# timeTransME: Transformation of Time to Match Two Time Series In qualV: Qualitative Validation Methods

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31``` ```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 class 'timeTransME' print(x, ..., digits = 3) ## S3 method for class 'timeTransME' summary(object, ...) ## S3 method for class '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))) ```

## 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`

`plot.timeTransME`

shows a picture visualising the fit of the transformed dataset. This can be used as a diagnostic.

## 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.

## Note

The deviance calculated by `timeTransME(..., time = "fixed")` and the corresponding deviance measure are different because the timeTransME does an interpolation and compares time sequences at different spacing, while a simple deviance measure compares values only.
The CPU usage of the calculation of the minimum, when `trans = "transform"` is very high, because the optimization is done a hundred times with random starting values for the parameters. This is necessary since with the given objective the general purpose optimizers often run into local minima and/or do not converge. The number of iterations can be controlled with the parameter `trials`. Setting `debug = TRUE` gives an impression how long it takes to find an improved optimum.

`transBeta`, `transBezier`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68``` ```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) ## Not run: ttsimplex <- timeTransME(o, p, trans = transSimplex, trials=5) plot(ttsimplex) ttbezier <- timeTransME(o, p, trans = transBezier, trials=5) plot(ttbezier) ## End(Not run) ## 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) ## Not run: 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) ## End(Not run) ```