| title | author | institute | output | bibliography | nocite | ||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Biomedical Applications of Time Series Analysis |
Tamás Ferenci$^1$ |
|
TamasFerenci_BiomedicalApplicationsOfTimeSeriesAnalysis.bib |
@fitzmaurice2012, @hedeker2006, @diggle2002, @pinheiro2009, @hardin2003, @madsen2008, @hamilton1994, @shumway2017, @cameron2013
|
\tableofcontents[subsubsectionstyle=hide/hide/hide]
- ,,Observations made over time'' (i.e. they are ordered)
- As a sample vs. in the population (stochastic process)
- Many-many (and important) biomedical data are available as time series
- Traditional methods can be applied -- but the nature of time series must be taken into account
- Many special methods too
- Maximum number of areas with minimum detail on each
- Practical, real-life examples for all methods
- All calculation is made with R
- Free and open source (http://www.r-project.org/)
- Enthusiastic, extremely active community; incredible number of packages at CRAN
- (There is an R package for any statistical task you can think of\dots{} and for many that you can't even think of)
- It includes packages making complex operations one-liners, streamlining entire analysis workflows (like Frank Harrell's wonderful \texttt{rms} for regression)
- A powerful IDE called RStudio (http://www.rstudio.org/) is freely available
- Extremely good at visualization (this presentation will use \texttt{lattice}), report generation, reproducible research too (just like this presentation!)
- Whole source code of this presentation is available at https://github.com/tamas-ferenci/BiomedicalApplicationsOfTimeSeriesAnalysis
- It is somewhat ill-defined what can be considered ,,time series analysis''
- I now try to be as broad as possible
- Therefore, a rough (and very subjective) categorization:
- Analysis of data that are only meaningful when collected over time: typically biomedical signals such as ECG or EEG
- Analysis of data that are meaningful cross-sectionally, but measurements are repeated to obtain information on the time dimension too: typical in longitudinal studies, analysis of growth curves
- Analysis of epidemiologic data with time dimension: typically incidence of diseases
As with any statistical model:
- Understanding phenomena (analysis, interpretation of the model, answering medical questions based on the results)
- Forecasting
The ts is the basic time series object, it can be multivariate, but it can only handle evenly spaced time series (see zoo for unevenly spaced time series for example):
ts( rnorm( 20 ), frequency = 4, start = c( 2010, 2 ) )## Qtr1 Qtr2 Qtr3 Qtr4
## 2010 -0.7959257 -0.1512653 -0.3575611
## 2011 -0.7956175 -0.1880978 0.2254061 0.8224560
## 2012 1.3941882 -0.8977673 1.0242012 1.1818423
## 2013 -2.1265545 -1.3074036 -0.9922208 -0.3795757
## 2014 -0.1981558 0.9191271 0.5872626 -0.8322504
## 2015 1.3388344
ts( rnorm( 30 ), frequency = 12, start = c( 2010, 2 ) )## Jan Feb Mar Apr May
## 2010 -1.34905845 -1.08165926 -0.26791001 -0.33187780
## 2011 -0.98080258 1.06546991 1.72213959 -0.32019389 -0.09044677
## 2012 -0.60040047 1.55106769 0.63542068 0.28087855 1.63617577
## Jun Jul Aug Sep Oct
## 2010 -0.17594537 -0.78469569 -1.62952706 1.65900243 -0.29385689
## 2011 1.37086585 -0.18889592 -1.00193031 1.84303233 -0.85811827
## 2012 -0.28285983 0.93290711
## Nov Dec
## 2010 0.48908285 -0.07639366
## 2011 -0.89801476 -0.17180304
## 2012
ldeaths## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 1974 3035 2552 2704 2554 2014 1655 1721 1524 1596 2074 2199 2512
## 1975 2933 2889 2938 2497 1870 1726 1607 1545 1396 1787 2076 2837
## 1976 2787 3891 3179 2011 1636 1580 1489 1300 1356 1653 2013 2823
## 1977 3102 2294 2385 2444 1748 1554 1498 1361 1346 1564 1640 2293
## 1978 2815 3137 2679 1969 1870 1633 1529 1366 1357 1570 1535 2491
## 1979 3084 2605 2573 2143 1693 1504 1461 1354 1333 1492 1781 1915
plot( ldeaths )
- Fundamental idea: every periodic function can be represented as a weighted sum of sinusoidals (sine waves)
- We may need infinite number of sinusoidals, but still countable many, the frequency of which are all multiples of a fundamental frequency
- If the function is non-periodic, it still works (quite universally), but we will need uncountably many terms
- Intuitive interpretation: how important is a certain frequency in making up the time series (what periodicites are present, and with what weight?)
It gives a picture of what frequencies ''create'' the signal:
SimDataFourier <- data.frame( t = 1:1000 )
SimDataFourier <- transform( SimDataFourier, y = 0.5*sin( t*2 ) + sin( t/10*2 ) +
rnorm( length( t ), 0, 0.1 ) )
xyplot( y ~ t, data = SimDataFourier, type = "l", xlim = c( 0, 200 ) )
It gives a picture of what frequencies ''create'' the signal:
xyplot( spec ~ freq, data = spectrum( SimDataFourier$y, plot = FALSE ), type = "l",
scales = list( y = list( log = 10 ) ) )
(Sidenote) Custom plotting:
locmaxpanel <- function( x, y, width, maxmeddiff = 1, rounddigit = 2, ... ) {
if( width%%2==0 )
width <- width+1
panel.xyplot( x, y, ... )
maxs <- zoo::rollapply( y, width, function(x) (which.max(x)==(width+1)/2)&
(max(x)-median(x)>maxmeddiff),
align = "center", fill = NA )
panel.abline( v = x[ maxs ], col = "gray" )
panel.points( x[ maxs ], y[ maxs ] )
panel.text( x[ maxs ], y[ maxs ], round( x[ maxs ], rounddigit ), pos = 4 )
}xyplot( spec ~ freq, data = spectrum( SimDataFourier$y, plot = FALSE ), type = "l",
scales = list( y = list( log = 10 ) ), panel = locmaxpanel, width = 21, maxmeddiff = 2 )
## require( tuneR ) ## require( pastecs ) ## devtools::install_github( "mkfs/r-physionet-ptb" )
## https://www.physionet.org/physiobank/database/ptbdb/
## system2( system.file( "exec", "download_ptb.sh", package = "r.physionet.ptb" ) )
## system2( system.file( "exec", "ptb_patient_to_json.rb", package = "r.physionet.ptb" ),
## args="patient001" )
library( r.physionet.ptb )
ptb <- r.physionet.ptb::ptb.from.file( "patient001.json" )
ptbecg <- r.physionet.ptb::ptb.extract.lead( ptb, "i" )$`1-10010`
xyplot( ptbecg~seq_along( ptbecg ), type = "l", xlim = c( 0, 5000 ), xlab = "Time", ylab = "" )
xyplot( spec ~ freq, data = spectrum( ptbecg, plot = FALSE, span = rep( 201, 3 ) ), type = "l",
scales = list( y = list( log = 10 ) ), panel = locmaxpanel, width = 21,
maxmeddiff = 2e-4 )
- Assumes that the spectrum is constant over time: no change in this sense
- One possible way to relax this: windowed analyis (short-term Fourier transform, STFT)
- Trade-off between time-resolution and frequency resolution
- An alternative modern method: wavelet analysis
- Roughly speaking: we perform (a) a local search (b) everywhere (c) with many different frequencies
SimDataWavelet <- data.frame( t = 1:2000 )
SimDataWavelet <- transform( SimDataWavelet,
y = ifelse( t<=1000, sin( t*2 ), sin( t/10*2 ) ) +
rnorm( length( t ), 0, 0.1 ) )
xyplot( y ~ t, data = SimDataWavelet, type = "l" )
xyplot( spec ~ freq, data = spectrum( SimDataWavelet$y, plot = FALSE ), type = "l",
scales = list( y = list( log = 10 ) ) )
WaveletComp::wt.image( WaveletComp::analyze.wavelet( SimDataWavelet, "y",
verbose = FALSE, make.pval = FALSE ) )
SimDataWavelet <- data.frame( t = 1:2000 )
SimDataWavelet <- transform( SimDataWavelet,
y = WaveletComp::periodic.series( start.period = 20,
end.period = 200,
length = length( t ) ) +
0.1*rnorm( length( t ) ) )
xyplot( y ~ t, data = SimDataWavelet, type = "l" )
xyplot( spec ~ freq, data = spectrum( SimDataWavelet$y, plot = FALSE ), type = "l",
scales = list( y = list( log = 10 ) ) )
WaveletComp::wt.image( WaveletComp::analyze.wavelet( SimDataWavelet, "y",
verbose = FALSE, make.pval = FALSE ) )
(Sidenote) A bit of data scraping:
tmpfile <- tempfile( fileext = ".xlsx" )
download.file( url = paste0( "https://www.gov.uk/government/uploads/system/uploads/",
"attachment_data/file/339410/NoidsHistoricAnnualTotals.xlsx" ),
destfile = tmpfile, mode = "wb" )
res1 <- XLConnect::loadWorkbook( tmpfile )
XLConnect::setMissingValue( res1, value = c( "*" ) )
res1 <- do.call( plyr::rbind.fill, lapply( XLConnect::getSheets( res1 ), function( s ) {
temp <- XLConnect::readWorksheet( res1, sheet = s, startRow = 4 )
temp <- temp[ , grep( "Disease", colnames( temp ) ):ncol( temp ) ]
temp <- temp[ 1:( if( sum( is.na( temp$Disease ) )==0 ) nrow( temp ) else
which( is.na( temp$Disease ) )[ 1 ]-1 ), ]
for( i in 2:ncol( temp ) )
temp[ , i ] <- as.numeric( gsub( "[[:space:].,‡†]", "", temp[ , i ] ) )
temp2 <- as.data.frame( t( temp[ , - 1 ] ) )
colnames( temp2 ) <- temp[ , 1 ]
temp2$Year <- as.numeric( substring( rownames( temp2 ), 2, 5 ) )
temp2
} ) )
unlink( tmpfile )(Sidenote) A bit of data scraping:
tmpfile <- tempfile( fileext = ".xlsx" )
download.file( url = paste0( "https://www.gov.uk/government/uploads/system/uploads/",
"attachment_data/file/664864/",
"Annual_totals_from_1982_to_2016.xlsx" ),
destfile = tmpfile, mode = "wb" )
res2 <- XLConnect::loadWorkbook( tmpfile )
XLConnect::setMissingValue( res2, value = c( "--" ) )
res2 <- do.call( plyr::rbind.fill, lapply( XLConnect::getSheets( res2 )[ -1 ], function( s ) {
temp <- XLConnect::readWorksheet( res2, sheet = s, startRow = 5 )
temp <- temp[ 1:( nrow( temp )-1 ), ]
temp2 <- as.data.frame( t( temp[ , - 1 ] ) )
colnames( temp2 ) <- temp[ , 1 ]
temp2$Year <- as.numeric( substring( rownames( temp2 ), 2, 5 ) )
temp2
} ) )
unlink( tmpfile )(Sidenote) A bit of data scraping:
tmpfile <- tempfile( fileext = ".xls" )
download.file( url = paste0( "https://www.ons.gov.uk/file?uri=/",
"peoplepopulationandcommunity/populationandmigration/",
"populationestimates/adhocs/",
"004358englandandwalespopulationestimates1838to2014/",
"englandandwalespopulationestimates18382014tcm77409914.xls" ),
destfile = tmpfile, mode = "wb" )
res3 <- XLConnect::readWorksheetFromFile( tmpfile, sheet = "EW Total Pop 1838-2014", startRow = 2,
endRow = 179 )
unlink( tmpfile )
names( res3 )[ 1 ] <- "Year"
res3$Persons <- ifelse( res3$Persons < 100000, res3$Persons*1000, res3$Persons )
res3 <- res3[ , c( "Year", "Persons" ) ]
res4 <- read.csv( paste0( "https://www.ons.gov.uk/generator?format=csv&uri=/",
"peoplepopulationandcommunity/populationandmigration/",
"populationestimates/timeseries/ewpop/pop" ), skip = 7 )
names( res4 ) <- c( "Year", "Persons" )
res4 <- res4[ res4$Year>=2015, ]
UKEpid <- merge( plyr::rbind.fill( res1, res2 ), rbind( res3, res4 ) )
UKPertussis <- UKEpid[ , c( "Year", "Whooping cough", "Persons" ) ]
UKPertussis$Inc <- UKPertussis$`Whooping cough`/UKPertussis$Persons*100000
UKPertussis <- UKPertussis[ !is.na( UKPertussis$`Whooping cough` ), ]xyplot( Inc ~ Year, data = UKPertussis, type = "l", ylab = "Incidence [/100 000/year]" )
WaveletComp::wt.image( WaveletComp::analyze.wavelet( UKPertussis, "Inc",
verbose = FALSE, make.pval = FALSE ) )
- Regression is perhaps \emph{the} most powerful tool for the analysis of time series in the time domain
- With appropriate measures taken to account for the nature of the data
- This of course gives rise to all usual issues of regression models (model specification such as the question of non-linearities, model diagnostics etc.)
- Mostly models with exogeneous regressors are used, stochastic models are employed much less often
- Count data are typical, giving rise to Generalized Linear Models
- Further complications within GLMs, such as overdispersion
- Need to take changing age- and sex composition into account
- Traditionally: standardization, but in the modern approach they're just confounders!
- Models can include many resolutions (scales) in time
data( "CVDdaily", package = "season" )
rownames( CVDdaily ) <- NULL
xyplot( cvd ~ date, data = CVDdaily, type = "l", xlab = "Time", ylab = "Number of deaths" )
CVDdaily$year <- lubridate::year( CVDdaily$date )
CVDdaily$wday <- as.factor( lubridate::wday( CVDdaily$date, week_start = 1 ) )
CVDdaily$yday <- lubridate::yday( CVDdaily$date )/yearDays( CVDdaily$date )
head( CVDdaily[ , c( "date", "year", "wday", "yday", "cvd" ) ] )date year wday yday cvd
1987-01-01 1987 4 0.0027397 55 1987-01-02 1987 5 0.0054795 73 1987-01-03 1987 6 0.0082192 64 1987-01-04 1987 7 0.0109589 57 1987-01-05 1987 1 0.0136986 56 1987-01-06 1987 2 0.0164384 65
library( mgcv )
fit <- gam( cvd ~ s( as.numeric( date ) ) + wday + s( yday, bs = "cc" ), data = CVDdaily,
family = nb( link = log ) )
summary( fit )##
## Family: Negative Binomial(177.091)
## Link function: log
##
## Formula:
## cvd ~ s(as.numeric(date)) + wday + s(yday, bs = "cc")
##
## Parametric coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 3.820888 0.006137 622.550 < 2e-16 ***
## wday2 -0.007799 0.008687 -0.898 0.369335
## wday3 -0.028719 0.008724 -3.292 0.000995 ***
## wday4 -0.025035 0.008714 -2.873 0.004065 **
## wday5 -0.015468 0.008697 -1.778 0.075323 .
## wday6 -0.022458 0.008709 -2.579 0.009920 **
## wday7 -0.038679 0.008738 -4.427 9.57e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df Chi.sq p-value
## s(as.numeric(date)) 7.696 8.568 254.4 <2e-16 ***
## s(yday) 7.771 8.000 2732.5 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.377 Deviance explained = 37.6%
## -REML = 17546 Scale est. = 1 n = 5114
plot( fit, select = 1, scale = 0, rug = FALSE, trans = exp, shade = TRUE,
col = trellis.par.get()$superpose.line$col[1], xaxt = "n", xlab = "Year", ylab = "IRR" )
axis( 1, at = seq( CVDdaily$date[1], tail( CVDdaily$date, 1 )+1, by = "year" ),
labels = year( CVDdaily$date[1] ):year( tail( CVDdaily$date, 1 )+1 ) )
plot( fit, select = 2, scale = 0, rug = FALSE, trans = exp, shade = TRUE,
col = trellis.par.get()$superpose.line$col[1], xaxt = "n", xlab = "Day of year",
ylab = "IRR" )
axis( 1, at = seq( 0, 1, 1/12 ), labels = seq( 0, 1, 1/12 )*30*12 )
source( "https://pastebin.com/raw/hBmStX4Y" )
termplot2( fit, terms = "wday", se = TRUE, yscale = "exponential",
col.term = trellis.par.get()$superpose.line$col[1],
col.se = "gray80", se.type = "polygon", xlab = "Day of week" )
- Filter: we create another time series from the investigated one
- Consider the well-known moving average filter: [ y'\left(t\right)=\frac{y_t+y_{t-1}+y_{t-2}+\ldots+y_{t-\left(p-1\right)}}{p} ]
- Traditionally used to ''filter noise'' or to separate components of the time series (decompose the time series)
- This can be achieved using deterministic time series regression (''model-based decomposition''), see the previous example
- But filters like the above moving average allows us to decompose the time series without assuming a parametric model
Its operation can actually be best understood in frequency domain: it filters out high-frequency components (and retains low-frequency):
do.call( gridExtra::grid.arrange, lapply( c( 2, 6, 12, 24 ), function( o ) {
xyplot( y ~ t, groups = grp, data = rbind( data.frame( grp = "data", SimDataFourier ),
data.frame( grp = "smooth", t = SimDataFourier$t,
y = forecast::ma( SimDataFourier$y,
o ) ) ),
type = "l", xlim = c( 0, 200 ), main = paste0( "Order: ", o ) )
} ) )
plot( decompose( ldeaths ) )
plot( stl( ldeaths, s.window = "periodic" ) )
plot( forecast::seasadj( stl( ldeaths, s.window = "periodic" ) ) )
- Same variables measured again and again over time, for the same subjects
- Typical questions: effect of an intervention, or natural history (growth curve)
- Usual tool: regression models, usual problem: intra-individual correlation (clustered data)
- Mostly obsolote solutions: RM-ANOVA (has many assumptions that are hard to test, and are usually not met in practice), pairwise tests (multiple comparisons problem, no interpolation possible, etc.), summary statistics (data are reduced to a few parameters in the first step, dramatic loss of information among others)
- Cluster-robust standard errors or GLS (works only for continuous responses)
- Mixed effects models (can handle hiearchical models, parameters can be different for each subject)
- Generalized Estimating Equations (marginal model)
- OLS assumes no autocorrelation (and homoscedasticity)
- This can be relaxed if the correlation structure is known
- Can be estimated from the data
- The OLS parameter estimates are unbiased and consistent even with autocorrelation, but the estimates of the error covariance matrix (thus the estimates of the standard errors) are biased
- GLS changes the standard errors to account for the correlation
TLCData <- read.table( "https://content.sph.harvard.edu/fitzmaur/ala2e/tlc-data.txt",
col.names = c( "ID", "Trt", paste0( "Wk", c( 0, 1, 4, 6 ) ) ) )
TLCData <- reshape( TLCData, varying = paste0( "Wk", c( 0, 1, 4, 6 ) ), v.names = "LeadLevel",
timevar = "Week", times = c( 0, 1, 4, 6 ), idvar = "ID", direction = "long" )
TLCData$Trt <- relevel( TLCData$Trt, ref = "P" )
TLCData$Week.f <- as.factor( TLCData$Week )
xyplot( LeadLevel ~ Week | Trt, groups = ID, data = TLCData, type = "b" )
TLCData <- data.table( TLCData )
TLCData$Time <- as.numeric( TLCData$Week.f )
dd <- datadist( TLCData )
options( datadist = "dd" )
xYplot( Cbind( mean, lwr, upr ) ~ Week, groups = Trt, type = "b",
data = TLCData[ , .( mean = mean( LeadLevel ),
lwr = t.test( LeadLevel )$conf.int[1],
upr = t.test( LeadLevel )$conf.int[2] ) , .( Trt, Week ) ],
ylim = c( 10, 30 ), ylab = "Mean lead level" )
ols( LeadLevel ~ Week.f*Trt, data = TLCData )## Linear Regression Model
##
## ols(formula = LeadLevel ~ Week.f * Trt, data = TLCData)
##
## Model Likelihood Discrimination
## Ratio Test Indexes
## Obs 400 LR chi2 159.22 R2 0.328
## sigma6.6257 d.f. 7 R2 adj 0.316
## d.f. 392 Pr(> chi2) 0.0000 g 4.920
##
## Residuals
##
## Min 1Q Median 3Q Max
## -16.662 -4.620 -0.993 3.673 43.138
##
##
## Coef S.E. t Pr(>|t|)
## Intercept 26.2720 0.9370 28.04 <0.0001
## Week.f=1 -1.6120 1.3251 -1.22 0.2245
## Week.f=4 -2.2020 1.3251 -1.66 0.0974
## Week.f=6 -2.6260 1.3251 -1.98 0.0482
## Trt=A 0.2680 1.3251 0.20 0.8398
## Week.f=1 * Trt=A -11.4060 1.8740 -6.09 <0.0001
## Week.f=4 * Trt=A -8.8240 1.8740 -4.71 <0.0001
## Week.f=6 * Trt=A -3.1520 1.8740 -1.68 0.0934
##
fit <- Gls( LeadLevel ~ Week.f*Trt, data = TLCData, corr = nlme::corSymm( form = ~ Time | ID ),
weights = nlme::varIdent( form = ~ 1 | Week.f ) )
fit## Generalized Least Squares Fit by REML
##
## Gls(model = LeadLevel ~ Week.f * Trt, data = TLCData, correlation = nlme::corSymm(form = ~Time |
## ID), weights = nlme::varIdent(form = ~1 | Week.f))
##
##
## Obs 400 Log-restricted-likelihood-1208.04
## Clusters100 Model d.f. 7
## g 4.920 sigma 5.0225
## d.f. 392
##
## Coef S.E. t Pr(>|t|)
## Intercept 26.2720 0.7103 36.99 <0.0001
## Week.f=1 -1.6120 0.7919 -2.04 0.0425
## Week.f=4 -2.2020 0.8149 -2.70 0.0072
## Week.f=6 -2.6260 0.8885 -2.96 0.0033
## Trt=A 0.2680 1.0045 0.27 0.7898
## Week.f=1 * Trt=A -11.4060 1.1199 -10.18 <0.0001
## Week.f=4 * Trt=A -8.8240 1.1525 -7.66 <0.0001
## Week.f=6 * Trt=A -3.1520 1.2566 -2.51 0.0125
##
## Correlation Structure: General
## Formula: ~Time | ID
## Parameter estimate(s):
## Correlation:
## 1 2 3
## 2 0.571
## 3 0.570 0.775
## 4 0.577 0.582 0.581
## Variance function:
## Structure: Different standard deviations per stratum
## Formula: ~1 | Week.f
## Parameter estimates:
## 0 1 4 6
## 1.000000 1.325884 1.370443 1.524804
##
temp <- Predict( fit, Trt, Week.f )
temp$Week.f <- as.numeric( levels( temp$Week.f ) )[ temp$Week.f ]
xYplot( Cbind( yhat, lower, upper ) ~ Week.f, groups = Trt, data = temp, type = "b",
ylim = c( 10, 30 ) )
- Two extremes: fitting a single model to all subjects vs. fitting an own model to each
- The latter is not efficient (a few observations to estimate the parameters, and this doesn't improve with sample size), and we are often not interested in each subject individually
- A mixed model is a compromise: assumes not a single parameter (for all sujects), nor many (one for each), but rather it assumes that the parameter is a random draw from a population (i.e. a population distribution is assumed for the parameter), presumed to be normally distributed, so only two parameters are estimated, a mean and a variance
- We infer on the population (in which we are really interested), and also it is more efficient, as we have a fixed number of parameters (2)
- Yet we can calculate individual curves (parameters calculated like the residuals in a regression)
- Induces a correlation structure
data( "Orthodont", package = "nlme" )
OrthoFem <- Orthodont[ Orthodont$Sex=="Female", ]
plot( OrthoFem )
fit1 <- nlme::lmList( distance ~ I( age - 11 ), data = OrthoFem )
plot( nlme::intervals( fit1 ) )
fit2 <- nlme::lme( distance ~ age, data = OrthoFem, random = ~1|Subject )
xyplot( distance + fitted( fit1 ) + fitted( fit2, level = 1 ) +
fitted( fit2, level = 0 ) ~ age | Subject, data = OrthoFem,
type = c( "p", "l", "l", "l" ), distribute.type = TRUE, ylab = "Distance", grid = TRUE,
auto.key = list( text = c( "Measured", "Fitted (individual models)",
"Fitted (mixed effects)", "Fitted (fixed effects only)" ),
columns = 4, points = FALSE, lines = TRUE ) )
- (Non-parametric) filtering and smoothing (LOESS, weighted moving average, Holt-Winters, exponential smoothing etc.)
- Tools of stochastic modelling (stationarity, autocorrelation function, ARIMA models etc.)
- Multivariate time series (coherence, cross-correlation, VAR models etc.)
- Questions of forecasting, quantifying forecast accuracy, comparing forecasts, validation
- Long-range memory
- State-space models
- Regime switching models
- etc. etc. etc.
- The biomedical application of time series data is getting more and more intensive
- They have role from basic science through clinical investigations to policymaking
- Understanding and -- sound! -- application of time series methods is of huge importance therefore
- This is not a problem of a selected few specialists: everyone working on biomedical field benefits from having basic knowledge about time series analysis