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itsmr: Time Series Analysis Using the Innovations Algorithm
Page 1
Package ‘itsmr’
October 13, 2022
Type Package
Title Time Series Analysis Using the Innovations Algorithm
Version 1.10
Date 2022-07-27
Author George Weigt
Maintainer George Weigt <g808391@icloud.com>
Description
Provides functions for modeling and forecasting time series data. Forecasting is based on the in-
novations algorithm. A description of the innovations algorithm can be found in the text-
book ``Introduction to Time Series and Forecasting'' by Peter J. Brock-
well and Richard A. Davis. <https://link.springer.com/book/10.1007/b97391>.
License FreeBSD
LazyLoad yes
NeedsCompilation no
URL https://georgeweigt.github.io/itsmr-refman.pdf
Repository CRAN
Date/Publication 2022-08-06 06:10:02 UTC
R topics documented:
itsmr-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
aacvf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
acvf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
airpass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
ar.inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
arar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
arma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
autofit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
burg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
deaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1
Page 2
2
itsmr-package
dowj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
hannan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
hr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
ia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
ma.inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
periodogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
plota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
plotc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Resid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
season . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
selftest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
sim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
smooth.exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
smooth.fft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
smooth.ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
smooth.rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
specify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
strikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Sunspots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
wine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
yw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Index
29
itsmr-package
Time Series Analysis Using the Innovations Algorithm
Description
Provides functions for modeling and forecasting time series data. Forecasting is based on the inno-
vations algorithm. A description of the innovations algorithm can be found in the textbook Intro-
duction 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
Page 3
aacvf
3
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)
aacvf
Autocovariance of ARMA model
Description
Autocovariance of ARMA model
Usage
aacvf(a, h)
Arguments
a
ARMA model
h
Maximum lag
Page 4
4
acvf
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)
acvf
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)
Page 5
ar.inf
5
airpass
Number of international airline passengers, 1949 to 1960
Description
Number of international airline passengers, 1949 to 1960
Examples
plotc(airpass)
ar.inf
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
Page 6
6
arar
Examples
a = yw(Sunspots,2)
ar.inf(a)
arar
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)
Page 7
arma
7
arma
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)
Page 8
8
autofit
autofit
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)
Page 9
burg
9
burg
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)
Page 10
10
deaths
check
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)
deaths
USA accidental deaths, 1973 to 1978
Description
USA accidental deaths, 1973 to 1978
Examples
plotc(deaths)
Page 11
forecast
11
dowj
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
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.)
Page 12
12
hannan
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)
hannan
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.
Page 13
hr
13
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)
hr
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)
Page 14
14
ia
ia
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)
Page 15
ma.inf
15
lake
Level of Lake Huron, 1875 to 1972
Description
Level of Lake Huron, 1875 to 1972
Examples
plotc(lake)
ma.inf
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
Page 16
16
periodogram
Examples
M = c("diff",12)
e = Resid(deaths,M)
a = arma(e,1,1)
ma.inf(a,10)
periodogram
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))
Page 17
plota
17
plota
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)
plotc
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
Page 18
18
Resid
Examples
plotc(uspop)
y = trend(uspop,2)
plotc(uspop,y)
plots
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)
Resid
Compute residuals
Description
Compute residuals
Usage
Resid(x, M = NULL, a = NULL)
Page 19
season
19
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)
season
Estimate seasonal component
Description
Estimate seasonal component
Page 20
20
selftest
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)
selftest
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()
Page 21
smooth.exp
21
sim
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)
smooth.exp
Apply an exponential filter
Description
Apply an exponential filter
Usage
smooth.exp(x, alpha)
Page 22
22
smooth.fft
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)
smooth.fft
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)
Page 23
smooth.ma
23
smooth.ma
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)
smooth.rank
Apply a spectral filter
Description
Apply a spectral filter
Usage
smooth.rank(x, k)
Arguments
x
Time series data
k
Number of frequencies
Page 24
24
specify
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
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))
Page 25
strikes
25
strikes
USA union strikes, 1951-1980
Description
USA union strikes, 1951-1980
Examples
plotc(strikes)
Sunspots
Number of sunspots, 1770 to 1869
Description
Number of sunspots, 1770 to 1869
Examples
plotc(Sunspots)
test
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.
Page 26
26
trend
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?
trend
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)
Page 27
wine
27
wine
Australian red wine sales, January 1980 to October 1991
Description
Australian red wine sales, January 1980 to October 1991
Examples
plotc(wine)
yw
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
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28
yw
Examples
M = c("diff",1)
e = Resid(dowj,M)
a = yw(e,1)
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Index
∗ datasets
airpass, 5
deaths, 10
dowj, 11
lake, 15
strikes, 25
Sunspots, 25
wine, 27
∗ package
itsmr-package, 2
aacvf, 3
acvf, 4
airpass, 5
ar.inf, 5, 15
arar, 6
arma, 4, 7, 8, 9, 12–14, 27
autofit, 7, 8
burg, 7, 9, 13, 14, 27
check, 10
deaths, 10
dowj, 11
forecast, 6, 11
hannan, 7, 9, 12, 14, 27
hr, 13
ia, 7, 9, 13, 14, 27
itsmr (itsmr-package), 2
itsmr-package, 2
lake, 15
ma.inf, 5, 15
periodogram, 16, 18
plota, 4, 17
plotc, 17
plots, 16, 18
Resid, 12, 18, 26
season, 19, 26
selftest, 20
sim, 21
smooth.exp, 21
smooth.fft, 22
smooth.ma, 23
smooth.rank, 23
specify, 24
strikes, 25
Sunspots, 25
test, 12, 19, 25
trend, 20, 26
wine, 27
yw, 7, 9, 13, 14, 27
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