| Type: | Package |
| Title: | Time Series Analysis Using the Innovations Algorithm |
| Version: | 1.11 |
| Date: | 2025-05-15 |
| Maintainer: | George Weigt <9634295@gmail.com> |
| 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. |
| License: | FreeBSD |
| LazyLoad: | yes |
| NeedsCompilation: | no |
| URL: | https://georgeweigt.github.io/itsmr-refman.pdf |
| Packaged: | 2025-05-15 19:03:42 UTC; georgeweigt |
| Author: | George Weigt [aut, cre] |
| Repository: | CRAN |
| Date/Publication: | 2025-05-16 08:50:17 UTC |
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: | 1.11 |
| Date: | 2025-05-15 |
| License: | FreeBSD |
| LazyLoad: | yes |
| URL: | https://georgeweigt.github.io/itsmr-refman.pdf |
Author(s)
George Weigt
Maintainer: George Weigt <g808391@icloud.com>
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)
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
Examples
M = c("log","season",12,"trend",1)
e = Resid(wine,M)
a = arma(e,1,1)
ee = Resid(wine,M,a)
Number of sunspots, 1770 to 1869
Description
Number of sunspots, 1770 to 1869
Examples
plotc(Sunspots)
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
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
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
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
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
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 |
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
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 |
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
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
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 |
p |
AR order |
q |
MA 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 |
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
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 |
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
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
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
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
Examples
a = specify(ar=c(0,0,.99))
plots(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
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)
Test residuals for stationarity and randomness
Description
Test residuals for stationarity and randomness
Usage
test(e)
Arguments
e |
Time series data (typically residuals from |
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
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
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 |
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
Examples
M = c("diff",1)
e = Resid(dowj,M)
a = yw(e,1)