timeTransME: Transformation of Time to Match Two Time Series

Description

Transforming the time of predicted values by means of a monotonic mapping.

Usage

timeTransME(o, p,
            o.t      = seq(0, 1, length.out = length(o)),
            p.t      = seq(0, 1, length.out = length(p)),
            ignore   = "scaled",
            geometry = "real",
            measure  = "mad",
            type     = c("dissimilarity", "normalized",
                         "similarity", "reference"),
            interval = range(c(o.t, p.t)),
            time     = c("transformed", "fixed"),
            trans    = transBeta,
            p0       = eval(formals(trans)$p0),
            pmin     = eval(formals(trans)$pmin, list(p = p0)),
            pmax     = eval(formals(trans)$pmax, list(p = p0)),
            timeMEFactor = 0,
            timeME       = MAE,
            timeMEtype   = "normalized",
            timeScale    = 1,
            ME     = generalME(o, p, ignore, geometry, measure,
                               type = "function"),
            MEtype = c("dissimilarity", "normalized"),
            trials = 100,
            debug  = FALSE)
# S3 method for timeTransME
print(x, ..., digits = 3)
# S3 method for timeTransME
summary(object, ...)
# S3 method for timeTransME
plot(x, y = NULL, ..., col.obs = "black", col.pred = "green",
     col.map = "red", sub = x$call, xlab = "t",
     xlim = range(x$x), ylim = range(c(0, x$yo, x$yp)))

Value

The result is an object of type timeTransME with the following entries:

totalME

the requested measure with specified type,

criterium

the "dissimilarity" measure, which was calculated as a minimum of

ME(o(x), p(trans(x, timep)), MEtype) + timeMEFactor *
  timeME(x * timeScale, trans(x, timep) * timeScale,
  timeMEtype)

reference

the reference value of this criterium achieved without time deformation and full dissimilarity.

call

the call used to generate this deviance.

x

the times at which the series were compared from the perspective of the observations.

xp

the transformed times at which the series were compared from the perspective of the prediction.

yo

the interpolated values of the observations at times x.

yp

the interpolated values of the time transformed predictions at times x.

timeME

the deviance of the time transformation:
timeME(x, trans(x, ME), timeMEtype)).

timeMEref

the reference value of timeME

timeMEFactor

the factor to be used for timeME in the weighting with respect to ME.

timeScale

the scaling to time to account for an other unit.

p

the parameter of trans minimizing the criterium.

interval

the interval of time under consideration

trans

the transformation function used for the time.

optim

contains informations about the convergence of the optimization procedure and a list of secondary minima found. This additional list element occurs only if there is actually a minimisation performed.

Arguments

x

a result from a call to timeTransME

object

a result from a call to timeTransME

o

vector of observed values

p

vector of predicted values

o.t

vector of observation times

p.t

vector of times for predicted values

ignore

one of "raw", "centered", "scaled" or "ordered" as defined in generalME to specify the aspects of the data to be ignored.

geometry

one of "real", "logarithmic", "geometric", "ordinal" as defined in generalME to specify the geometry of the observed data.

measure

one of "mad", "sd", "var" to specify the type of error to be measured.

type

one of "dissimilarity", "normalized", "similarity" or "reference" as defined in generalME to specify the type of deviance measure to be used.

interval

a vector with two entries giving start and end time of the experiment.

time

indicates wether the time should actually be transformed. LCS is currently not implemented. Use the LCS method directly.

trans

the model function for the time transformation. See transBezier for possible alternatives.

p0

the identity parameters for the time-transformation. A non identity value can be given to force specific parameters for the transformation with time = "fixed".

pmin

number or vector providing the minimal allowed values for the parameters of the transformation.

pmax

number or vector providing the minimal allowed values for the parameters of the transformation.

timeME

The timeTransME minimizes a weighted sum of the deformation of the time scale and of the data values according to totalME = minimum of

ME(o(x), p(trans(x, timep)), MEtype) +
  timeMEFactor * timeME(x * timeScale,
  trans(x, timep) * timeScale, timeMEtype)

over p for x = c(ot, trans(pt, timep, inv = TRUE)).
timeME specifies the function to be used to quantify the temporal deformation.

timeMEtype

the type of deviance measure (``dissimilarity'' or ``normalized'') to be used for timeME.

timeMEFactor

a real value specifying the weighting of the time deformation against the value deformation. A value of 0 avoids penalty for time deformation.

timeScale

a scaling applied to the time values before timeME is applied. This can be used to change the units of measurement for the time.

ME

the deviance function to be used for the data. See MSE for alternatives.

MEtype

the type of Mean Error to be used in the calculations. This is not the type of Measure to be reported.

trials

The number of random starting values that should be used during the optimization of the time transformation. The optimization of the time transformation is a very critical task of this procedure and it had been shown by practical tests that a single local optimization typically fails to find the globally best fit. Depending on the number of parameters a value between 100 and 10000 seems reasonable for this parameter.

debug

a logical. If true some diagnostic information for the optimization step is printed.

...

further parameters to be passed to plot

col.obs

color to plot the observations

col.pred

color to plot the predictions

col.map

color to plot the mapped predictions

sub

the sub-headline of the plot

xlab

the label of the x-axis of the plot

xlim

the size of the plot in x-direction

ylim

the size of the plot in y-direction

y

y unused

digits

number of significant digits displayed

Details

Common quantitative deviance measures underestimate the similarity of patterns if there are shifts in time between measurement and simulation. An alternative to measure model performance independent of shifts in time is to transform the time of the simulation, i.e. to run the time faster or slower, and to compare the performance before and after the transformation. The applied transformation function must be monotonic. timeTransME minimizes the joint criterium
ME(o(x), p(trans(x, timep)), MEtype) +
timeMEFactor * timeME(x * timeScale, trans(x, timep) * timeScale, timeMEtype) to find a best fitting time transformation.

print.timeTransME: prints only the requested value, without additional information.
summary.timeTransME: prints all the additional information.
plot.timeTransME: shows a picture visualising the fit of the transformed dataset. This can be used as a diagnostic.

Examples

Run this code

set.seed(123)
## a constructed example
x <- seq(0, 2*pi, length=10)
o <- 5 + sin(x) + rnorm(x, sd=0.2) # observation with random error
p <- 5 + sin(x-1)                  # simulation with time shift

# timeTransME(o, p) # reasonably accurate but takes very long!
# timeTransME(o, p, trials=5, debug=TRUE)

ttbeta <- timeTransME(o, p, trials=5)
plot(ttbeta)
if (FALSE) {
ttsimplex <- timeTransME(o, p, trans = transSimplex, trials=5)
plot(ttsimplex)

ttbezier <- timeTransME(o, p, trans = transBezier, trials=5)
plot(ttbezier)
}

## observed and measured data with non-matching time intervals
data(phyto)
bbobs    <- dpill(obs$t, obs$y)
n        <- diff(range(obs$t)) + 1
obss     <- ksmooth(obs$t, obs$y, kernel = "normal", bandwidth = bbobs,
            n.points = n)
names(obss) <- c("t", "y")
obss     <- as.data.frame(obss)[match(sim$t, obss$t), ]

tt       <- timeTransME(obss$y, sim$y, obss$t, sim$t, ME = SMSE,
            timeMEFactor = 0, time = "transform", type = "n", trials = 5)
round(tt$totalME, digits = 3)

basedate <- as.Date("1960/1/1")
plot(basedate + sim$t, sim$y, type="l", ylim = c(min(obs$y, sim$y),
  max(obs$y, sim$y)), xlab = "time", ylab = "Phytoplankton (mg/L)",
  col = 2, font = 2, lwd = 2, cex.lab = 1.2, las = 1)
lines(basedate + obss$t, obss$y, lwd = 2)
points(basedate + obs$t, obs$y, lwd = 2)
lines(basedate + tt$x, tt$yp, lwd = 2, col = 2, lty = 2)
legend(basedate + 12600, 50, c("measurement", "smoothed measurement",
"simulation", "transformed simulation"), lty = c(0, 1, 1, 2),
pch = c(1, NA, NA, NA), lwd = 2, col = c(1, 1, 2, 2))

tt1 <- timeTransME(obs$y, sim$y, obs$t, sim$t, ME = SMSLE, type = "n",
  time = "fixed")
tt1
plot(tt1)
summary(tt1)

if (FALSE) {
tt2 <- timeTransME(obss$y, sim$y, obss$t, sim$t, ME = SMSLE, type = "n",
  time = "trans", debug = TRUE)
tt2
plot(tt2)  # logarithm (SMSLE) is not appropriate for the example
summary(tt2)
tt3 <- timeTransME(obss$y, sim$y, obss$t, sim$t, ME = SMSE, type = "n",
  time = "trans", trans = transBezier, debug = TRUE)
tt3
plot(tt3)
summary(tt3)
tt4 <- timeTransME(obss$y, sim$y, obss$t, sim$t, ME = MSOE, type = "n",
  time = "trans", trans = transBezier, debug = TRUE)
tt4
plot(tt4)
summary(tt4)
}