usage

## S4 method for signature 'PMM2fit,missing'
plot(x, y, which = 1:4, ...)

Arguments

Value

Invisibly returns the input object

Plot diagnostic plots for PMM3fit object

Description

Plot diagnostic plots for PMM3fit object

Usage

## S4 method for signature 'PMM3fit,missing'
plot(x, y, which = 1:4, ...)

Arguments

Value

Invisibly returns the input object

Build diagnostic plots for TS2fit objects

Description

Build diagnostic plots for TS2fit objects

Usage

## S4 method for signature 'TS2fit,missing'
plot(x, y, which = c(1:4), ...)

Arguments

Value

Invisibly returns x

Plot diagnostic plots for TS3fit object

Description

Plot diagnostic plots for TS3fit object

Usage

## S4 method for signature 'TS3fit,missing'
plot(x, y, which = 1:4, ...)

Arguments

Value

Invisibly returns the input object

Plot bootstrap distributions for PMM2 fit

Description

Plot bootstrap distributions for PMM2 fit

Usage

plot_pmm2_bootstrap(object, coefficients = NULL)

Arguments

Value

Invisibly returns histogram information

Universal PMM2 algorithm for all model types

Description

Universal PMM2 algorithm for all model types

Usage

pmm2_algorithm(
  b_init,
  X,
  y,
  m2,
  m3,
  m4,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE,
  poly_terms = NULL
)

Arguments

Value

List with estimation results

Bootstrap inference for PMM2 fit

Description

Bootstrap inference for PMM2 fit

Usage

pmm2_inference(
  object,
  formula,
  data,
  B = 200,
  seed = NULL,
  parallel = FALSE,
  cores = NULL
)

Arguments

Value

data.frame with columns: Estimate, Std.Error, t.value, p.value

Monte Carlo comparison of PMM2 estimation methods

Description

Function generates time series for given models, repeatedly estimates parameters using different methods and compares their accuracy by MSE criterion. Additionally outputs theoretical and empirical characteristics of the innovation distribution (skewness, excess kurtosis, theoretical gain of PMM2).

Usage

pmm2_monte_carlo_compare(
  model_specs,
  methods = c("css", "pmm2"),
  n,
  n_sim,
  innovations = list(type = "gaussian"),
  seed = NULL,
  include.mean = TRUE,
  progress = interactive(),
  verbose = FALSE
)

Arguments

Value

List with three components:

parameter_results

MSE and relative MSE for each parameter

summary

Averaged MSE over parameters for each model/method

gain

Comparison of theoretical and empirical PMM2 gain

Universal PMM2 estimator (Iterative)

Description

Universal PMM2 estimator (Iterative)

Usage

pmm2_nonlinear_iterative(
  theta_init,
  fn_residuals,
  fn_jacobian = NULL,
  max_iter = 100,
  tol = 1e-06,
  verbose = FALSE
)

Arguments

Value

List with results (theta, residuals, convergence, etc.)

Universal PMM2 estimator (One-step Global)

Description

Applies a single PMM2 correction to classical estimation results.

Usage

pmm2_nonlinear_onestep(
  theta_classical,
  fn_residuals,
  fn_jacobian = NULL,
  verbose = FALSE
)

Arguments

Value

List with results (theta, residuals, etc.)

Calculate theoretical skewness, kurtosis coefficients and variance reduction factor

Description

Calculate theoretical skewness, kurtosis coefficients and variance reduction factor

Usage

pmm2_variance_factor(m2, m3, m4)

Arguments

Value

List with fields c3, c4 and g

Calculate theoretical variance matrices for OLS and PMM2

Description

Calculate theoretical variance matrices for OLS and PMM2

Usage

pmm2_variance_matrices(X, m2, m3, m4)

Arguments

Value

List with fields ols, pmm2, c3, c4, g

Calculate PMM3 theoretical variance reduction factor

Description

Computes the standardised cumulant coefficients gamma4 and gamma6 from the raw central moments, and derives the PMM3 efficiency factor g_3 = 1 - \gamma_4^2 / (6 + 9\gamma_4 + \gamma_6).

Usage

pmm3_variance_factor(m2, m4, m6)

Arguments

Value

A list with components:

Automatic PMM method selection

Description

Analyses OLS residual cumulants to recommend the best estimation method: OLS (Gaussian errors), PMM2 (asymmetric errors), or PMM3 (symmetric platykurtic errors).

Usage

pmm_dispatch(
  residuals,
  symmetry_threshold = 0.3,
  kurtosis_threshold = -0.7,
  g2_threshold = 0.95,
  verbose = TRUE
)

Arguments

Value

A list with components:

Examples

set.seed(42)
x <- rnorm(200); eps <- runif(200, -1, 1)
y <- 1 + 2 * x + eps
fit_ols <- lm(y ~ x)
pmm_dispatch(residuals(fit_ols))

Compute sixth-order cumulant coefficient gamma6

Description

Calculates \gamma_6 = m_6/m_2^3 - 15 m_4/m_2^2 + 30 from a numeric vector. For a Gaussian distribution gamma6 equals zero.

Usage

pmm_gamma6(x)

Arguments

Value

Numeric scalar: the sixth-order cumulant coefficient

Calculate kurtosis from data

Description

Calculate kurtosis from data

Usage

pmm_kurtosis(x, excess = TRUE)

Arguments

Value

Kurtosis value

Calculate skewness from data

Description

Calculate skewness from data

Usage

pmm_skewness(x)

Arguments

Value

Skewness value

Prediction method for PMM2fit objects

Description

Computes predictions for new data using a fitted PMM2 model. The method extracts the formula from the fitted model, constructs a design matrix from the new data, and computes predictions via matrix multiplication. This approach works with arbitrary variable names and model specifications.

Usage

## S4 method for signature 'PMM2fit'
predict(object, newdata = NULL, debug = FALSE, ...)

Arguments

Details

The prediction is computed as X %*% beta where X is the design matrix constructed from newdata using the model formula, and beta are the estimated coefficients. The method automatically handles:

• Models with intercepts and without

• Arbitrary variable names (not limited to "x1", "x2", etc.)

• Interaction terms and transformations specified in the formula

• Automatic coefficient reordering to match design matrix columns

Value

Numeric vector of predicted values for the observations in newdata

Examples

# Fit model
fit <- lm_pmm2(mpg ~ wt + hp, data = mtcars)

# Predict on new data
newdata <- data.frame(wt = c(2.5, 3.0), hp = c(100, 150))
predictions <- predict(fit, newdata = newdata)

Predict method for PMM3fit objects

Description

Predict method for PMM3fit objects

Usage

## S4 method for signature 'PMM3fit'
predict(object, newdata = NULL, ...)

Arguments

Value

Numeric vector of predicted values

Prediction method for TS2fit objects

Description

Prediction method for TS2fit objects

Usage

## S4 method for signature 'TS2fit'
predict(object, n.ahead = 1, ...)

Arguments

Value

Vector or list of predictions, depending on model type

Predict method for TS3fit objects

Description

Predict method for TS3fit objects

Usage

## S4 method for signature 'TS3fit'
predict(object, n.ahead = 1, ...)

Arguments

Value

Numeric vector of predicted values

Extract residuals from PMM2fit object

Description

Extract residuals from PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
residuals(object, ...)

Arguments

Value

Vector of residuals

Extract residuals from PMM3fit object

Description

Extract residuals from PMM3fit object

Usage

## S4 method for signature 'PMM3fit'
residuals(object, ...)

Arguments

Value

Vector of residuals

Extract residuals from TS2fit object

Description

Extract residuals from TS2fit object

Usage

## S4 method for signature 'TS2fit'
residuals(object, ...)

Arguments

Value

Vector of residuals (innovations)

Extract residuals from TS3fit object

Description

Extract residuals from TS3fit object

Usage

## S4 method for signature 'TS3fit'
residuals(object, ...)

Arguments

Value

Vector of residuals

Fit Seasonal AR model using PMM2 method

Description

Fits a Seasonal Autoregressive (SAR) model using the Polynomial Maximization Method (PMM2). The model can include both non-seasonal and seasonal AR components.

Usage

sar_pmm2(
  x,
  order = c(0, 1),
  season = list(period = 12),
  method = "pmm2",
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  include.mean = TRUE,
  multiplicative = FALSE,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

The SAR model has the form

y_t = \phi_1 y_{t-1} + \dots + \phi_p y_{t-p} + \Phi_1 y_{t-s} + \dots + \Phi_P y_{t-Ps} + \mu + \epsilon_t.

Where:

• p is the non-seasonal AR order

• P is the seasonal AR order

• s is the seasonal period

• epsilon_t are innovations (errors)

The PMM2 method provides more efficient parameter estimates than OLS when the innovation distribution is asymmetric (non-Gaussian). The expected variance reduction is given by g = 1 - c3^2 / (2 + c4), where c3 and c4 are the skewness and excess kurtosis coefficients.

Variant Recommendations for SAR:

• "unified_global" (default): Fast one-step correction, suitable for most SAR models

• "unified_iterative": Best accuracy for complex seasonal patterns

• "linearized": Not recommended for SAR (designed for MA/SMA)

Value

S4 object of class SARPMM2 containing:

• coefficients: Estimated AR and SAR parameters

• residuals: Model residuals/innovations (padded with zeros to match original length)

• m2, m3, m4: Central moments of residuals

• convergence: Convergence status

• iterations: Number of iterations performed

See Also

ar_pmm2, ts_pmm2, compare_sar_methods

Examples

# Generate synthetic seasonal data
n <- 120
y <- arima.sim(n = n, list(ar = 0.7, seasonal = list(sar = 0.5, period = 12)))

# Fit SAR(1,1)_12 model with PMM2
fit <- sar_pmm2(y, order = c(1, 1), season = list(period = 12))
summary(fit)

# Simple seasonal model (no non-seasonal component)
fit_pure_sar <- sar_pmm2(y, order = c(0, 1), season = list(period = 12))

# Compare with OLS
fit_ols <- sar_pmm2(y, order = c(1, 1), season = list(period = 12), method = "ols")

Fit a Seasonal ARIMA model using PMM2 method

Description

Fits a Seasonal Autoregressive Integrated Moving Average (SARIMA) model with both non-seasonal and seasonal differencing operators.

Usage

sarima_pmm2(
  x,
  order = c(0, 1, 0, 1),
  seasonal = list(order = c(1, 1), period = 12),
  method = "pmm2",
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  include.mean = NULL,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  ma_method = c("mle", "pmm2"),
  verbose = FALSE,
  multiplicative = TRUE
)

Arguments

Details

The SARIMA(p,d,q) x (P,D,Q)_s model satisfies

(1 - \phi_1 B - \dots - \phi_p B^p)(1 - \Phi_1 B^s - \dots - \Phi_P B^{Ps})(1 - B)^d (1 - B^s)^D y_t = (1 + \theta_1 B + \dots + \theta_q B^q)(1 + \Theta_1 B^s + \dots + \Theta_Q B^{Qs}) \epsilon_t.

Where B is the backshift operator. The model combines:

• Non-seasonal differencing: (1-B)^d

• Seasonal differencing: (1-B^s)^D

• Non-seasonal ARMA components

• Seasonal ARMA components

Variant Recommendations for SARIMA:

• "unified_iterative" (recommended): Monte Carlo experiments showed 16.4\ seasonal dynamics

• "unified_global" (default): Faster alternative with good accuracy

• "linearized": Not recommended for SARIMA (designed for MA/SMA)

Value

S4 object of class SARIMAPMM2 containing:

• coefficients: Estimated parameters

• residuals: Model residuals/innovations (padded with zeros to match original length)

• m2, m3, m4: Central moments of residuals

• convergence: Convergence status

• iterations: Number of iterations performed

References

Monte Carlo validation (R=50, n=200): Unified Iterative achieved 16.4\ improvement for SARIMA models. See NEWS.md (Version 0.2.0).

See Also

sarma_pmm2, arima_pmm2

Examples

set.seed(123)
n <- 200
y <- arima.sim(n = n,
  model = list(order = c(1, 0, 1), ar = 0.5, ma = 0.3,
    seasonal = list(order = c(1, 0, 0), ar = 0.4, period = 12)))
fit <- sarima_pmm2(y,
  order = c(1, 0, 1, 0),
  seasonal = list(order = c(1, 0), period = 12))
summary(fit)

Fit a Seasonal ARMA model using PMM2 method

Description

Fits a Seasonal Autoregressive Moving Average (SARMA) model that combines both seasonal AR and seasonal MA components, optionally with non-seasonal AR and MA terms as well.

Usage

sarma_pmm2(
  x,
  order = c(0, 1, 0, 1),
  season = list(period = 12),
  method = "pmm2",
  include.mean = TRUE,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

The SARMA model combines four components:

• AR(p): \phi_1 y_{t-1} + \dots + \phi_p y_{t-p}

• Seasonal AR: \Phi_1 y_{t-s} + \dots + \Phi_P y_{t-Ps}

• MA(q): \theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q}

• Seasonal MA: \Theta_1 \epsilon_{t-s} + \dots + \Theta_Q \epsilon_{t-Qs}

The PMM2 method provides more efficient parameter estimates than ML/CSS when the innovation distribution is asymmetric (non-Gaussian).

Value

S4 object of class SARMAPMM2 containing:

• coefficients: Estimated parameters

• residuals: Model residuals/innovations (padded with zeros to match original length)

• m2, m3, m4: Central moments of residuals

• convergence: Convergence status

• iterations: Number of iterations performed

See Also

sar_pmm2, sma_pmm2, sarima_pmm2

Examples

# Generate synthetic seasonal data with SARMA structure
set.seed(123)
n <- 200
y <- arima.sim(n = n, list(
  ar = 0.5, ma = 0.3,
  seasonal = list(sar = 0.6, sma = 0.4, period = 12)
))

# Fit SARMA(1,1,1,1)_12 model with PMM2
fit <- sarma_pmm2(y, order = c(1, 1, 1, 1), season = list(period = 12))
summary(fit)

# Pure seasonal model (no non-seasonal components)
fit_pure <- sarma_pmm2(y, order = c(0, 1, 0, 1), season = list(period = 12))

Build design matrix for SMA

Description

Build design matrix for SMA

Usage

sma_build_design(intercept, residuals, x, Q, s)

Arguments

Value

List with X (design matrix) and y (response)

Compute SMA innovations (forward recursion)

Description

Compute SMA innovations (forward recursion)

Usage

sma_compute_innovations(x, Theta, Q, s)

Arguments

Value

Vector of innovations

CSS fit for Seasonal MA model

Description

CSS fit for Seasonal MA model

Usage

sma_css_fit(x, Q, s, include_mean = TRUE, verbose = FALSE)

Arguments

Value

List with coefficients, intercept, residuals, convergence info

Fit a Seasonal MA model using PMM2

Description

Fits a Seasonal Moving Average (SMA) model using the Polynomial Maximization Method (PMM2). This is particularly effective when the innovations have asymmetric or non-Gaussian distributions.

Usage

sma_pmm2(
  x,
  order = 1,
  season = list(period = 12),
  method = "pmm2",
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

The SMA(Q)_s model has the form

y_t = \mu + \epsilon_t + \Theta_1 \epsilon_{t-s} + \Theta_2 \epsilon_{t-2s} + \dots + \Theta_Q \epsilon_{t-Qs}.

Where:

• Q is the seasonal MA order

• s is the seasonal period (12 for monthly, 4 for quarterly)

• epsilon_t are innovations (errors)

The PMM2 method provides more efficient parameter estimates than ML/CSS when the innovation distribution is asymmetric (non-Gaussian). The expected variance reduction is given by g = 1 - c3^2 / (2 + c4), where c3 and c4 are the skewness and excess kurtosis coefficients.

Variant Recommendations for SMA:

• "linearized" (recommended): Fastest and most stable for SMA models, avoids complex Jacobian computation while maintaining accuracy

• "unified_global" (default): Good balance of speed and accuracy

• "unified_iterative": Best accuracy but slower, use for complex SMA patterns

Value

An S4 object of class SMAPMM2 containing:

• coefficients: Seasonal MA coefficients (Theta_1, Theta_2, ..., Theta_Q)

• innovations: Estimated innovations (residuals)

• m2, m3, m4: Central moments of innovations

• convergence: Convergence status

• iterations: Number of iterations performed

• intercept: Model intercept

• original_series: Original time series

• order: Model order list(Q, s)

See Also

ma_pmm2, sar_pmm2, arima_pmm2

Examples

# Generate synthetic seasonal data
set.seed(123)
n <- 120
s <- 12
theta <- 0.6

# Gamma innovations (asymmetric)
innov <- rgamma(n, shape = 2, scale = 1) - 2
y <- numeric(n)
for (t in 1:n) {
  ma_term <- if (t > s) theta * innov[t - s] else 0
  y[t] <- innov[t] + ma_term
}

# Fit SMA(1)_12 model with PMM2
fit <- sma_pmm2(y, order = 1, season = list(period = 12))
summary(fit)

# Compare with CSS
fit_css <- sma_pmm2(y, order = 1, season = list(period = 12), method = "css")

Fit Seasonal MA model using PMM2

Description

Fit Seasonal MA model using PMM2

Usage

sma_pmm2_fit(x, Q, s, css_fit, max_iter = 50, tol = 1e-06, verbose = FALSE)

Arguments

Value

List with coefficients, intercept, innovations, convergence info

Universal solver for PMM2 system of equations

Description

Universal solver for PMM2 system of equations

Usage

solve_pmm2(
  b_init,
  X,
  y,
  m2,
  m3,
  m4,
  max_iter = 1000,
  tol = 1e-05,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Value

Vector of PMM2 parameter estimates

PMM2 step solver

Description

Solves the linearized system to find the parameter update. Based on the Taylor expansion: e(theta) ~ e(theta_k) - J * delta

Usage

solve_pmm2_step(residuals, J, pmm_stats)

Arguments

Value

Update vector delta

Generic summary method for PMM2fit objects

Description

Generic summary method for PMM2fit objects

Usage

## S4 method for signature 'PMM2fit'
summary(object, formula = NULL, data = NULL, B = 100, ...)

Arguments

Value

Prints summary to console; returns object (invisibly).

Summary method for PMM3fit objects

Description

Summary method for PMM3fit objects

Usage

## S4 method for signature 'PMM3fit'
summary(object, ...)

Arguments

Value

Prints summary to console; returns object (invisibly)

Generic summary method for SARIMAPMM2 objects

Description

Generic summary method for SARIMAPMM2 objects

Usage

## S4 method for signature 'SARIMAPMM2'
summary(object, ...)

Arguments

Value

Prints summary to console; returns object (invisibly).

Generic summary method for SARMAPMM2 objects

Description

Generic summary method for SARMAPMM2 objects

Usage

## S4 method for signature 'SARMAPMM2'
summary(object, ...)

Arguments

Value

Prints summary to console; returns object (invisibly).

Summary method for SARPMM2 objects

Description

Summary method for SARPMM2 objects

Usage

## S4 method for signature 'SARPMM2'
summary(object, ...)

Arguments

Value

Invisibly returns the object

Summary method for SMAPMM2 objects

Description

Summary method for SMAPMM2 objects

Usage

## S4 method for signature 'SMAPMM2'
summary(object, ...)

Arguments

Value

Invisibly returns the object

Generic summary method for TS2fit objects

Description

Generic summary method for TS2fit objects

Usage

## S4 method for signature 'TS2fit'
summary(object, ...)

Arguments

Value

Prints summary to console; returns object (invisibly).

Summary method for TS3fit objects

Description

Summary method for TS3fit objects

Usage

## S4 method for signature 'TS3fit'
summary(object, ...)

Arguments

Value

Prints summary to console; returns object (invisibly)

Test whether residuals are sufficiently symmetric for PMM3

Description

Computes the skewness coefficient gamma3 and checks whether its absolute value falls below a given threshold. This helps decide between PMM2 (asymmetric) and PMM3 (symmetric platykurtic) estimation.

Usage

test_symmetry(x, threshold = 0.3)

Arguments

Value

A list with components:

Fit a time series model using the PMM2 method

Description

Fit a time series model using the PMM2 method

Usage

ts_pmm2(
  x,
  order,
  model_type = c("ar", "ma", "arma", "arima"),
  method = "pmm2",
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

Details

The PMM2 algorithm works as follows:

1. Fits an initial model using a standard method (OLS, Yule-Walker, CSS or ML)

2. Computes central moments (m2, m3, m4) from initial residuals/innovations

3. Uses these moments with a specialized solver (pmm2_algorithm) to find robust parameter estimates

Value

An S4 object TS2fit of the corresponding subclass

Bootstrap inference for PMM2 time series models

Description

Bootstrap inference for PMM2 time series models

Usage

ts_pmm2_inference(
  object,
  x = NULL,
  B = 200,
  seed = NULL,
  block_length = NULL,
  method = c("residual", "block"),
  parallel = FALSE,
  cores = NULL,
  debug = FALSE
)

Arguments

Value

data.frame with columns: Estimate, Std.Error, t.value, p.value

Fit a time series model using PMM3

Description

Core function that fits AR, MA, ARMA, or ARIMA models using the Polynomial Maximization Method of order 3 (PMM3). Designed for symmetric platykurtic innovations.

Usage

ts_pmm3(
  x,
  order,
  model_type = c("ar", "ma", "arma", "arima"),
  max_iter = 100,
  tol = 1e-06,
  adaptive = FALSE,
  step_max = 5,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  verbose = FALSE
)

Arguments

Details

The PMM3 time series algorithm:

1. Obtains initial estimates via classical methods (OLS/YW for AR, CSS for MA/ARMA/ARIMA)

2. Computes moments m2, m4, m6 and kappa from initial residuals

3. Checks symmetry: warns if |gamma3| > 0.3

4. Applies Newton-Raphson with PMM3 score equations

PMM3 is beneficial when innovations are symmetric and platykurtic (gamma4 < 0), e.g. uniform, truncated normal.

Value

An S4 object of the appropriate TS3fit subclass

Unified PMM2 Wrapper for Time Series Models

Description

Universal wrapper that dispatches to appropriate PMM2 variant based on model structure and user preference.

Usage

unified_pmm2_wrapper(
  x,
  order,
  model_type,
  pmm2_variant = c("unified_global", "unified_iterative", "linearized"),
  seasonal = NULL,
  include.mean = TRUE,
  max_iter = 100,
  tol = 1e-06,
  verbose = FALSE
)

Arguments

Value

List with PMM2 estimation results

Update MA model innovations

Description

Update MA model innovations

Usage

update_ma_innovations(x, ma_coef)

Arguments

Value

Vector of innovations

Validate and prepare time series parameters

Description

Validate and prepare time series parameters

Usage

validate_ts_parameters(x, order, model_type, include.mean)

Arguments

Value

List of validated parameters and model information