gmwm {simts}

Generalized Method of Wavelet Moments (GMWM)

Description

Performs estimation of time series models by using the GMWM estimator.

Usage

gmwm(
  model,
  data,
  model.type = "imu",
  compute.v = "auto",
  robust = FALSE,
  eff = 0.6,
  alpha = 0.05,
  seed = 1337,
  G = NULL,
  K = 1,
  H = 100,
  freq = 1
)

Arguments

model

A ts.model object containing one of the allowed models.

data

A matrix or data.frame object with only column (e.g. N \times 1), a lts object, or a gts object.

model.type

A string containing the type of GMWM needed: "imu" or "ssm".

compute.v

A string indicating the type of covariance matrix solver. Valid values are: "fast", "bootstrap", "diag" (asymptotic diag), "full" (asymptotic full). By default, the program will fit a "fast" model.

robust

A boolean indicating whether to use the robust computation (TRUE) or not (FALSE).

eff

A double between 0 and 1 that indicates the efficiency.

alpha

A double between 0 and 1 that correspondings to the \frac{\alpha}{2} value for the wavelet confidence intervals.

seed

An integer that controls the reproducibility of the auto model selection phase.

G

An integer to sample the space for IMU and SSM models to ensure optimal identitability.

K

An integer that controls how many times the bootstrapping procedure will be initiated.

H

An integer that indicates how many different samples the bootstrap will be collect.

freq

A double that indicates the sampling frequency. By default, this is set to 1 and only is important if GM() is in the model

Details

This function is under work. Some of the features are active. Others... Not so much.

The V matrix is calculated by: diag\left[ {{{\left( {Hi - Lo} \right)}^2}} \right].

The function is implemented in the following manner: 1. Calculate MODWT of data with levels = floor(log2(data)) 2. Apply the brick.wall of the MODWT (e.g. remove boundary values) 3. Compute the empirical wavelet variance (WV Empirical). 4. Obtain the V matrix by squaring the difference of the WV Empirical's Chi-squared confidence interval (hi - lo)^2 5. Optimize the values to obtain \hat{\theta} 6. If FAST = TRUE, return these results. Else, continue.

Loop k = 1 to K Loop h = 1 to H 7. Simulate xt under F_{\hat{\theta}} 8. Compute WV Empirical END 9. Calculate the covariance matrix 10. Optimize the values to obtain \hat{\theta} END 11. Return optimized values.

The function estimates a variety of time series models. If type = "imu" or "ssm", then parameter vector should indicate the characters of the models that compose the latent or state-space model. The model options are:

• "AR1": a first order autoregressive process with parameters (\phi,\sigma^2)

• "GM": a guass-markov process (\beta,\sigma_{gm}^2)

• "ARMA": an autoregressive moving average process with parameters (\phi _p, \theta _q, \sigma^2)

• "DR": a drift with parameter \omega

• "QN": a quantization noise process with parameter Q

• "RW": a random walk process with parameter \sigma^2

• "WN": a white noise process with parameter \sigma^2

If only an ARMA() term is supplied, then the function takes conditional least squares as starting values If robust = TRUE the function takes the robust estimate of the wavelet variance to be used in the GMWM estimation procedure.

Value

A gmwm object with the structure:

• estimate: Estimated Parameters Values from the GMWM Procedure

• init.guess: Initial Starting Values given to the Optimization Algorithm

• wv.empir: The data's empirical wavelet variance

• ci_low: Lower Confidence Interval

• ci_high: Upper Confidence Interval

• orgV: Original V matrix

• V: Updated V matrix (if bootstrapped)

• omega: The V matrix inversed

• obj.fun: Value of the objective function at Estimated Parameter Values

• theo: Summed Theoretical Wavelet Variance

• decomp.theo: Decomposed Theoretical Wavelet Variance by Process

• scales: Scales of the GMWM Object

• robust: Indicates if parameter estimation was done under robust or classical

• eff: Level of efficiency of robust estimation

• model.type: Models being guessed

• compute.v: Type of V matrix computation

• augmented: Indicates moments have been augmented

• alpha: Alpha level used to generate confidence intervals

• expect.diff: Mean of the First Difference of the Signal

• N: Length of the Signal

• G: Number of Guesses Performed

• H: Number of Bootstrap replications

• K: Number of V matrix bootstraps

• model: ts.model supplied to gmwm

• model.hat: A new value of ts.model object supplied to gmwm

• starting: Indicates whether the procedure used the initial guessing approach

• seed: Randomization seed used to generate the guessing values

• freq: Frequency of data