Package 'simts'

Title: 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.
Authors: Stéphane Guerrier [aut, cre, cph], James Balamuta [aut, cph], Roberto Molinari [aut, cph], Justin Lee [aut], Lionel Voirol [aut], Yuming Zhang [aut], Wenchao Yang [ctb], Nathanael Claussen [ctb], Yunxiang Zhang [ctb], Christian Gunning [cph], Romain Francois [cph], Ross Ihaka [cph], R Core Team [cph]
Maintainer: Stéphane Guerrier <[email protected]>
License: AGPL-3 | file LICENSE
Version: 0.2.2
Built: 2024-11-23 06:54:42 UTC
Source: CRAN

Help Index

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

x

A imu object
i

A integer vector that specifies the rows to subset. If blank, all rows are selected.
j

A integer vector that specifies the columns to subset. Special rules apply see details.
drop

A boolean indicating whether the structure should be preserved or simplified.

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)

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

object

A fitsimts object.
k

The penalty per parameter to be used; the default k = 2 is the classical AIC.
...

Optionally more fitted model objects.

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)))

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

phi

A vector with double values for the ϕ\phi of an AR(P) process (see Note for details).
sigma2

A double value for the variance, σ2\sigma ^2, of an AR(P) process. (see Note for details).

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

ϕ1\phi_1, ϕ2\phi_2, ..., ϕp\phi_p, σ2\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:

Xt=∑j=1pϕjXt−1+εtX_t = \sum_{j = 1}^p \phi_j X_{t-1} + \varepsilon_t

, where εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2\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

phi

A double value for the parameter ϕ\phi (see Note for details).
sigma2

A double value for the variance parameter σ2\sigma ^2 (see Note for details).

Value

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

process.desc

Used in summary: "AR1","SIGMA2"

theta

Parameter vector including ϕ\phi, σ2\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:

Xt=ϕXt−1+εtX_t = \phi X_{t-1} + \varepsilon_t

, where εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2\sigma^2.

Author(s)

James Balamuta

Examples

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

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

phi

A double that is the phi term of the AR(1) process
sigma2

A double corresponding to variance of AR(1) process
tau

A vec containing the scales e.g. 2τ2^{\tau}

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 11 (AR(11)) process has a Haar Wavelet Variance given by:

2σ2(4ϕτj2+1−ϕτj+1−12ϕ2τj+τj2−3ϕ)(1−ϕ)2(1−ϕ2)τj2\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

ar

A vector or integer containing either the coefficients for ϕ\phi's or the process number pp for the Autoregressive (AR) term.
i

An integer containing the number of differences to be done.
ma

A vector or integer containing either the coefficients for θ\theta's or the process number qq for the Moving Average (MA) term.
sigma2

A double value for the standard deviation, σ\sigma, of the ARIMA process.

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∗pAR*p, MA∗qMA*q

theta

σ\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:

ΔiXt=∑j=1pϕjΔiXt−j+∑j=1qθjεt−j+εt\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 εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2\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

ar

A vector or integer containing either the coefficients for ϕ\phi's or the process number pp for the Autoregressive (AR) term.
ma

A vector or integer containing either the coefficients for θ\theta's or the process number qq for the Moving Average (MA) term.
sigma2

A double value for the standard deviation, σ\sigma, of the ARMA process.

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∗pAR*p, MA∗qMA*q

theta

σ\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:

Xt=∑j=1pϕjXt−j+∑j=1qθjεt−j+εtX_t = \sum_{j = 1}^p \phi_j X_{t-j} + \sum_{j = 1}^q \theta_j \varepsilon_{t-j} + \varepsilon_t

, where εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2\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 process to WV

Description

This function computes the Haar Wavelet Variance of an ARMA process

Usage

arma_to_wv(ar, ma, sigma2, tau)

Arguments

ar

A vec containing the coefficients of the AR process
ma

A vec containing the coefficients of the MA process
sigma2

A double containing the residual variance
tau

A vec containing the scales e.g. 2τ2^{\tau}

Details

The function is a generic implementation that requires a stationary theoretical autocorrelation function (ACF) and the ability to transform an ARMA(pp,qq) 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 pp and Moving Average Order qq (ARMA(pp,qq)) process has a Haar Wavelet Variance given by:

τj[1−ρ(τj2)]+2∑i=1τj2−1i[2ρ(τj2−i)−ρ(i)−ρ(τj−i)]τj2σX2\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 σX2\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

phi

A double containing the parameter ϕ1\phi _1 (see Note for details).
theta

A double containing the parameter θ1\theta _1 (see Note for details).
sigma2

A double value for the parameter σ2\sigma^2 (see Note for details).

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

AR1AR1, MA1MA1, SIGMA2SIGMA2

theta

ϕ\phi, θ\theta, σ2\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:

Xt=ϕXt−1+θ1εt−1+εt,X_t = \phi X_{t-1} + \theta_1 \varepsilon_{t-1} + \varepsilon_t,

where εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2\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

phi

A double corresponding to the autoregressive term.
theta

A double corresponding to the moving average term.
sigma2

A double the variance of the process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

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 11 and Moving Average Order 11 (ARMA(11,11)) process has a Haar Wavelet Variance given by:

νj2(ϕ,θ,σ2)=−2σ2(−12(θ+1)2(ϕ2−1)τj−(θ+ϕ)(θϕ+1)(ϕτj−4ϕτj2+3))(ϕ−1)3(ϕ+1)τj2\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}}

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

x

A vector or ts object (of length N>1N > 1).
lag.max

An integer indicating the maximum lag up to which to compute the empirical ACF / PACF.
pacf

A boolean indicating whether to output the PACF. If it's TRUE, then the function will only estimate the empirical PACF. If it's FALSE (the default), then the function will only estimate the empirical ACF.
type

A character string giving the type of acf to be computed. Allowed values are "correlation" (the default) and "covariance".
demean

A boolean indicating whether the data should be detrended (TRUE) or not (FALSE). Defaults to TRUE.
robust

A boolean indicating whether a robust estimator should be used (TRUE) or not (FALSE). Defaults to FALSE. This only works when the function is estimating ACF.

Details

lagmax default is 10∗log10(N/m)10*log10(N/m) where NN is the number of observations and mm 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×1×1N \times 1 \times 1.

Author(s)

Yuming Zhang

Examples

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

Select the Best Model

Description

This function retrieves the best model from a selection procedure.

Usage

best_model(x, ic = "aic")

Arguments

x

An object of class select_arma, select_ar or select_ma.
ic

A string indicating the type of criterion to use in selecting the best model. Supported criteria include "aic" (AIC), "bic" (BIC) and "hq" (HQ).

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")

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

model

A fitsimts, lm or gam object.
resids

A vector of residuals for diagnostics.
simple

A boolean indicating whether to return simple diagnostic plots or not.

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)

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

x

A vector or "ts" object (of length N>1N > 1).
lag.max

A integer indicating the maximum lag up to which to compute the ACF and PACF functions.
demean

A bool indicating whether the data should be detrended (TRUE) or not (FALSE). Defaults to TRUE.
show.ci

A bool indicating whether to compute and show the confidence region. Defaults to TRUE.
alpha

A double indicating the level of significance for the confidence interval. By default alpha = 0.05 which gives a 1 - alpha = 0.95 confidence interval.
plot

A bool indicating whether a plot of the computed quantities should be produced. Defaults to TRUE.
...

Additional parameters.

Author(s)

Yunxiang Zhang

Examples

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

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

x

A vector or "ts" object (of length N>1N > 1).
lag.max

A integer indicating the maximum lag up to which to compute the ACF and PACF functions.
type

A character string giving the type of acf to be computed. Allowed values are "correlation" (the default) and "covariance".
demean

A bool indicating whether the data should be detrended (TRUE) or not (FALSE). Defaults to TRUE.
show.ci

A bool indicating whether to compute and show the confidence region. Defaults to TRUE.
alpha

A double indicating the level of significance for the confidence interval. By default alpha = 0.05 which gives a 1 - alpha = 0.95 confidence interval.
plot

A bool indicating whether a plot of the computed quantities should be produced. Defaults to TRUE.
...

Additional parameters.

Value

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

Author(s)

Yunxiang Zhang

Examples

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

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

phi

A double corresponding to the phi coefficient of an AR(1) process.
sigma2

A double corresponding to the error term of an AR(1) process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the second partial derivative with respect to ϕ\phi and the second column contains the second partial derivative with respect to σ2\sigma ^2

Process Haar WV Second Derivative

Taking the second derivative with respect to ϕ\phi yields:

∂2∂ϕ2νj2(ϕ,σ2)=2σ2((ϕ2−1)τj(2(ϕ(7ϕ+4)+1)ϕτj2−1−(ϕ(7ϕ+4)+1)ϕτj−1+3(ϕ+1)2)+(ϕ2−1)2τj2(ϕτj2−1)ϕτj2−1+4(3ϕ+1)(ϕ2+ϕ+1)(ϕτj−4ϕτj2+3))(ϕ−1)5(ϕ+1)3τj2\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 σ2\sigma^2 yields:

∂2∂σ4νj2(σ2)=0\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( \sigma ^2 \right) = 0

Taking the derivative with respect to ϕ\phi and σ2\sigma ^2 yields:

∂2∂ϕ∂σ2νj2(ϕ,σ2)=2((ϕ2−1)τj(ϕτj−2ϕτj2−ϕ−1)−(ϕ(3ϕ+2)+1)(ϕτj−4ϕτj2+3))(ϕ−1)4(ϕ+1)2τj2\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

phi

A double corresponding to the phi coefficient of an ARMA(1,1) process.
theta

A double corresponding to the theta coefficient of an ARMA(1,1) process.
sigma2

A double corresponding to the error term of an ARMA(1,1) process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with:

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

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

  • The third column contains the second partial derivative with respect to σ2\sigma ^2.

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

  • The fiveth column contains the partial derivative with respect to σ2\sigma ^2 and ϕ\phi.

  • The sixth column contains the partial derivative with respect to σ2\sigma ^2 and θ\theta.

Process Haar WV Second Derivative

Taking the second derivative with respect to ϕ\phi yields:

∂2∂ϕ2νj2(ϕ,θ,σ2)=2σ2(ϕ−1)5(ϕ+1)3τj2((ϕ−1)2((ϕ+1)2(θ2ϕ+θϕ2+θ+ϕ)τj2(ϕτj2−1)ϕτj2−2+(ϕ2−1)(θ2(−ϕ)+θ(ϕ2+4ϕ+1)−ϕ)τj(ϕτj2−2)ϕτj2−2−2(θ−1)2(ϕτj−4ϕτj2+3))−12(ϕ+1)2(−12(θ+1)2(ϕ2−1)τj−(θ+ϕ)(θϕ+1)(ϕτj−4ϕτj2+3))+6(ϕ+1)(ϕ−1)(12(θ+1)2(ϕ2−1)τj+(θ+ϕ)(θϕ+1)(ϕτj−4ϕτj2+3)+(ϕ+1)(−(θ+ϕ)(θϕ+1)τj(ϕτj2−2)ϕτj2−1−θ(θ+ϕ)(ϕτj−4ϕτj2+3)−(θϕ+1)(ϕτj−4ϕτj2+3)−(θ+1)2ϕτj)))\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 θ\theta yields:

∂2∂θ2νj2(ϕ,θ,σ2)=2σ2((ϕ2−1)τj+2ϕ(ϕτj−4ϕτj2+3))(ϕ−1)3(ϕ+1)τj2\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 σ2\sigma ^2 yields:

∂2∂σ4νj2(ϕ,θ,σ2)=0\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\phi ,\theta ,{\sigma ^2}} \right) = 0

Taking the derivative with respect to σ2\sigma^2 and θ\theta yields:

∂∂θ∂∂σ2νj2(ϕ,θ,σ2)=2(ϕ−1)3(ϕ+1)τj2((θ+1)(ϕ2−1)τj+(2θϕ+ϕ2+1)(ϕτj−4ϕτj2+3))\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 σ2\sigma^2 and ϕ\phi yields:

∂∂ϕ∂∂σ2νj2(ϕ,θ,σ2)=2(ϕ−1)4(ϕ+1)2τj2(−(ϕ−1)(ϕ+1)(−(θ+ϕ)(θϕ+1)τj(ϕτj2−2)ϕτj2−1−θ(θ+ϕ)(ϕτj−4ϕτj2+3)−(θϕ+1)(ϕτj−4ϕτj2+3)−(θ+1)2ϕτj)+(ϕ−1)(−12(θ+1)2(ϕ2−1)τj−(θ+ϕ)(θϕ+1)(ϕτj−4ϕτj2+3))+3(ϕ+1)(−12(θ+1)2(ϕ2−1)τj−(θ+ϕ)(θϕ+1)(ϕτj−4ϕτj2+3)))\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 ϕ\phi and θ\theta yields:

∂∂θ∂∂ϕνj2(ϕ,θ,σ2)=−2σ2(ϕ−1)4(ϕ+1)2τj2(τj(2(θ+1)(ϕ−1)(ϕ+1)2+2(ϕ2−1)(2θϕ+ϕ2+1)ϕτj2−1−(ϕ2−1)(2θϕ+ϕ2+1)ϕτj−1)+2(θ(ϕ(3ϕ+2)+1)+ϕ(ϕ2+ϕ+3)+1)(ϕτj−4ϕτj2+3))\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

tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

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

theta

A double corresponding to the theta coefficient of an MA(1) process.
sigma2

A double corresponding to the error term of an MA(1) process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the second partial derivative with respect to θ\theta, the second column contains the partial derivative with respect to θ\theta and σ2\sigma ^2, and lastly we have the second partial derivative with respect to σ2\sigma ^2.

Process Haar WV Second Derivative

Taking the second derivative with respect to θ\theta yields:

∂2∂θ2νj2(θ,σ2)=2σ2τj\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 σ2\sigma^2 yields:

∂2∂σ4νj2(θ,σ2)=0\frac{{{\partial ^2}}}{{\partial {\sigma ^4}}}\nu _j^2\left( {\theta ,{\sigma ^2}} \right) = 0

Taking the first derivative with respect to θ\theta and σ2\sigma^2 yields:

∂∂θ∂∂σ2νj2(θ,σ2)=2(θ+1)τj−6τj2\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

phi

A double corresponding to the phi coefficient of an AR(1) process.
sigma2

A double corresponding to the error term of an AR(1) process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the partial derivative with respect to ϕ\phi and the second column contains the partial derivative with respect to σ2\sigma ^2

Process Haar WV First Derivative

Taking the derivative with respect to ϕ\phi yields:

∂∂ϕνj2(ϕ,σ2)=2σ2((ϕ2−1)τj(−2ϕτj2+ϕτj−ϕ−1)−(ϕ(3ϕ+2)+1)(−4ϕτj2+ϕτj+3))(ϕ−1)4(ϕ+1)2τj2\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 σ2\sigma ^2 yields:

∂∂σ2νj2(ϕ,σ2)=(ϕ2−1)τj+2ϕ(−4ϕτj2+ϕτj+3)(ϕ−1)3(ϕ+1)τj2\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

phi

A double corresponding to the phi coefficient of an ARMA(1,1) process.
theta

A double corresponding to the theta coefficient of an ARMA(1,1) process.
sigma2

A double corresponding to the error term of an ARMA(1,1) process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with:

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

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

  • The third column contains the partial derivative with respect to σ2\sigma ^2.

Process Haar WV First Derivative

Taking the derivative with respect to ϕ\phi yields:

∂∂ϕνj2(ϕ,θ,σ2)=2σ2(ϕ−1)4(ϕ+1)2τj2(τj(−(θ+1)2(ϕ−1)(ϕ+1)2−2(ϕ2−1)(θ+ϕ)(θϕ+1)ϕτj2−1+(ϕ2−1)(θϕ+1)(θ+ϕ)ϕτj−1)−(θ2((3ϕ+2)ϕ+1)+2θ((ϕ2+ϕ+3)ϕ+1)+(3ϕ+2)ϕ+1)(ϕτj−4ϕτj2+3))\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 θ\theta yields:

∂∂θνj2(ϕ,θ,σ2)=2σ2((θ+1)(ϕ2−1)τj+(2θϕ+ϕ2+1)(ϕτj−4ϕτj2+3))(ϕ−1)3(ϕ+1)τj2\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 σ2\sigma^2 yields:

∂∂σ2νj2(ϕ,θ,σ2)=2σ2((ϕ2−1)τj+2ϕ(ϕτj−4ϕτj2+3))(ϕ−1)3(ϕ+1)τj2\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

omega

A double that is the slope of the drift.
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

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

Process Haar WV First Derivative

Taking the derivative with respect to ω\omega yields:

∂∂ωνj2(ω)=τj2ω8\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 ω\omega and not ω2\omega^2 as the ω\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

theta

A double corresponding to the theta coefficient of an MA(1) process.
sigma2

A double corresponding to the error term of an MA(1) process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the partial derivative with respect to θ\theta and the second column contains the partial derivative with respect to σ2\sigma ^2

Process Haar WV First Derivative

Taking the derivative with respect to θ\theta yields:

∂∂θνj2(θ,σ2)=σ2(2(θ+1)τj−6)τj2\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 σ2\sigma^2 yields:

∂∂σ2νj2(θ,σ2)=(θ+1)2τj−6θτj2\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

tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the partial derivative with respect to Q2Q^2.

Process Haar WV First Derivative

Taking the derivative with respect to Q2Q^2 yields:

∂∂Q2νj2(Q2)=6τj2\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

tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the partial derivative with respect to γ2\gamma^2.

Process Haar WV First Derivative

Taking the derivative with respect to γ2\gamma ^2 yields:

∂∂γ2νj2(γ2)=τj2+212τj\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

tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A matrix with the first column containing the partial derivative with respect to σ2\sigma^2.

Process Haar WV First Derivative

Taking the derivative with respect to σ2\sigma^2 yields:

∂∂σ2νj2(σ2)=1τj\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

theta

A vec containing the list of estimated parameters.
desc

A vector<string> containing a list of descriptors.
objdesc

A field<vec> containing a list of object descriptors.
tau

A vec containing the scales e.g. 2τ2^{\tau}

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)

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

x

An arima or data set.
order

An integer indicating the degrees of freedom. If 'x' is not a series of residuals, then set equal to 0.
stop_lag

An integer indicating the length of lags that should be calculated.
stdres

A boolean indicating whether to standardize the residualizes (e.g. res/sd(res)res/sd(res)) or not.
plot

A logical. If TRUE (the default) a plot should be produced.

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

x

An arima or data set.
order

An integer indicating the degrees of freedom. If 'x' is not a series of residuals, then set equal to 0.
stop_lag

An integer indicating the length of lags that should be calculated.
stdres

A boolean indicating whether to standardize the residualizes (e.g. res/sd(res)res/sd(res)) or not.
plot

A logical. If TRUE (the default) a plot should be produced.

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

Xt

The data used to construct said model.
model

A fitsimts, lm or gam object.
resids

A vector of residuals for diagnostics.
std

A boolean indicating whether we use standardized residuals for (1) residuals plot and (8) Box test results.

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

x

An arima or data set.
order

An integer indicating the degrees of freedom. If 'x' is not a series of residuals, then set equal to 0.
stop_lag

An integer indicating the length of lags that should be calculated.
stdres

A boolean indicating whether to standardize the residualizes (e.g. res/sd(res)res/sd(res)) or not.
test

A string indicating whether to perform Ljung-Box test or Box-Pierce test.
plot

A logical. If TRUE (the default) a plot should be produced.

Author(s)

James Balamuta, Stéphane Guerrier, Yuming Zhang

Create an Drift (DR) Process

Description

Sets up the necessary backend for the DR process.

Usage

DR(omega = NULL)

Arguments

omega

A double value for the slope of a DR process (see Note for details).

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:

Yt=ωtY_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

omega

A double corresponding to the slope of the drift
tau

A vec containing the scales e.g. 2τ2^{\tau}

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:

νj2(ω)=τj2ω216\nu _j^2\left( {{\omega }} \right) = \frac{{\tau _j^2{\omega ^2}}}{{16}}

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

model

A time series model.
Xt

A vector of time series data.
method

A string indicating the method used for model fitting. Supported methods include mle, yule-walker, gmwm and rgmwm.
demean

A boolean indicating whether the model includes a mean / intercept term or not.

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

models

A time series model or a list of time series models.
Xt

A time series (i.e gts object).
criterion

Either "IC" for AIC, BIC and HQ or "MAPE" for MAPE.
start

A numeric indicating the starting proportion of the data that is used for prediction (assuming criterion = "MAPE").
demean

A boolean indicating whether the model includes a mean / intercept term or not.
print

logical. If TRUE (the default) results are printed.

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")

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

sigma2

A double.
H

A double.

Value

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

process.desc

Used in summary: "SIGMA2","H"

theta

Parameter vector including σ2\sigma^2, HH

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)

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

phi

A double value for the autocorrection parameter ϕ\phi.
sigma2

A double value for the variance parameter σ2\sigma ^2.
n_total

An integer indicating the length of the simulated AR(1) block process.
n_block

An integer indicating the length of each block of the AR(1) block process.
scale

An integer indicating the number of levels of decomposition. The default value is 10.
title

A string indicating the name of the time series data.
seed

An integer defined for simulation replication purposes.
...

Additional parameters.

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 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

sigma2

A double value for the variance parameter σ2\sigma ^2.
n_total

An integer indicating the length of the simulated bias-instability process.
n_block

An integer indicating the length of each block of the bias-instability process.
title

A string defining the name of the time series data.
seed

An integer defined for simulation replication purposes.
...

Additional parameters.

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)

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

n

An integer containing the length of the time series.
model

A ts.model or simts object containing the available models in the simts package.
start

A numeric that provides the time of the first observation.
end

A numeric that provides the time of the last observation.
freq

A numeric that provides the rate of samples. Default value is 1.
unit_ts

A string that contains the unit expression of the time series. Default value is NULL.
unit_time

A string that contains the unit expression of the time. Default value is NULL.
name_ts

A string that provides an identifier for the time series data. Default value is NULL.
name_time

A string that provides an identifier for the time. Default value is NULL.

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

n

An interger indicating the amount of observations generated in this function.
model

A ts.model or simts object containing one of the allowed models.
start

A numeric that provides the time of the first observation.
end

A numeric that provides the time of the last observation.
freq

A numeric that provides the rate/frequency at which the time series is sampled. The default value is 1.
unit_ts

A string that contains the unit of measure of the time series. The default value is NULL.
unit_time

A string that contains the unit of measure of the time. The default value is NULL.
name_ts

A string that provides an identifier for the time series data. Default value is NULL.
name_time

A string that provides an identifier for the time. Default value is NULL.
process

A vector that contains model names of each column in the data object where the last name is the sum of the previous names.

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 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

n_total

An integer indicating the length of the simulated non-stationary white noise process.
title

A string defining the name of the time series data.
seed

An integer defined for simulation replication purposes.
...

Additional parameters.

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)

Create a Gauss-Markov (GM) Process

Description

Sets up the necessary backend for the GM process.

Usage

GM(beta = NULL, sigma2_gm = 1)

Arguments

beta

A double value for the β\beta of an GM process (see Note for details).
sigma2_gm

A double value for the variance, σgm2\sigma ^2_{gm}, of a GM process (see Note for details).

Details

When supplying values for β\beta and σgm2\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 ϕ\phi, σ2\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

β\beta, σgm2\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:

Xt=e(−β)Xt−1+εtX_t = e^{(-\beta)} X_{t-1} + \varepsilon_t

, where εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2(1−e2β)\sigma^2(1-e^{2\beta}).

Author(s)

James Balamuta

Examples

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

Generalized Method of Wavelet Moments (GMWM)

Description

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

Usage

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

Arguments

model

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

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

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

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

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

A double between 0 and 1 that indicates the efficiency.
alpha

A double between 0 and 1 that correspondings to the α2\frac{\alpha}{2} value for the wavelet confidence intervals.
seed

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

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

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

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

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

Details

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

The V matrix is calculated by: diag[(Hi−Lo)2]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 θ^\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 Fθ^F_{\hat{\theta}} 8. Compute WV Empirical END 9. Calculate the covariance matrix 10. Optimize the values to obtain θ^\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 (ϕ,σ2)(\phi,\sigma^2)

  • "GM": a guass-markov process (β,σgm2)(\beta,\sigma_{gm}^2)

  • "ARMA": an autoregressive moving average process with parameters (ϕp,θq,σ2)(\phi _p, \theta _q, \sigma^2)

  • "DR": a drift with parameter ω\omega

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

  • "RW": a random walk process with parameter σ2\sigma^2

  • "WN": a white noise process with parameter σ2\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

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

model

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

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

A string indicating the type of covariance matrix solver. "fast", "bootstrap", "asymp.diag", "asymp.comp", "fft"
robust

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

A double between 0 and 1 that indicates the efficiency.
...

Other arguments passed to the main gmwm function

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

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

data

A one-column matrix, data.frame, or a numeric vector.
start

A numeric that provides the time of the first observation.
end

A numeric that provides the time of the last observation.
freq

A numeric that provides the rate/frequency at which the time series is sampled. The default value is 1.
unit_ts

A string that contains the unit of measure of the time series. The default value is NULL.
unit_time

A string that contains the unit of measure of the time. The default value is NULL.
name_ts

A string that provides an identifier for the time series data. Default value is NULL.
name_time

A string that provides an identifier for the time. Default value is NULL.
data_name

A string that contains the name of the time series data.
Time

A numeric or character vector containing the times of observations. Default value is NULL. See x object in as.Date function.
time_format

A string specifiying the format of 'Time'. If not provided, 'Time' is assumed to be all integers. Default value is NULL. See format argument in as.Date function.

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)

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

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

data

A vector which contains data, or a matrix or data.frame which contains the data in each column.
gyros

A vector that contains the index of columns where gyroscope data (such as Gyro. X, Gyro. Y and Gyro. Z) is placed.
accels

A vector that contains the index of columns where accelerometer data (such as Accel. X, Accel. Y and Accel. Z) is placed.
axis

A vector that indicates the axises, such as 'X', 'Y', 'Z'. Please supply the axises for gyroscope data before that for accelerometer data, if gyroscope data exists.
freq

An integer that provides the frequency for the data.
unit

A string that contains the unit expression of the frequency. Default value is NULL.
name

A string that provides an identifier to the data. Default value is NULL.

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

x

A imu object

Value

A vector with numeric information.

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

x

A gts, imu, lts object.

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

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

data

A multiple-column matrix or data.frame. It must contain at least 3 columns of which the last represents the latent time series obtained through the sum of the previous columns.
start

A numeric that provides the time of the first observation.
end

A numeric that provides the time of the last observation.
freq

A numeric that provides the rate/frequency at which the time series is sampled. The default value is 1.
unit_ts

A string that contains the unit of measure of the time series. The default value is NULL.
unit_time

A string that contains the unit of measure of the time. The default value is NULL.
name_ts

A string that provides an identifier for the time series data. Default value is NULL.
name_time

A string that provides an identifier for the time. Default value is NULL.
process

A vector that contains model names of each column in the data object where the last name is the sum of the previous names.

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 XX and vector β\beta

Description

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

Usage

M(X, beta)

Arguments

X

A Matrix with dimension n*p.
beta

A vector with dimension p*1

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

theta

A double value for the parameter θ\theta (see Note for details).
sigma2

A double value for the variance parameter σ2\sigma ^2 (see Note for details).

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

θ1\theta_1, θ2\theta_2, ..., θq\theta_q, σ2\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:

Xt=∑j=1qθjεt−1+εtX_t = \sum_{j = 1}^q \theta_j \varepsilon_{t-1} + \varepsilon_t

, where εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2\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.

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

theta

A double value for the parameter θ\theta (see Note for details).
sigma2

A double value for the variance parameter σ2\sigma ^2 (see Note for details).

Value

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

process.desc

Used in summary: "MA1","SIGMA2"

theta

θ\theta, σ2\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:

Xt=θεt−1+εtX_t = \theta \varepsilon_{t-1} + \varepsilon_t

, where εt\varepsilon_t is iid from a zero mean normal distribution with variance σ2\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

theta

A double corresponding to the moving average term.
sigma2

A double the variance of the process.
tau

A vec containing the scales e.g. 2τ2^{\tau}

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 11 (MA(11)) process has a Haar Wavelet Variance given by:

νj2(θ,σ2)=((θ+1)2τj−6θ)σ2τj2\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

x_range

A numeric providing the range of values for the x-axis.
y_range

A numeric providing the range of values for the y-axis.
xlab

A string that gives a title for the x-axis.
ylab

A string that gives a title for the y-axis.
main

A string that gives an overall title for the plot. Default is an empty string.
mar

A vector indicating overall margin values for the plot.
add_axis_x

A boolean indicating whether a x-axis should be added.
add_axis_y

A boolean indicating whether a y-axis should be added.
col_box

A string indicating the color for the title box.
col_grid

A string indicating the color of the grid for the plot.
col_band

A string indicating the color of the band.
col_title

A string indicating the color of the plot title.
add_band

A boolean indicating whether there should be a band.
title_band_width

A double providing the value of the band width. Default is 0.09.
grid_lty

A integer indicating the line type of the grid lines.

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

model

A time series model.
Xt

A vector of time series data.
start

A numeric indicating the starting proportion of the data that is used for prediction.
plot

A boolean indicating whether a model accuracy plot based on MAPE is returned or not.

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

sigma2

A double.
lambda

A double.
alpha

A double.

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 σ2\sigma^2, λ\lambda,α\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)

Bootstrap standard error for the median

Description

Non-parametric bootstrap to obtain the standard of the median of iid data.

Usage

np_boot_sd_med(x, B = 5000)

Arguments

x

A vector of data.
B

A numeric indicating the number of simulations.

Value

Bootstrap standard error for the median

Plot Time Series Forecast Function

Description

This function plots the time series output from a forecast method with approximate 68

Usage

plot_pred(
  x,
  model,
  n.ahead,
  level = NULL,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  ...
)

Arguments

x

A gts object
model

A ts model
n.ahead

An integer indicating number of units of time ahead for which to make forecasts
level

A double or vector indicating confidence level of prediction interval. By default, it uses the levels of 0.50 and 0.95.
xlab

A string for the title of x axis
ylab

A string for the title of y axis
main

A string for the over all title of the plot
...

Additional parameters

Author(s)

Yuming Zhang

Plot the GMWM with the Wavelet Variance

Description

Displays a plot of the Wavelet Variance (WV) with the CI values and the WV implied by the estimated parameters.

Usage

## S3 method for class 'gmwm'
plot(
  x,
  decomp = FALSE,
  units = NULL,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  col_wv = NULL,
  col_ci = NULL,
  nb_ticks_x = NULL,
  nb_ticks_y = NULL,
  legend_position = NULL,
  ci_wv = NULL,
  point_cex = NULL,
  point_pch = NULL,
  ...
)

Arguments

x

A gmwm object.
decomp

A boolean that determines whether the contributions of each individual model are plotted.
units

A string that specifies the units of time plotted on the x axis.
xlab

A string that gives a title for the x axis.
ylab

A string that gives a title for the y axis.
main

A string that gives an overall title for the plot.
col_wv

A string that specifies the color of the wavelet variance line.
col_ci

A string that specifies the color of the shaded area covered by the confidence intervals.
nb_ticks_x

An integer that specifies the maximum number of ticks for the x-axis.
nb_ticks_y

An integer that specifies the maximum number of ticks for the y-axis.
legend_position

A string that specifies the position of the legend (use legend_position = NA to remove legend).
ci_wv

A boolean that determines whether to plot the confidence interval shaded area.
point_cex

A double that specifies the size of each symbol to be plotted.
point_pch

A double that specifies the symbol type to be plotted.
...

Additional arguments affecting the plot.

Value

Plot of WV and relative confidence intervals for each scale.

Author(s)

Stephane Guerrier and Yuming Zhang

Plot Partial Auto-Covariance and Correlation Functions

Description

The function plots the output of the theo_pacf and auto_corr functions (partial autocovariance or autocorrelation functions).

Usage

## S3 method for class 'PACF'
plot(
  x,
  xlab = NULL,
  ylab = NULL,
  show.ci = TRUE,
  alpha = NULL,
  col_ci = NULL,
  transparency = NULL,
  main = NULL,
  parValue = NULL,
  ...
)

Arguments

x

A "PACF" object output from theo_pacf or auto_corr.
xlab

A string indicating the label of the x axis: the default name is 'Lags'.
ylab

A string indicating the label of the y axis: the default name is 'PACF'.
show.ci

A bool indicating whether to show the confidence region. Defaults to TRUE.
alpha

A double indicating the level of significance for the confidence interval. By default alpha = 0.05 which gives a 1 - alpha = 0.95 confidence interval.
col_ci

A string that specifies the color of the region covered by the confidence intervals (confidence region).
transparency

A double between 0 and 1 indicating the transparency level of the color defined in col_ci. Defaults to 0.25.
main

A string indicating the title of the plot. Default name is "Variable name PACF plot'.
parValue

A vector defining the margins for the plot.
...

Additional parameters

Author(s)

Yunxiang Zhang and Yuming Zhang

Examples

# Plot the Partial Autocorrelation
m = auto_corr(datasets::AirPassengers, pacf = TRUE)
plot(m)

# More customized CI
plot(m, xlab = "my xlab", ylab = "my ylab", show.ci = TRUE, 
alpha = NULL, col_ci = "grey", transparency = 0.5, main = "my main")

Plot Auto-Covariance and Correlation Functions

Description

The function plots the output of the theo_acf and auto_corr functions (autocovariance or autocorrelation functions).

Usage

## S3 method for class 'simtsACF'
plot(
  x,
  xlab = NULL,
  ylab = NULL,
  show.ci = TRUE,
  alpha = NULL,
  col_ci = NULL,
  transparency = NULL,
  main = NULL,
  parValue = NULL,
  ...
)

Arguments

x

An "ACF" object output from theo_acf and auto_corr.
xlab

A string indicating the label of the x axis: the default name is 'Lags'.
ylab

A string indicating the label of the y axis: the default name is 'ACF'.
show.ci

A bool indicating whether to show the confidence region. Defaults to TRUE.
alpha

A double indicating the level of significance for the confidence interval. By default alpha = 0.05 which gives a 1 - alpha = 0.95 confidence interval.
col_ci

A string that specifies the color of the region covered by the confidence intervals (confidence region).
transparency

A double between 0 and 1 indicating the transparency level of the color defined in col_ci. Defaults to 0.25.
main

A string indicating the title of the plot. Default name is "Variable name ACF plot'.
parValue

A vector defining the margins for the plot.
...

Additional parameters

Author(s)

Yunxiang Zhang, Stéphane Guerrier and Yuming Zhang

Examples

# Calculate the Autocorrelation
m = auto_corr(datasets::AirPassengers)

# Plot with 95% CI
plot(m) 

# Plot with 90% CI
plot(m, alpha = 0.1) 

# Plot without 95% CI
plot(m, show.ci = FALSE)

# More customized CI
plot(m, xlab = "my xlab", ylab = "my ylab", show.ci = TRUE,
alpha = NULL, col_ci = "grey", transparency = 0.5, main = "my main")

Definition of a Power Law Process

Description

Definition of a Power Law Process

Usage

PLP(sigma2 = 1, d = 0.4)

Arguments

sigma2

A double.
d

A double.

Value

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

process.desc

Used in summary: "SIGMA2","d"

theta

Parameter vector including σ2\sigma^2, dd

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)

Time Series Prediction

Description

This function plots the time series forecast.

Usage

## S3 method for class 'fitsimts'
predict(
  object,
  n.ahead = 10,
  show_last = 100,
  level = NULL,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  plot = TRUE,
  ...
)

Arguments

object

A fitsimts object obtained from estimate function.
n.ahead

An integer indicating number of units of time ahead for which to make forecasts.
show_last

A integer indicating the number of last observations to show in the forecast plot.
level

A double or vector indicating confidence level of prediction interval. By default, it uses the levels of 0.50 and 0.95.
xlab

A string for the title of x axis.
ylab

A string for the title of y axis.
main

A string for the over all title of the plot.
plot

A logical value. logical. If TRUE(the default) the predictions are plotted.
...

Additional 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)
predict(model)
predict(model, level = 0.95)

x = gts(as.vector(lynx), start = 1821, end = 1934, freq = 1, 
unit_ts = bquote(paste(10^8," ",m^3)), name_ts = "Numbers", 
unit_time = "year", data_name = "Annual Numbers of Lynx Trappings")
model = estimate(AR(1), x)
predict(model, n.ahead = 20)
predict(model, n.ahead = 20, level = 0.95)
predict(model, n.ahead = 20, level = c(0.50, 0.80, 0.95))

Predict future points in the time series using the solution of the Generalized Method of Wavelet Moments

Description

Creates a prediction using the estimated values of GMWM through the ARIMA function within R.

Usage

## S3 method for class 'gmwm'
predict(object, data.in.gmwm, n.ahead = 1, ...)

Arguments

object

A gmwm object
data.in.gmwm

The data SAME EXACT DATA used in the GMWM estimation
n.ahead

Number of observations to forecast
...

Additional parameters passed to ARIMA Predict

Value

A predict.gmwm object with:

  • predPredictions

  • seStandard Errors

  • residResiduals from ARIMA ML Fit

Create an Quantisation Noise (QN) Process

Description

Sets up the necessary backend for the QN process.

Usage

QN(q2 = NULL)

Arguments

q2

A double value for the Q2Q^2 of a QN process.

Value

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

process.desc

Used in summary: "QN"

theta

Q2Q^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)

Quantisation Noise (QN) to WV

Description

This function compute the Haar WV of a Quantisation Noise (QN) process

Usage

qn_to_wv(q2, tau)

Arguments

q2

A double corresponding to variance of drift
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A vec containing the wavelet variance of the QN.

Process Haar Wavelet Variance Formula

The Quantization Noise (QN) process has a Haar Wavelet Variance given by:

νj2(Q2)=6Q2τj2\nu _j^2\left( {{Q^2}} \right) = \frac{{6{Q^2}}}{{\tau _j^2}}

Read an IMU Binary File into R

Description

Process binary files within the

Usage

read.imu(file, type, unit = NULL, name = NULL)

Arguments

file

A string containing file names or paths.
type

A string that contains a supported IMU type given below.
unit

A string that contains the unit expression of the frequency. Default value is NULL.
name

A string that provides an identifier to the data. Default value is NULL.

Details

Currently supports the following IMUs:

  • IMAR

  • LN200

  • LN200IG

  • IXSEA

  • NAVCHIP_INT

  • NAVCHIP_FLT

Value

An imu object that contains 3 gyroscopes and 3 accelerometers in that order.

Author(s)

James Balamuta We hope to soon be able to support delimited files.

References

Thanks goes to Philipp Clausen of Labo TOPO, EPFL, Switzerland, topo.epfl.ch, Tel:+41(0)21 693 27 55 for providing a matlab function that reads in IMUs. This function is a heavily modified port of MATLAB code into Armadillo/C++.

Examples

## Not run: 
# Relative
setwd("F:/")

a = read.imu(file = "Documents/James/short_test_data.imu", type = "IXSEA")

# Fixed path
b = read.imu(file = "F:/Desktop/short_test_data.imu", type = "IXSEA")

## End(Not run)

Plot the Distribution of (Standardized) Residuals

Description

This function plots a histogram (with kernel density function and normal distribution) of the standardized residuals or a basic plot the (standardized) residuals, or both.

Usage

resid_plot(res, std = FALSE, type = "hist", ...)

Arguments

res

A vector of residuals.
std

A boolean indicating whether the residuals plot is for standardized residuals or original residuals.
type

A string indicating either: "hist" (standardized residual histogram with superimposed kernel density estimator and normal distribution), "resid" (standard residual plot), or "both"
...

Additional parameters

Author(s)

Yuming Zhang

GMWM for Robust/Classical Comparison

Description

Creates a rgmwm object to compare the results generated by robust/classical method.

Usage

rgmwm(model, data, eff = c(0.9, 0.8, 0.6), ...)

Arguments

model

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

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

A double vector between 0 and 1 that indicates the efficiency.
...

Other arguments passed to the main gmwm function.

Details

By default, the rgmwm function will fit a classical gmwm object. From there, the user has the ability to specify any eff that is less than or equal to 0.99.

Value

A rgmwm object

Truncated Normal Distribution Sampling Algorithm

Description

Enables sampling from a truncated normal

Usage

rtruncated_normal(n, mu, sigma, a, b)

Arguments

n

An unsigned int indicating the number of observations to generate.
mu

A double indicating the mean of the normal.
sigma

A double indicating the standard deviation of the normal.
a

A double that is the lower bound of the truncated normal.
b

A double that is the upper bound of the truncated normal.

Create an Random Walk (RW) Process

Description

Sets up the necessary backend for the RW process.

Usage

RW(gamma2 = NULL)

Arguments

gamma2

A double value for the variance γ2\gamma ^2

Value

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

process.desc

Used in summary: "RW"

theta

σ\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:

Yt=∑t=0Tγ0∗ZtY_t = \sum\nolimits_{t=0}^{T} \gamma_0*Z_t

where ZtZ_t is iid and follows a standard normal distribution.

Author(s)

James Balamuta

Examples

RW()
RW(gamma2=3.4)

Random Walk to WV

Description

This function compute the WV (haar) of a Random Walk process

Usage

rw_to_wv(gamma2, tau)

Arguments

gamma2

A double corresponding to variance of RW
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A vec containing the wavelet variance of the random walk.

Process Haar Wavelet Variance Formula

The Random Walk (RW) process has a Haar Wavelet Variance given by:

νj2(γ2)=(τj2+2)γ212τj\nu _j^2\left( {{\gamma ^2}} \right) = \frac{{\left( {\tau _j^2 + 2} \right){\gamma ^2}}}{{12{\tau _j}}}

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

steps

An integer that counts the number of steps of the random walk.
probs

A vector of double that specifies the probabilities to choose each direction.

Author(s)

Stéphane Guerrier

Examples

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

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

ar

A vector or integer containing either the coefficients for ϕ\phi's or the process number pp for the Autoregressive (AR) term.
i

An integer containing the number of differences to be done.
ma

A vector or integer containing either the coefficients for θ\theta's or the process number qq for the Moving Average (MA) term.
sar

A vector or integer containing either the coefficients for Φ\Phi's or the process number PP for the Seasonal Autoregressive (SAR) term.
si

An integer containing the number of seasonal differences to be done.
sma

A vector or integer containing either the coefficients for Θ\Theta's or the process number QQ for the Seasonal Moving Average (SMA) term.
s

An integer containing the seasonality.
sigma2

A double value for the standard deviation, σ\sigma, of the SARMA process.

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∗pAR*p, MA∗qMA*q, SAR∗PSAR*P, SMA∗QSMA*Q

theta

σ\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) process
SARIMA(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

ar

A vector or integer containing either the coefficients for ϕ\phi's or the process number pp for the Autoregressive (AR) term.
ma

A vector or integer containing either the coefficients for θ\theta's or the process number qq for the Moving Average (MA) term.
sar

A vector or integer containing either the coefficients for Φ\Phi's or the process number PP for the Seasonal Autoregressive (SAR) term.
sma

A vector or integer containing either the coefficients for Θ\Theta's or the process number QQ for the Seasonal Moving Average (SMA) term.
s

A integer indicating the seasonal value of the data.
sigma2

A double value for the standard deviation, σ\sigma, of the SARMA process.

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∗pAR*p, MA∗qMA*q, SAR∗PSAR*P, SMA∗QSMA*Q

theta

σ\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) process
SARMA(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)

Personal Saving Rate

Description

Personal saving as a percentage of disposable personal income (DPI), frequently referred to as "the personal saving rate," is calculated as the ratio of personal saving to DPI.

Usage

savingrt

Format

A gts time series object with frequency 12 starting at 1959 and going to 2016 for a total of 691 observations.

Source

https://fred.stlouisfed.org/series/PSAVERT

Time Series Model Selection

Description

This function performs model fitting and calculates the model selection criteria to be plotted.

Usage

select(model, Xt, include.mean = TRUE, criterion = "aic", plot = TRUE)

Arguments

model

A time series model (only ARIMA are currently supported).
Xt

A vector of time series data.
include.mean

A boolean indicating whether to fit ARIMA with the mean or not.
criterion

A string indicating which model selection criterion should be used (possible values: "aic" (default), "bic", "hq").
plot

A boolean indicating whether a model selection plot is returned or not.

Author(s)

Stéphane Guerrier and Yuming Zhang

Examples

set.seed(763)
Xt = gen_gts(100, AR(phi = c(0.2, -0.5, 0.4), sigma2 = 1))
select(AR(5), Xt, include.mean = FALSE)

Xt = gen_gts(100, MA(theta = c(0.2, -0.5, 0.4), sigma2 = 1))
select(MA(5), Xt, include.mean = FALSE)

Xt = gen_gts(500, ARMA(ar = 0.5, ma = c(0.5, -0.5, 0.4), sigma2 = 1))
select(ARMA(5,3), Xt, criterion = "hq", include.mean = FALSE)

Run Model Selection Criteria on ARIMA Models

Description

This function performs model fitting and calculates the model selection criteria to be plotted or used in best_model function.

Usage

select_arima(
  xt,
  p.min = 0L,
  p.max = 3L,
  d = 0L,
  q.min = 0L,
  q.max = 3L,
  include.mean = TRUE,
  plot = TRUE
)

select_arma(
  xt,
  p.min = 0L,
  p.max = 3L,
  q.min = 0L,
  q.max = 3L,
  include.mean = TRUE,
  plot = TRUE
)

select_ar(xt, p.min = 0L, p.max = 3L, include.mean = TRUE, plot = TRUE)

select_ma(xt, q.min = 0L, q.max = 3L, include.mean = TRUE, plot = TRUE)

Arguments

xt

A vector of univariate time series.
p.min

An integer indicating the lowest order of AR(p) process to search.
p.max

An integer indicating the highest order of AR(p) process to search.
d

An integer indicating the differencing order for the data.
q.min

An integer indicating the lowest order of MA(q) process to search.
q.max

An integer indicating the highest order of MA(q) process to search.
include.mean

A bool indicating whether to fit ARIMA with the mean or not.
plot

A logical. If TRUE (the default) a plot should be produced.

Examples

xt = gen_arima(N=100, ar=0.3, d=1, ma=0.3)
x = select_arima(xt, d=1L)

xt = gen_ma1(100, 0.3, 1)
x = select_ma(xt, q.min=2L, q.max=5L)
best_model(x)

xt = gen_arma(10, c(.4,.5), c(.1), 1, 0)  
x = select_arma(xt, p.min = 1L, p.max = 4L,
                q.min = 1L, q.max = 3L)

Basic Diagnostic Plot of Residuals

Description

This function will plot four diagnostic plots to assess how well the model fits the data. These plots are: (1) residuals plot, (2) histogram of (standardized) residuals, (3) normal Q-Q plot of residuals and (4) residuals vs fitted values plot.

Usage

simple_diag_plot(Xt, model, std = FALSE)

Arguments

Xt

The original time series data.
model

The arima model fit to the data.
std

A boolean indicating whether we use standardized residuals for the (1) residuals plot and the (2) histogram of (standardized) residuals.

Author(s)

Yuming Zhang

Simplify and print SARIMA model

Description

Simplify and print SARIMA model

Usage

simplified_print_SARIMA(p, i, q, P, si, Q, s)

Arguments

p

An integer denoting the length of ar.
i

An integer containing the number of differences to be done.
q

An integer denoting the length of ma.
P

An integer denoting the length of sma.
si

An integer containing the number of seasonal differences to be done.
Q

An integer denoting the length of sar.
s

An integer indicating the seasonal value of the data.

Value

An S3 object with the following structure:

print

String containing simplified model

simplified

Type of model (after simplification)

Author(s)

Stephane Guerrier

Definition of a Sinusoidal (SIN) Process

Description

Definition of a Sinusoidal (SIN) Process

Usage

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

Arguments

alpha2

A double value for the squared amplitude parameter α2\alpha^2 (see Note for details).
beta

A double value for the angular frequency parameter β\beta (see Note for details).
U

A double value for the phase parameter UU (see Note for details).

Value

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

process.desc

Used in summary: "ALPHA2","BETA"

theta

Parameter vector including α2\alpha^2, β\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 :

Xt=αsin⁡(βt+U)X_t = \alpha \sin(\beta t + U)

, where U∼U(0,2π)U \sim \mathcal{U}(0, 2\pi) and β∈(0,π2)\beta \in (0, \frac{\pi}{2})

Author(s)

Lionel Voirol

Examples

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

Summary of fitsimts object

Description

Displays summary information about fitsimts object

Usage

## S3 method for class 'fitsimts'
summary(object, ...)

Arguments

object

A fitsimts object
...

Other arguments passed to specific methods

Value

Estimated parameters values with confidence intervals and standard errors.

Author(s)

Stéphane Guerrier

Summary of GMWM object

Description

Displays summary information about GMWM object

Usage

## S3 method for class 'gmwm'
summary(
  object,
  inference = NULL,
  bs.gof = NULL,
  bs.gof.p.ci = NULL,
  bs.theta.est = NULL,
  bs.ci = NULL,
  B = 100,
  ...
)

Arguments

object

A GMWM object
inference

A value containing either: NULL (auto), TRUE, or FALSE
bs.gof

A value containing either: NULL (auto), TRUE, FALSE
bs.gof.p.ci

A value containing either: NULL (auto), TRUE, FALSE
bs.theta.est

A value containing either: NULL (auto), TRUE, FALSE
bs.ci

A value containing either: NULL (auto), TRUE, FALSE
B

An int that indicates how many bootstraps should be performed.
...

Other arguments passed to specific methods

Value

A summary.gmwm object with:

  • estimateEstimated Theta Values

  • testinfoGoodness of Fit Information

  • inferenceInference performed? T/F

  • bs.gofBootstrap GOF? T/F

  • bs.gof.p.ciBootstrap GOF P-Value CI? T/F

  • bs.theta.estBootstrap Theta Estimates? T/F

  • bs.ciBootstrap CI? T/F

  • startingIndicates if program supplied initial starting values

  • seedSeed used during guessing / bootstrapping

  • obj.funValue of obj.fun at minimized theta

  • NLength of Time Series

Author(s)

JJB

Theoretical Autocorrelation (ACF) of an ARMA process

Description

This function computes the theoretical Autocorrelation (ACF) of an ARMA process.

Usage

theo_acf(ar, ma = NULL, lagmax = 20)

Arguments

ar

A vector containing the AR coefficients.
ma

A vector containing the MA coefficients.
lagmax

An integer indicating the maximum lag up to which to compute the theoretical ACF.

Author(s)

Yuming Zhang

Examples

# Compute the theoretical ACF for an ARMA(1,0) (i.e. a first-order autoregressive model: AR(1))
theo_acf(ar = -0.25, ma = NULL)
# Computes the theoretical ACF for an ARMA(2, 1)
theo_acf(ar = c(.50, -0.25), ma = 0.20, lagmax = 10)

Theoretical Partial Autocorrelation (PACF) of an ARMA process

Description

This function computes the theoretical Partial Autocorrelation (PACF) of an ARMA process.

Usage

theo_pacf(ar, ma = NULL, lagmax = 20)

Arguments

ar

A vector containing the AR coefficients.
ma

A vector containing the MA coefficients.
lagmax

An integer indicating the maximum lag up to which to compute the theoretical PACF.

Author(s)

Yuming Zhang

Examples

# Computes the theoretical ACF for an ARMA(1,0) (i.e. a first-order autoregressive model: AR(1))
theo_pacf(ar = -0.25, ma = NULL, lagmax = 7)
# Computes the theoretical ACF for an ARMA(2, 1)
theo_pacf(ar = c(.50, -0.25), ma = .20, lagmax = 10)

Update (Robust) GMWM object for IMU or SSM

Description

Provides a way to estimate different models over the previously estimated wavelet variance values and covariance matrix.

Usage

## S3 method for class 'gmwm'
update(object, model, ...)

Arguments

object

A gmwm object.
model

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

Additional parameters (not used)

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

Update Object Attribute

Description

Update the attributes of lts, gts and imu object

Usage

## S3 method for class 'lts'
update(object, type, new, keep.start = T, ...)

## S3 method for class 'gts'
update(object, type, new, keep.start = T, ...)

## S3 method for class 'imu'
update(object, type, new, ...)

Arguments

object

A lts, gts or imu object
type

A string that contains the attribute to be updated
new

The updated value for the attribute
keep.start

A boolean value that indicates whether 'start' or 'end' should remain the same when 'freq' is updated
...

Further arguments passed to or from other methods.

Details

This function is able to update some attributes for gts, lts and imu objects. For lts object, the attributes that can be updated are 'start', 'end', 'freq', 'unit_time', 'name_ts' and 'process'. For gts object, the attributes that can be updated are 'start', 'end', 'freq', 'unit_time' and 'name_ts'. For imu object, the attributes that can be updated are 'axis', 'freq', 'unit_time' and 'name_ts'.

If one between 'start' and 'end' is updated, the other one will also be updated, since end-start == (N-1)/freq must be TRUE, where N is the number of observations in the object.

If 'freq' is updated, by default 'start' will remain the same, and 'end' will be updated at the same time, unless you set 'keep.start = F'.

If 'unit_time' is updated, the old unit_time will be replaced by the new one, and other attributes will remain the same. It is different from the unit_time conversion feature.

Value

An object with the updated attribute.

Examples

gts1 = gts(rnorm(50), freq = 1, unit_time = 'sec', name_ts = 'test1')
gts2 = update(gts1, 'unit_time', 'min')
attr(gts2, 'unit_time')

gts3 = update(gts1, 'name_ts', 'test2')
attr(gts3, 'name_ts')

Obtain the value of an object's properties

Description

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

Usage

value(x, type)

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

Arguments

x

A gts, imu, or lts object.
type

A string indicating the field to be retrieved.

Details

To access information about imu properties use:

"accel"

Returns the number of accelerometers

"gyro"

Returns the number of gyroscopes

"sensors"

Returns total number of sensors

Value

The method will return a single numeric or string result depending on the slot being accessed.

Methods (by class)

  • value(imu): Access imu object properties

Author(s)

James Balamuta

Create an White Noise (WN) Process

Description

Sets up the necessary backend for the WN process.

Usage

WN(sigma2 = NULL)

Arguments

sigma2

A double value for the variance, σ2\sigma ^2, of a WN process.

Value

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

process.desc

Used in summary: "WN"

theta

σ\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, YtY_t is iid from a zero mean normal distribution with variance σ2\sigma^2

Author(s)

James Balamuta

Examples

WN()
WN(sigma2=3.4)

Gaussian White Noise to WV

Description

This function compute the Haar WV of a Gaussian White Noise process

Usage

wn_to_wv(sigma2, tau)

Arguments

sigma2

A double corresponding to variance of WN
tau

A vec containing the scales e.g. 2τ2^{\tau}

Value

A vec containing the wavelet variance of the white noise.

Process Haar Wavelet Variance Formula

The Gaussian White Noise (WN) process has a Haar Wavelet Variance given by:

νj2(σ2)=σ2τj2\nu _j^2\left( {{\sigma ^2}} \right) = \frac{{{\sigma ^2}}}{{\tau _j^2}}