Package 'itsmr'

Title: Time Series Analysis Using the Innovations Algorithm
Description: Provides functions for modeling and forecasting time series data. Forecasting is based on the innovations algorithm. A description of the innovations algorithm can be found in the textbook "Introduction to Time Series and Forecasting" by Peter J. Brockwell and Richard A. Davis. <https://link.springer.com/book/10.1007/b97391>.
Authors: George Weigt
Maintainer: George Weigt <[email protected]>
License: FreeBSD
Version: 1.10
Built: 2025-02-28 05:31:56 UTC
Source: https://github.com/cran/itsmr

Help Index

Time Series Analysis Using the Innovations Algorithm

Description

Provides functions for modeling and forecasting time series data. Forecasting is based on the innovations algorithm. A description of the innovations algorithm can be found in the textbook Introduction to Time Series and Forecasting by Peter J. Brockwell and Richard A. Davis.

Details

Package: itsmr
Type: Package
Version: 1.10
Date: 2022-07-27
License: FreeBSD
LazyLoad: yes
URL: https://georgeweigt.github.io/itsmr-refman.pdf

Author(s)

George Weigt
Maintainer: George Weigt <[email protected]>

References

Brockwell, Peter J., and Richard A. Davis. Introduction to Time Series and Forecasting. 2nd ed. Springer, 2002.

Examples

plotc(wine)

## Define a suitable data model
M = c("log","season",12,"trend",1)

## Obtain residuals and check for stationarity
e = Resid(wine,M)
test(e)

## Define a suitable ARMA model
a = arma(e,p=1,q=1)

## Obtain residuals and check for white noise
ee = Resid(wine,M,a)
test(ee)

## Forecast future values
forecast(wine,M,a)

Autocovariance of ARMA model

Description

Autocovariance of ARMA model

Usage

aacvf(a, h)

Arguments

a

ARMA model
h

Maximum lag

Details

The ARMA model is a list with the following components.

phi Vector of AR coefficients (index number equals coefficient subscript)
theta Vector of MA coefficients (index number equals coefficient subscript)
sigma2 White noise variance

Value

Returns a vector of length h+1 to accomodate lag 0 at index 1.

See Also

arma

Examples

a = arma(Sunspots,2,0)
aacvf(a,40)

Autocovariance of data

Description

Autocovariance of data

Usage

acvf(x, h = 40)

Arguments

x

Time series data
h

Maximum lag

Value

Returns a vector of length h+1 to accomodate lag 0 at index 1.

See Also

plota

Examples

acvf(Sunspots)

Number of international airline passengers, 1949 to 1960

Description

Number of international airline passengers, 1949 to 1960

Examples

plotc(airpass)

Compute AR infinity coefficients

Description

Compute AR infinity coefficients

Usage

ar.inf(a, n = 50)

Arguments

a

ARMA model
n

Order

Details

The ARMA model is a list with the following components.

phi Vector of AR coefficients (index number equals coefficient subscript)
theta Vector of MA coefficients (index number equals coefficient subscript)
sigma2 White noise variance

Value

Returns a vector of length n+1 to accomodate coefficient 0 at index 1.

See Also

ma.inf

Examples

a = yw(Sunspots,2)
ar.inf(a)

Forecast using ARAR algorithm

Description

Forecast using ARAR algorithm

Usage

arar(y, h = 10, opt = 2)

Arguments

y

Time series data
h

Steps ahead
opt

Display option (0 silent, 1 tabulate, 2 plot and tabulate)

Value

Returns the following list invisibly.

pred

Predicted values
se

Standard errors
l

Lower bounds (95% confidence interval)
u

Upper bounds

See Also

forecast

Examples

arar(airpass)

Estimate ARMA model coefficients using maximum likelihood

Description

Estimate ARMA model coefficients using maximum likelihood

Usage

arma(x, p = 0, q = 0)

Arguments

x

Time series data
p

AR order
q

MA order

Details

Calls the standard R function arima to estimate AR and MA coefficients. The innovations algorithm is used to estimate white noise variance.

Value

Returns an ARMA model consisting of a list with the following components.

phi

Vector of AR coefficients (index number equals coefficient subscript)
theta

Vector of MA coefficients (index number equals coefficient subscript)
sigma2

White noise variance
aicc

Akaike information criterion corrected
se.phi

Standard errors for the AR coefficients
se.theta

Standard errors for the MA coefficients

See Also

autofit burg hannan ia yw

Examples

M = c("diff",1)
e = Resid(dowj,M)
a = arma(e,1,0)
print(a)

Find the best model from a range of possible ARMA models

Description

Find the best model from a range of possible ARMA models

Usage

autofit(x, p = 0:5, q = 0:5)

Arguments

x

Time series data (typically residuals from Resid)
p

Range of AR orders
q

Range of MA orders

Details

Tries all combinations of p and q and returns the model with the lowest AICC. The arguments p and q should be small ranges as this function can be slow otherwise. The innovations algorithm is used to estimate white noise variance.

Value

Returns an ARMA model consisting of a list with the following components.

phi

Vector of AR coefficients (index number equals coefficient subscript)
theta

Vector of MA coefficients (index number equals coefficient subscript)
sigma2

White noise variance
aicc

Akaike information criterion corrected
se.phi

Standard errors for the AR coefficients
se.theta

Standard errors for the MA coefficients

See Also

arma

Examples

M = c("diff",1)
e = Resid(dowj,M)
a = autofit(e)
print(a)

Estimate AR coefficients using the Burg method

Description

Estimate AR coefficients using the Burg method

Usage

burg(x, p)

Arguments

x

Time series data (typically residuals from Resid)
p

AR order

Details

The innovations algorithm is used to estimate white noise variance.

Value

Returns an ARMA model consisting of a list with the following components.

phi

Vector of AR coefficients (index number equals coefficient subscript)
theta

0
sigma2

White noise variance
aicc

Akaike information criterion corrected
se.phi

Standard errors for the AR coefficients
se.theta

0

See Also

arma hannan ia yw

Examples

M = c("diff",1)
e = Resid(dowj,M)
a = burg(e,1)
print(a)

Check for causality and invertibility

Description

Check for causality and invertibility

Usage

check(a)

Arguments

a

ARMA model

Details

The ARMA model is a list with the following components.

phi Vector of AR coefficients (index number equals coefficient subscript)
theta Vector of MA coefficients (index number equals coefficient subscript)
sigma2 White noise variance

Value

None

Examples

a = specify(ar=c(0,0,.99))
check(a)

USA accidental deaths, 1973 to 1978

Description

USA accidental deaths, 1973 to 1978

Examples

plotc(deaths)

Dow Jones utilities index, August 28 to December 18, 1972

Description

Dow Jones utilities index, August 28 to December 18, 1972

Examples

plotc(dowj)

Forecast future values

Description

Forecast future values

Usage

forecast(x, M, a, h = 10, opt = 2, alpha = 0.05)

Arguments

x

Time series data
M

Data model
a

ARMA model
h

Steps ahead
opt

Display option (0 silent, 1 tabulate, 2 plot and tabulate)
alpha

Level of significance

Details

The data model can be NULL for none. Otherwise M is a vector of function names and arguments.

Example:

M = c("log","season",12,"trend",1)

The above model takes the log of the data, then subtracts a seasonal component of period 12, then subtracts a linear trend component.

These are the available functions:

diff Difference the data. Has a single argument, the lag.
hr Subtract harmonic components. Has one or more arguments, each specifying the number of observations per harmonic.
log Take the log of the data, has no arguments.
season Subtract a seasonal component. Has a single argument, the number of observations per season.
trend Subtract a trend component. Has a single argument, the order of the trend (1 linear, 2 quadratic, etc.)

At the end of the model there is an implicit subtraction of the mean operation. Hence the resulting time series always has zero mean.

All of the functions are inverted before the forecast results are displayed.

Value

Returns the following list invisibly.

pred

Predicted values
se

Standard errors (not returned for data models with log)
l

Lower bounds (95% confidence interval)
u

Upper bounds

See Also

arma Resid test

Examples

M = c("log","season",12,"trend",1)
e = Resid(wine,M)
a = arma(e,1,1)
forecast(wine,M,a)

Estimate ARMA coefficients using the Hannan-Rissanen algorithm

Description

Estimate ARMA coefficients using the Hannan-Rissanen algorithm

Usage

hannan(x, p, q)

Arguments

x

Time series data (typically residuals from Resid)
p

AR order
q

MA order (q > 0)

Details

The innovations algorithm is used to estimate white noise variance.

Value

Returns an ARMA model consisting of a list with the following components.

phi

Vector of AR coefficients (index number equals coefficient subscript)
theta

Vector of MA coefficients (index number equals coefficient subscript)
sigma2

White noise variance
aicc

Akaike information criterion corrected
se.phi

Standard errors for the AR coefficients
se.theta

Standard errors for the MA coefficients

See Also

arma burg ia yw

Examples

M = c("diff",12)
e = Resid(deaths,M)
a = hannan(e,1,1)
print(a)

Estimate harmonic components

Description

Estimate harmonic components

Usage

hr(x, d)

Arguments

x

Time series data
d

Vector of harmonic periods

Value

Returns a vector the same length as x. Subtract from x to obtain residuals.

Examples

y = hr(deaths,c(12,6))
plotc(deaths,y)

Estimate MA coefficients using the innovations algorithm

Description

Estimate MA coefficients using the innovations algorithm

Usage

ia(x, q, m = 17)

Arguments

x

Time series data (typically residuals from Resid)
q

MA order
m

Recursion level

Details

Normally m should be set to the default value. The innovations algorithm is used to estimate white noise variance.

Value

Returns an ARMA model consisting of a list with the following components.

phi

0
theta

Vector of MA coefficients (index number equals coefficient subscript)
sigma2

White noise variance
aicc

Akaike information criterion corrected
se.phi

0
se.theta

Standard errors for the MA coefficients

See Also

arma burg hannan yw

Examples

M = c("diff",1)
e = Resid(dowj,M)
a = ia(e,1)
print(a)

Level of Lake Huron, 1875 to 1972

Description

Level of Lake Huron, 1875 to 1972

Examples

plotc(lake)

Compute MA infinity coefficients

Description

Compute MA infinity coefficients

Usage

ma.inf(a, n = 50)

Arguments

a

ARMA model
n

Order

Details

The ARMA model is a list with the following components.

phi Vector of AR coefficients (index number equals coefficient subscript)
theta Vector of MA coefficients (index number equals coefficient subscript)
sigma2 White noise variance

Value

Returns a vector of length n+1 to accomodate coefficient 0 at index 1.

See Also

ar.inf

Examples

M = c("diff",12)
e = Resid(deaths,M)
a = arma(e,1,1)
ma.inf(a,10)

Plot a periodogram

Description

Plot a periodogram

Usage

periodogram(x, q = 0, opt = 2)

Arguments

x

Time series data
q

MA filter order
opt

Plot option (0 silent, 1 periodogram only, 2 periodogram and filter)

Details

The filter q can be a vector in which case the overall filter is the composition of MA filters of the designated orders.

Value

The periodogram vector divided by 2pi is returned invisibly.

See Also

plots

Examples

periodogram(Sunspots,c(1,1,1,1))

Plot data and/or model ACF and PACF

Description

Plot data and/or model ACF and PACF

Usage

plota(u, v = NULL, h = 40)

Arguments

u, v

Data and/or ARMA model in either order
h

Maximum lag

Value

None

Examples

plota(Sunspots)
a = yw(Sunspots,2)
plota(Sunspots,a)

Plot one or two time series

Description

Plot one or two time series

Usage

plotc(y1, y2 = NULL)

Arguments

y1

Data vector (plotted in blue with knots)
y2

Data vector (plotted in red, no knots)

Value

None

Examples

plotc(uspop)
y = trend(uspop,2)
plotc(uspop,y)

Plot spectrum of data or ARMA model

Description

Plot spectrum of data or ARMA model

Usage

plots(u)

Arguments

u

Data vector or an ARMA model

Value

None

See Also

periodogram

Examples

a = specify(ar=c(0,0,.99))
plots(a)

Compute residuals

Description

Compute residuals

Usage

Resid(x, M = NULL, a = NULL)

Arguments

x

Time series data
M

Data model
a

ARMA model

Details

The data model can be NULL for none. Otherwise M is a vector of function names and arguments.

Example:

M = c("log","season",12,"trend",1)

The above model takes the log of the data, then subtracts a seasonal component of period 12, then subtracts a linear trend component.

These are the available functions:

diff Difference the data. Has a single argument, the lag.
hr Subtract harmonic components. Has one or more arguments, each specifying the number of observations per harmonic.
log Take the log of the data, has no arguments.
season Subtract a seasonal component. Has a single argument, the number of observations per season.
trend Subtract a trend component. Has a single argument, the order of the trend (1 linear, 2 quadratic, etc.)

At the end of the model there is an implicit subtraction of the mean operation. Hence the resulting time series always has zero mean.

Value

Returns a vector of residuals the same length as x.

See Also

test

Examples

M = c("log","season",12,"trend",1)
e = Resid(wine,M)

a = arma(e,1,1)
ee = Resid(wine,M,a)

Estimate seasonal component

Description

Estimate seasonal component

Usage

season(x, d)

Arguments

x

Time series data
d

Number of observations per season

Value

Returns a vector the same length as x. Subtract from x to obtain residuals.

See Also

trend

Examples

y = season(deaths,12)
plotc(deaths,y)

Run a self test

Description

Run a self test

Usage

selftest()

Details

This function is a useful check if the code is modified.

Value

None

Examples

selftest()

Generate synthetic observations

Description

Generate synthetic observations

Usage

sim(a, n = 100)

Arguments

a

ARMA model
n

Number of synthetic observations required

Details

The ARMA model is a list with the following components.

phi Vector of AR coefficients (index number equals coefficient subscript)
theta Vector of MA coefficients (index number equals coefficient subscript)
sigma2 White noise variance

Value

Returns a vector of n synthetic observations.

Examples

a = specify(ar=c(0,0,.99))
x = sim(a,60)
plotc(x)

Apply an exponential filter

Description

Apply an exponential filter

Usage

smooth.exp(x, alpha)

Arguments

x

Time series data
alpha

Smoothness setting, 0-1

Details

Zero is maximum smoothness.

Value

Returns a vector of smoothed data the same length as x.

Examples

y = smooth.exp(strikes,.4)
plotc(strikes,y)

Apply a low pass filter

Description

Apply a low pass filter

Usage

smooth.fft(x, f)

Arguments

x

Time series data
f

Cut-off frequency, 0-1

Details

The cut-off frequency is specified as a fraction. For example, c=.25 passes the lowest 25% of the spectrum.

Value

Returns a vector the same length as x.

Examples

y = smooth.fft(deaths,.1)
plotc(deaths,y)

Apply a moving average filter

Description

Apply a moving average filter

Usage

smooth.ma(x, q)

Arguments

x

Time series data
q

Filter order

Details

The averaging function uses 2q+1 values.

Value

Returns a vector the same length as x.

Examples

y = smooth.ma(strikes,2)
plotc(strikes,y)

Apply a spectral filter

Description

Apply a spectral filter

Usage

smooth.rank(x, k)

Arguments

x

Time series data
k

Number of frequencies

Details

Passes the mean and the k frequencies with the highest amplitude. The remainder of the spectrum is filtered out.

Value

Returns a vector the same length as x.

Examples

y = smooth.rank(deaths,2)
plotc(deaths,y)

Specify an ARMA model

Description

Specify an ARMA model

Usage

specify(ar = 0, ma = 0, sigma2 = 1)

Arguments

ar

Vector of AR coefficients (index number equals coefficient subscript)
ma

Vector of MA coefficients (index number equals coefficient subscript)
sigma2

White noise variance

Value

Returns an ARMA model consisting of a list with the following components.

phi

Vector of AR coefficients (index number equals coefficient subscript)
theta

Vector of MA coefficients (index number equals coefficient subscript)
sigma2

White noise variance

Examples

specify(ar=c(0,0,.99))

USA union strikes, 1951-1980

Description

USA union strikes, 1951-1980

Examples

plotc(strikes)

Number of sunspots, 1770 to 1869

Description

Number of sunspots, 1770 to 1869

Examples

plotc(Sunspots)

Test residuals for stationarity and randomness

Description

Test residuals for stationarity and randomness

Usage

test(e)

Arguments

e

Time series data (typically residuals from Resid)

Details

Plots ACF, PACF, residuals, and QQ. Displays results for Ljung-Box, McLeod-Li, turning point, difference-sign, and rank tests. The plots can be used to check for stationarity and the other tests check for white noise.

Value

None

See Also

Resid

Examples

M = c("log","season",12,"trend",1)
e = Resid(wine,M)
test(e) ## Is e stationary?
a = arma(e,1,1)
ee = Resid(wine,M,a)
test(ee) ## Is ee white noise?

Estimate trend component

Description

Estimate trend component

Usage

trend(x, p)

Arguments

x

Time series data
p

Polynomial order (1 linear, 2 quadratic, etc.)

Value

Returns a vector the same length as x. Subtract from x to obtain residuals. The returned vector is the least squares fit of a polynomial to the data.

See Also

season

Examples

y = trend(uspop,2)
plotc(uspop,y)

Australian red wine sales, January 1980 to October 1991

Description

Australian red wine sales, January 1980 to October 1991

Examples

plotc(wine)

Estimate AR coefficients using the Yule-Walker method

Description

Estimate AR coefficients using the Yule-Walker method

Usage

yw(x, p)

Arguments

x

Time series data (typically residuals from Resid)
p

AR order

Details

The innovations algorithm is used to estimate white noise variance.

Value

Returns an ARMA model consisting of a list with the following components.

phi

Vector of AR coefficients (index number equals coefficient subscript)
theta

0
sigma2

White noise variance
aicc

Akaike information criterion corrected
se.phi

Standard errors for the AR coefficients
se.theta

0

See Also

arma burg hannan ia

Examples

M = c("diff",1)
e = Resid(dowj,M)
a = yw(e,1)