In tuckermcelroy/sigex: Ecce Signum: Multivariate Signal Extraction
New Zealand Arrivals
This illustration considers a daily immigration series (imm for short): New
Zealand residents arriving in New Zealand after an absence of less than 12
months, covering the period January 1, 2008 through July 31, 2012. The series
consists of New Zealand residents arriving in New Zealand after an absence of
less than 12 months. These are public use data produced by Statistics New
Zealand via a customized extract, and correspond to a portion of the
"daily border crossings - arrivals" tab (Total) of the Travel category in the
Covid-19 portal: https://www.stats.govt.nz/experimental/covid-19-data-portal.
Loading Packages
This code installs and loads the packages, and loads the data into a variable
named imm.
library(devtools)
library(Rcpp)
devtools::install_github("tuckermcelroy/sigex")
library(sigex)
- We do some initial processing.
- The original data is grouped into "months" of 31 days, with extra days of
value zero added for months of shorter length; our code removes these features,
as well as a monthly summation.
n.months <- dim(imm)[1]/32
imm <- imm[-seq(1,n.months)*32,]
imm <- matrix(imm[imm != 0],ncol=6)
- Also we load a set of regressors that will be used in later analysis.
- The first column is a regressor that captures a moving holiday effect for
Easter, and the next six columns correspond to school holidays for public
schools in New Zealand: an effect is estimated for the start and end of the
holiday for the first three school vacations in each year. The last three
columns correspond to Y2K effects (but will not be used in this script).
nz.path <- system.file('extdata','NZregressors.txt',package='sigex')
NZregs <- read.table(nz.path)
Enter Metadata
- We perform an analysis of one series by using an embedding to the weekly
frequency.
- First enter metadata, including starting date and frequency.
start.date <- c(9,1,1997)
end.date <- day2date(dim(imm)[1]-1,start.date)
period <- 365
- Next we do some calendar calculations.
start.day <- date2day(start.date[1],start.date[2],start.date[3])
end.day <- date2day(end.date[1],end.date[2],end.date[3])
begin <- c(start.date[3],start.day)
end <- c(end.date[3],end.day)
- Finally, there is a call to sigex.load, which associates labels to the
data.
- The last argument of sigex.load generates a time series plot.
dataALL.ts <- sigex.load(imm,begin,period,
c("NZArr","NZDep","VisArr","VisDep","PLTArr","PLTDep"),TRUE)
Select Spans and Transforms
- On the basis of the above plot, we decide to utilize a log transformation for
the data.
- The aggregate option will sum across the indicated series, in case we want
to analyze an aggregate. Here we set it to FALSE.
- We can also select a subcomponent of series with subseries. Here we focus
on the first series only.
- The range argument of sigex.prep will select a subset of dates for the
time series to be analyzed.
transform <- "log"
aggregate <- FALSE
subseries <- 1
range <- list(c(2008,1),end)
dataONE.ts <- sigex.prep(dataALL.ts,transform,aggregate,subseries,range,TRUE)
Spectral Exploratory Analysis
- In order to get an idea about model specification, we can examine spectral
estimates, using sigex.specar.
- We can look at raw data or differenced data (growth rates).
par(mfrow=c(1,1))
for(i in 1:length(subseries)) { sigex.specar(dataONE.ts,FALSE,i,7) }
par(mfrow=c(1,1))
for(i in 1:length(subseries)) { sigex.specar(dataONE.ts,TRUE,i,7) }
Embed to Weekly
- We set the first day of the week to be Sunday, and embed the time series as a
$7$-variate weekly time series via a call to sigex.daily2weekly.
first.day <- 1
data.ts <- sigex.daily2weekly(dataONE.ts,first.day,start.date)
plot(data.ts)
- As a result $N= 7$ and $T = 240$, which yields an initial NA (because the
first data value is on Monday, but the week begins on Sunday). Also the data
ends on a Sunday, so the last week has six NA values.
- This is a classic example of a ragged edge data pattern.
Model Declaration
- Now we want to define our time series model.
- We begin by defining dimension $N$ and sample size $T$.
- Note that although the data analyzed is univariate, because of the embedding
now $N=7$.
N <- dim(data.ts)[2]
T <- dim(data.ts)[1]
- Next, we define appropriate subspans of the holiday regressors.
- The easter regressor is based on the Easter day itself and also the day
before Easter.
- The school1 regressor corresponds to the beginning of the first school
holiday, with a window for the day of and the day after.
- The school2 and school3 regressors are analogous for the second and third
holidays.
- The school1e regressor corresponds to the end of the first school holiday,
with a window for the day of and the day before.
- The school2e and school3e regressors are analogous for the second and
third holidays.
times <- seq((range[[1]][1]-begin[1])*period+(range[[1]][2]-begin[2])+1,
(range[[2]][1]-begin[1])*period+(range[[2]][2]-begin[2])+1,1)
easter.reg <- NZregs[times,1]
school1.reg <- NZregs[times,2]
school1e.reg <- NZregs[times,3]
school2.reg <- NZregs[times,4]
school2e.reg <- NZregs[times,5]
school3.reg <- NZregs[times,6]
school3e.reg <- NZregs[times,7]
- These are the regressors for daily data, and need to be embedded as weekly
series.
easter.reg <- sigex.daily2weekly(easter.reg,first.day,start.date)
school1.reg <- sigex.daily2weekly(school1.reg,first.day,start.date)
school1e.reg <- sigex.daily2weekly(school1e.reg,first.day,start.date)
school2.reg <- sigex.daily2weekly(school2.reg,first.day,start.date)
school2e.reg <- sigex.daily2weekly(school2e.reg,first.day,start.date)
school3.reg <- sigex.daily2weekly(school3.reg,first.day,start.date)
school3e.reg <- sigex.daily2weekly(school3e.reg,first.day,start.date)
- Next, we replace any ragged edge NA values (arising from embedding) with zero,
which is appropriate because the corresponding data values are also missing.
easter.reg[is.na(easter.reg)] <- 0
school1.reg[is.na(school1.reg)] <- 0
school1e.reg[is.na(school1e.reg)] <- 0
school2.reg[is.na(school2.reg)] <- 0
school2e.reg[is.na(school2e.reg)] <- 0
school3.reg[is.na(school3.reg)] <- 0
school3e.reg[is.na(school3e.reg)] <- 0
Model Specification
- The data has a slight upwards trend drift, and there is strong annual
correlation present.
- We will fit a SVARMA model for data differenced to stationarity by applying
annual differencing $1 - L^{52}$. (Here $L$ is the weekly lag operator, and
there are approximately 52 weeks per year.)
- We set the model order to a nonseasonal $q=1$ and seasonal $P = 1$.
mdl <- NULL
mdl <- sigex.add(mdl,seq(1,N),"svarma",c(0,1,1,0,52),NULL,"process",
c(1,rep(0,51),-1))
mdl <- sigex.meaninit(mdl,data.ts,0)
- The call to sigex.add indicates there is one main component.
- The second argument indicates the rank configuration for the process; setting
vrank equal to seq(1,N) ensures a full rank covariance matrix.
- The third argument gives the model type as svarma; the order $q=1$, $P=1$ of
the SVARMA is given in the fourth argument.
- The fifth argument provides parameter bounds, and does not apply to this type
of model, and hence is set to NULL.
- The sixth argument associates a label.
- The last argument is for the scalar differencing operator $\delta^{(1)} (z)$,
expressed as a vector of coefficients.
- The specification of the regressors takes more coding, as we loop over the
days of the week.
for(i in 1:N) {
mdl <- sigex.reg(mdl,i,ts(as.matrix(easter.reg[,i]),
start=start(easter.reg),
frequency=frequency(easter.reg),
names="Easter-day"))
mdl <- sigex.reg(mdl,i,ts(as.matrix(school1.reg[,i]),
start=start(school1.reg),
frequency=frequency(school1.reg),
names="School1-Start"))
mdl <- sigex.reg(mdl,i,ts(as.matrix(school1e.reg[,i]),
start=start(school1e.reg),
frequency=frequency(school1e.reg),
names="School1-End"))
mdl <- sigex.reg(mdl,i,ts(as.matrix(school2.reg[,i]),
start=start(school2.reg),
frequency=frequency(school2.reg),
names="School2-Start"))
mdl <- sigex.reg(mdl,i,ts(as.matrix(school2e.reg[,i]),
start=start(school2e.reg),
frequency=frequency(school2e.reg),
names="School2-End"))
mdl <- sigex.reg(mdl,i,ts(as.matrix(school3.reg[,i]),
start=start(school3.reg),
frequency=frequency(school3.reg),
names="School3-Start"))
mdl <- sigex.reg(mdl,i,ts(as.matrix(school3e.reg[,i]),
start=start(school3e.reg),
frequency=frequency(school3e.reg),
names="School3-End"))
}
Model Estimation
- Next we consider model fitting.
- Here we make use of parameter constraints for the regressors: because the same
holiday effect is present for each day of the week (recall that the weekly
regressors are derived from a single daily regressor), the parameter estimates
should be constrained to be identical.
- We use the sigex.constrainreg function to do this: the regindex argument
specifies indices $J_1, J_2, \ldots, J_N$ that delineate which regressors are
being considered for each series.
- For instance, $J_1 = { 1, 3, 4 }$ would correspond to the trend,
school1, and school1e regressors for the first (Sunday) series.
- The combos argument provides the linear combinations of these regressors,
but when set to NULL instead it is enforced that all the regression parameters
are the same.
constraint <- NULL
constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts,
list(2,2,2,2,2,2,2),NULL))
constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts,
list(3,3,3,3,3,3,3),NULL))
constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts,
list(4,4,4,4,4,4,4),NULL))
constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts,
list(5,5,5,5,5,5,5),NULL))
constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts,
list(6,6,6,6,6,6,6),NULL))
constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts,
list(7,7,7,7,7,7,7),NULL))
- So the first call here identifies in regindex the easter regressor for
each of the seven series, and combos set to NULL indicates that these
parameters will be equal.
- The next call does the same for the school1 regressor, and so forth; the
first regressor is the trend, and this remains unconstrained.
- The output of sigex.constrainreg is a constraint matrix, which can be
then passed into the MLE fitting routine.
par.mle <- sigex.default(mdl,data.ts,constraint)
psi.mle <- sigex.par2psi(par.mle,mdl)
- Having initialized the pre-parameter vector, we can start nonlinear
optimization.
- In this case the nonlinear optimization can take a long time. So we just read
the final MLE for psi in below.
psi.mle <- c(0.195543136661102, 0.210896311604314, 0.224884839193806,
0.297710366252992, 0.318076964322756, 0.288167112704856,
0.130848490409544, 0.17299456406149, 0.213720675984967,
0.0780486278573135, 0.188624612762698, 0.467926733192453,
0.434759495960342, 0.314822770386916, 0.280937951321511,
0.484722644882768, 0.413444260303994, 0.193778981158949,
0.550345260075344, 0.621238984502903, 0.563243057038342,
-3.99208014979805, -4.19642305454813, -3.8415818035668,
-4.32856255035426, -3.95499509133263, -3.96279211490641,
-4.03911188215587, -0.369778097197869, -0.0318374973193093,
-0.0459307709775452, -0.138903858423006, -0.10135335156786,
-0.2142068950075, -0.112796685064521, -0.140796608637905,
-0.237750837664742, -0.194243215055276, 0.0219933280900727,
-0.0185736546749142, 0.139055815995136, 0.185830059881372,
0.0297317536468041, -0.0155812148508496, -0.0671396386183212,
0.0476284405203113, -0.0570660189385589, -0.155367001953893,
-0.198330717820409, -0.0295063266207021, -0.073765478114995,
-0.0829227752708303, -0.187804918878073, -0.0422869655675992,
-0.0251387557650662, 0.0656914410289482, -0.0840920843255883,
-0.145935243196311, -0.0687523983229335, -0.117014892923535,
-0.144653364980642, -0.0790943505687107, 0.0611829711400193,
-0.104109662533708, 0.112359996172669, -0.124553026055195,
-0.163431409482439, -0.0283323592215797, -0.0312605011210402,
-0.178946106450384, -0.266581746889425, -0.164720047360709,
0.00959770743708815, 0.159928359498417, -0.0132411806278071,
-0.0642101491748187, -0.143487318464752, -0.086691994085389,
0.0814467471639927, -0.0123455935133012, 0.00908782055333108,
-0.0441428053848971, -0.0194126102991047, -0.00524075702955244,
-0.0105279430631441, -0.00420628604690744, 0.0789699162821441,
-0.015850906929479, -0.043499933859938, -0.0062803013365077,
-0.0207414587537556, -0.0459309858718756, 0.0445914128582114,
-0.0946876132299148, 0.00588335338136514, 0.00922656520387984,
-0.0140866870525241, -0.0157842160720905, -0.0319375488373468,
0.0514126088016708, 0.0141484170525309, -0.124101882618575,
-0.00774709495827467, 0.0255995670388491, -0.0364850207049338,
0.0230572817360244, 0.0220446070005591, -0.000478476269822034,
-0.0168616246820038, -0.102427194747844, 0.00112763406309242,
-0.0259316390382219, -0.010758329862067, -0.00198961225224573,
-0.0464823007621206, -0.0165414789720724, 0.0104091697972505,
-0.14827440476309, 0.0505648610397286, -0.0183900737977979,
0.0311301005304512, 0.00518380184796121, 0.0104592851480601,
-0.0109208667318225, -0.0637859237889965, -0.0656126331052389,
0.00033010004554818, -0.191533893764455, -0.157111537080312,
0.141669136870683, -0.0225792281088362, 0.157100658532765,
-0.08441684545665, 0.1847632577291, 0.000561604448005,
-0.191533893764455, -0.157111537080311, 0.141669136870683,
-0.0225792281088361, 0.157100658532765, -0.0844168454566499,
0.183411740706148, 0.000312708939566006, -0.191533893764455,
-0.157111537080312, 0.141669136870683, -0.0225792281088362,
0.157100658532765, -0.08441684545665, -0.00163601810705088,
0.00100594355939265, -0.191533893764455, -0.157111537080311,
0.141669136870683, -0.0225792281088362, 0.157100658532765,
-0.08441684545665, 0.0156601320968592, 9.68417444876468e-05,
-0.191533893764455, -0.157111537080312, 0.141669136870683,
-0.0225792281088361, 0.157100658532765, -0.08441684545665,
0.00328023110252021, 0.000461006050610564, -0.191533893764455,
-0.157111537080312, 0.141669136870683, -0.0225792281088361,
0.157100658532765, -0.08441684545665, 0.00430421751741802,
0.000496876409071807, -0.191533893764455, -0.157111537080311,
0.141669136870683, -0.0225792281088362, 0.157100658532765,
-0.08441684545665, 0.0023532873975757)
par.mle <- sigex.psi2par(psi.mle,mdl,data.ts)
- We obtain, format, and store the residuals.
- Then we compute the autocorrelations, and plot.
resid.mle <- sigex.resid(psi.mle,mdl,data.ts)[[1]]
resid.mle <- sigex.load(t(resid.mle),start(data.ts),frequency(data.ts),
colnames(data.ts),TRUE)
resid.acf <- acf(resid.mle,lag.max=4*53,plot=FALSE)$acf
par(mfrow=c(N,N),mar=c(3,2,2,0)+0.1,cex.lab=.8,cex.axis=.5,bty="n")
for(j in 1:N)
{
for(k in 1:N)
{
plot.ts(resid.acf[,j,k],ylab="",xlab="Lag",ylim=c(-1,1),cex=.5)
abline(h=1.96/sqrt(T),lty=3)
abline(h=-1.96/sqrt(T),lty=3)
}
}
- We can also examine the condition numbers, to see if there is rank reduction
possible.
- Typically there won't be any rank reduction with a single component model.
log(sigex.conditions(data.ts,psi.mle,mdl))
- We can examine the Portmanteau statistic for residual serial correlation.
- We use 2 years of lags.
sigex.portmanteau(resid.mle,2*52,length(psi.mle))
- The output indicates there is moderate residual autocorrelation, which is
confirmed by inspection of the sample autocorrelation.
- We can also inspect normality via the Shapiro-Wilks normality test.
sigex.gausscheck(resid.mle)
- Having completed our model analysis, we store our results in a single list
object.
analysis.mle <- sigex.bundle(data.ts,transform,mdl,psi.mle)
Signal Extraction
- Now we consider applying a daily seasonal adjustment filter that we embed as a
$7$-variate weekly filter.
- First we load up the fitted model for signal extraction.
data.ts <- analysis.mle[[1]]
mdl <- analysis.mle[[3]]
psi <- analysis.mle[[4]]
param <- sigex.psi2par(psi,mdl,data.ts)
- The daily seasonal adjustment filter sa.hifilter is an extension of the
classic crude trend filter of X11 from monthly to daily frequency.
- It should suppress all daily frequencies of the form $2\pi j/365$ for
$1 \leq j \leq 183$, while also passing linear trends. This is embedded with a
call to sigex.hi2low, where the high and low frequencies are indicated and
the shift.hi parameter is $c = 183$, corresponding to a symmetric filter.
- The output is the $7$-variate weekly filter sa.lowfilter, with shift
parameter $C = 27$ given by shift.low.
sa.hifilter <- c(1,rep(2,365),1)/(2*365)
len <- 183
hi.freq <- 7
low.freq <- 1
shift.hi <- len
out <- sigex.hi2low(sa.hifilter,hi.freq,low.freq,shift.hi)
sa.lowfilter <- out[[1]]
shift.low <- out[[2]]
- This filter is applied using a standard call of sigex.adhocextract.
- This function automatically supplies missing value imputations for the ragged
edges, and does forecast and aftcast extension for application of the filter.
However, this should be transformed back to a daily series, so that we can view
the daily series with its seasonal adjustment.
- The function sigex.weekly2daily transforms the series, using the
indicated value of first.day.
- We repeat this for the lower and upper values of the confidence intervals.
sa.low <- sigex.adhocextract(psi,mdl,data.ts,sa.lowfilter,shift.low,0,TRUE)
sa.hi.daily <- list()
sa.hi.daily[[1]] <- sigex.weekly2daily(ts(sa.low[[1]],start=start(dataONE.ts),
frequency=frequency(dataONE.ts)),first.day)
sa.hi.daily[[2]] <- sigex.weekly2daily(ts(sa.low[[2]],start=start(dataONE.ts),
frequency=frequency(dataONE.ts)),first.day)
sa.hi.daily[[3]] <- sigex.weekly2daily(ts(sa.low[[3]],start=start(dataONE.ts),
frequency=frequency(dataONE.ts)),first.day)
- For comparison we can design another filter that only removes the weekly
dynamics.
- This is done by considering a simple filter annihilating weekly effects from
the daily data, namely $(1 + B + \ldots + B^6)/7$, which we call the Trading
Day (TD) filter.
td.hifilter <- rep(1,7)/7
len <- 3
hi.freq <- 7
low.freq <- 1
shift.hi <- len
out <- sigex.hi2low(td.hifilter,hi.freq,low.freq,shift.hi)
td.lowfilter <- out[[1]]
shift.low <- out[[2]]
- Now $c = 3$ and $C = 1$ for the TD filter.
td.low <- sigex.adhocextract(psi,mdl,data.ts,td.lowfilter,shift.low,0,TRUE)
td.hi.daily <- list()
td.hi.daily[[1]] <- sigex.weekly2daily(ts(td.low[[1]],start=start(dataONE.ts),
frequency=frequency(dataONE.ts)),first.day)
td.hi.daily[[2]] <- sigex.weekly2daily(ts(td.low[[2]],start=start(dataONE.ts),
frequency=frequency(dataONE.ts)),first.day)
td.hi.daily[[3]] <- sigex.weekly2daily(ts(td.low[[3]],start=start(dataONE.ts),
frequency=frequency(dataONE.ts)),first.day)
- Before plotting, the trend must be computed.
- A subtlety is that we have $7$ trends, one for each day of week, which
therefore combine long-term trend effects and day-of-week effects. The mean of
the $7$ parameter values corresponds to the overall mean of each week, and the
residual is the day-of-week effect in excess of the mean.
- First reg.td is computed, which is the trend means for each day-of-week
expressed as a daily time series. We augment by one prior week's values, which
would be all zero, and then convert to a daily series.
- Next, reg.trend is obtained by filtering with $(1 + B + \ldots + B^6)/7$,
and finally remove the NA value (resulting from using filter, which does not
extend the input series).
reg.td <- NULL
for(i in 1:N){reg.td <- cbind(reg.td,sigex.fixed(data.ts,mdl,i,param,"Trend"))}
reg.td <- rbind(rep(0,N),reg.td)
reg.td <- ts(sigex.weekly2daily(reg.td,first.day),
start=start(dataONE.ts),frequency=period)
reg.trend <- stats::filter(reg.td,td.hifilter,method="convolution",sides=1)
reg.trend <- as.matrix(reg.trend[8:length(reg.td)])
- The illustration is completed with a display: the seasonal adjustment contains
some mild weekly effects (because the filter is designed to remove daily, and
not weekly frequencies), whereas the TD-adjusted component has annual movements
but not weekly effects.
trendcol <- "tomato"
cyccol <- "orchid"
seascol <- "seagreen"
sacol <- "navyblue"
fade <- 60
plot(dataONE.ts,xlab="Year")
sigex.graph(sa.hi.daily,reg.trend,start(sa.hi.daily[[1]]),
period,1,0,trendcol,fade)
sigex.graph(td.hi.daily,reg.trend,start(td.hi.daily[[1]]),
period,1,0,sacol,fade)
tuckermcelroy/sigex documentation built on Nov. 14, 2024, 3:28 p.m.
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New Zealand Arrivals
This illustration considers a daily immigration series (imm for short): New Zealand residents arriving in New Zealand after an absence of less than 12 months, covering the period January 1, 2008 through July 31, 2012. The series consists of New Zealand residents arriving in New Zealand after an absence of less than 12 months. These are public use data produced by Statistics New Zealand via a customized extract, and correspond to a portion of the "daily border crossings - arrivals" tab (Total) of the Travel category in the Covid-19 portal: https://www.stats.govt.nz/experimental/covid-19-data-portal.
Loading Packages
This code installs and loads the packages, and loads the data into a variable named imm.
library(devtools) library(Rcpp) devtools::install_github("tuckermcelroy/sigex") library(sigex)
- We do some initial processing.
- The original data is grouped into "months" of 31 days, with extra days of value zero added for months of shorter length; our code removes these features, as well as a monthly summation.
n.months <- dim(imm)[1]/32 imm <- imm[-seq(1,n.months)*32,] imm <- matrix(imm[imm != 0],ncol=6)
- Also we load a set of regressors that will be used in later analysis.
- The first column is a regressor that captures a moving holiday effect for Easter, and the next six columns correspond to school holidays for public schools in New Zealand: an effect is estimated for the start and end of the holiday for the first three school vacations in each year. The last three columns correspond to Y2K effects (but will not be used in this script).
nz.path <- system.file('extdata','NZregressors.txt',package='sigex') NZregs <- read.table(nz.path)
Enter Metadata
- We perform an analysis of one series by using an embedding to the weekly frequency.
- First enter metadata, including starting date and frequency.
start.date <- c(9,1,1997) end.date <- day2date(dim(imm)[1]-1,start.date) period <- 365
- Next we do some calendar calculations.
start.day <- date2day(start.date[1],start.date[2],start.date[3]) end.day <- date2day(end.date[1],end.date[2],end.date[3]) begin <- c(start.date[3],start.day) end <- c(end.date[3],end.day)
- Finally, there is a call to sigex.load, which associates labels to the data.
- The last argument of sigex.load generates a time series plot.
dataALL.ts <- sigex.load(imm,begin,period, c("NZArr","NZDep","VisArr","VisDep","PLTArr","PLTDep"),TRUE)
Select Spans and Transforms
- On the basis of the above plot, we decide to utilize a log transformation for the data.
- The aggregate option will sum across the indicated series, in case we want to analyze an aggregate. Here we set it to FALSE.
- We can also select a subcomponent of series with subseries. Here we focus on the first series only.
- The range argument of sigex.prep will select a subset of dates for the time series to be analyzed.
transform <- "log" aggregate <- FALSE subseries <- 1 range <- list(c(2008,1),end) dataONE.ts <- sigex.prep(dataALL.ts,transform,aggregate,subseries,range,TRUE)
Spectral Exploratory Analysis
- In order to get an idea about model specification, we can examine spectral estimates, using sigex.specar.
- We can look at raw data or differenced data (growth rates).
par(mfrow=c(1,1)) for(i in 1:length(subseries)) { sigex.specar(dataONE.ts,FALSE,i,7) }
par(mfrow=c(1,1)) for(i in 1:length(subseries)) { sigex.specar(dataONE.ts,TRUE,i,7) }
Embed to Weekly
- We set the first day of the week to be Sunday, and embed the time series as a $7$-variate weekly time series via a call to sigex.daily2weekly.
first.day <- 1 data.ts <- sigex.daily2weekly(dataONE.ts,first.day,start.date) plot(data.ts)
- As a result $N= 7$ and $T = 240$, which yields an initial NA (because the first data value is on Monday, but the week begins on Sunday). Also the data ends on a Sunday, so the last week has six NA values.
- This is a classic example of a ragged edge data pattern.
Model Declaration
- Now we want to define our time series model.
- We begin by defining dimension $N$ and sample size $T$.
- Note that although the data analyzed is univariate, because of the embedding now $N=7$.
N <- dim(data.ts)[2] T <- dim(data.ts)[1]
- Next, we define appropriate subspans of the holiday regressors.
- The easter regressor is based on the Easter day itself and also the day before Easter.
- The school1 regressor corresponds to the beginning of the first school holiday, with a window for the day of and the day after.
- The school2 and school3 regressors are analogous for the second and third holidays.
- The school1e regressor corresponds to the end of the first school holiday, with a window for the day of and the day before.
- The school2e and school3e regressors are analogous for the second and third holidays.
times <- seq((range[[1]][1]-begin[1])*period+(range[[1]][2]-begin[2])+1, (range[[2]][1]-begin[1])*period+(range[[2]][2]-begin[2])+1,1) easter.reg <- NZregs[times,1] school1.reg <- NZregs[times,2] school1e.reg <- NZregs[times,3] school2.reg <- NZregs[times,4] school2e.reg <- NZregs[times,5] school3.reg <- NZregs[times,6] school3e.reg <- NZregs[times,7]
- These are the regressors for daily data, and need to be embedded as weekly series.
easter.reg <- sigex.daily2weekly(easter.reg,first.day,start.date) school1.reg <- sigex.daily2weekly(school1.reg,first.day,start.date) school1e.reg <- sigex.daily2weekly(school1e.reg,first.day,start.date) school2.reg <- sigex.daily2weekly(school2.reg,first.day,start.date) school2e.reg <- sigex.daily2weekly(school2e.reg,first.day,start.date) school3.reg <- sigex.daily2weekly(school3.reg,first.day,start.date) school3e.reg <- sigex.daily2weekly(school3e.reg,first.day,start.date)
- Next, we replace any ragged edge NA values (arising from embedding) with zero, which is appropriate because the corresponding data values are also missing.
easter.reg[is.na(easter.reg)] <- 0 school1.reg[is.na(school1.reg)] <- 0 school1e.reg[is.na(school1e.reg)] <- 0 school2.reg[is.na(school2.reg)] <- 0 school2e.reg[is.na(school2e.reg)] <- 0 school3.reg[is.na(school3.reg)] <- 0 school3e.reg[is.na(school3e.reg)] <- 0
Model Specification
- The data has a slight upwards trend drift, and there is strong annual correlation present.
- We will fit a SVARMA model for data differenced to stationarity by applying annual differencing $1 - L^{52}$. (Here $L$ is the weekly lag operator, and there are approximately 52 weeks per year.)
- We set the model order to a nonseasonal $q=1$ and seasonal $P = 1$.
mdl <- NULL mdl <- sigex.add(mdl,seq(1,N),"svarma",c(0,1,1,0,52),NULL,"process", c(1,rep(0,51),-1)) mdl <- sigex.meaninit(mdl,data.ts,0)
- The call to sigex.add indicates there is one main component.
- The second argument indicates the rank configuration for the process; setting vrank equal to seq(1,N) ensures a full rank covariance matrix.
- The third argument gives the model type as svarma; the order $q=1$, $P=1$ of the SVARMA is given in the fourth argument.
- The fifth argument provides parameter bounds, and does not apply to this type of model, and hence is set to NULL.
- The sixth argument associates a label.
- The last argument is for the scalar differencing operator $\delta^{(1)} (z)$, expressed as a vector of coefficients.
- The specification of the regressors takes more coding, as we loop over the days of the week.
for(i in 1:N) { mdl <- sigex.reg(mdl,i,ts(as.matrix(easter.reg[,i]), start=start(easter.reg), frequency=frequency(easter.reg), names="Easter-day")) mdl <- sigex.reg(mdl,i,ts(as.matrix(school1.reg[,i]), start=start(school1.reg), frequency=frequency(school1.reg), names="School1-Start")) mdl <- sigex.reg(mdl,i,ts(as.matrix(school1e.reg[,i]), start=start(school1e.reg), frequency=frequency(school1e.reg), names="School1-End")) mdl <- sigex.reg(mdl,i,ts(as.matrix(school2.reg[,i]), start=start(school2.reg), frequency=frequency(school2.reg), names="School2-Start")) mdl <- sigex.reg(mdl,i,ts(as.matrix(school2e.reg[,i]), start=start(school2e.reg), frequency=frequency(school2e.reg), names="School2-End")) mdl <- sigex.reg(mdl,i,ts(as.matrix(school3.reg[,i]), start=start(school3.reg), frequency=frequency(school3.reg), names="School3-Start")) mdl <- sigex.reg(mdl,i,ts(as.matrix(school3e.reg[,i]), start=start(school3e.reg), frequency=frequency(school3e.reg), names="School3-End")) }
Model Estimation
- Next we consider model fitting.
- Here we make use of parameter constraints for the regressors: because the same holiday effect is present for each day of the week (recall that the weekly regressors are derived from a single daily regressor), the parameter estimates should be constrained to be identical.
- We use the sigex.constrainreg function to do this: the regindex argument specifies indices $J_1, J_2, \ldots, J_N$ that delineate which regressors are being considered for each series.
- For instance, $J_1 = { 1, 3, 4 }$ would correspond to the trend, school1, and school1e regressors for the first (Sunday) series.
- The combos argument provides the linear combinations of these regressors, but when set to NULL instead it is enforced that all the regression parameters are the same.
constraint <- NULL constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts, list(2,2,2,2,2,2,2),NULL)) constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts, list(3,3,3,3,3,3,3),NULL)) constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts, list(4,4,4,4,4,4,4),NULL)) constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts, list(5,5,5,5,5,5,5),NULL)) constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts, list(6,6,6,6,6,6,6),NULL)) constraint <- rbind(constraint,sigex.constrainreg(mdl,data.ts, list(7,7,7,7,7,7,7),NULL))
- So the first call here identifies in regindex the easter regressor for each of the seven series, and combos set to NULL indicates that these parameters will be equal.
- The next call does the same for the school1 regressor, and so forth; the first regressor is the trend, and this remains unconstrained.
- The output of sigex.constrainreg is a constraint matrix, which can be then passed into the MLE fitting routine.
par.mle <- sigex.default(mdl,data.ts,constraint) psi.mle <- sigex.par2psi(par.mle,mdl)
- Having initialized the pre-parameter vector, we can start nonlinear optimization.
- In this case the nonlinear optimization can take a long time. So we just read the final MLE for psi in below.
psi.mle <- c(0.195543136661102, 0.210896311604314, 0.224884839193806, 0.297710366252992, 0.318076964322756, 0.288167112704856, 0.130848490409544, 0.17299456406149, 0.213720675984967, 0.0780486278573135, 0.188624612762698, 0.467926733192453, 0.434759495960342, 0.314822770386916, 0.280937951321511, 0.484722644882768, 0.413444260303994, 0.193778981158949, 0.550345260075344, 0.621238984502903, 0.563243057038342, -3.99208014979805, -4.19642305454813, -3.8415818035668, -4.32856255035426, -3.95499509133263, -3.96279211490641, -4.03911188215587, -0.369778097197869, -0.0318374973193093, -0.0459307709775452, -0.138903858423006, -0.10135335156786, -0.2142068950075, -0.112796685064521, -0.140796608637905, -0.237750837664742, -0.194243215055276, 0.0219933280900727, -0.0185736546749142, 0.139055815995136, 0.185830059881372, 0.0297317536468041, -0.0155812148508496, -0.0671396386183212, 0.0476284405203113, -0.0570660189385589, -0.155367001953893, -0.198330717820409, -0.0295063266207021, -0.073765478114995, -0.0829227752708303, -0.187804918878073, -0.0422869655675992, -0.0251387557650662, 0.0656914410289482, -0.0840920843255883, -0.145935243196311, -0.0687523983229335, -0.117014892923535, -0.144653364980642, -0.0790943505687107, 0.0611829711400193, -0.104109662533708, 0.112359996172669, -0.124553026055195, -0.163431409482439, -0.0283323592215797, -0.0312605011210402, -0.178946106450384, -0.266581746889425, -0.164720047360709, 0.00959770743708815, 0.159928359498417, -0.0132411806278071, -0.0642101491748187, -0.143487318464752, -0.086691994085389, 0.0814467471639927, -0.0123455935133012, 0.00908782055333108, -0.0441428053848971, -0.0194126102991047, -0.00524075702955244, -0.0105279430631441, -0.00420628604690744, 0.0789699162821441, -0.015850906929479, -0.043499933859938, -0.0062803013365077, -0.0207414587537556, -0.0459309858718756, 0.0445914128582114, -0.0946876132299148, 0.00588335338136514, 0.00922656520387984, -0.0140866870525241, -0.0157842160720905, -0.0319375488373468, 0.0514126088016708, 0.0141484170525309, -0.124101882618575, -0.00774709495827467, 0.0255995670388491, -0.0364850207049338, 0.0230572817360244, 0.0220446070005591, -0.000478476269822034, -0.0168616246820038, -0.102427194747844, 0.00112763406309242, -0.0259316390382219, -0.010758329862067, -0.00198961225224573, -0.0464823007621206, -0.0165414789720724, 0.0104091697972505, -0.14827440476309, 0.0505648610397286, -0.0183900737977979, 0.0311301005304512, 0.00518380184796121, 0.0104592851480601, -0.0109208667318225, -0.0637859237889965, -0.0656126331052389, 0.00033010004554818, -0.191533893764455, -0.157111537080312, 0.141669136870683, -0.0225792281088362, 0.157100658532765, -0.08441684545665, 0.1847632577291, 0.000561604448005, -0.191533893764455, -0.157111537080311, 0.141669136870683, -0.0225792281088361, 0.157100658532765, -0.0844168454566499, 0.183411740706148, 0.000312708939566006, -0.191533893764455, -0.157111537080312, 0.141669136870683, -0.0225792281088362, 0.157100658532765, -0.08441684545665, -0.00163601810705088, 0.00100594355939265, -0.191533893764455, -0.157111537080311, 0.141669136870683, -0.0225792281088362, 0.157100658532765, -0.08441684545665, 0.0156601320968592, 9.68417444876468e-05, -0.191533893764455, -0.157111537080312, 0.141669136870683, -0.0225792281088361, 0.157100658532765, -0.08441684545665, 0.00328023110252021, 0.000461006050610564, -0.191533893764455, -0.157111537080312, 0.141669136870683, -0.0225792281088361, 0.157100658532765, -0.08441684545665, 0.00430421751741802, 0.000496876409071807, -0.191533893764455, -0.157111537080311, 0.141669136870683, -0.0225792281088362, 0.157100658532765, -0.08441684545665, 0.0023532873975757) par.mle <- sigex.psi2par(psi.mle,mdl,data.ts)
- We obtain, format, and store the residuals.
- Then we compute the autocorrelations, and plot.
resid.mle <- sigex.resid(psi.mle,mdl,data.ts)[[1]] resid.mle <- sigex.load(t(resid.mle),start(data.ts),frequency(data.ts), colnames(data.ts),TRUE) resid.acf <- acf(resid.mle,lag.max=4*53,plot=FALSE)$acf par(mfrow=c(N,N),mar=c(3,2,2,0)+0.1,cex.lab=.8,cex.axis=.5,bty="n") for(j in 1:N) { for(k in 1:N) { plot.ts(resid.acf[,j,k],ylab="",xlab="Lag",ylim=c(-1,1),cex=.5) abline(h=1.96/sqrt(T),lty=3) abline(h=-1.96/sqrt(T),lty=3) } }
- We can also examine the condition numbers, to see if there is rank reduction possible.
- Typically there won't be any rank reduction with a single component model.
log(sigex.conditions(data.ts,psi.mle,mdl))
- We can examine the Portmanteau statistic for residual serial correlation.
- We use 2 years of lags.
sigex.portmanteau(resid.mle,2*52,length(psi.mle))
- The output indicates there is moderate residual autocorrelation, which is confirmed by inspection of the sample autocorrelation.
- We can also inspect normality via the Shapiro-Wilks normality test.
sigex.gausscheck(resid.mle)
- Having completed our model analysis, we store our results in a single list object.
analysis.mle <- sigex.bundle(data.ts,transform,mdl,psi.mle)
Signal Extraction
- Now we consider applying a daily seasonal adjustment filter that we embed as a $7$-variate weekly filter.
- First we load up the fitted model for signal extraction.
data.ts <- analysis.mle[[1]] mdl <- analysis.mle[[3]] psi <- analysis.mle[[4]] param <- sigex.psi2par(psi,mdl,data.ts)
- The daily seasonal adjustment filter sa.hifilter is an extension of the classic crude trend filter of X11 from monthly to daily frequency.
- It should suppress all daily frequencies of the form $2\pi j/365$ for $1 \leq j \leq 183$, while also passing linear trends. This is embedded with a call to sigex.hi2low, where the high and low frequencies are indicated and the shift.hi parameter is $c = 183$, corresponding to a symmetric filter.
- The output is the $7$-variate weekly filter sa.lowfilter, with shift parameter $C = 27$ given by shift.low.
sa.hifilter <- c(1,rep(2,365),1)/(2*365) len <- 183 hi.freq <- 7 low.freq <- 1 shift.hi <- len out <- sigex.hi2low(sa.hifilter,hi.freq,low.freq,shift.hi) sa.lowfilter <- out[[1]] shift.low <- out[[2]]
- This filter is applied using a standard call of sigex.adhocextract.
- This function automatically supplies missing value imputations for the ragged edges, and does forecast and aftcast extension for application of the filter. However, this should be transformed back to a daily series, so that we can view the daily series with its seasonal adjustment.
- The function sigex.weekly2daily transforms the series, using the indicated value of first.day.
- We repeat this for the lower and upper values of the confidence intervals.
sa.low <- sigex.adhocextract(psi,mdl,data.ts,sa.lowfilter,shift.low,0,TRUE) sa.hi.daily <- list() sa.hi.daily[[1]] <- sigex.weekly2daily(ts(sa.low[[1]],start=start(dataONE.ts), frequency=frequency(dataONE.ts)),first.day) sa.hi.daily[[2]] <- sigex.weekly2daily(ts(sa.low[[2]],start=start(dataONE.ts), frequency=frequency(dataONE.ts)),first.day) sa.hi.daily[[3]] <- sigex.weekly2daily(ts(sa.low[[3]],start=start(dataONE.ts), frequency=frequency(dataONE.ts)),first.day)
- For comparison we can design another filter that only removes the weekly dynamics.
- This is done by considering a simple filter annihilating weekly effects from the daily data, namely $(1 + B + \ldots + B^6)/7$, which we call the Trading Day (TD) filter.
td.hifilter <- rep(1,7)/7 len <- 3 hi.freq <- 7 low.freq <- 1 shift.hi <- len out <- sigex.hi2low(td.hifilter,hi.freq,low.freq,shift.hi) td.lowfilter <- out[[1]] shift.low <- out[[2]]
- Now $c = 3$ and $C = 1$ for the TD filter.
td.low <- sigex.adhocextract(psi,mdl,data.ts,td.lowfilter,shift.low,0,TRUE) td.hi.daily <- list() td.hi.daily[[1]] <- sigex.weekly2daily(ts(td.low[[1]],start=start(dataONE.ts), frequency=frequency(dataONE.ts)),first.day) td.hi.daily[[2]] <- sigex.weekly2daily(ts(td.low[[2]],start=start(dataONE.ts), frequency=frequency(dataONE.ts)),first.day) td.hi.daily[[3]] <- sigex.weekly2daily(ts(td.low[[3]],start=start(dataONE.ts), frequency=frequency(dataONE.ts)),first.day)
- Before plotting, the trend must be computed.
- A subtlety is that we have $7$ trends, one for each day of week, which therefore combine long-term trend effects and day-of-week effects. The mean of the $7$ parameter values corresponds to the overall mean of each week, and the residual is the day-of-week effect in excess of the mean.
- First reg.td is computed, which is the trend means for each day-of-week expressed as a daily time series. We augment by one prior week's values, which would be all zero, and then convert to a daily series.
- Next, reg.trend is obtained by filtering with $(1 + B + \ldots + B^6)/7$, and finally remove the NA value (resulting from using filter, which does not extend the input series).
reg.td <- NULL for(i in 1:N){reg.td <- cbind(reg.td,sigex.fixed(data.ts,mdl,i,param,"Trend"))} reg.td <- rbind(rep(0,N),reg.td) reg.td <- ts(sigex.weekly2daily(reg.td,first.day), start=start(dataONE.ts),frequency=period) reg.trend <- stats::filter(reg.td,td.hifilter,method="convolution",sides=1) reg.trend <- as.matrix(reg.trend[8:length(reg.td)])
- The illustration is completed with a display: the seasonal adjustment contains some mild weekly effects (because the filter is designed to remove daily, and not weekly frequencies), whereas the TD-adjusted component has annual movements but not weekly effects.
trendcol <- "tomato" cyccol <- "orchid" seascol <- "seagreen" sacol <- "navyblue" fade <- 60 plot(dataONE.ts,xlab="Year") sigex.graph(sa.hi.daily,reg.trend,start(sa.hi.daily[[1]]), period,1,0,trendcol,fade) sigex.graph(td.hi.daily,reg.trend,start(td.hi.daily[[1]]), period,1,0,sacol,fade)
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