simts: Time Series Analysis Tools

Description

A system contains easy-to-use tools as a support for time series analysis courses. In particular, it incorporates a technique called Generalized Method of Wavelet Moments (GMWM) as well as its robust implementation for fast and robust parameter estimation of time series models which is described, for example, in Guerrier et al. (2013) doi: 10.1080/01621459.2013.799920. More details can also be found in the paper linked to via the URL below.

A system contains easy-to-use tools as a support for time series analysis courses. In particular, it incorporates a technique called Generalized Method of Wavelet Moments (GMWM) as well as its robust implementation for fast and robust parameter estimation of time series models which is described, for example, in Guerrier et al. (2013) doi: 10.1080/01621459.2013.799920. More details can also be found in the paper linked to via the URL below.

Details

The data sets in this package may change at a moments notice.

Author(s)

Maintainer: Stéphane Guerrier stef.guerrier@gmail.com [copyright holder]

Authors:

• James Balamuta [copyright holder]

• Roberto Molinari [copyright holder]

• Justin Lee

• Lionel Voirol

• Yuming Zhang

Other contributors:

• Wenchao Yang [contributor]

• Nathanael Claussen [contributor]

• Yunxiang Zhang [contributor]

• Christian Gunning [copyright holder]

• Romain Francois [copyright holder]

• Ross Ihaka [copyright holder]

• R Core Team [copyright holder]

James Balamuta balamut2@illinois.edu, Stephane Guerrier stef.guerrier@gmail.com, Roberto Molinari roberto.molinari@hotmail.it, Wenchao Yang wyang40@illinois.edu Justin Lee munsheet93@gmail.com Yuming Zhang yuming.zhang@unige.ch

See Also

Useful links:

• https://github.com/SMAC-Group/simts

• https://arxiv.org/pdf/1607.04543.pdf

• Report bugs at https://github.com/SMAC-Group/simts/issues

Useful links:

• https://github.com/SMAC-Group/simts

• https://arxiv.org/pdf/1607.04543.pdf

• Report bugs at https://github.com/SMAC-Group/simts/issues

Auto-Covariance and Correlation Functions

Description

The acf function computes the estimated autocovariance or autocorrelation for both univariate and multivariate cases.

Usage

.acf(x, lagmax = 0L, cor = TRUE, demean = TRUE)

Arguments

Subset an IMU Object

Description

Enables the IMU object to be subsettable. That is, you can load all the data in and then select certain properties.

Usage

## S3 method for class 'imu'
x[i, j, drop = FALSE]

Arguments

Details

When using the subset operator, note that all the Gyroscopes are placed at the front of object and, then, the Accelerometers are placed.

Value

An imu object class.

Examples

## Not run: 
if(!require("imudata")){
install_imudata()
library("imudata")
}

data(imu6)

# Create an IMU Object that is full. 
ex = imu(imu6, gyros = 1:3, accels = 4:6, axis = c('X', 'Y', 'Z', 'X', 'Y', 'Z'), freq = 100)

# Create an IMU object that has only gyros. 
ex.gyro = ex[,1:3]
ex.gyro2 = ex[,c("Gyro. X","Gyro. Y","Gyro. Z")]

# Create an IMU object that has only accels. 
ex.accel = ex[,4:6]
ex.accel2 = ex[,c("Accel. X","Accel. Y","Accel. Z")]

# Create an IMU object with both gyros and accels on axis X and Y
ex.b = ex[,c(1,2,4,5)]
ex.b2 = ex[,c("Gyro. X","Gyro. Y","Accel. X","Accel. Y")]


## End(Not run)

Multiple a ts.model by constant

Description

Sets up the necessary backend for creating multiple model objects.

Usage

## S3 method for class 'ts.model'
x * y

Arguments

Value

An S3 object with called ts.model with the following structure:

process.desc

y desc replicated x times

theta

y theta replicated x times

plength

Number of Parameters

desc

y desc replicated x times

obj.desc

Depth of Parameters e.g. list(c(1,1),c(1,1))

starting

Guess Starting values? TRUE or FALSE (e.g. specified value)

Author(s)

James Balamuta and Stephane Guerrier

Examples

4*DR()+2*WN()
DR()*4 + WN()*2
AR1(phi=.3,sigma=.2)*3

Add ts.model objects together

Description

Sets up the necessary backend for combining ts.model objects.

Usage

## S3 method for class 'ts.model'
x + y

Arguments

Value

An S3 object with called ts.model with the following structure:

• process.desccombined x, y desc

• thetacombined x, y theta

• plengthNumber of Parameters

• descAdd process to queue e.g. c("AR1","WN")

• obj.descDepth of Parameters e.g. list(1, c(1,1), c(length(ar),length(ma),1) )

• startingGuess Starting values? TRUE or FALSE (e.g. specified value)

Author(s)

James Balamuta

Examples

DR()+WN()
AR1(phi=.3,sigma=.2)

Helper Function for ARMA to WV Approximation

Description

Indicates where the minimum ARMAacf value is and returns that as an index.

Usage

acf_sum(ar, ma, last_tau, alpha = 0.99)

Arguments

Value

A vec containing the wavelet variance of the ARMA process.

Akaike's Information Criterion

Description

This function calculates AIC, BIC or HQ for a fitsimts object. This function currently only supports models estimated by the MLE.

Usage

## S3 method for class 'fitsimts'
AIC(object, k = 2, ...)

Arguments

Value

AIC, BIC or HQ

Author(s)

Stéphane Guerrier

Examples

set.seed(1)
n = 300
Xt = gen_gts(n, AR(phi = c(0, 0, 0.8), sigma2 = 1))
mod = estimate(AR(3), Xt)

# AIC
AIC(mod)

# BIC
AIC(mod, k = log(n))

# HQ
AIC(mod, k = 2*log(log(n)))

Bootstrap for Everything!

Description

Using the bootstrap approach, we simulate a model based on user supplied parameters, obtain the wavelet variance, and then V.

Usage

all_bootstrapper(
  theta,
  desc,
  objdesc,
  scales,
  model_type,
  N,
  robust,
  eff,
  alpha,
  H
)

Arguments

Details

Expand in detail...

Value

A vec that contains the parameter estimates from GMWM estimator.

Author(s)

JJB

Create an Autoregressive P [AR(P)] Process

Description

Sets up the necessary backend for the AR(P) process.

Usage

AR(phi = NULL, sigma2 = 1)

Arguments

Value

An S3 object with called ts.model with the following structure:

process.desc

Used in summary: "AR-1","AR-2", ..., "AR-P", "SIGMA2"

theta

\phi_1, \phi_2, ..., \phi_p, \sigma^2

plength

Number of Parameters

desc

"AR"

print

String containing simplified model

obj.desc

Depth of Parameters e.g. list(p,1)

starting

Guess starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

X_t = \sum_{j = 1}^p \phi_j X_{t-1} + \varepsilon_t

, where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2.

Author(s)

James Balamuta

Examples

AR(1) # Slower version of AR1()
AR(phi=.32, sigma=1.3) # Slower version of AR1()
AR(2) # Equivalent to ARMA(2,0).

Definition of an Autoregressive Process of Order 1

Description

Definition of an Autoregressive Process of Order 1

Usage

AR1(phi = NULL, sigma2 = 1)

Arguments

Value

An S3 object containing the specified ts.model with the following structure:

process.desc

Used in summary: "AR1","SIGMA2"

theta

Parameter vector including \phi, \sigma^2

plength

Number of parameters

print

String containing simplified model

desc

"AR1"

obj.desc

Depth of Parameters e.g. list(1,1)

starting

Find starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following AR(1) model:

X_t = \phi X_{t-1} + \varepsilon_t

, where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2.

Author(s)

James Balamuta

Examples

AR1()
AR1(phi=.32, sigma2 = 1.3)

Randomly guess starting parameters for AR1

Description

Sets starting parameters for each of the given parameters.

Usage

ar1_draw(draw_id, last_phi, sigma2_total, model_type)

Arguments

Value

A vec containing smart parameter starting guesses to be iterated over.

Transform AR1 to GM

Description

Takes AR1 values and transforms them to GM

Usage

ar1_to_gm(theta, freq)

Arguments

Details

The function takes a vector of AR1 values \phi and \sigma ^2 and transforms them to GM values \beta and \sigma ^2_{gm} using the formulas: \beta = - \frac{{\ln \left( \phi \right)}}{{\Delta t}} \sigma _{gm}^2 = \frac{{{\sigma ^2}}}{{1 - {\phi ^2}}}

Value

A vec containing GM values.

Author(s)

James Balamuta

AR(1) process to WV

Description

This function computes the Haar WV of an AR(1) process

Usage

ar1_to_wv(phi, sigma2, tau)

Arguments

Details

This function is significantly faster than its generalized counter part arma_to_wv.

Value

A vec containing the wavelet variance of the AR(1) process.

Process Haar Wavelet Variance Formula

The Autoregressive Order 1 (AR(1)) process has a Haar Wavelet Variance given by:

\frac{{2{\sigma ^2}\left( {4{\phi ^{\frac{{{\tau _j}}}{2} + 1}} - {\phi ^{{\tau _j} + 1}} - \frac{1}{2}{\phi ^2}{\tau _j} + \frac{{{\tau _j}}}{2} - 3\phi } \right)}}{{{{\left( {1 - \phi } \right)}^2}\left( {1 - {\phi ^2}} \right)\tau _j^2}}

Create an Autoregressive Integrated Moving Average (ARIMA) Process

Description

Sets up the necessary backend for the ARIMA process.

Usage

ARIMA(ar = 1, i = 0, ma = 1, sigma2 = 1)

Arguments

Details

A variance is required since the model generation statements utilize randomization functions expecting a variance instead of a standard deviation like R.

Value

An S3 object with called ts.model with the following structure:

process.desc

AR*p, MA*q

theta

\sigma

plength

Number of parameters

print

String containing simplified model

obj.desc

y desc replicated x times

obj

Depth of parameters e.g. list(c(length(ar),length(ma),1) )

starting

Guess starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

\Delta^i X_t = \sum_{j = 1}^p \phi_j \Delta^i X_{t-j} + \sum_{j = 1}^q \theta_j \varepsilon_{t-j} + \varepsilon_t

, where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2.

Author(s)

James Balamuta

Examples

# Create an ARMA(1,2) process
ARIMA(ar=1,2)
# Creates an ARMA(3,2) process with predefined coefficients.
ARIMA(ar=c(0.23,.43, .59), ma=c(0.4,.3))

# Creates an ARMA(3,2) process with predefined coefficients and standard deviation
ARIMA(ar=c(0.23,.43, .59), ma=c(0.4,.3), sigma2 = 1.5)

Create an Autoregressive Moving Average (ARMA) Process

Description

Sets up the necessary backend for the ARMA process.

Usage

ARMA(ar = 1, ma = 1, sigma2 = 1)

Arguments

Details

A variance is required since the model generation statements utilize randomization functions expecting a variance instead of a standard deviation like R.

Value

An S3 object with called ts.model with the following structure:

process.desc

AR*p, MA*q

theta

\sigma

plength

Number of Parameters

print

String containing simplified model

obj.desc

y desc replicated x times

obj

Depth of Parameters e.g. list(c(length(ar),length(ma),1) )

starting

Guess Starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

X_t = \sum_{j = 1}^p \phi_j X_{t-j} + \sum_{j = 1}^q \theta_j \varepsilon_{t-j} + \varepsilon_t

, where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2.

Author(s)

James Balamuta

Examples

# Create an ARMA(1,2) process
ARMA(ar=1,2)
# Creates an ARMA(3,2) process with predefined coefficients.
ARMA(ar=c(0.23,.43, .59), ma=c(0.4,.3))

# Creates an ARMA(3,2) process with predefined coefficients and standard deviation
ARMA(ar=c(0.23,.43, .59), ma=c(0.4,.3), sigma2 = 1.5)

ARMA Adapter to ARMA to WV Process function

Description

Molds the data so that it works with the arma_to_wv function.

Usage

arma_adapter(theta, p, q, tau)

Arguments

Details

The data conversion is more or less a rearrangement of values without using the obj desc.

Value

A vec containing the ARMA to WV results

Author(s)

James Joseph Balamuta (JJB)

Randomly guess starting parameters for ARMA

Description

Sets starting parameters for each of the given parameters.

Usage

arma_draws(p, q, sigma2_total)

Arguments

Value

A vec containing smart parameter starting guesses to be iterated over.

ARMA process to WV

Description

This function computes the Haar Wavelet Variance of an ARMA process

Usage

arma_to_wv(ar, ma, sigma2, tau)

Arguments

Details

The function is a generic implementation that requires a stationary theoretical autocorrelation function (ACF) and the ability to transform an ARMA(p,q) process into an MA(\infty) (e.g. infinite MA process).

Value

A vec containing the wavelet variance of the ARMA process.

Process Haar Wavelet Variance Formula

The Autoregressive Order p and Moving Average Order q (ARMA(p,q)) process has a Haar Wavelet Variance given by:

\frac{{{\tau _j}\left[ {1 - \rho \left( {\frac{{{\tau _j}}}{2}} \right)} \right] + 2\sum\limits_{i = 1}^{\frac{{{\tau _j}}}{2} - 1} {i\left[ {2\rho \left( {\frac{{{\tau _j}}}{2} - i} \right) - \rho \left( i \right) - \rho \left( {{\tau _j} - i} \right)} \right]} }}{{\tau _j^2}}\sigma _X^2

where \sigma _X^2 is given by the variance of the ARMA process. Furthermore, this assumes that stationarity has been achieved as it directly

ARMA process to WV Approximation

Description

This function computes the (haar) WV of an ARMA process

Usage

arma_to_wv_app(ar, ma, sigma2, tau, alpha = 0.9999)

Arguments

Details

This function provides an approximation to the arma_to_wv as computation times were previously a concern. However, this is no longer the case and, thus, this has been left in for the curious soul to discover...

Value

A vec containing the wavelet variance of the ARMA process.

Process Haar Wavelet Variance Formula

The Autoregressive Order p and Moving Average Order q (ARMA(p,q)) process has a Haar Wavelet Variance given by:

\frac{{{\tau _j}\left[ {1 - \rho \left( {\frac{{{\tau _j}}}{2}} \right)} \right] + 2\sum\limits_{i = 1}^{\frac{{{\tau _j}}}{2} - 1} {i\left[ {2\rho \left( {\frac{{{\tau _j}}}{2} - i} \right) - \rho \left( i \right) - \rho \left( {{\tau _j} - i} \right)} \right]} }}{{\tau _j^2}}\sigma _X^2

where \sigma _X^2 is given by the variance of the ARMA process. Furthermore, this assumes that stationarity has been achieved as it directly

Definition of an ARMA(1,1)

Description

Definition of an ARMA(1,1)

Usage

ARMA11(phi = NULL, theta = NULL, sigma2 = 1)

Arguments

Details

A variance is required since the model generation statements utilize randomization functions expecting a variance instead of a standard deviation like R.

Value

An S3 object with called ts.model with the following structure:

process.desc

AR1, MA1, SIGMA2

theta

\phi, \theta, \sigma^2

plength

Number of Parameters: 3

print

String containing simplified model

obj.desc

Depth of Parameters e.g. list(c(1,1,1))

starting

Guess Starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

X_t = \phi X_{t-1} + \theta_1 \varepsilon_{t-1} + \varepsilon_t,

where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2.

Author(s)

James Balamuta

Examples

# Creates an ARMA(1,1) process with predefined coefficients.
ARMA11(phi = .23, theta = .1, sigma2 = 1)

# Creates an ARMA(1,1) process with values to be guessed on callibration.
ARMA11()

ARMA(1,1) to WV

Description

This function computes the WV (haar) of an Autoregressive Order 1 - Moving Average Order 1 (ARMA(1,1)) process.

Usage

arma11_to_wv(phi, theta, sigma2, tau)

Arguments

Details

This function is significantly faster than its generalized counter part arma_to_wv

Value

A vec containing the wavelet variance of the ARMA(1,1) process.

Process Haar Wavelet Variance Formula

The Autoregressive Order 1 and Moving Average Order 1 (ARMA(1,1)) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = - \frac{{2{\sigma ^2}\left( { - \frac{1}{2}{{(\theta + 1)}^2}\left( {{\phi ^2} - 1} \right){\tau _j} - (\theta + \phi )(\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right)}}{{{{(\phi - 1)}^3}(\phi + 1)\tau _j^2}}

Compute Theoretical ACF for an ARMA Process

Description

Compute the theoretical autocorrelation function for an ARMA process.

Usage

ARMAacf_cpp(ar,ma,lag_max)

Arguments

Details

This is an implementaiton of the ARMAacf function in R. It is approximately 40x times faster. The benchmark was done on iMac Late 2013 using vecLib as the BLAS.

Value

x A matrix listing values from 1...nx in one column and 1...1, 2...2,....,n...n, in the other

Author(s)

JJB

Converting an ARMA Process to an Infinite MA Process

Description

Takes an ARMA function and converts it to an infinite MA process.

Usage

ARMAtoMA_cpp(ar, ma, lag_max)

Arguments

Details

This function is a port of the base stats package's ARMAtoMA. There is no significant speed difference between the two.

Value

A column vector containing coefficients

Author(s)

R Core Team and JJB

Quarterly Increase in Stocks Non-Farm Total, Australia

Description

A dataset containing the quarterly increase in stocks non-farm total in Australia, with frequency 4 starting from September 1959 to March 1991 with a total of 127 observations.

Usage

australia

Format

A data frame with 127 rows and 2 variables:

Quarter

year and quarter

Increase

quarterly increase in stocks non-farm total

Source

Time Series Data Library (citing: Australian Bureau of Statistics) datamarket

Empirical ACF and PACF

Description

This function can estimate either the autocovariance / autocorrelation for univariate time series, or the partial autocovariance / autocorrelation for univariate time series.

Usage

auto_corr(
  x,
  lag.max = NULL,
  pacf = FALSE,
  type = "correlation",
  demean = TRUE,
  robust = FALSE
)

Arguments

Details

lagmax default is 10*log10(N/m) where N is the number of observations and m is the number of time series being compared. If lagmax supplied is greater than the number of observations N, then one less than the total will be taken (i.e. N - 1).

Value

An array of dimensions N \times 1 \times 1.

Author(s)

Yuming Zhang

Examples

m = auto_corr(datasets::AirPassengers)
m = auto_corr(datasets::AirPassengers, pacf = TRUE)

Find the auto imu result

Description

Provides the core material to create an S3 object for auto.imu

Usage

auto_imu_cpp(
  data,
  combs,
  full_model,
  alpha,
  compute_v,
  model_type,
  K,
  H,
  G,
  robust,
  eff,
  bs_optimism,
  seed
)

Arguments

Value

A field<field<field<mat>>> that contains the model score matrix and the best GMWM model object.

B Matrix

Description

B Matrix

Usage

B_matrix(A, at_omega)

Arguments

Value

A mat

Computes the MO/DWT wavelet variance for multiple processes

Description

Calculates the MO/DWT wavelet variance

Usage

batch_modwt_wvar_cpp(
  signal,
  nlevels,
  robust,
  eff,
  alpha,
  ci_type,
  strWavelet,
  decomp
)

Arguments

Details

This function processes the decomposition of multiple signals quickly

Value

A field<mat> with the structure:

• "variance"Wavelet Variance

• "low"Lower CI

• "high"Upper CI

Select the Best Model

Description

This function retrieves the best model from a selection procedure.

Usage

best_model(x, ic = "aic")

Arguments

Examples

 
set.seed(18)
xt = gen_arima(N=100, ar=0.3, d=1, ma=0.3)
x = select_arima(xt, d=1L)
best_model(x, ic = "aic")

set.seed(19)
xt = gen_ma1(100, 0.3, 1)
x = select_ma(xt, q.min=2L, q.max=5L)
best_model(x, ic = "bic")

set.seed(20)
xt = gen_arma(100, c(.3,.5), c(.1), 1, 0)  
x = select_arma(xt, p.min = 1L, p.max = 4L,
                q.min = 1L, q.max = 3L)
best_model(x, ic = "hq")

bl14 filter construction

Description

Creates the bl14 filter

Usage

bl14_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

bl20 filter construction

Description

Creates the bl20 filter

Usage

bl20_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

Generate the Confidence Interval for GOF Bootstrapped

Description

yaya

Usage

boot_pval_gof(obj, obj_boot, B = 1000L, alpha = 0.05)

Arguments

Value

A vec that has the alpha/2.0 quantile and then the 1-alpha/2.0 quantile.

Compute the Bootstrapped GoF Test

Description

Handles the bootstrap computation and the bootstrapped p-value.

Usage

bootstrap_gof_test(obj_value, bs_obj_values, alpha, bs_gof_p_ci)

Arguments

Value

A vec that has

• Test Statistic

• Low CI

• Upper CI - BS

Removal of Boundary Wavelet Coefficients

Description

Removes the first n wavelet coefficients.

Usage

brick_wall(x, wave_filter, method)

Arguments

Details

The vectors are truncated by removing the first n wavelet coefficients. These vectors are then stored into the field that is returned. Note: As a result, there are no NA's introduced and hence the na.omit is not needed.

Value

A field<vec> with boundary modwt or dwt taken care of.

Build List of Unique Models

Description

Creates a set containing unique strings.

Usage

build_model_set(combs, x)

Arguments

Value

A set<string> that contains the list of unique models.

Calculate the Psi matrix

Description

Computes the Psi matrix using supplied parameters

Usage

calculate_psi_matrix(A, v_hat, omega)

Arguments

Value

A mat that has the first column

Time Series Convolution Filters

Description

Applies a convolution filter to a univariate time series.

Usage

cfilter(x, filter, sides, circular)

Arguments

Details

This is a port of the cfilter function harnessed by the filter function in stats. It is about 5-7 times faster than R's base function. The benchmark was done on iMac Late 2013 using vecLib as the BLAS.

Value

A column vec that contains the results of the filtering process.

Author(s)

R Core Team and JJB

Diagnostics on Fitted Time Series Model

Description

This function can perform (simple) diagnostics on the fitted time series model. It can output 6 diagnostic plots to assess the model, including (1) residuals plot, (2) histogram of distribution of standardized residuals, (3) Normal Q-Q plot of residuals, (4) ACF plot, (5) PACF plot, (6) Box test results.

Usage

check(model = NULL, resids = NULL, simple = FALSE)

Arguments

Author(s)

Stéphane Guerrier and Yuming Zhang

Examples

Xt = gen_gts(300, AR(phi = c(0, 0, 0.8), sigma2 = 1))
model = estimate(AR(3), Xt)
check(model)

check(resids = rnorm(100))

Xt = gen_gts(1000, SARIMA(ar = c(0.5, -0.25), i = 0, ma = 0.5, sar = -0.8, 
si = 1, sma = 0.25, s = 24, sigma2 = 1))
model = estimate(SARIMA(ar = 2, i = 0, ma = 1, sar = 1, si = 1, sma = 1, s = 24), 
Xt, method = "rgmwm")
check(model)
check(model, simple=TRUE)

Generate eta3 confidence interval

Description

Computes the eta3 CI

Usage

ci_eta3(y, dims, alpha_ov_2)

Arguments

Value

A matrix with the structure:

• Column 1Wavelet Variance

• Column 2Chi-squared Lower Bounds

• Column 3Chi-squared Upper Bounds

Generate eta3 robust confidence interval

Description

Computes the eta3 robust CI

Usage

ci_eta3_robust(wv_robust, wv_ci_class, alpha_ov_2, eff)

Arguments

Details

Within this function we are scaling the classical

Value

A matrix with the structure:

• Column 1Robust Wavelet Variance

• Column 2Chi-squared Lower Bounds

• Column 3Chi-squared Upper Bounds

Generate a Confidence intervval for a Univariate Time Series

Description

Computes an estimate of the multiscale variance and a chi-squared confidence interval

Usage

ci_wave_variance(
  signal_modwt_bw,
  wv,
  type = "eta3",
  alpha_ov_2 = 0.025,
  robust = FALSE,
  eff = 0.6
)

Arguments

Details

This function can be expanded to allow for other confidence interval calculations.

Value

A matrix with the structure:

• Column 1Wavelet Variance

• Column 2Chi-squared Lower Bounds

• Column 3Chi-squared Upper Bounds

Optim loses NaN

Description

This function takes numbers that are very small and sets them to the minimal tolerance for C++. Doing this prevents NaN from entering the optim routine.

Usage

code_zero(theta)

Arguments

Value

A vec that contains safe theta values.

Combine math expressions

Description

Combine math expressions

Usage

comb(...)

Arguments

Value

A combined expression.

Author(s)

Stephane Guerrier

Comparison of Classical and Robust Correlation Analysis Functions

Description

Compare classical and robust ACF of univariate time series.

Usage

compare_acf(
  x,
  lag.max = NULL,
  demean = TRUE,
  show.ci = TRUE,
  alpha = 0.05,
  plot = TRUE,
  ...
)

Arguments

Author(s)

Yunxiang Zhang

Examples

# Estimate both the ACF and PACF functions
compare_acf(datasets::AirPassengers)

Computes the (MODWT) wavelet covariance matrix

Description

Calculates the (MODWT) wavelet covariance matrix

Usage

compute_cov_cpp(
  signal_modwt,
  nb_level,
  compute_v = "diag",
  robust = TRUE,
  eff = 0.6
)

Arguments

Value

A field<mat> containing the covariance matrix.

GM Conversion

Description

Convert from AR1 to GM and vice-versa

Usage

conv.ar1.to.gm(theta, process.desc, freq)

conv.gm.to.ar1(theta, process.desc, freq)

Arguments

Author(s)

James Balamuta

Correlation Analysis Functions

Description

Correlation Analysis function computes and plots both empirical ACF and PACF of univariate time series.

Usage

corr_analysis(
  x,
  lag.max = NULL,
  type = "correlation",
  demean = TRUE,
  show.ci = TRUE,
  alpha = 0.05,
  plot = TRUE,
  ...
)

Arguments

Value

Two array objects (ACF and PACF) of dimension N \times S \times S.

Author(s)

Yunxiang Zhang

Examples

# Estimate both the ACF and PACF functions
corr_analysis(datasets::AirPassengers)

Count Models

Description

Count the amount of models that exist.

Usage

count_models(desc)

Arguments

Value

A map<string, int> containing how frequent the model component appears.

Bootstrap for Matrix V

Description

Using the bootstrap approach, we simulate a model based on user supplied parameters, obtain the wavelet variance, and then V.

Usage

cov_bootstrapper(theta, desc, objdesc, N, robust, eff, H, diagonal_matrix)

Arguments

Details

Expand in detail...

Value

A vec that contains the parameter estimates from GMWM estimator.

Author(s)

JJB

Internal IMU Object Construction

Description

Internal quick build for imu object.

Usage

create_imu(
  data,
  ngyros,
  nacces,
  axis,
  freq,
  unit = NULL,
  name = NULL,
  stype = NULL
)

Arguments

Value

An imu object class.

Author(s)

James Balamuta

Custom legend function

Description

Legend placement function

Usage

custom_legend(x, usr = par("usr"), ...)

Analytic D matrix of Processes

Description

This function computes each process to WV (haar) in a given model.

Usage

D_matrix(theta, desc, objdesc, tau, omegadiff)

Arguments

Details

Function returns the matrix effectively known as "D"

Value

A matrix with the process derivatives going down the column

Author(s)

James Joseph Balamuta (JJB)

d16 filter construction

Description

Creates the d16 filter

Usage

d16_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

d4 filter construction

Description

Creates the d4 filter

Usage

d4_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

d6 filter construction

Description

Creates the d6 filter

Usage

d6_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

d8 filter construction

Description

Creates the d8 filter

Usage

d8_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

Each Models Process Decomposed to WV

Description

This function computes each process to WV (haar) in a given model.

Usage

decomp_theoretical_wv(theta, desc, objdesc, tau)

Arguments

Value

A mat containing the wavelet variance of each process in the model

Decomposed WV to Single WV

Description

This function computes the combined processes to WV (haar) in a given model.

Usage

decomp_to_theo_wv(decomp)

Arguments

Value

A vec containing the wavelet variance of the process for the overall model

Analytic second derivative matrix for AR(1) process

Description

Calculates the second derivative for the AR(1) process and places it into a matrix form. The matrix form in this case is for convenience of the calculation.

Usage

deriv_2nd_ar1(phi, sigma2, tau)

Arguments

Value

A matrix with the first column containing the second partial derivative with respect to \phi and the second column contains the second partial derivative with respect to \sigma ^2

Process Haar WV Second Derivative

Taking the second derivative with respect to \phi yields:

\frac{{{\partial ^2}}}{{\partial {\phi ^2}}}\nu _j^2\left( \phi, \sigma ^2 \right) = \frac{2 \sigma ^2 \left(\left(\phi ^2-1\right) \tau _j \left(2 (\phi (7 \phi +4)+1) \phi ^{\frac{\tau _j}{2}-1}-(\phi (7 \phi +4)+1) \phi ^{\tau _j-1}+3 (\phi +1)^2\right)+\left(\phi ^2-1\right)^2 \tau _j^2 \left(\phi ^{\frac{\tau _j}{2}}-1\right) \phi ^{\frac{\tau _j}{2}-1}+4 (3 \phi +1) \left(\phi ^2+\phi +1\right) \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2}}+3\right)\right)}{(\phi -1)^5 (\phi +1)^3 \tau _j^2}

Taking the second derivative with respect to \sigma^2 yields:

\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( \sigma ^2 \right) = 0

Taking the derivative with respect to \phi and \sigma ^2 yields:

\frac{{{\partial ^2}}}{{\partial {\phi } \partial {\sigma ^2}}}\nu _j^2\left( \phi, \sigma ^2 \right) = \frac{2 \left(\left(\phi ^2-1\right) \tau _j \left(\phi ^{\tau _j}-2 \phi ^{\frac{\tau _j}{2}}-\phi -1\right)-(\phi (3 \phi +2)+1) \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2}}+3\right)\right)}{(\phi -1)^4 (\phi +1)^2 \tau _j^2}

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix for ARMA(1,1) process

Description

Obtain the second derivative of the ARMA(1,1) process.

Usage

deriv_2nd_arma11(phi, theta, sigma2, tau)

Arguments

Value

A matrix with:

• The first column containing the second partial derivative with respect to \phi;

• The second column containing the second partial derivative with respect to \theta;

• The third column contains the second partial derivative with respect to \sigma ^2.

• The fourth column contains the partial derivative with respect to \phi and \theta.

• The fiveth column contains the partial derivative with respect to \sigma ^2 and \phi.

• The sixth column contains the partial derivative with respect to \sigma ^2 and \theta.

Process Haar WV Second Derivative

Taking the second derivative with respect to \phi yields:

\frac{{{\partial ^2}}}{{\partial {\phi ^2}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}}}{{{{(\phi - 1)}^5}{{(\phi + 1)}^3}\tau _j^2}}\left( \begin{array}{cc} &{(\phi - 1)^2}\left( {{{(\phi + 1)}^2}\left( {{\theta ^2}\phi + \theta {\phi ^2} + \theta + \phi } \right)\tau _j^2\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 1} \right){\phi ^{\frac{{{\tau _j}}}{2} - 2}} + \left( {{\phi ^2} - 1} \right)\left( {{\theta ^2}( - \phi ) + \theta \left( {{\phi ^2} + 4\phi + 1} \right) - \phi } \right){\tau _j}\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 2} \right){\phi ^{\frac{{{\tau _j}}}{2} - 2}} - 2{{(\theta - 1)}^2}\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right) \\ &- 12{(\phi + 1)^2}\left( { - \frac{1}{2}{{(\theta + 1)}^2}\left( {{\phi ^2} - 1} \right){\tau _j} - (\theta + \phi )(\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right) \\ &+ 6(\phi + 1)(\phi - 1)\left( {\frac{1}{2}{{(\theta + 1)}^2}\left( {{\phi ^2} - 1} \right){\tau _j} + (\theta + \phi )(\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) + (\phi + 1)\left( { - (\theta + \phi )(\theta \phi + 1){\tau _j}\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 2} \right){\phi ^{\frac{{{\tau _j}}}{2} - 1}} - \theta (\theta + \phi )\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) - (\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) - {{(\theta + 1)}^2}\phi {\tau _j}} \right)} \right) \\ \end{array} \right)

Taking the second derivative with respect to \theta yields:

\frac{{{\partial ^2}}}{{\partial {\theta ^2}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}\left( {\left( {{\phi ^2} - 1} \right){\tau _j} + 2\phi \left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right)}}{{{{(\phi - 1)}^3}(\phi + 1)\tau _j^2}}

Taking the second derivative with respect to \sigma ^2 yields:

\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = 0

Taking the derivative with respect to \sigma^2 and \theta yields:

\frac{\partial }{{\partial \theta }}\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{2}{{{{(\phi - 1)}^3}(\phi + 1)\tau _j^2}}\left( {(\theta + 1)\left( {{\phi ^2} - 1} \right){\tau _j} + \left( {2\theta \phi + {\phi ^2} + 1} \right)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right)

Taking the derivative with respect to \sigma^2 and \phi yields:

\frac{\partial }{{\partial \phi }}\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{2}{{{{(\phi - 1)}^4}{{(\phi + 1)}^2}\tau _j^2}}\left( \begin{array}{ll} &- (\phi - 1)(\phi + 1)\left( \begin{array}{ll} &- (\theta + \phi )(\theta \phi + 1){\tau _j}\left( {{\phi ^{\frac{{{\tau _j}}}{2}}} - 2} \right){\phi ^{\frac{{{\tau _j}}}{2} - 1}} \\ &- \theta (\theta + \phi )\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) \\ &- (\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) \\ &- {(\theta + 1)^2}\phi {\tau _j} \\ \end{array} \right) \\ &+ (\phi - 1)\left( { - \frac{1}{2}{{(\theta + 1)}^2}\left( {{\phi ^2} - 1} \right){\tau _j} - (\theta + \phi )(\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right) \\ &+ 3(\phi + 1)\left( { - \frac{1}{2}{{(\theta + 1)}^2}\left( {{\phi ^2} - 1} \right){\tau _j} - (\theta + \phi )(\theta \phi + 1)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right) \\ \end{array} \right)

Taking the derivative with respect to \phi and \theta yields:

\frac{\partial }{{\partial \theta }}\frac{\partial }{{\partial \phi }}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = - \frac{{2{\sigma ^2}}}{{{{(\phi - 1)}^4}{{(\phi + 1)}^2}\tau _j^2}}\left( \begin{array}{cc} &{\tau _j}\left( \begin{array}{cc} &2(\theta + 1)(\phi - 1){(\phi + 1)^2} \\ &+ 2\left( {{\phi ^2} - 1} \right)\left( {2\theta \phi + {\phi ^2} + 1} \right){\phi ^{\frac{{{\tau _j}}}{2} - 1}} \\ &- \left( {{\phi ^2} - 1} \right)\left( {2\theta \phi + {\phi ^2} + 1} \right){\phi ^{{\tau _j} - 1}} \\ \end{array} \right) \\ &+ 2\left( {\theta (\phi (3\phi + 2) + 1) + \phi \left( {{\phi ^2} + \phi + 3} \right) + 1} \right)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) \\ \end{array} \right)

Author(s)

James Joseph Balamuta (JJB)

Analytic second derivative matrix for drift process

Description

To ease a later calculation, we place the result into a matrix structure.

Usage

deriv_2nd_dr(tau)

Arguments

Value

A matrix with the first column containing the second partial derivative with respect to \omega.

Author(s)

James Joseph Balamuta (JJB)

Analytic second derivative for MA(1) process

Description

To ease a later calculation, we place the result into a matrix structure.

Usage

deriv_2nd_ma1(theta, sigma2, tau)

Arguments

Value

A matrix with the first column containing the second partial derivative with respect to \theta, the second column contains the partial derivative with respect to \theta and \sigma ^2, and lastly we have the second partial derivative with respect to \sigma ^2.

Process Haar WV Second Derivative

Taking the second derivative with respect to \theta yields:

\frac{{{\partial ^2}}}{{\partial {\theta ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}}}{{{\tau _j}}}

Taking the second derivative with respect to \sigma^2 yields:

\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = 0

Taking the first derivative with respect to \theta and \sigma^2 yields:

\frac{\partial }{{\partial \theta }}\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{2(\theta + 1){\tau _j} - 6}}{{\tau _j^2}}

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix for AR(1) process

Description

Obtain the first derivative of the AR(1) process.

Usage

deriv_ar1(phi, sigma2, tau)

Arguments

Value

A matrix with the first column containing the partial derivative with respect to \phi and the second column contains the partial derivative with respect to \sigma ^2

Process Haar WV First Derivative

Taking the derivative with respect to \phi yields:

\frac{\partial }{{\partial \phi }}\nu _j^2\left( {\phi ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}\left( {\left( {{\phi ^2} - 1} \right){\tau _j}\left( { - 2{\phi ^{\frac{{{\tau _j}}}{2}}} + {\phi ^{{\tau _j}}} - \phi - 1} \right) - \left( {\phi \left( {3\phi + 2} \right) + 1} \right)\left( { - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + {\phi ^{{\tau _j}}} + 3} \right)} \right)}}{{{{\left( {\phi - 1} \right)}^4}{{\left( {\phi + 1} \right)}^2}\tau _j^2}}

Taking the derivative with respect to \sigma ^2 yields:

\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\phi ,{\sigma ^2}} \right) = \frac{{\left( {{\phi ^2} - 1} \right){\tau _j} + 2\phi \left( { - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + {\phi ^{{\tau _j}}} + 3} \right)}}{{{{\left( {\phi - 1} \right)}^3}\left( {\phi + 1} \right)\tau _j^2}}

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix for ARMA(1,1) process

Description

Obtain the first derivative of the ARMA(1,1) process.

Usage

deriv_arma11(phi, theta, sigma2, tau)

Arguments

Value

A matrix with:

• The first column containing the partial derivative with respect to \phi;

• The second column containing the partial derivative with respect to \theta;

• The third column contains the partial derivative with respect to \sigma ^2.

Process Haar WV First Derivative

Taking the derivative with respect to \phi yields:

\frac{\partial }{{\partial \phi }}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}}}{{{{(\phi - 1)}^4}{{(\phi + 1)}^2}\tau _j^2}}\left( \begin{array}{cc} &{\tau _j}\left( { - {{(\theta + 1)}^2}(\phi - 1){{(\phi + 1)}^2} - 2\left( {{\phi ^2} - 1} \right)(\theta + \phi )(\theta \phi + 1){\phi ^{\frac{{{\tau _j}}}{2} - 1}} + \left( {{\phi ^2} - 1} \right)(\theta \phi + 1)(\theta + \phi ){\phi ^{{\tau _j} - 1}}} \right) \\ &- \left( {{\theta ^2}((3\phi + 2)\phi + 1) + 2\theta \left( {\left( {{\phi ^2} + \phi + 3} \right)\phi + 1} \right) + (3\phi + 2)\phi + 1} \right)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right) \\ \end{array} \right)

Taking the derivative with respect to \theta yields:

\frac{\partial }{{\partial \theta }}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{{2{\sigma ^2}\left( {(\theta + 1)\left( {{\phi ^2} - 1} \right){\tau _j} + \left( {2\theta \phi + {\phi ^2} + 1} \right)\left( {{\phi ^{{\tau _j}}} - 4{\phi ^{\frac{{{\tau _j}}}{2}}} + 3} \right)} \right)}}{{{{(\phi - 1)}^3}(\phi + 1)\tau _j^2}}

Taking the derivative with respect to \sigma^2 yields:

\frac{\partial }{{\partial \sigma ^2 }}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = \frac{2 \sigma ^2 \left(\left(\phi ^2-1\right) \tau _j+2 \phi \left(\phi ^{\tau _j}-4 \phi ^{\frac{\tau _j}{2}}+3\right)\right)}{(\phi -1)^3 (\phi +1) \tau _j^2}

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix for Drift (DR) Process

Description

Obtain the first derivative of the Drift (DR) process.

Usage

deriv_dr(omega, tau)

Arguments

Value

A matrix with the first column containing the partial derivative with respect to \omega.

Process Haar WV First Derivative

Taking the derivative with respect to \omega yields:

\frac{\partial }{{\partial \omega }}\nu _j^2\left( \omega \right) = \frac{{\tau _j^2\omega }}{8}

Note: We are taking the derivative with respect to \omega and not \omega^2 as the \omega relates to the slope of the process and not the processes variance like RW and WN. As a result, a second derivative exists and is not zero.

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix for MA(1) process

Description

Obtain the first derivative of the MA(1) process.

Usage

deriv_ma1(theta, sigma2, tau)

Arguments

Value

A matrix with the first column containing the partial derivative with respect to \theta and the second column contains the partial derivative with respect to \sigma ^2

Process Haar WV First Derivative

Taking the derivative with respect to \theta yields:

\frac{\partial }{{\partial \theta }}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{{\sigma ^2}\left( {2\left( {\theta + 1} \right){\tau _j} - 6} \right)}}{{\tau _j^2}}

Taking the derivative with respect to \sigma^2 yields:

\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{{{\left( {\theta + 1} \right)}^2}{\tau _j} - 6\theta }}{{\tau _j^2}}

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix for Quantization Noise (QN) Process

Description

Obtain the first derivative of the Quantization Noise (QN) process.

Usage

deriv_qn(tau)

Arguments

Value

A matrix with the first column containing the partial derivative with respect to Q^2.

Process Haar WV First Derivative

Taking the derivative with respect to Q^2 yields:

\frac{\partial }{{\partial {Q^2}}}\nu _j^2\left( {{Q^2}} \right) = \frac{6}{{\tau _j^2}}

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix Random Walk (RW) Process

Description

Obtain the first derivative of the Random Walk (RW) process.

Usage

deriv_rw(tau)

Arguments

Value

A matrix with the first column containing the partial derivative with respect to \gamma^2.

Process Haar WV First Derivative

Taking the derivative with respect to \gamma ^2 yields:

\frac{\partial }{{\partial {\gamma ^2}}}\nu _j^2\left( {{\gamma ^2}} \right) = \frac{{\tau _j^2 + 2}}{{12{\tau _j}}}

Author(s)

James Joseph Balamuta (JJB)

Analytic D Matrix for a Gaussian White Noise (WN) Process

Description

Obtain the first derivative of the Gaussian White Noise (WN) process.

Usage

deriv_wn(tau)

Arguments

Value

A matrix with the first column containing the partial derivative with respect to \sigma^2.

Process Haar WV First Derivative

Taking the derivative with respect to \sigma^2 yields:

\frac{\partial }{{\partial {\sigma ^2}}}\nu _j^2\left( {{\sigma ^2}} \right) = \frac{1}{{{\tau _j}}}

Author(s)

James Joseph Balamuta (JJB)

Analytic D matrix of Processes

Description

This function computes each process to WV (haar) in a given model.

Usage

derivative_first_matrix(theta, desc, objdesc, tau)

Arguments

Details

Function returns the matrix effectively known as "D"

Value

A matrix with the process derivatives going down the column

Author(s)

James Joseph Balamuta (JJB)

Create a ts.model from desc string

Description

Sets up the necessary backend for using Cpp functions to build R ts.model objects

Usage

desc.to.ts.model(desc)

Arguments

Value

An S3 object with called ts.model with the following structure:

• desc

• theta

Author(s)

James Balamuta

Examples

desc.to.ts.model(c("AR1","WN"))

Discrete Fourier Transformation for Autocovariance Function

Description

Calculates the autovariance function (ACF) using Discrete Fourier Transformation.

Usage

dft_acf(x)

Arguments

Details

This implementation is 2x as slow as Rs. Two issues: 1. memory resize and 2. unoptimized fft algorithm in arma. Consider piping back into R and rewrapping the object. (Decrease of about 10 microseconds.)

Value

A vec containing the ACF.

Box-Pierce

Description

Performs the Box-Pierce test to assess the Null Hypothesis of Independence in a Time Series

Usage

diag_boxpierce(x, order = NULL, stop_lag = 20, stdres = FALSE, plot = TRUE)

Arguments

Author(s)

James Balamuta, Stéphane Guerrier, Yuming Zhang

Ljung-Box

Description

Performs the Ljung-Box test to assess the Null Hypothesis of Independence in a Time Series

Usage

diag_ljungbox(x, order = NULL, stop_lag = 20, stdres = FALSE, plot = TRUE)

Arguments

Author(s)

James Balamuta, Stéphane Guerrier, Yuming Zhang

Diagnostic Plot of Residuals

Description

This function will plot 8 diagnostic plots to assess the model used to fit the data. These include: (1) residuals plot, (2) residuals vs fitted values, (3) histogram of distribution of standardized residuals, (4) Normal Q-Q plot of residuals, (5) ACF plot, (6) PACF plot, (7) Haar Wavelet Variance Representation, (8) Box test results.

Usage

diag_plot(Xt = NULL, model = NULL, resids = NULL, std = FALSE)

Arguments

Author(s)

Yuming Zhang

Portmanteau Tests

Description

Performs the Portmanteau test to assess the Null Hypothesis of Independence in a Time Series

Usage

diag_portmanteau_(
  x,
  order = NULL,
  stop_lag = 20,
  stdres = FALSE,
  test = "Ljung-Box",
  plot = TRUE
)

Arguments

Author(s)

James Balamuta, Stéphane Guerrier, Yuming Zhang

Lagged Differences in Armadillo

Description

Returns the ith difference of a time series of rth lag.

Usage

diff_cpp(x, lag, differences)

Arguments

Value

A vector containing the differenced time series.

Author(s)

JJB

Root Finding C++

Description

Used to interface with Armadillo

Usage

do_polyroot_arma(z)

Arguments

Root Finding C++

Description

Vroom Vroom

Usage

do_polyroot_cpp(z)

Arguments

Create an Drift (DR) Process

Description

Sets up the necessary backend for the DR process.

Usage

DR(omega = NULL)

Arguments

Value

An S3 object with called ts.model with the following structure:

process.desc

Used in summary: "DR"

theta

slope

print

String containing simplified model

plength

Number of parameters

obj.desc

y desc replicated x times

obj

Depth of parameters e.g. list(1)

starting

Guess starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

Y_t = \omega t

Author(s)

James Balamuta

Examples

DR()
DR(omega=3.4)

Drift to WV

Description

This function compute the WV (haar) of a Drift process

Usage

dr_to_wv(omega, tau)

Arguments

Value

A vec containing the wavelet variance of the drift.

Process Haar Wavelet Variance Formula

The Drift (DR) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {{\omega }} \right) = \frac{{\tau _j^2{\omega ^2}}}{{16}}

Discrete Wavelet Transform

Description

Calculation of the coefficients for the discrete wavelet transformation.

Usage

dwt_cpp(x, filter_name, nlevels, boundary, brickwall)

Arguments

Details

Performs a level J decomposition of the time series using the pyramid algorithm

Value

y A field<vec> that contains the wavelet coefficients for each decomposition level

Author(s)

JJB

Expected value DR

Description

This function computes the expected value of a drift process.

Usage

e_drift(omega, n_ts)

Arguments

Value

A vec containing the expected value of the drift.

Fit a Time Series Model to Data

Description

This function can fit a time series model to data using different methods.

Usage

estimate(model, Xt, method = "mle", demean = TRUE)

Arguments

Author(s)

Stéphane Guerrier and Yuming Zhang

Examples

Xt = gen_gts(300, AR(phi = c(0, 0, 0.8), sigma2 = 1))
plot(Xt)
estimate(AR(3), Xt)

Xt = gen_gts(300, MA(theta = 0.5, sigma2 = 1))
plot(Xt)
estimate(MA(1), Xt, method = "gmwm")

Xt = gen_gts(300, ARMA(ar = c(0.8, -0.5), ma = 0.5, sigma2 = 1))
plot(Xt)
estimate(ARMA(2,1), Xt, method = "rgmwm")

Xt = gen_gts(300, ARIMA(ar = c(0.8, -0.5), i = 1, ma = 0.5, sigma2 = 1))
plot(Xt)
estimate(ARIMA(2,1,1), Xt, method = "mle")

Xt = gen_gts(1000, SARIMA(ar = c(0.5, -0.25), i = 0, ma = 0.5, sar = -0.8, 
si = 1, sma = 0.25, s = 24, sigma2 = 1))
plot(Xt)
estimate(SARIMA(ar = 2, i = 0, ma = 1, sar = 1, si = 1, sma = 1, s = 24), Xt, 
method = "rgmwm")

Evalute a time series or a list of time series models

Description

This function calculates AIC, BIC and HQ or the MAPE for a list of time series models. This function currently only supports models estimated by the MLE.

Usage

evaluate(
  models,
  Xt,
  criterion = "IC",
  start = 0.8,
  demean = TRUE,
  print = TRUE
)

Arguments

Value

AIC, BIC and HQ or MAPE

Author(s)

Stéphane Guerrier

Examples

set.seed(18)
n = 300
Xt = gen_gts(n, AR(phi = c(0, 0, 0.8), sigma2 = 1))
evaluate(AR(1), Xt)
evaluate(list(AR(1), AR(3), MA(3), ARMA(1,2), 
SARIMA(ar = 1, i = 0, ma = 1, sar = 1, si = 1, sma = 1, s = 12)), Xt)
evaluate(list(AR(1), AR(3)), Xt, criterion = "MAPE")

Computes the (MODWT) wavelet covariance matrix using Chi-square confidence interval bounds

Description

Calculates the (MODWT) wavelet covariance matrix using Chi-square confidence interval bounds

Usage

fast_cov_cpp(ci_hi, ci_lo)

Arguments

Value

A diagonal matrix.

Definition of a Fractional Gaussian Noise (FGN) Process

Description

Definition of a Fractional Gaussian Noise (FGN) Process

Usage

FGN(sigma2 = 1, H = 0.9999)

Arguments

Value

An S3 object containing the specified ts.model with the following structure:

process.desc

Used in summary: "SIGMA2","H"

theta

Parameter vector including \sigma^2, H

plength

Number of parameters

print

String containing simplified model

desc

"FGN"

obj.desc

Depth of Parameters e.g. list(1,1)

starting

Find starting values? TRUE or FALSE (e.g. specified value)

Author(s)

Lionel Voirol, Davide Cucci

Examples

FGN()
FGN(sigma2 = 1, H = 0.9999)

Transform an Armadillo field<vec> to a matrix

Description

Unlists vectors in a field and places them into a matrix

Usage

field_to_matrix(x)

Arguments

Value

A mat containing the field elements within a column.

Author(s)

JJB

Find the Common Denominator of the Models

Description

Determines the common denominator among models

Usage

find_full_model(x)

Arguments

Value

A vector<string> that contains the terms of the common denominator of all models

fk14 filter construction

Description

Creates the fk14 filter

Usage

fk14_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

fk22 filter construction

Description

Creates the fk22 filter

Usage

fk22_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

fk4 filter construction

Description

Creates the fk4 filter

Usage

fk4_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

fk6 filter construction

Description

Creates the fk6 filter

Usage

fk6_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

fk8 filter construction

Description

Creates the fk8 filter

Usage

fk8_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

Format the Confidence Interval for Estimates

Description

Creates hi and lo confidence based on SE and alpha.

Usage

format_ci(theta, se, alpha)

Arguments

Value

A mat that has:

• Column 1: Lo CI

• Column 2: Hi CI

• Column 3: SE

Generate an Autoregressive Order 1 ( AR(1) ) sequence

Description

Generate an Autoregressive Order 1 sequence given \phi and \sigma^2.

Usage

gen_ar1(N, phi = 0.3, sigma2 = 1)

Arguments

Details

The function implements a way to generate the AR(1)'s x_t values without calling the general ARMA function. Thus, the function is able to generate values much faster than gen_arma.

Value

A vec containing the AR(1) process.

Process Definition

The Autoregressive order 1 (AR1) process with non-zero parameter \phi \in (-1,+1) and \sigma^2 \in {\rm I\!R}^{2}. This process is defined as:

{X_t} = {\phi _1}{X_{t - 1}} + {\varepsilon_t}

, where

{\varepsilon_t}\mathop \sim \limits^{iid} N\left( {0,\sigma^2} \right)

AR(1) processes are sometimes used as an approximation for Bias Instability noises.

Generation Algorithm

The function first generates a vector of White Noise with length N+1 using gen_wn and then obtains the autoregressive values under the above process definition.

The X_0 (first value of X_t) is discarded.

Generate AR(1) Block Process

Description

This function allows us to generate a non-stationary AR(1) block process.

Usage

gen_ar1blocks(phi, sigma2, n_total, n_block, scale = 10, 
title = NULL, seed = 135, ...)

Arguments

Value

A vector containing the AR(1) block process.

Note

This function generates a non-stationary AR(1) block process whose theoretical maximum overlapping allan variance (MOAV) is different from the theoretical MOAV of a stationary AR(1) process. This difference in the value of the allan variance between stationary and non-stationary processes has been shown through the calculation of the theoretical allan variance given in "A Study of the Allan Variance for Constant-Mean Non-Stationary Processes" by Xu et al. (IEEE Signal Processing Letters, 2017), preprint available: https://arxiv.org/abs/1702.07795.

Author(s)

Yuming Zhang and Haotian Xu

Examples

Xt = gen_ar1blocks(phi = 0.9, sigma2 = 1, 
n_total = 1000, n_block = 10, scale = 100)
plot(Xt)

Yt = gen_ar1blocks(phi = 0.5, sigma2 = 5, n_total = 800, 
n_block = 20, scale = 50)
plot(Yt)

Generate Autoregressive Order p, Integrated d, Moving Average Order q (ARIMA(p,d,q)) Model

Description

Generate an ARIMA(p,d,q) process with supplied vector of Autoregressive Coefficients (\phi), Integrated d, Moving Average Coefficients (\theta), and \sigma^2.

Usage

gen_arima(N, ar, d, ma, sigma2 = 1.5, n_start = 0L)

Arguments

Details

The innovations are generated from a normal distribution. The \sigma^2 parameter is indeed a variance parameter. This differs from R's use of the standard deviation, \sigma.

Value

A vec that contains the generated observations.

Warning

Please note, this function will generate a sum of N + d number of observations, where d denotes the number of differences necessary.

Generate Autoregressive Order p - Moving Average Order q (ARMA(p,q)) Model

Description

Generate an ARMA(p,q) process with supplied vector of Autoregressive Coefficients (\phi), Moving Average Coefficients (\theta), and \sigma^2.

Usage

gen_arma(N, ar, ma, sigma2 = 1.5, n_start = 0L)

Arguments

Details

For AR(1), MA(1), and ARMA(1,1) please use their functions if speed is important as this function is designed to generate generic ARMA processes.

Value

A vec that contains the generated observations.

Process Definition

The Autoregressive order p and Moving Average order q (ARMA(p,q)) process with non-zero parameters \phi_i \in (-1,+1) for the AR components, \theta_j \in (-1,+1) for the MA components, and \sigma^2 \in {\rm I\!R}^{+}. This process is defined as:

{X_t} = \sum\limits_{i = 1}^p {{\phi _i}{X_{t - i}}} + \sum\limits_{i = 1}^q {{\theta _i}{\varepsilon _{t - i}}} + {\varepsilon _t}

where

{\varepsilon_t}\mathop \sim \limits^{iid} N\left( {0,\sigma^2} \right)

Generation Algorithm

The innovations are generated from a normal distribution. The \sigma^2 parameter is indeed a variance parameter. This differs from R's use of the standard deviation, \sigma.

Generate an ARMA(1,1) sequence

Description

Generate an ARMA(1,1) sequence given \phi, \theta, and \sigma^2.

Usage

gen_arma11(N, phi = 0.1, theta = 0.3, sigma2 = 1)

Arguments

Details

The function implements a way to generate the x_t values without calling the general ARMA function.

Value

A vec containing the MA(1) process.

Process Definition

The Autoregressive order 1 and Moving Average order 1 (ARMA(1,1)) process with non-zero parameters \phi \in (-1,+1) for the AR component, \theta \in (-1,+1) for the MA component, and \sigma^2 \in {\rm I\!R}^{+}. This process is defined as:

{X_t} = {\phi _1}{X_{t - 1}} + {\theta _1}{\varepsilon_{t - 1}} + {\varepsilon_t}

, where

{\varepsilon_t}\mathop \sim \limits^{iid} N\left( {0,\sigma^2} \right)

Generation Algorithm

The function first generates a vector of white noise using gen_wn and then obtains the ARMA values under the above equation.

The X_0 (first value of X_t) is discarded.

Generate Bias-Instability Process

Description

This function allows to generate a non-stationary bias-instability process.

Usage

gen_bi(sigma2, n_total, n_block, title = NULL, seed = 135, ...)

Arguments

Value

A vector containing the bias-instability process.

Note

This function generates a non-stationary bias-instability process whose theoretical maximum overlapping allan variance (MOAV) is close to the theoretical MOAV of the best approximation of this process through a stationary AR(1) process over some scales. However, this approximation is not good enough when considering the logarithmic representation of the allan variance. Therefore, the exact form of the allan variance of this non-stationary process allows us to better interpret the signals characterized by bias-instability, as shown in "A Study of the Allan Variance for Constant-Mean Non-Stationary Processes" by Xu et al. (IEEE Signal Processing Letters, 2017), preprint available: https://arxiv.org/abs/1702.07795.

Author(s)

Yuming Zhang

Examples

Xt = gen_bi(sigma2 = 1, n_total = 1000, n_block = 10)
plot(Xt)

Yt = gen_bi(sigma2 = 0.8, n_total = 800, n_block = 20,
title = "non-stationary bias-instability process")
plot(Yt)

Generate a Drift Process

Description

Simulates a Drift Process with a given slope, \omega.

Usage

gen_dr(N, omega = 5)

Arguments

Value

A vec containing the drift.

Process Definition

Drift with parameter \omega \in \Omega where \Omega is in {\rm I\!R}^{+} or {\rm I\!R}^{-}. This process is defined as: X_t = \omega t and is occasionally referred to as Rate Ramp.

Generation Algorithm

To generate the Drift process, we first fill a vector with the \omega parameter. After, we take the cumulative sum along the vector.

Generate a Fractional Gaussian noise given \sigma^2 and H.

Description

Simulates a Fractional Gaussian noise given \sigma^2 and H.

Usage

gen_fgn(N, sigma2 = 1, H = 0.9)

Arguments

Value

fgn A vec containing the Fractional Gaussian noise process.

Generate Generic Seasonal Autoregressive Order P - Moving Average Order Q (SARMA(p,q)x(P,Q)) Model

Description

Generate an ARMA(P,Q) process with supplied vector of Autoregressive Coefficients (\phi), Moving Average Coefficients (\theta), and \sigma^2.

Usage

gen_generic_sarima(N, theta_values, objdesc, sigma2 = 1.5, n_start = 0L)

Arguments

Details

The innovations are generated from a normal distribution. The \sigma^2 parameter is indeed a variance parameter. This differs from R's use of the standard deviation, \sigma.

Value

A vec that contains the generated observations.

Simulate a simts TS object using a theoretical model

Description

Create a gts object based on a time series model.

Usage

gen_gts(
  n,
  model,
  start = 0,
  end = NULL,
  freq = 1,
  unit_ts = NULL,
  unit_time = NULL,
  name_ts = NULL,
  name_time = NULL
)

Arguments

Details

This function accepts either a ts.model object (e.g. AR1(phi = .3, sigma2 =1) + WN(sigma2 = 1)) or a simts object.

Value

A gts object

Author(s)

James Balamuta and Wenchao Yang

Examples

# Set seed for reproducibility
set.seed(1336)
n = 1000

# AR1 + WN
model = AR1(phi = .5, sigma2 = .1) + WN(sigma2=1)
x = gen_gts(n, model)
plot(x)

# Reset seed
set.seed(1336)

# GM + WN
# Convert from AR1 to GM values
m = ar1_to_gm(c(.5,.1),10)

# Beta = 6.9314718, Sigma2_gm = 0.1333333
model = GM(beta = m[1], sigma2_gm = m[2]) + WN(sigma2=1)
x2 = gen_gts(n, model, freq = 10, unit_time = 'sec')
plot(x2)

# Same time series
all.equal(x, x2, check.attributes = FALSE)

Generate a Latent Time Series Object Based on a Model

Description

Simulate a lts object based on a supplied time series model.

Usage

gen_lts(
  n,
  model,
  start = 0,
  end = NULL,
  freq = 1,
  unit_ts = NULL,
  unit_time = NULL,
  name_ts = NULL,
  name_time = NULL,
  process = NULL
)

Arguments

Details

This function accepts either a ts.model object (e.g. AR1(phi = .3, sigma2 =1) + WN(sigma2 = 1)) or a simts object.

Value

A lts object with the following attributes:

start

The time of the first observation.

end

The time of the last observation.

freq

Numeric representation of the sampling frequency/rate.

unit

A string reporting the unit of measurement.

name

Name of the generated dataset.

process

A vector that contains model names of decomposed and combined processes

Author(s)

James Balamuta, Wenchao Yang, and Justin Lee

Examples

# AR
set.seed(1336)
model = AR1(phi = .99, sigma2 = 1) + WN(sigma2 = 1)
test = gen_lts(1000, model)
plot(test)

Generate Latent Time Series based on Model (Internal)

Description

Create a latent time series based on a supplied time series model.

Usage

gen_lts_cpp(N, theta, desc, objdesc)

Arguments

Value

A mat containing data for each decomposed and combined time series.

Generate an Moving Average Order 1 (MA(1)) Process

Description

Generate an MA(1) Process given \theta and \sigma^2.

Usage

gen_ma1(N, theta = 0.3, sigma2 = 1)

Arguments

Details

The function implements a way to generate the x_t values without calling the general ARMA function.

Value

A vec containing the MA(1) process.

Process Definition

The Moving Average order 1 (MA(1)) process with non-zero parameter \theta \in (-1,+1) and \sigma^2 \in {\rm I\!R}^{+}. This process is defined as:

{x_t} = {\varepsilon_t} + {\theta _1}{\varepsilon_{t - 1}}

, where

{\varepsilon_t}\mathop \sim \limits^{iid} N\left( {0,\sigma^2} \right)

Generation Algorithm

The function first generates a vector of white noise using gen_wn and then obtains the MA values under the above equation.

The X_0 (first value of X_t) is discarded.

Generate a Matern Process given \sigma^2, \lambda and \alpha.

Description

Simulates a Matern Process given \sigma^2, \lambda and \alpha.

Usage

gen_matern(N, sigma2 = 1, lambda = 0.35, alpha = 0.9)

Arguments

Value

mtp A vec containing the Matern Process.

Generate a determinist vector returned by the matrix by vector product of matrix X and vector \beta.

Description

Generate a determinist vector returned by the matrix by vector product of matrix X and vector \beta.

Usage

gen_mean(X, beta)

Arguments

Value

mean_vec A vec containing the determinist vector.

Generate Time Series based on Model (Internal)

Description

Create a time series process based on a supplied ts.model.

Usage

gen_model(N, theta, desc, objdesc)

Arguments

Value

A vec that contains combined time series.

Generate Non-Stationary White Noise Process

Description

This function allows to generate a non-stationary white noise process.

Usage

gen_nswn(n_total, title = NULL, seed = 135, ...)

Arguments

Value

A vector containing the non-stationary white noise process.

Note

This function generates a non-stationary white noise process whose theoretical maximum overlapping allan variance (MOAV) corresponds to the theoretical MOAV of the stationary white noise process. This example confirms that the allan variance is unable to distinguish between a stationary white noise process and a white noise process whose second-order behavior is non-stationary, as pointed out in the paper "A Study of the Allan Variance for Constant-Mean Non-Stationary Processes" by Xu et al. (IEEE Signal Processing Letters, 2017), preprint available: https://arxiv.org/abs/1702.07795.

Author(s)

Yuming Zhang

Examples

Xt = gen_nswn(n_total = 1000)
plot(Xt)

Yt = gen_nswn(n_total = 2000, title = "non-stationary 
white noise process", seed = 1960)
plot(Yt)

Generate a Power Law Process given \sigma^2 and d.

Description

Simulates a a Power Law Process given \sigma^2 and d.

Usage

gen_powerlaw(N, sigma2 = 1, d = 0.9)

Arguments

Value

plp A vec containing the Power Law Process.

Generate a Quantisation Noise (QN) or Rounding Error Sequence

Description

Simulates a QN sequence given Q^2.

Usage

gen_qn(N, q2 = 0.1)

Arguments

Value

A vec containing the QN process.

Process Definition

Quantization Noise (QN) with parameter Q^2 \in R^{+}. With i.i.d Y_t \sim U(0,1) (i.e. a standard uniform variable), this process is defined as:

X_t = \sqrt{12Q^2}(Y_{t}-Y_{t-1})

Generation Algorithm

To generate the quantisation noise, we follow this recipe: First, we generate using a random uniform distribution:

U_k^*\sim U\left[ {0,1} \right]

Then, we multiple the sequence by \sqrt{12} so:

{U_k} = \sqrt{12} U_k^*

Next, we find the derivative of {U_k}

{{\dot U}_k} = \frac{{{U_{k + \Delta t}} - {U_k}}}{{\Delta t}}

In this case, we modify the derivative such that: {{\dot U}_k}\Delta t = {U_{k + \Delta t}} - {U_k}

Thus, we end up with:

{x_k} = \sqrt Q {{\dot U}_k}\Delta t

{x_k} = \sqrt Q \left( {{U_{k + 1}} - {U_k}} \right)

Generate a Random Walk without Drift

Description

Generates a random walk without drift.

Usage

gen_rw(N, sigma2 = 1)

Arguments

Value

grw A vec containing the random walk without drift.

Process Definition

Random Walk (RW) with parameter \gamma^2 \in {\rm I\!R}^{+}. This process is defined as:

{X_t} = \sum\limits_{t = 1}^T {\gamma {Z_t}}

and is often called Rate Random Walk in the engineering literature.

Generation Algorithm

To generate we first obtain the standard deviation from the variance by taking a square root. Then, we sample N times from a N(0,\sigma^2) distribution. Lastly, we take the cumulative sum over the vector.

Generate Seasonal Autoregressive Order P - Moving Average Order Q (SARMA(p,q)x(P,Q)) Model

Description

Generate an ARMA(P,Q) process with supplied vector of Autoregressive Coefficients (\phi), Moving Average Coefficients (\theta), and \sigma^2.

Usage

gen_sarima(N, ar, d, ma, sar, sd, sma, sigma2 = 1.5, s = 12L, n_start = 0L)

Arguments

Details

The innovations are generated from a normal distribution. The \sigma^2 parameter is indeed a variance parameter. This differs from R's use of the standard deviation, \sigma.

Value

A vec that contains the generated observations.

Generate Seasonal Autoregressive Order P - Moving Average Order Q (SARMA(p,q)x(P,Q)) Model

Description

Generate an ARMA(P,Q) process with supplied vector of Autoregressive Coefficients (\phi), Moving Average Coefficients (\theta), and \sigma^2.

Usage

gen_sarma(N, ar, ma, sar, sma, sigma2 = 1.5, s = 12L, n_start = 0L)

Arguments

Details

The innovations are generated from a normal distribution. The \sigma^2 parameter is indeed a variance parameter. This differs from R's use of the standard deviation, \sigma.

Value

A vec that contains the generated observations.

Generate a Sinusoidal Process given \alpha^2 and \beta.

Description

Simulates a Sinusoidal Process Process with parameter \alpha^2 and \beta

Usage

gen_sin(N, alpha2 = 9e-04, beta = 0.06, U = 1)

Arguments

Value

sn A vec containing the sinusoidal process.

Generation Algorithm

The function first generates a initial cycle oscillation at t=0 from a Uniform law with parameter a = 0 and b = 2 * pi and then compute the signal from its definition

X_t = \alpha \sin(\beta t + U)

.

Generate a Gaussian White Noise Process (WN(\sigma ^2))

Description

Simulates a Gaussian White Noise Process with variance parameter \sigma ^2.

Usage

gen_wn(N, sigma2 = 1)

Arguments

Value

wn A vec containing the white noise.

Process Definition

Gaussian White Noise (WN) with parameter \sigma^2 \in {\rm I\!R}^{+}. This process is defined as X_t\sim N(0,\sigma^2) and is sometimes referred to as Angle (Velocity) Random Walk.

Generation Algorithm

To generate the Gaussian White Noise (WN) process, we first obtain the standard deviation from the variance by taking a square root. Then, we sample N times from a N(0,\sigma ^2) distribution.

Routing function for summary info

Description

Gets all the data for the summary.gmwm function.

Usage

get_summary(
  theta,
  desc,
  objdesc,
  model_type,
  wv_empir,
  theo,
  scales,
  V,
  omega,
  obj_value,
  N,
  alpha,
  robust,
  eff,
  inference,
  fullV,
  bs_gof,
  bs_gof_p_ci,
  bs_theta_est,
  bs_ci,
  B
)

Arguments

Value

A field<mat> that contains bootstrapped / asymptotic GoF results as well as CIs.

Retrieve GMWM starting value from Yannick's objective function

Description

Obtains the GMWM starting value given by Yannick's objective function optimization

Usage

getObjFun(theta, desc, objdesc, model_type, omega, wv_empir, tau)

Arguments

Value

A double that is the value of the Objective function under Yannick's starting algorithm

Retrieve GMWM starting value from Yannick's objective function

Description

Obtains the GMWM starting value given by Yannick's objective function optimization

Usage

getObjFunStarting(theta, desc, objdesc, model_type, wv_empir, tau)

Arguments

Value

A double that is the value of the Objective function under Yannick's starting algorithm

Create a Gauss-Markov (GM) Process

Description

Sets up the necessary backend for the GM process.

Usage

GM(beta = NULL, sigma2_gm = 1)

Arguments

Details

When supplying values for \beta and \sigma ^2_{gm}, these parameters should be of a GM process and NOT of an AR1. That is, do not supply AR1 parameters such as \phi, \sigma^2.

Internally, GM parameters are converted to AR1 using the 'freq' supplied when creating data objects (gts) or specifying a 'freq' parameter in simts or simts.imu.

The 'freq' of a data object takes precedence over the 'freq' set when modeling.

Value

An S3 object with called ts.model with the following structure:

process.desc

Used in summary: "BETA","SIGMA2"

theta

\beta, \sigma ^2_{gm}

plength

Number of parameters

print

String containing simplified model

desc

"GM"

obj.desc

Depth of parameters e.g. list(1,1)

starting

Guess starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

X_t = e^{(-\beta)} X_{t-1} + \varepsilon_t

, where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2(1-e^{2\beta}).

Author(s)

James Balamuta

Examples

GM()
GM(beta=.32, sigma2_gm=1.3)

Transform GM to AR1

Description

Takes GM values and transforms them to AR1

Usage

gm_to_ar1(theta, freq)

Arguments

Value

A vec containing GM values.

Author(s)

James Balamuta The function takes a vector of GM values \beta and \sigma ^2_{gm} and transforms them to AR1 values \phi and \sigma ^2 using the formulas: \phi = \exp \left( { - \beta \Delta t} \right) {\sigma ^2} = \sigma _{gm}^2\left( {1 - \exp \left( { - 2\beta \Delta t} \right)} \right)

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

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

Engine for obtaining the GMWM Estimator

Description

This function uses the Generalized Method of Wavelet Moments (GMWM) to estimate the parameters of a time series model.

Usage

gmwm_engine(
  theta,
  desc,
  objdesc,
  model_type,
  wv_empir,
  omega,
  scales,
  starting
)

Arguments

Details

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)

• "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 model_type = "imu" or type = "ssm" then starting values pass through an initial bootstrap and pseudo-optimization before being passed to the GMWM optimization. If robust = TRUE the function takes the robust estimate of the wavelet variance to be used in the GMWM estimation procedure.

Value

A vec that contains the parameter estimates from GMWM estimator.

Author(s)

JJB

References

Wavelet variance based estimation for composite stochastic processes, S. Guerrier and Robust Inference for Time Series Models: a Wavelet-Based Framework, S. Guerrier

GMWM for (Robust) Inertial Measurement Units (IMUs)

Description

Performs the GMWM estimation procedure using a parameter transform and sampling scheme specific to IMUs.

Usage

gmwm_imu(model, data, compute.v = "fast", robust = F, eff = 0.6, ...)

Arguments

Details

This version of the gmwm function has customized settings ideal for modeling with an IMU object. If you seek to model with an Gauss Markov, GM, object. Please note results depend on the freq specified in the data construction step within the imu. If you wish for results to be stable but lose the ability to interpret with respect to freq, then use AR1 terms.

Value

A gmwm object with the structure:

• estimateEstimated Parameters Values from the GMWM Procedure

• init.guessInitial Starting Values given to the Optimization Algorithm

• wv.empirThe data's empirical wavelet variance

• ci_lowLower Confidence Interval

• ci_highUpper Confidence Interval

• orgVOriginal V matrix

• VUpdated V matrix (if bootstrapped)

• omegaThe V matrix inversed

• obj.funValue of the objective function at Estimated Parameter Values

• theoSummed Theoretical Wavelet Variance

• decomp.theoDecomposed Theoretical Wavelet Variance by Process

• scalesScales of the GMWM Object

• robustIndicates if parameter estimation was done under robust or classical

• effLevel of efficiency of robust estimation

• model.typeModels being guessed

• compute.vType of V matrix computation

• augmentedIndicates moments have been augmented

• alphaAlpha level used to generate confidence intervals

• expect.diffMean of the First Difference of the Signal

• NLength of the Signal

• GNumber of Guesses Performed

• HNumber of Bootstrap replications

• KNumber of V matrix bootstraps

• modelts.model supplied to gmwm

• model.hatA new value of ts.model object supplied to gmwm

• startingIndicates whether the procedure used the initial guessing approach

• seedRandomization seed used to generate the guessing values

• freqFrequency of data

Master Wrapper for the GMWM Estimator

Description

This function generates WV, GMWM Estimator, and an initial test estimate.

Usage

gmwm_master_cpp(
  data,
  theta,
  desc,
  objdesc,
  model_type,
  starting,
  alpha,
  compute_v,
  K,
  H,
  G,
  robust,
  eff
)

Arguments

Value

A field<mat> that contains a list of ever-changing estimates...

Author(s)

JJB

References

Wavelet variance based estimation for composite stochastic processes, S. Guerrier and Robust Inference for Time Series Models: a Wavelet-Based Framework, S. Guerrier

Bootstrap for Estimating Both Theta and Theta SD

Description

Using the bootstrap approach, we simulate a model based on user supplied parameters, obtain the wavelet variance, and then V.

Usage

gmwm_param_bootstrapper(
  theta,
  desc,
  objdesc,
  scales,
  model_type,
  N,
  robust,
  eff,
  alpha,
  H
)

Arguments

Details

Expand in detail...

Value

A vec that contains the parameter estimates from GMWM estimator.

Author(s)

JJB

Bootstrap for Standard Deviations of Theta Estimates

Description

Using the bootstrap approach, we simulate a model based on user supplied parameters

Usage

gmwm_sd_bootstrapper(
  theta,
  desc,
  objdesc,
  scales,
  model_type,
  N,
  robust,
  eff,
  alpha,
  H
)

Arguments

Details

Expand in detail...

Value

A vec that contains the parameter estimates from GMWM estimator.

Author(s)

JJB

Update Wrapper for the GMWM Estimator

Description

This function uses information obtained previously (e.g. WV covariance matrix) to re-estimate a different model parameterization

Usage

gmwm_update_cpp(
  theta,
  desc,
  objdesc,
  model_type,
  N,
  expect_diff,
  ranged,
  orgV,
  scales,
  wv,
  starting,
  compute_v,
  K,
  H,
  G,
  robust,
  eff
)

Arguments

Value

A field<mat> that contains the parameter estimates from GMWM estimator.

Author(s)

JJB

References

Wavelet variance based estimation for composite stochastic processes, S. Guerrier and Robust Inference for Time Series Models: a Wavelet-Based Framework, S. Guerrier

Compute the GOF Test

Description

yaya

Usage

gof_test(theta, desc, objdesc, model_type, tau, v_hat, wv_empir)

Arguments

Value

A vec that has

• Test Statistic

• P-Value

• DF

Create a simts TS object using time series data

Description

Takes a time series and turns it into a time series oriented object that can be used for summary and graphing functions in the simts package.

Usage

gts(
  data,
  start = 0,
  end = NULL,
  freq = 1,
  unit_ts = NULL,
  unit_time = NULL,
  name_ts = NULL,
  name_time = NULL,
  data_name = NULL,
  Time = NULL,
  time_format = NULL
)

Arguments

Value

A gts object

Author(s)

James Balamuta and Wenchao Yang

Examples

m = data.frame(rnorm(50))
x = gts(m, unit_time = 'sec', name_ts = 'example')
plot(x)

x = gen_gts(50, WN(sigma2 = 1))
x = gts(x, freq = 100, unit_time = 'sec')
plot(x)

Time of a gts object

Description

Extracting the time of a gts object

Usage

gts_time(x)

Value

Time vector of a gts object.

Author(s)

Stéphane Guerrier

Randomly guess a starting parameter

Description

Sets starting parameters for each of the given parameters.

Usage

guess_initial(
  desc,
  objdesc,
  model_type,
  num_param,
  expect_diff,
  N,
  wv,
  tau,
  ranged,
  G
)

Arguments

Value

A vec containing smart parameter starting guesses to be iterated over.

Randomly guess a starting parameter

Description

Sets starting parameters for each of the given parameters.

Usage

guess_initial_old(
  desc,
  objdesc,
  model_type,
  num_param,
  expect_diff,
  N,
  wv_empir,
  tau,
  B
)

Arguments

Value

A vec containing smart parameter starting guesses to be iterated over.

Haar filter construction

Description

Creates the haar filter

Usage

haar_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

Obtain the value of an object's properties

Description

Used to access different properties of the gts, imu, or lts object.

Usage

has(x, type)

## S3 method for class 'imu'
has(x, type)

Arguments

Details

To access information about imu properties use:

"accel"

Returns whether accelerometers have been specified

"gyro"

Returns whether accelerometers have been specified

"sensors"

Returns whether there exists both types of sensors

Value

The method will return a single TRUE or FALSE response

Methods (by class)

• has(imu): Access imu object properties

Author(s)

James Balamuta

Mean Monthly Precipitation, from 1907 to 1972

Description

Hydrology data that indicates a robust approach may be preferred to a classical approach when estimating time series.

Usage

hydro

Format

A time series object with frequency 12 starting at 1907 and going to 1972 for a total of 781 observations.

Source

datamarket, mean-monthly-precipitation-1907-1972

Indirect Inference for ARMA

Description

Option for indirect inference

Usage

idf_arma(ar, ma, sigma2, N, robust, eff, H)

Arguments

Value

A vec with the indirect inference results.

Indirect Inference for ARMA

Description

Option for indirect inference

Usage

idf_arma_total(ar, ma, sigma2, N, robust, eff, H)

Arguments

Value

A vec with the indirect inference results.

Create an IMU Object

Description

Builds an IMU object that provides the program with gyroscope, accelerometer, and axis information per column in the dataset.

Usage

imu(
  data,
  gyros = NULL,
  accels = NULL,
  axis = NULL,
  freq = NULL,
  unit = NULL,
  name = NULL
)

Arguments

Details

data can be a numeric vector, matrix or data frame.

gyros and accels cannot be NULL at the same time, but it will be fine if one of them is NULL. In the new implementation, the length of gyros and accels do not need to be equal.

In axis, duplicate elements are not alowed for each sensor. In the new implementation, please specify the axis for each column of data. axis will be automatically generated if there are less than or equal to 3 axises for each sensor.

Value

An imu object in the following attributes:

sensor

A vector that indicates whether data contains gyroscope sensor, accelerometer sensor, or both.

num.sensor

A vector that indicates how many columns of data are for gyroscope sensor and accelerometer sensor.

axis

Axis value such as 'X', 'Y', 'Z'.

freq

Observations per second.

unit

String representation of the unit.

name

Name of the dataset.

Author(s)

James Balamuta and Wenchao Yang

Examples

## Not run: 
if(!require("imudata")){
   install_imudata()
   library("imudata")
}

data(imu6)

# Example 1 - Only gyros
test1 = imu(imu6, gyros = 1:3, axis = c('X', 'Y', 'Z'), freq = 100)
df1 = wvar.imu(test1)
plot(df1)

# Example 2 - One gyro and one accelerometer
test2 = imu(imu6, gyros = 1, accels = 4, freq = 100)
df2 = wvar.imu(test2)
plot(df2)

# Example 3 - 3 gyros and 3 accelerometers
test3 = imu(imu6, gyros = 1:3, accels = 4:6, axis = 
                       c('X', 'Y', 'Z', 'X', 'Y', 'Z'), freq = 100)
df3 = wvar.imu(test3)
plot(df3)

# Example 4 - Custom axis
test4 = imu(imu6, gyros = 1:2, accels = 4:6, axis = 
                       c('X', 'Y', 'X', 'Y', 'Z'), freq = 100)
df4 = wvar.imu(test4)
plot(df4)

## End(Not run)

Pulls the IMU time from the IMU object

Description

Helper function for the IMU object to access rownames() with a numeric conversion.

Usage

imu_time(x)

Arguments

Value

A vector with numeric information.

Discrete Intergral: Inverse Difference

Description

Takes the inverse difference (e.g. goes from diff() result back to previous vector)

Usage

intgr_vec(x, xi, lag)

diff_inv_values(x, lag, d, xi)

diff_inv(x, lag, d)

Arguments

Check Invertibility Conditions

Description

Checks the invertiveness of series of coefficients.

Usage

invert_check(x)

Arguments

Value

True (if outside unit circle) || False (if inside unit circle)

Is simts Object

Description

Is the object a gts, imu, or lts object?

Usage

is.gts(x)

is.imu(x)

is.lts(x)

is.ts.model(x)

Arguments

Details

Uses inherits over is for speed.

Value

A logical value that indicates whether the object is of that class (TRUE) or not (FALSE).

Author(s)

James Balamuta

Integer Check

Description

Checks whether the submitted value is an integer

Usage

is.whole(x)

Arguments

Value

A boolean value indicating whether the value is an integer or not.

Author(s)

James Balamuta

Examples

is.whole(2.3)
is.whole(4)
is.whole(c(1,2,3))
is.whole(c(.4,.5,.6))
is.whole(c(7,.8,9))

Calculates the Jacobian for the ARMA process

Description

Take the numerical derivative for the first derivative of an ARMA using the 2 point rule.

Usage

jacobian_arma(theta, p, q, tau)

Arguments

Value

A mat that returns the first numerical derivative of the ARMA process.

Author(s)

James Joseph Balamuta (JJB)

la16 filter construction

Description

Creates the la16 filter

Usage

la16_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

la20 filter construction

Description

Creates the la20 filter

Usage

la20_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

la8 filter construction

Description

Creates the la8 filter

Usage

la8_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

MLR in Armadillo

Description

Perform Multiple Linear Regression using armadillo in C++

Usage

lm_arma(y, X)

Arguments

Value

A field<vec> with:

coef

Coefficients

resid

Residuals

sigma2

Sigma^2

Linear Regression with Drift

Description

Perform a linear regression with drift.

Usage

lm_dr(x)

Arguments

Value

A field<vec> with:

coef

Coefficients

resid

Residuals

sigma2

Sigma^2

Logit Function

Description

This function computes the logit link function.

Usage

logit(x)

Arguments

Value

A vec containing logit terms.

Author(s)

James Joseph Balamuta (JJB)

Logit Inverse Function

Description

This function computes the inverse of a logit transformation of the parameters.

Usage

logit_inv(x)

Arguments

Value

A vec containing logit probabilities.

Author(s)

James Joseph Balamuta (JJB)

Logit2 Function

Description

This function computes the logit2 link function.

Usage

logit2(x)

Arguments

Value

A vec containing logit terms.

Author(s)

James Joseph Balamuta (JJB)

Logit2 Inverse Function

Description

This function computes the inverse of a logit transformation of the parameters.

Usage

logit2_inv(x)

Arguments

Value

A vec containing logit probabilities.

Author(s)

James Joseph Balamuta (JJB)

Generate a Latent Time Series Object from Data

Description

Create a lts object based on a supplied matrix or data frame. The latent time series is obtained by the sum of underlying time series.

Usage

lts(
  data,
  start = 0,
  end = NULL,
  freq = 1,
  unit_ts = NULL,
  unit_time = NULL,
  name_ts = NULL,
  name_time = NULL,
  process = NULL
)

Arguments

Value

A lts object

Author(s)

Wenchao Yang and Justin Lee

Examples

model1 = AR1(phi = .99, sigma2 = 1) 
model2 = WN(sigma2 = 1)
col1 = gen_gts(1000, model1)
col2 = gen_gts(1000, model2)
testMat = cbind(col1, col2, col1+col2)
testLts = lts(testMat, unit_time = 'sec', process = c('AR1', 'WN', 'AR1+WN'))
plot(testLts)

Definition of a Mean deterministic vector returned by the matrix by vector product of matrix X and vector \beta

Description

Definition of a Mean deterministic vector returned by the matrix by vector product of matrix X and vector \beta

Usage

M(X, beta)

Arguments

Value

An S3 object containing the specified ts.model with the following structure:

process.desc

Used in summary: "X","BETA"

theta

Matrix X, vector beta

plength

Number of parameters

print

String containing simplified model

desc

"M"

obj.desc

Depth of Parameters e.g. list(1,1)

starting

Find starting values? TRUE or FALSE (e.g. specified value)

Author(s)

Lionel Voirol, Davide Cucci

Examples

X = matrix(rnorm(15*5), nrow = 15, ncol = 5)
beta=seq(5)
M(X = X, beta = beta)

Second moment DR

Description

This function computes the second moment of a drift process.

Usage

m2_drift(omega, n_ts)

Arguments

Value

A vec containing the second moment of the drift.

Create an Moving Average Q [MA(Q)] Process

Description

Sets up the necessary backend for the MA(Q) process.

Usage

MA(theta = NULL, sigma2 = 1)

Arguments

Value

An S3 object with called ts.model with the following structure:

process.desc

Used in summary: "MA-1","MA-2", ..., "MA-Q", "SIGMA2"

theta

\theta_1, \theta_2, ..., \theta_q, \sigma^2

plength

Number of parameters

desc

"MA"

print

String containing simplified model

obj.desc

Depth of parameters e.g. list(q,1)

starting

Guess starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

X_t = \sum_{j = 1}^q \theta_j \varepsilon_{t-1} + \varepsilon_t

, where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2.

Author(s)

James Balamuta

Examples

MA(1) # One theta
MA(2) # Two thetas!

MA(theta=.32, sigma=1.3) # 1 theta with a specific value.
MA(theta=c(.3,.5), sigma=.3) # 2 thetas with specific values.

Ma function.

Description

Ma function.

Usage

Ma_cpp(x, alpha)

Arguments

Ma vectorized function.

Description

Ma vectorized function.

Usage

Ma_cpp_vec(x, alpha)

Arguments

Definition of an Moving Average Process of Order 1

Description

Definition of an Moving Average Process of Order 1

Usage

MA1(theta = NULL, sigma2 = 1)

Arguments

Value

An S3 object with called ts.model with the following structure:

process.desc

Used in summary: "MA1","SIGMA2"

theta

\theta, \sigma^2

plength

Number of parameters

print

String containing simplified model

desc

"MA1"

obj.desc

Depth of parameters e.g. list(1,1)

starting

Guess starting values? TRUE or FALSE (e.g. specified value)

Note

We consider the following model:

X_t = \theta \varepsilon_{t-1} + \varepsilon_t

, where \varepsilon_t is iid from a zero mean normal distribution with variance \sigma^2.

Author(s)

James Balamuta

Examples

MA1()
MA1(theta = .32, sigma2 = 1.3)

Moving Average Order 1 (MA(1)) to WV

Description

This function computes the WV (haar) of a Moving Average order 1 (MA1) process.

Usage

ma1_to_wv(theta, sigma2, tau)

Arguments

Details

This function is significantly faster than its generalized counter part arma_to_wv.

Value

A vec containing the wavelet variance of the MA(1) process.

Process Haar Wavelet Variance Formula

The Moving Average Order 1 (MA(1)) process has a Haar Wavelet Variance given by:

\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = \frac{{\left( {{{\left( {\theta + 1} \right)}^2}{\tau _j} - 6\theta } \right){\sigma ^2}}}{{\tau _j^2}}

Default utility function for various plots titles

Description

Adds title, grid, and required x- and y-axes.

Usage

make_frame(
  x_range,
  y_range,
  xlab,
  ylab,
  main = "",
  mar = c(5.1, 5.1, 1, 2.1),
  add_axis_x = TRUE,
  add_axis_y = TRUE,
  col_box = "black",
  col_grid = "grey95",
  col_band = "grey95",
  col_title = "black",
  add_band = TRUE,
  title_band_width = 0.09,
  grid_lty = 1
)

Arguments

Value

Added title, grid, and axes.

Author(s)

Stephane Guerrier and Justin Lee

Examples

make_frame(x_range = c(0, 1), y_range = c(0, 1), xlab = "my xlab", 
           ylab = "my ylab", main = "my title")
           
make_frame(x_range = c(0, 1), y_range = c(0, 1), xlab = "my xlab", 
           ylab = "my ylab", add_band = FALSE)
           
make_frame(x_range = c(0, 1), y_range = c(0, 1), xlab = "my xlab", 
           ylab = "my ylab", main = "my title", col_band = "blue3", 
           col_title = "white", col_grid = "lightblue", grid_lty = 3)

make_frame(x_range = c(0, 1), y_range = c(0, 1), xlab = "my xlab", 
           ylab = "my ylab", main = "my title", col_band = "blue3", 
           col_title = "white", col_grid = "lightblue", grid_lty = 3,
           title_band_width = 0.18)

Median Absolute Prediction Error

Description

This function calculates Median Absolute Prediction Error (MAPE), which assesses the prediction performance with respect to point forecasts of a given model. It is calculated based on one-step ahead prediction and reforecasting.

Usage

MAPE(model, Xt, start = 0.8, plot = TRUE)

Arguments

Value

The MAPE calculated based on one-step ahead prediction and reforecasting is returned along with its standard deviation.

Author(s)

Stéphane Guerrier and Yuming Zhang

Definition of a Matérn Process

Description

Definition of a Matérn Process

Usage

MAT(sigma2 = 1, lambda = 0.35, alpha = 0.9)

Arguments

Value

An S3 object containing the specified ts.model with the following structure:

process.desc

Used in summary: "SIGMA2","LAMBDA""ALPHA"

theta

Parameter vector including \sigma^2, \lambda,\alpha

plength

Number of parameters

print

String containing simplified model

desc

"MAT"

obj.desc

Depth of Parameters e.g. list(1,1,1)

starting

Find starting values? TRUE or FALSE (e.g. specified value)

Author(s)

Lionel Voirol, Davide Cucci

Examples

MAT()
MAT(sigma2 = 1, lambda = 0.35, alpha = 0.9)

mb16 filter construction

Description

Creates the mb16 filter

Usage

mb16_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

mb24 filter construction

Description

Creates the mb24 filter

Usage

mb24_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

mb4 filter construction

Description

Creates the mb4 filter

Usage

mb4_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

mb8 filter construction

Description

Creates the mb8 filter

Usage

mb8_filter()

Details

This template can be used to increase the amount of filters available for selection.

Value

A field<vec> that contains:

• "L"A integer specifying the length of the filter

• "h"A vector containing the coefficients for the wavelet filter

• "g"A vector containing the coefficients for the scaling filter

Author(s)

JJB

Mean of the First Difference of the Data

Description

The mean of the first difference of the data

Usage

mean_diff(x)

Arguments

Value

A double that contains the mean of the first difference of the data.

Obtain the smallest polynomial root

Description

Calculates all the roots of a polynomial and returns the root that is the smallest.

Usage

minroot(x)

Arguments

Value

A double with the minimum root value.

Absolute Value or Modulus of a Complex Number.

Description

Computes the value of the Modulus.

Usage

Mod_cpp(x)

Arguments

Details

Consider a complex number defined as: z = x + i y with real x and y, The modulus is defined as: r = Mod(z) = \sqrt{(x^2 + y^2)}