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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 Guerrierstef.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 Balamutabalamut2@illinois.edu,Stephane Guerrierstef.guerrier@gmail.com,Roberto Molinariroberto.molinari@hotmail.it,Wenchao Yangwyang40@illinois.eduJustin Leemunsheet93@gmail.comYuming Zhangyuming.zhang@unige.ch

See Also

Useful links:

• https://github.com/SMAC-Group/simts

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

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

Useful links:

• https://github.com/SMAC-Group/simts

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

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

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()*2AR1(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:

Author(s)

James Balamuta

Examples

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

Auto-Covariance and Correlation Functions

Description

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

Usage

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

Arguments

Akaike's Information Criterion

Description

This function calculates AIC, BIC or HQ for a fitsimts object. This function currentlyonly 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 = 300Xt = gen_gts(n, AR(phi = c(0, 0, 0.8), sigma2 = 1))mod = estimate(AR(3), Xt)# AICAIC(mod)# BICAIC(mod, k = log(n))# HQAIC(mod, k = 2*log(log(n)))

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)

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) processARIMA(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 deviationARIMA(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) processARMA(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 deviationARMA(ar=c(0.23,.43, .59), ma=c(0.4,.3), sigma2 = 1.5)

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 orFALSE (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()

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 Amatrix 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

Acolumn vector containing coefficients

Author(s)

R Core Team and JJB

B Matrix

Description

B Matrix

Usage

B_matrix(A, at_omega)

Arguments

Value

Amat

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)

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

Amatrix with the process derivatives going down the column

Author(s)

James Joseph Balamuta (JJB)

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)

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)

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

Description

Definition of a Mean deterministic vector returned by the matrix by vector product of matrixX 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)

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 thetaMA(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.

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)

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 reforecastingis 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)

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

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 realx andy,The modulus is defined as:r = Mod(z) = \sqrt{(x^2 + y^2)}

Value

Avec containing the modulus for each element.

Definition of a Power Law Process

Description

Definition of a Power Law Process

Usage

PLP(sigma2 = 1, d = 0.4)

Arguments

Value

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

process.desc

Used in summary: "SIGMA2","d"

theta

Parameter vector including\sigma^2,d

plength

Number of parameters

print

String containing simplified model

desc

"PLP"

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

PLP()PLP(sigma2 = 1, d = 0.4)

Create an Quantisation Noise (QN) Process

Description

Sets up the necessary backend for the QN process.

Usage

QN(q2 = NULL)

Arguments

Value

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

process.desc

Used in summary: "QN"

theta

Q^2

plength

Number of parameters

print

String containing simplified model

desc

y desc replicated x times

obj.desc

Depth of parameters e.g. list(1)

starting

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

Author(s)

James Balamuta

Examples

QN()QN(q2=3.4)

Create an Random Walk (RW) Process

Description

Sets up the necessary backend for the RW process.

Usage

RW(gamma2 = NULL)

Arguments

Value

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

process.desc

Used in summary: "RW"

theta

\sigma

plength

Number of parameters

print

String containing simplified model

desc

y desc replicated x times

obj.desc

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 = \sum\nolimits_{t=0}^{T} \gamma_0*Z_t

whereZ_t is iid and follows a standard normal distribution.

Author(s)

James Balamuta

Examples

RW()RW(gamma2=3.4)

Function to Compute Direction Random Walk Moves

Description

The RW2dimension function computes direction random walk moves.

Usage

RW2dimension(steps = 100, probs = c(0.25, 0.5, 0.75))

Arguments

Author(s)

Stéphane Guerrier

Examples

RW2dimension(steps = 50, probs = c(0.2, 0.5, 0.6))

Hook into R's ARIMA function

Description

Uses R's ARIMA function to obtain CSS values for starting condition

Usage

Rcpp_ARIMA(data, params)

Arguments

Value

Avec containing the CSS of the ARMA parameters.

Create a Seasonal Autoregressive Integrated Moving Average (SARIMA) Process

Description

Sets up the necessary backend for the SARIMA process.

Usage

SARIMA(ar = 1, i = 0, ma = 1, sar = 1, si = 0, sma = 1, s = 12, sigma2 = 1)

Arguments

Details

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

Value

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

process.desc

AR*p,MA*q,SAR*P,SMA*Q

theta

\sigma

plength

Number of parameters

desc

Type of model

desc.simple

Type of model (after simplification)

print

String containing simplified model

obj.desc

y desc replicated x times

obj

Depth of Parameters e.g. list(c(length(ar), length(ma), length(sar), length(sma), 1, i, si) )

starting

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

Author(s)

James Balamuta

Examples

# Create an SARIMA(1,1,2)x(1,0,1) processSARIMA(ar = 1, i = 1, ma = 2, sar = 1, si = 0, sma =1)# Creates an SARMA(1,0,1)x(1,1,1) process with predefined coefficients.SARIMA(ar=0.23, i = 0, ma=0.4, sar = .3,  sma = .3)

Create a Seasonal Autoregressive Moving Average (SARMA) Process

Description

Sets up the necessary backend for the SARMA process.

Usage

SARMA(ar = 1, ma = 1, sar = 1, sma = 1, s = 12, sigma2 = 1)

Arguments

Details

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

Value

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

process.desc

AR*p,MA*q,SAR*P,SMA*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), length(sar), length(sma), 1) )

starting

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

Author(s)

James Balamuta

Examples

# Create an SARMA(1,2)x(1,1) processSARMA(ar = 1, ma = 2,sar = 1, sma =1)# Creates an SARMA(1,1)x(1,1) process with predefined coefficients.SARMA(ar=0.23, ma=0.4, sar = .3, sma = .3)

Definition of a Sinusoidal (SIN) Process

Description

Definition of a Sinusoidal (SIN) Process

Usage

SIN(alpha2 = 9e-04, beta = 0.06, U = NULL)

Arguments

Value

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

process.desc

Used in summary: "ALPHA2","BETA"

theta

Parameter vector including\alpha^2,\beta

plength

Number of parameters

print

String containing simplified model

desc

"SIN"

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 sinusoidal process :

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

, whereU \sim \mathcal{U}(0, 2\pi)and\beta \in (0, \frac{\pi}{2})

Author(s)

Lionel Voirol

Examples

SIN()SIN(alpha2 = .5, beta = .05)

Create an White Noise (WN) Process

Description

Sets up the necessary backend for the WN process.

Usage

WN(sigma2 = NULL)

Arguments

Value

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

process.desc

Used in summary: "WN"

theta

\sigma

plength

Number of Parameters

print

String containing simplified model

desc

y desc replicated x times

obj.desc

Depth of Parameters e.g. list(1)

starting

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

Note

In this process,Y_t is iid from a zero mean normal distribution with variance\sigma^2

Author(s)

James Balamuta

Examples

WN()WN(sigma2=3.4)

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

Animu 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 Yex.b = ex[,c(1,2,4,5)]ex.b2 = ex[,c("Gyro. X","Gyro. Y","Accel. X","Accel. Y")]## End(Not run)

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

Avec containing the wavelet variance of the ARMA process.

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

Avec that contains the parameter estimates from GMWM estimator.

Author(s)

JJB

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

Avec 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 ^2and 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

Avec 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 partarma_to_wv.

Value

Avec containing the wavelet variance of the AR(1) process.

Process Haar Wavelet Variance Formula

The Autoregressive Order1 (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}}

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 partarma_to_wv

Value

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

Process Haar Wavelet Variance Formula

The Autoregressive Order1 and Moving Average Order1 (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}}

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

Avec 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

Avec 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

Avec containing the wavelet variance of the ARMA process.

Process Haar Wavelet Variance Formula

The Autoregressive Orderp and Moving Average Orderq (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 thearma_to_wv as computation timeswere previously a concern. However, this is no longer the case and, thus, this has been leftin for the curious soul to discover...

Value

Avec containing the wavelet variance of the ARMA process.

Process Haar Wavelet Variance Formula

The Autoregressive Orderp and Moving Average Orderq (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

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 is10*log10(N/m) whereN is the number ofobservations andm is the number of time series being compared. Iflagmax supplied is greater than the number of observations N, then oneless than the total will be taken (i.e. N - 1).

Value

Anarray of dimensionsN \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

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

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

Afield<mat> with the structure:

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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Avec 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

Avec 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

Afield<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

Aset<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

Amat 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

Acolumn 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

Amatrix with the structure:

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

Amatrix with the structure:

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

Amatrix with the structure:

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

Avec 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 ACFof 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 functionscompare_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

Afield<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 PACFof univariate time series.

Usage

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

Arguments

Value

Twoarray objects (ACF and PACF) of dimensionN \times S \times S.

Author(s)

Yunxiang Zhang

Examples

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

Count Models

Description

Count the amount of models that exist.

Usage

count_models(desc)

Arguments

Value

Amap<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

Avec 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

Animu object class.

Author(s)

James Balamuta

Custom legend function

Description

Legend placement function

Usage

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

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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Amat 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

Avec 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

Amatrix with the first column containing thesecond partial derivative with respect to\phi andthe 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

Amatrix with:

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

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

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

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

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

• Thesixth 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

Amatrix 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

Amatrix 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

Amatrix 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

Amatrix with:

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

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

• Thethird 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

Amatrix 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\omegarelates 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

Amatrix with the first column containing the partial derivative with respect to\thetaand 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

Amatrix with the first column containing the partial derivative with respect toQ^2.

Process Haar WV First Derivative

Taking the derivative with respect toQ^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

Amatrix with the first column containingthe 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

Amatrix 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

Amatrix 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

Avec containing the ACF.

Box-Pierce

Description

Performs the Box-Pierce test to assess the Null Hypothesis of Independencein 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 Independencein 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 Independencein 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

Avector 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

Drift to WV

Description

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

Usage

dr_to_wv(omega, tau)

Arguments

Value

Avec 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 Afield<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

Avec 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 seriesmodels. 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 = 300Xt = 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.

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

Amat 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

Avector<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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Afield<vec> that contains:

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

Amat 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)'sx_t valueswithout calling the general ARMA function.Thus, the function is able to generate values much faster thangen_arma.

Value

Avec 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 lengthN+1 usinggen_wn and then obtains theautoregressive values under the above process definition.

TheX_0 (first value ofX_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

Avector 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), Integratedd, 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

Avec that contains the generated observations.

Warning

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

Generate Autoregressive Orderp - Moving Average Orderq (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

ForAR(1),MA(1), andARMA(1,1) please use their functions if speed is importantas this function is designed to generate generic ARMA processes.

Value

Avec that contains the generated observations.

Process Definition

The Autoregressive orderp and Moving Average orderq (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 thex_t values without calling the general ARMA function.

Value

Avec 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 usinggen_wn and then obtains theARMA values under the above equation.

TheX_0 (first value ofX_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

Avector containing the bias-instability process.

Note

This function generates a non-stationary bias-instability processwhose theoretical maximum overlapping allan variance (MOAV) is close to the theoreticalMOAV 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

Avec 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 avector with the\omega parameter.After, we take the cumulative sum along the vector.

Generate a Fractional Gaussian noise given\sigma^2 andH.

Description

Simulates a Fractional Gaussian noise given\sigma^2 andH.

Usage

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

Arguments

Value

fgn Avec 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

Avec that contains the generated observations.

Simulate a simts TS object using a theoretical model

Description

Create agts 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 ats.model object (e.g. AR1(phi = .3, sigma2 =1) + WN(sigma2 = 1)) or asimts object.

Value

Agts object

Author(s)

James Balamuta and Wenchao Yang

Examples

# Set seed for reproducibilityset.seed(1336)n = 1000# AR1 + WNmodel = AR1(phi = .5, sigma2 = .1) + WN(sigma2=1)x = gen_gts(n, model)plot(x)# Reset seedset.seed(1336)# GM + WN# Convert from AR1 to GM valuesm = ar1_to_gm(c(.5,.1),10)# Beta = 6.9314718, Sigma2_gm = 0.1333333model = 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 seriesall.equal(x, x2, check.attributes = FALSE)

Generate a Latent Time Series Object Based on a Model

Description

Simulate alts 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 ats.model object (e.g. AR1(phi = .3, sigma2 =1) + WN(sigma2 = 1)) or asimts object.

Value

Alts 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

Avector that contains model names of decomposed and combined processes

Author(s)

James Balamuta, Wenchao Yang, and Justin Lee

Examples

# ARset.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

Amat 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 thex_t values without calling the general ARMA function.

Value

Avec 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 usinggen_wn and then obtains theMA values under the above equation.

TheX_0 (first value ofX_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 Avec containing the Matern Process.

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

Description

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

Usage

gen_mean(X, beta)

Arguments

Value

mean_vec Avec containing the determinist vector.

Generate Time Series based on Model (Internal)

Description

Create a time series process based on a suppliedts.model.

Usage

gen_model(N, theta, desc, objdesc)

Arguments

Value

Avec 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

Avector 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 andd.

Description

Simulates a a Power Law Process given\sigma^2 andd.

Usage

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

Arguments

Value

plp Avec containing the Power Law Process.

Generate a Quantisation Noise (QN) or Rounding Error Sequence

Description

Simulates a QN sequence givenQ^2.

Usage

gen_qn(N, q2 = 0.1)

Arguments

Value

Avec containing the QN process.

Process Definition

Quantization Noise (QN) with parameterQ^2 \in R^{+}. With i.i.dY_t \sim U(0,1) (i.e. a standard uniform variable), this process isdefined 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 Avec 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 calledRate 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 sampleN times from aN(0,\sigma^2) distribution. Lastly, we take thecumulative 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

Avec 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

Avec 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 Avec 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 Avec containing the white noise.

Process Definition

Gaussian White Noise (WN) with parameter\sigma^2 \in {\rm I\!R}^{+}.This process is defined asX_t\sim N(0,\sigma^2) andis 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 sampleN times from aN(0,\sigma ^2) distribution.

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

Adouble 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

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

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

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

Transform GM to AR1

Description

Takes GM values and transforms them to AR1

Usage

gm_to_ar1(theta, freq)

Arguments

Value

Avec containing GM values.

Author(s)

James BalamutaThe function takes a vector of GM values\beta and\sigma ^2_{gm}and transforms them to AR1 values\phi and\sigma ^2using 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)^25. Optimize the values to obtain\hat{\theta}6. If FAST = TRUE, return these results. Else, continue.

Loop k = 1 to KLoop h = 1 to H7. Simulate xt underF_{\hat{\theta}}8. Compute WV EmpiricalEND9. Calculate the covariance matrix10. Optimize the values to obtain\hat{\theta}END11. Return optimized values.

The function estimates a variety of time series models. If type = "imu" or "ssm", thenparameter vector should indicate the characters of the models that compose the latent or state-space model. The modeloptions 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 parameterQ

• "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 valuesIf robust = TRUE the function takes the robust estimate of the wavelet variance to be used in the GMWM estimation procedure.

Value

Agmwm 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 ofts.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 parameterQ

"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" thenstarting 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

Avec 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 samplingscheme 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 settingsideal for modeling with an IMU object. If you seek to model with an GaussMarkov,GM, object. Please note results depend on thefreq specified in the data construction step within theimu. If you wish for results to be stable but lose theability to interpret with respect tofreq, then useAR1 terms.

Value

Agmwm object with the structure:

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

Afield<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

Avec 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

Avec 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

Afield<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

Avec 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 thesimts 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

Agts 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

Avec 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

Avec 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

Afield<vec> that contains:

Author(s)

JJB

Obtain the value of an object's properties

Description

Used to access different properties of thegts,imu, orlts object.

Usage

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

Arguments

Details

To access information aboutimu 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): Accessimu 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 aclassical approach when estimating time series.

Usage

hydro

Format

A time series object with frequency 12 starting at 1907 and going to 1972for 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

Avec 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

Avec 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 andaccels cannot beNULL at the same time, but it will be fine if one of them isNULL.In the new implementation, the length ofgyros andaccels do not need to be equal.

Inaxis, 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

Animu object in the following attributes:

sensor

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

num.sensor

Avector 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)