gmwm: Generalized Method of Wavelet Moments (GMWM) for IMUs, ARMA,...
In SMAC-Group/gmwm: Generalized Method of Wavelet Moments

Description Usage Arguments Details Value Examples

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

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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)

`model`	A `ts.model` object containing one of the allowed models.
`data`	A `matrix` or `data.frame` object with only column (e.g. N x 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 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

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

The V matrix is calculated by: diag[(Hi-Lo)^2].

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 theta^hat 6. If FAST = TRUE, return these results. Else, continue.

Loop k = 1 to K Loop h = 1 to H 7. Simulate xt under F_theta^hat 8. Compute WV Empirical END 9. Calculate the covariance matrix 10. Optimize the values to obtain theta^hat 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.

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

# AR
set.seed(1336)
n = 200
data = gen_gts(n, AR1(phi = .99, sigma2 = 0.01) + WN(sigma2 = 1))

# Models can contain specific parameters e.g.
adv.model = gmwm(AR1(phi = .99, sigma2 = 0.01) + WN(sigma2 = 0.01),
                            data)

# Or we can guess the parameters:
guided.model = gmwm(AR1() + WN(), data) 

# Want to try different models? 
guided.ar1 = gmwm(AR1(), data)

# Faster:
guided.ar1.wn.prev = update(guided.ar1, AR1()+WN())

# OR 

# Create new GMWM object. 
# Note this is SLOWER since the Covariance Matrix is recalculated.
guided.ar1.wn.new = gmwm(AR1()+WN(), data)
 
# ARMA case
set.seed(1336)
data = gen_gts(n, ARMA(ar = c(0.8897, -0.4858), ma = c(-0.2279, 0.2488),
              sigma2 = 0.1796))
#guided.arma = gmwm(ARMA(2,2), data, model.type="ssm")
adv.arma = gmwm(ARMA(ar=c(0.8897, -0.4858), ma = c(-0.2279, 0.2488), sigma2=0.1796),
                data, model.type="ssm")