Online Appendix to Mixed Modeling for Irregularly Sampled and Correlated Functional Data: Speech Science Spplications

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1 Online Appendix to Mixed Modeling for Irregularly Sampled and Correlated Functional Data: Speech Science Spplications Marianne Pouplier, Jona Cederbaum, Philip Hoole, Stefania Marin, Sonja Greven R Syntax and Plotting Commands for Example Datasets This Online Appendix illustrates how the sparseflmm package is used to run the case studies discussed in the paper. Plotting commands are exemplified in order to help the reader understand the structure of the output. A generally adaptable plotting function is provided for one of the datasets on the basis of which the reader should, mutatis mutandis, be able to generate plots for their own data in many cases. We provide with this Appendix the input data for each case study and a results file so that the reader can examine the output without having to run the models. Installation The R package sparseflmm can be downloaded from CRAN. For installation, R version or later is required, as are the following packages: mgcv (>= ) data.table (>= 1.9-5) refund (>= ) First load the package: require(sparseflmm) Here is how the citation information for the package can be retrieved: citation("sparseflmm") Romanian Vowel Sequences Model Specification The Romanian vowel F2 data were analysed with a model specifying a single covariate and three functional random effects (Subject (B), Item (C), Subject-Item-Repetition (E)). 1

2 The input to the model is an R data table. Any data frame can be easily transformed to a data.table by applying the function data.table() on an existing data frame. The advantage of data table is that it is faster and more memory efficient. The data table must contain the following columns: n_long (integer): unique identification number for each curve. In this dataset n_long ranges from 1:163, since there are 163 curves for analysis. subject_long (integer): unique identification number for each level of the first functional random effect (term B in equation (1) of the main paper), here Subject. Since we have 5 speakers, subject_long ranges from 1:5 word_long (integer): unique identification number for each level of the second functional random effect (term C in equation (1) of the main paper). This study has 6 stimulus items (3 diphthong, 3 hiatus items), therefore word_long takes on the range 1:6 combi_long (integer): repetition of each subject_long word_long combination. Here this ranges from 1:6 since each speaker recorded each stimulus item six times. y_vec (numeric): F2 value of each sample t (numeric): the normalized time points at which the F2 values were sampled covariate.1 (numeric): covariate values, here dummy coded levels 0 = diphthong, 1 = hiatus Note that the data table has to be in long format, in which each row represents a single data point. This means that, except for the coordinates of each sample, the coding is redundant within a given curve. The first step is to load the data table into the workspace. In our case it is called RoDiphF2: load("rodiphf2.rdata") head(rodiphf2) ## n_long subject_long word_long combi_long y_vec t covariate.1 ## 1: ## 2: ## 3: ## 4: ## 5: ## 6: When calling sparseflmm the parameters for which there are no defaults or for which we want to depart from the default settings have to be specified (see?sparseflmm for information on defaults). The data table with the input data is given to the function as variable curve_info: curve_info = RoDiphF2 We have a single covariate and specify the covariate form. The covariate form needs one entry for each covariate (here, one). Since the covariate is discrete (dummy coded), the only 2

3 option is by covariate = TRUE num_covariates = 1 covariate_form = by We have no interaction, hence interaction = FALSE We specify in bf_covs the number of basis functions for each functional random effect (here, three: Subject, Item, Subject-Item-Repetition). It is important to choose a sufficiently high number of basis functions. Which number can be considered sufficient depends on the complexity and smoothness of the curves. Since during the estimation the curves are represented as a linear combination of the given number of basis functions, this number basically determines the amount of detail retained from the original curves. While a high number is desirable (over-fitting is avoided by penalty terms) it is computationally very costly. One way to determine the appropriate number of basis functions is to successively increase the number of basis functions until the output stabilizes. The term m_covs is a list that specifies for each functional random intercept the order of the spline and of the penalty for covariance estimation (here, cubic B-Splines which corresponds to order = 2, and the third derivative). bf_covs = c(10, 10, 10) m_covs = list(c(2,3), c(2,3), c(2,3)) For the number of basis functions for the mean (bf_mean) we use the default of 8 and thus no further specification in the call to the function is needed for this parameter. Generally, as for the functional random effects, a higher number of basis functions is advantageous in the estimation since more detailed information on curve shape is preserved. Over-fitting is again avoided due to the penalty term. A high number of basis functions is, however, computationally very costly (see main paper for more discussion and references). Using the third derivative (jerk) for the penalty of the covariance estimation can be motivated additionally for speech data under the consideration that jerk minimization is commonly considered to be an important factor in movement optimization. Since we want confidence bands to be computed, we require functional additive mixed modelling (FAMM) to be used use_famm = TRUE Here is the call to the function: results <- sparseflmm(curve_info = RoDiphF2, covariate = TRUE, num_covariates = 1, covariate_form = "by", interaction = FALSE, bf_covs = c(10,10,10), m_covs = list(c(2,3), c(2,3), c(2,3)), use_famm = TRUE) This dataset is small and the analysis can be run on an ordinary desktop computer. 3

4 Output Effects Plots For present purposes we first load a file with the results so that we can look at the results without having to run the model. load("resultsrodiphf2.rdata") For plotting the reference mean, we extract the effect of the reference mean from the fpc_famm estimation and add it to the intercept. This recreates Figure 2 of the paper. intercept <- results$fpc_famm_hat_tri_constr$intercept y_mean <- results$fpc_famm_hat_tri_constr$famm_cb_mean$value se_mean <- results$fpc_famm_hat_tri_constr$famm_cb_mean$se my_grid <- results$my_grid plot(my_grid, y_mean+intercept, t = "l", lwd=3,xlab = "normalized time (t)", ylab = "delta Hz", ylim = c(1000,2000), col = "red", cex.axis = 1.2, main = expression(paste("reference Mean ", f[0](t))),cex.main = 1.2,cex.lab = 1.2) #add confidence bands to the plot lines(x = my_grid, y = (y_mean+2*se_mean+intercept), lty = 2, lwd = 3, col = "blue") lines(x = my_grid, y = (y_mean-2*se_mean+intercept), lty = 2, lwd = 3, col = "blue") Reference Mean f 0 (t) delta Hz normalized time (t) 4

5 In the same way we plot the effect of the covariate: y_cov1<-results$fpc_famm_hat_tri_constr$famm_cb_covariate.1$value se_cov1<-results$fpc_famm_hat_tri_constr$famm_cb_covariate.1$se plot(my_grid,y_cov1, t = "l", lwd = 3, xlab = " normalized time (t)", ylab = "delta Hz", ylim = c(-200,800), col = "red",cex.axis = 1.2, main = expression(paste("status ", f[1](t))), cex.main = 1.2, cex.lab = 1.2) #confidence bands lines(x = my_grid, y = (y_cov1+2*se_cov1), lty = 2, lwd = 3, col = "blue") lines(x = my_grid, y = (y_cov1-2*se_cov1), lty = 2, lwd = 3, col = "blue") #reference zero line abline(h = 0, lty = 3) Status f 1 (t) delta Hz normalized time (t) Summed Effect Curves To obtain the summed effect curves we sum the intercept, the mean, and the covariate effect. # define colors and scaling of the y-axis cols <- rainbow(12) miny <

6 maxy < # sum the intercept, mean and intercept, mean, covariate effect none <- intercept+y_mean Status <- intercept+y_mean+y_cov1 #plot first the reference mean and then the covariate effect plot(my_grid, none, col = cols[1], lty = 1, lwd = 4, t = "l", ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n") plot(my_grid, Status, col = cols[10], lty = 4, lwd = 4, ylim = c(miny, maxy), ylab = "delta Hz", xlab = "normalized time", main = paste0("summed Effect Curves"), t = "l", cex.axis = 1.2, cex.main = 1.2, font.main = 1, cex.lab = 1.2) #zero reference line abline(h = 0, lty = 3) # legend text leg.text <- c(expression(paste(f[0](t))), expression(paste(f[0](t)+f[1](t)))) legend("bottomleft", legend = leg.text, bty = "n", col = c(cols[1], cols[10]), lty = c(1,4), lwd = 4,cex = 1.2) Summed Effect Curves delta Hz f 0 (t) f 0 (t) + f 1 (t) normalized time 6

7 Variance Decomposition For variance decomposition, results$fpc_hat_tri_constr has the information needed to obtain the number of FPCs chosen for each functional random effect, the percent variance explained by each FPC, and the weights associated with each chosen FPC. results$fpc_hat_tri_constr$n_b/c/e contains the number of FPCs chosen for each functional random effect, here 3 for B (Speaker), 1 for C (Item), 3 for E (Speaker-Item-Repetition). The estimated total variance explained can be found in results$fpc_hat_tri_constr$var_explained ## [1] To calculate the amount of variance explained by each FPC we need the estimated total variance totvar_hat <- results$fpc_hat_tri_constr$total_var print(totvar_hat) ## [1] as well as the estimated eigenvalues for each functional random effect results$fpc_hat_tri_constr$nu_b/c/e_hat The estimated variance accounted for by the functional random effect Speaker (B) in total is then obtained like this: sum(results$fpc_hat_tri_constr$nu_b_hat)/totvar_hat*100 ## [1] This renders the variance accounted for by each of the FPCs selected for Speaker: results$fpc_hat_tri_constr$nu_b_hat/totvar_hat*100 ## [1] Accordingly, the estimated eigenvalue for the functional random effect C (Item) is results$fpc_hat_tri_constr$nu_c_hat/totvar_hat*100 ## [1] The variance accounted for by random error results$cov_hat_tri_constr$sigmasq/totvar_hat*100 ## [1] To obtain plots as given in Figures 3 and 4 of the paper, we need to calculate twice the square root of the relevant eigenvalue and retrieve the estimated eigenfunctions (FPCs) for 7

8 the given functional random effect. Here we illustrate this for Speaker (B), in correspondence to Figure 3 of the paper. mult <- sqrt(results$fpc_hat_tri_constr$nu_b_hat[1])*2 phi_b_hat_grid <- results$fpc_hat_tri_constr$phi_b_hat_grid The global mean is obtained by setting the covariates to their mean (0.5): intercept <- results$fpc_famm_hat_tri_constr$intercept gmean <- intercept+y_mean+y_cov1*0.5 The left part of Figure 3 of the paper which gives FPC1 chosen for functional random effect Speaker (B) is then obtained as follows: #percent variance accounted for (for legend) exvarb_sep <- results$fpc_hat_tri_constr$nu_b_hat/totvar_hat*100 # set margins so that the ylabel is not cut off par(mar = c (5, 5, 4, 2)) #plot the global mean plot(my_grid, gmean, type = "l", ylim = c(1200,2400), xlab = "normalized time",ylab = expression(paste(phi[1]^b, "(t)")), cex.axis = 1.2) #plot the suitable multiple (twice the square root of the eigenvalue) points(my_grid, gmean+mult*phi_b_hat_grid[, 1], pch = "+", t = "p") points(my_grid, gmean-mult*phi_b_hat_grid[, 1], pch = "-", t = "p") #legend text legend("topright", legend = paste0("explains ", round(exvarb_sep[1], 2), "%"), bty = "n", cex = 1.2) 8

9 φ 1 B (t) explains 37.64% normalized time To get each speaker s random intercept curve (right part of Figure 3), we proceed as follows: selected_use <- 1:5 # plot means for all 5 speakers color_use <- rainbow(length(selected_use)) #define colors #retrieve the random intercept curve per speaker B_hat <- results$fpc_famm_hat_tri_constr$famm_predict_b #add the random intercept curves to the global mean B_hat_plus_mean <- t(t(b_hat)+gmean) #plot the global mean plot(my_grid, gmean, ylim = c(1200,2400), lty = 2, lwd = 3, t = "l", xlab = "", ylab="") #plot the estimated random effects curve for each speaker for (ii in selected_use){ plot(my_grid, t(b_hat_plus_mean[ii,]), t = "l", lty = ii, lwd = 3, ylim = c(1200,2400), col = color_use[ii], xlab = "normalized time", ylab = expression(paste(mu(t)+b[i], "(t)"))) } 9

10 #legend text legend("topright",legend = (c("global mean", selected_use)), text.col = c("#000000",color_use), bty = "n", lty = c(2,1:5), col = c("#000000",color_use),lwd = 3) µ(t) + B i (t) global mean normalized time All of the above steps are summarized in the plotting functions plot_resultsrodiph.r and plot_summed_effectsrodiph.r provided as part of this Appendix. The functions are specific to the Romanian dataset but they can straightfowardly be adapted to the other example datasets and other model outputs. EPG Data of German Fricative Sequences Model Specification The EPG data were analysed using a model with two covariates and their interaction. Again this is a model with crossed functional random effects for Subject, Item, and Subject-Item- Repetition. 10

11 load("epgdata.rdata") Curve_info is set to the data table with the input data: curve_info = EPGdata We have two covariates. Form needs to be specified for each covariate. covariate = TRUE num_covariates = 2 covariate_form = rep( by,2) This time we have an interaction, hence we specify the symmetric interaction matrix with as many rows and columns as there are covariates: interaction = TRUE which_interaction = matrix(c(false, TRUE, TRUE, FALSE), byrow = TRUE, nrow = 2, ncol = 2) We specify the number of basis functions and the penalty in bf_covs and m_covs in the same way as for the Romanian vowel model since there are three functional random effects: bf_covs = c(10,10,10) m_covs = list(c(2,3), c(2,3), c(2,3)) Since we want confidence bands to be computed, we require FAMM to be used: use_famm = TRUE Here is the call to the function: results <- sparseflmm(curve_info = EPGData, covariate = TRUE, num_covariates = 2, covariate_form = rep("by",2), interaction = TRUE, which_interaction = matrix(c(false, TRUE,TRUE,FALSE), byrow = TRUE,nrow = 2,ncol = 2), bf_covs = c(10,10,10), m_covs = list(c(2,3),c(2,3),c(2,3)), use_famm = TRUE) Output We again provide a results file so that plots can be generated without having to run the model. load("resultsepgdata.rdata") 11

12 Effects Plots Plotting the effects is analogous to the previous example, we only add plotting commands for the second covariate and the interaction. #retrieve the intercept, reference mean and confidence bands intercept <- results$fpc_famm_hat_tri_constr$intercept y_mean <- results$fpc_famm_hat_tri_constr$famm_cb_mean$value se_mean <- results$fpc_famm_hat_tri_constr$famm_cb_mean$se my_grid <- results$my_grid plot(my_grid, y_mean+intercept, lwd = 3,t = "l", xlab = "normalized time (t)", ylab = "delta index", ylim = c(-1,1), col = "red", cex.axis = 1.2, main = expression(paste("reference Mean ", f[0](t))), cex.main = 1.2, cex.lab = 1.2) #confidence bands lines(x = my_grid, y = (y_mean+2*se_mean+intercept), lty = 2, lwd = 3,col = "blue") lines(x = my_grid, y = (y_mean-2*se_mean+intercept), lty = 2, lwd = 3,col = "blue") # zero reference line abline(h = 0, lty = 3) Reference Mean f 0 (t) delta index normalized time (t) 12

13 The plot for covariate.1 (C1): y_cov1 <- results$fpc_famm_hat_tri_constr$famm_cb_covariate.1$value se_cov1 <- results$fpc_famm_hat_tri_constr$famm_cb_covariate.1$se plot(my_grid, y_cov1, t = "l",lwd = 3, xlab = " normalized time (t)", ylab = "delta index", ylim = c(-1,1), col = "red", cex.axis = 1.2, main = expression(paste("c1 ", f[1](t))), cex.main = 1.2, cex.lab = 1.2) lines(x = my_grid, y = (y_cov1+2*se_cov1), lty = 2, lwd = 3, col = "blue") lines(x = my_grid, y = (y_cov1-2*se_cov1), lty = 2, lwd = 3, col = "blue") abline(h = 0, lty = 3) C1 f 1 (t) delta index normalized time (t) The plot for covariate.2 (Stress): y_cov2 <- results$fpc_famm_hat_tri_constr$famm_cb_covariate.2$value se_cov2 <- results$fpc_famm_hat_tri_constr$famm_cb_covariate.2$se plot(my_grid, y_cov2, t = "l",lwd = 3, xlab = " normalized time (t)", ylab = "delta index", ylim = c(-1,1), col = "red", cex.axis = 1.2, main = expression(paste("stress ", f[2](t))), cex.main = 1.2, cex.lab = 1.2) lines(x = my_grid, y = (y_cov2+2*se_cov2), lty = 2, lwd = 3, col = "blue") lines(x = my_grid, y = (y_cov2-2*se_cov2), lty = 2, lwd = 3, col = "blue") abline(h = 0, lty = 3) 13

14 Stress f 2 (t) delta index normalized time (t) The plot for the C1 * Stress Interaction is obtained analogously: y_inter_1_2 <- results$fpc_famm_hat_tri_constr$famm_cb_inter_1_2$value se_inter_1_2 <- results$fpc_famm_hat_tri_constr$famm_cb_inter_1_2$se plot(my_grid, y_inter_1_2, t = "l", lwd = 3, xlab = " normalized time (t)", ylab = "delta index", ylim = c(-1,1), col = "red", cex.axis = 1.2, main = expression(paste("c1 * Stress ", f[3](t))), cex.main = 1.2, cex.lab = 1.2) lines(x = my_grid, y = (y_inter_1_2+2*se_inter_1_2), lty = 2,lwd = 3,col = "blue") lines(x = my_grid, y = (y_inter_1_2-2*se_inter_1_2), lty = 2,lwd = 3,col = "blue") abline(h = 0, lty = 3) 14

15 C1 * Stress f 3 (t) delta index normalized time (t) Summed Effects Again we proceed in analogy to the Romanian vowel data, only adding a second covariate and the interaction. #define colors and scaling of y-axis cols <- rainbow(12) miny <- -1 maxy <- 1 #define the summed effects none <- intercept+y_mean C1 <- intercept+y_mean+y_cov1 Stress <- intercept+y_mean+y_cov2 Stress_C1 <- intercept+y_mean+y_cov1+y_cov2+y_inter_1_2 plot(my_grid, none, col = cols[1], lty = 1,lwd = 4,t = "l", ylim = c(miny, maxy), ylab = "", xlab = "", xaxt="n", yaxt = "n") plot(my_grid, C1, col = cols[8], lty = 4, lwd = 4,t = "l", ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n") plot(my_grid, Stress, col = cols[2], lty = 5,lwd = 4, t = "l", ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n") 15

16 plot(my_grid, Stress_C1, col = cols[9], lty = 3,lwd = 4, ylim = c(miny, maxy), ylab = "delta index",xlab = "normalized time", main = expression(paste("summed Effects: C1 ", f[1](t), ", Stress ", f[2](t), ", and Their Interaction ", f[3](t))), t = "l", cex.axis=1.2, cex.main = 1.2, cex.lab = 1.2) abline(h = 0,lty = 3) leg.text <- c(expression(paste(f[0](t))), expression(paste(f[0](t)+f[1](t))), expression(paste(f[0](t)+f[2](t))), expression(paste(f[0](t)+f[1](t)+f[2](t)+f[3](t)))) legend("topright", legend = leg.text, bty = "n", col = c(cols[1], cols[8], cols[2], cols[9]), lty = c(1,4,5,9,2,1,1,1), lwd = c(4,4,4,4)) Summed Effects: C1 f 1 (t), Stress f 2 (t), and Their Interaction f 3 (t) delta index f 0 (t) f 0 (t) + f 1 (t) f 0 (t) + f 2 (t) f 0 (t) + f 1 (t) + f 2 (t) + f 3 (t) normalized time 16

17 Variance Decomposition The procedures for plotting variance decomposition are analogous to the Romanian vowel example and will not be repeated here. We only point out that in the case of two covariates and an interaction the global mean is computed as follows: intercept <- results$fpc_famm_hat_tri_constr$intercept y_mean <- results$fpc_famm_hat_tri_constr$famm_cb_mean$value y_cov1 <- results$fpc_famm_hat_tri_constr$famm_cb_covariate.1$value y_cov2 <- results$fpc_famm_hat_tri_constr$famm_cb_covariate.2$value y_inter_1_2 <- results$fpc_famm_hat_tri_constr$famm_cb_inter_1_2$value gmean <- intercept+y_mean+y_cov1*0.5+y_cov2*0.5+y_inter_1_2*0.25 This is also implemented in the plot_results function provided with the Romanian data in case an interaction is specified. Frequency Effects in Articulography Data Model Specification For the Russian articulography data, we use a model with a functional random intercept for Subject and for Subject-Repetition. There is no functional random effect for Item, since each frequency is only represented by a single syllable. load("freqdata.rdata") First we specify the input data table as curve_info curve_info = FreqData The default setting is for crossed functional random effects. Since we only have a functional random effect for Speaker here, we specify use_ri = TRUE There is a single covariate and no interaction. Note that the continuous covariate has to be centered prior to running the model. covariate = TRUE num_covariate = 1 covariate_form = by interaction = FALSE The number of basis functions is specified for the functional random effects: bf_covs = c(10, 10) m_covs = list(c(2, 3), c(2, 3)) To obtain confidence bands, set 17

18 use_famm = TRUE Since this is a large dataset, we use two cores to parallelize the auto-covariance and FAMM estimation: para_estim_cov = TRUE para_estim_cov_nc = 2 para_estim_famm = TRUE para_estim_famm_nc = 2 results <- sparseflmm(curve_info = FreqData, use_ri = TRUE, covariate = TRUE, num_covariate = 1, covariate_form = "by", interaction = FALSE, bf_covs = c(10, 10), m_covs = list(c(2, 3), c(2, 3)), use_famm = TRUE, para_estim_cov = TRUE, para_estim_cov_nc = 2, para_estim_famm = TRUE, para_estim_famm_nc = 2) Output The effects plots are generated as before and the plotting commands are not repeated here. Specific to this dataset is, besides the use of only two functional random effects, the use of a continuous covariate. This means that in order to understand the estimated effect of frequency on the kinematics of C2, it is advisable to plot the summed effects for a few selected frequency values (alternatively, for instance the quantiles could be used for plotting). This is done by adding to the reference mean the estimated curve of covariate.1 (Frequency) multiplied by the given centered frequency value of interest. Again we load the provided results file load("resultsfreqdata.rdata") The centered frequency values are retrieved from the input data table. fs <- unique(freqdata$covariate.1) fs <- sort(fs) As in the previous cases, we retrieve the intercept, the reference mean, and the covariate.1 (Frequency) effect. intercept <- results$fpc_famm_hat_tri_constr$intercept my_grid <- results$my_grid y_mean <- results$fpc_famm_hat_tri_constr$famm_cb_mean$value y_cov1 <- results$fpc_famm_hat_tri_constr$famm_cb_covariate.1$value # 5 frequencies are chosen for the summed effects plot freq1 <- intercept+y_mean+y_cov1*fs[1] #lowest frequency (/mx/, freq=6.0) freq4 <- intercept+y_mean+y_cov1*fs[4] 18

19 freq8 <- intercept+y_mean+y_cov1*fs[8] freq9 <- intercept+y_mean+y_cov1*fs[9] freq12 <- intercept+y_mean+y_cov1*fs[12] #highest frequency (/gl/, freq=12.7) none <- intercept+y_mean miny <- -2 maxy <- 2 cols <- rainbow(12) plot(my_grid, my_grid, t = "n", ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt="n") plot(my_grid, none, col = "#000000", lty = 1,t = "l", ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n",lwd = 4) plot(my_grid, freq1, col = cols[2], lty = 2, ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n", t = "l",lwd = 4) plot(my_grid, freq4, col = cols[6], lty = 3, ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n", t = "l",lwd = 4) plot(my_grid, freq8, col = cols[7], lty = 4, ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n", t = "l",lwd = 4) plot(my_grid, freq9, col = cols[9], lty = 5, ylim = c(miny, maxy), ylab = "", xlab = "", xaxt = "n", yaxt = "n", t = "l",lwd=4) plot(my_grid, freq12, col = cols[12], lty = 6, ylim = c(miny, maxy), ylab = "delta position", xlab = "normalized time", t = "l",lwd=4, main = paste0("summed Effect Curves"), cex.axis = 1.2, cex.main = 1.2) leg.text <- c(expression(paste(f[0](t))), expression(paste(f[0](t)+f[1](t),"*",freq1)), expression(paste(f[0](t)+f[1](t),"*",freq4)), expression(paste(f[0](t)+f[1](t),"*",freq8)), expression(paste(f[0](t)+f[1](t),"*",freq9)), expression(paste(f[0](t)+f[1](t),"*",freq12))) legend("topleft", legend = leg.text, bty = "n", col = c("#000000", cols[2], cols[6], cols[7], cols[9], cols[12]), lty = 1:6, lwd = 4) 19

20 abline(h = 0, lty = 3) Summed Effect Curves delta position f 0 (t) f 0 (t) + f 1 (t)*freq1 f 0 (t) + f 1 (t)*freq4 f 0 (t) + f 1 (t)*freq8 f 0 (t) + f 1 (t)*freq9 f 0 (t) + f 1 (t)*freq normalized time 20