simil_DTW_metric() computes similarity metrics between two or more trajectories using
Dynamic Time Warping (DTW). It allows for different superposition methods
to align trajectories before calculating the DTW metric. The function also supports
testing with simulations to calculate p-values for the DTW distance metrics.
Arguments
- data
A
trackR object, which is a list consisting of two elements:Trajectories: A list of interpolated trajectories, where each trajectory is a series of midpoints between consecutive footprints.Footprints: A list of data frames containing footprint coordinates, metadata (e.g., image reference, ID), and a marker indicating whether the footprint is actual or inferred.
- test
Logical; if
TRUE, the function compares the observed DTW distances against simulated trajectories and calculates p-values. Default isFALSE.- sim
A
track simulationR object consisting of a list of simulated trajectories to use for comparison whentest = TRUE.- superposition
A character string indicating the method used to align trajectories. Options are
"None","Centroid", or"Origin". Default is"None".
Value
A track similarity R object consisting of a list containing the following elements:
- DTW_distance_metric
A numeric matrix of pairwise Dynamic Time Warping (DTW) distances between trajectories. Each entry represents the DTW distance between the corresponding pair of trajectories.
- DTW_distance_metric_p_values
(If
test = TRUE) A numeric matrix of raw pairwise p-values, computed by Monte Carlo tail tests with the (+1) correction (Phipson & Smyth, 2010): $$p = (1 + \#\{\text{extreme}\}) / (nsim + 1)$$. Each entry reflects the probability of observing a DTW distance as extreme as the observed one, given the null hypothesis of no difference.- DTW_distance_metric_p_values_BH
(If
test = TRUE) A numeric matrix of Benjamini–Hochberg (BH) adjusted p-values controlling the false discovery rate (FDR), applied across the set of unique pairwise tests (Benjamini & Hochberg, 1995).- DTW_metric_p_values_combined
(If
test = TRUE) A single numeric value representing the combined p-value across all DTW distances (based on the global statistic: the sum of pairwise distances). This indicates the overall significance of the observed DTW distances relative to simulations.- DTW_distance_metric_simulations
(If
test = TRUE) A list containing matrices of DTW distances for each simulation iteration, allowing for inspection of the distribution of DTW distances across randomized scenarios.
Details
The simil_DTW_metric() function calculates the similarity between trajectories using
the Dynamic Time Warping (DTW) algorithm from the dtw package. The dtw() function
is used with the dist.method argument set to "Euclidean" for computing the local distances
between points in the trajectories.
DTW aligns two time series by minimizing the cumulative distance between their points, creating an optimal alignment despite variations in length or temporal distortions. The algorithm constructs a distance matrix where each element represents the cost of aligning points between the two series and finds a warping path through this matrix that minimizes the total distance. The warping path is contiguous and monotonic, starting from the bottom-left corner and ending at the top-right corner (Cleasby et al., 2019).
DTW measures are non-negative and unbounded, with larger values indicating greater dissimilarity between the time series. This method has been used in various contexts, including ecological studies to analyze and cluster trajectory data (Cleasby et al., 2019).
Potential limitations and biases of DTW include sensitivity to noise and outliers, computational complexity, and the need for appropriate distance metrics. Additionally, DTW may not always account for all structural differences between trajectories and can be biased by the chosen alignment constraints. While DTW can handle trajectories of different lengths due to its elastic nature, having trajectories of similar lengths can improve the accuracy and interpretability of the similarity measure. Similar lengths result in a more meaningful alignment and can make the computation more efficient. When trajectories differ significantly in length, preprocessing or normalization might be necessary, and careful analysis is required to understand the alignment path. The function’s flexibility in handling different lengths allows it to be applied in various contexts. However, large differences in trajectory lengths might introduce potential biases that should be considered when interpreting the results.
The function offers three different superposition methods to align the trajectories
before DTW() is applied:
"None": No superposition is applied."Centroid": Trajectories are shifted to align based on their centroids."Origin": Trajectories are shifted to align based on their starting point.
If test = TRUE, the function can compute p-values by comparing the observed DTW
distances with those generated from a set of simulated trajectories. The p-values
are calculated for both individual trajectory pairs and for the entire set of trajectories.
Pairwise p-values are computed with a Monte Carlo tail test using the (+1) correction to avoid
zero-values (see Phipson & Smyth, 2010):
$$p = (1 + \#\{\text{extreme}\}) / (nsim + 1)$$
These raw p-values are then adjusted for multiple comparisons across the set of unique pairwise tests using the Benjamini–Hochberg (BH) procedure for false discovery rate (FDR) control (Benjamini & Hochberg, 1995). Both the raw and the BH-adjusted p-value matrices are returned in the output object, allowing users to inspect either uncorrected or corrected results. In addition, a global combined p-value is provided, summarizing the overall deviation from the null across all pairs.
References
Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal statistical society: series B (Methodological), 57(1), 289-300.
Cleasby, I. R., Wakefield, E. D., Morrissey, B. J., Bodey, T. W., Votier, S. C., Bearhop, S., & Hamer, K. C. (2019). Using time-series similarity measures to compare animal movement trajectories in ecology. Behavioral Ecology and Sociobiology, 73, 1-19.
Phipson, B., & Smyth, G. K. (2010). Permutation P-values should never be zero: calculating exact P-values when permutations are randomly drawn. Statistical applications in genetics and molecular biology, 9(1).
Author
Humberto G. Ferrón
humberto.ferron@uv.es
Macroevolution and Functional Morphology Research Group (www.macrofun.es)
Cavanilles Institute of Biodiversity and Evolutionary Biology
Calle Catedrático José Beltrán Martínez, nº 2
46980 Paterna - Valencia - Spain
Phone: +34 (9635) 44477
Examples
# Example 1: Simulating tracks using the "Directed" model and comparing DTW distance
# in the PaluxyRiver dataset
s1 <- simulate_track(PaluxyRiver, nsim = 3, model = "Directed")
simil_DTW_metric(PaluxyRiver, test = TRUE, sim = s1, superposition = "None")
#> 2025-12-30 00:07:29.750689 Iteration 1
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 17.58427
#> Track_2 NA NA
#> ------------------------------------
#> 2025-12-30 00:07:29.75338 Iteration 2
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 16.48403
#> Track_2 NA NA
#> ------------------------------------
#> 2025-12-30 00:07:29.755981 Iteration 3
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 24.70844
#> Track_2 NA NA
#> ------------------------------------
#> ANALYSIS COMPLETED
#> ------------------------------------
#>
#> $DTW_distance_metric
#> Track_1 Track_2
#> Track_1 NA 23.47722
#> Track_2 23.47722 NA
#>
#> $DTW_distance_metric_p_values
#> Track_1 Track_2
#> Track_1 NA 0.75
#> Track_2 NA NA
#>
#> $DTW_distance_metric_p_values_BH
#> Track_1 Track_2
#> Track_1 NA 0.75
#> Track_2 NA NA
#>
#> $DTW_metric_p_values_combined
#> [1] 0.6666667
#>
#> $DTW_distance_metric_simulations
#> $DTW_distance_metric_simulations[[1]]
#> Track_1 Track_2
#> Track_1 NA 17.58427
#> Track_2 NA NA
#>
#> $DTW_distance_metric_simulations[[2]]
#> Track_1 Track_2
#> Track_1 NA 16.48403
#> Track_2 NA NA
#>
#> $DTW_distance_metric_simulations[[3]]
#> Track_1 Track_2
#> Track_1 NA 24.70844
#> Track_2 NA NA
#>
#>
# Example 2: Simulating tracks using the "Constrained" model and comparing DTW distance
# in the PaluxyRiver dataset
s2 <- simulate_track(PaluxyRiver, nsim = 3, model = "Constrained")
simil_DTW_metric(PaluxyRiver, test = TRUE, sim = s2, superposition = "None")
#> 2025-12-30 00:07:29.771256 Iteration 1
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 252.5979
#> Track_2 NA NA
#> ------------------------------------
#> 2025-12-30 00:07:29.77386 Iteration 2
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 95.39256
#> Track_2 NA NA
#> ------------------------------------
#> 2025-12-30 00:07:29.776456 Iteration 3
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 170.2938
#> Track_2 NA NA
#> ------------------------------------
#> ANALYSIS COMPLETED
#> ------------------------------------
#>
#> $DTW_distance_metric
#> Track_1 Track_2
#> Track_1 NA 23.47722
#> Track_2 23.47722 NA
#>
#> $DTW_distance_metric_p_values
#> Track_1 Track_2
#> Track_1 NA 0.25
#> Track_2 NA NA
#>
#> $DTW_distance_metric_p_values_BH
#> Track_1 Track_2
#> Track_1 NA 0.25
#> Track_2 NA NA
#>
#> $DTW_metric_p_values_combined
#> [1] 0
#>
#> $DTW_distance_metric_simulations
#> $DTW_distance_metric_simulations[[1]]
#> Track_1 Track_2
#> Track_1 NA 252.5979
#> Track_2 NA NA
#>
#> $DTW_distance_metric_simulations[[2]]
#> Track_1 Track_2
#> Track_1 NA 95.39256
#> Track_2 NA NA
#>
#> $DTW_distance_metric_simulations[[3]]
#> Track_1 Track_2
#> Track_1 NA 170.2938
#> Track_2 NA NA
#>
#>
# Example 3: Simulating tracks using the "Unconstrained" model and comparing DTW distance
# in the PaluxyRiver dataset
s3 <- simulate_track(PaluxyRiver, nsim = 3, model = "Unconstrained")
simil_DTW_metric(PaluxyRiver, test = TRUE, sim = s3, superposition = "None")
#> 2025-12-30 00:07:29.791792 Iteration 1
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 717.4026
#> Track_2 NA NA
#> ------------------------------------
#> 2025-12-30 00:07:29.794413 Iteration 2
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 738.6735
#> Track_2 NA NA
#> ------------------------------------
#> 2025-12-30 00:07:29.797015 Iteration 3
#>
#> DTW metric
#> Track_1 Track_2
#> Track_1 NA 766.5934
#> Track_2 NA NA
#> ------------------------------------
#> ANALYSIS COMPLETED
#> ------------------------------------
#>
#> $DTW_distance_metric
#> Track_1 Track_2
#> Track_1 NA 23.47722
#> Track_2 23.47722 NA
#>
#> $DTW_distance_metric_p_values
#> Track_1 Track_2
#> Track_1 NA 0.25
#> Track_2 NA NA
#>
#> $DTW_distance_metric_p_values_BH
#> Track_1 Track_2
#> Track_1 NA 0.25
#> Track_2 NA NA
#>
#> $DTW_metric_p_values_combined
#> [1] 0
#>
#> $DTW_distance_metric_simulations
#> $DTW_distance_metric_simulations[[1]]
#> Track_1 Track_2
#> Track_1 NA 717.4026
#> Track_2 NA NA
#>
#> $DTW_distance_metric_simulations[[2]]
#> Track_1 Track_2
#> Track_1 NA 738.6735
#> Track_2 NA NA
#>
#> $DTW_distance_metric_simulations[[3]]
#> Track_1 Track_2
#> Track_1 NA 766.5934
#> Track_2 NA NA
#>
#>
# Example 4: Simulating and comparing DTW distance in the MountTom dataset using the
# "Centroid" superposition method
sbMountTom <- subset_track(MountTom, tracks = c(1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 18))
s4 <- simulate_track(sbMountTom, nsim = 3)
#> Warning: `model` is NULL. Defaulting to 'Unconstrained'.
simil_DTW_metric(sbMountTom, test = TRUE, sim = s4, superposition = "Centroid")
#> 2025-12-30 00:07:29.93031 Iteration 1
#>
#> DTW metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 34.77507 40.83665 28.97181 7.98321 39.801883 40.97504
#> Track_02 NA NA 33.15206 52.11918 20.13191 25.407164 18.04472
#> Track_03 NA NA NA 37.95267 23.53295 8.399035 19.73591
#> Track_04 NA NA NA NA 24.44538 40.943513 49.13389
#> Track_07 NA NA NA NA NA 21.926868 23.58801
#> Track_08 NA NA NA NA NA NA 11.16308
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 39.12781 26.669164 15.953157 39.613209
#> Track_02 40.79062 36.246694 21.116625 34.865976
#> Track_03 14.93785 20.189044 12.790502 5.126975
#> Track_04 25.34226 9.750902 15.373086 33.302578
#> Track_07 24.07498 16.958717 8.490390 22.822048
#> Track_08 20.46661 23.313536 12.973165 12.013410
#> Track_09 31.60224 30.167942 17.917873 23.002248
#> Track_13 NA 12.313347 12.653304 10.495460
#> Track_15 NA NA 9.390416 16.660829
#> Track_16 NA NA NA 11.736856
#> Track_18 NA NA NA NA
#> ------------------------------------
#> 2025-12-30 00:07:29.976346 Iteration 2
#>
#> DTW metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 12.45845 39.22490 66.70123 24.60561 8.035231 24.928465
#> Track_02 NA NA 35.58791 59.53738 17.91867 12.662517 16.324110
#> Track_03 NA NA NA 12.72239 18.52695 20.454961 24.738623
#> Track_04 NA NA NA NA 33.03691 42.737136 39.977566
#> Track_07 NA NA NA NA NA 14.446761 4.820698
#> Track_08 NA NA NA NA NA NA 17.287810
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 31.623826 42.359709 28.294056 23.937678
#> Track_02 22.775928 38.565890 25.353980 17.303799
#> Track_03 21.854177 4.311715 5.184952 17.583788
#> Track_04 34.014885 14.593099 19.033953 32.036203
#> Track_07 6.162870 22.039899 12.739054 1.269285
#> Track_08 20.025424 23.749981 12.542857 13.868949
#> Track_09 7.600545 27.684611 17.961896 4.639764
#> Track_13 NA 25.246732 16.600928 6.335406
#> Track_15 NA NA 7.889243 21.107408
#> Track_16 NA NA NA 12.047081
#> Track_18 NA NA NA NA
#> ------------------------------------
#> 2025-12-30 00:07:30.021863 Iteration 3
#>
#> DTW metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 19.51285 38.94657 60.13078 40.02200 36.102832 35.892140
#> Track_02 NA NA 33.11222 45.74144 33.69174 27.286147 23.896833
#> Track_03 NA NA NA 17.33850 3.67163 5.953785 15.294390
#> Track_04 NA NA NA NA 13.97407 8.426659 14.435758
#> Track_07 NA NA NA NA NA 5.192675 14.030102
#> Track_08 NA NA NA NA NA NA 8.529183
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 5.477892 11.894416 29.161582 39.077199
#> Track_02 12.051650 6.148277 25.281876 35.845563
#> Track_03 26.606883 22.020042 4.558738 7.450284
#> Track_04 46.078995 37.863686 20.287013 28.371299
#> Track_07 28.137917 23.189137 7.140877 10.253712
#> Track_08 23.745531 18.512185 6.477767 11.670501
#> Track_09 24.559188 18.657639 12.677109 21.094087
#> Track_13 NA 7.365335 18.682095 27.305054
#> Track_15 NA NA 15.022130 23.319641
#> Track_16 NA NA NA 5.848749
#> Track_18 NA NA NA NA
#> ------------------------------------
#> ANALYSIS COMPLETED
#> ------------------------------------
#>
#> $DTW_distance_metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 7.155827 6.889059 7.146449 8.550492 9.358030 41.27125
#> Track_02 7.155827 NA 7.367287 7.095034 8.511176 8.404712 44.46323
#> Track_03 6.889059 7.367287 NA 2.752378 5.017897 5.558007 23.61988
#> Track_04 7.146449 7.095034 2.752378 NA 3.748965 5.898481 21.55969
#> Track_07 8.550492 8.511176 5.017897 3.748965 NA 3.362680 27.87452
#> Track_08 9.358030 8.404712 5.558007 5.898481 3.362680 NA 27.11038
#> Track_09 41.271248 44.463229 23.619876 21.559693 27.874517 27.110379 NA
#> Track_13 23.690795 26.769160 11.380548 10.754453 16.245880 16.011649 12.44577
#> Track_15 10.006241 8.652232 4.011654 2.107453 4.076379 5.869619 21.45921
#> Track_16 8.956456 8.004512 6.187040 7.121107 5.378815 3.814577 36.64336
#> Track_18 11.561263 9.939214 4.835886 3.128265 5.173189 6.169448 21.46042
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 23.69080 10.006241 8.956456 11.561263
#> Track_02 26.76916 8.652232 8.004512 9.939214
#> Track_03 11.38055 4.011654 6.187040 4.835886
#> Track_04 10.75445 2.107453 7.121107 3.128265
#> Track_07 16.24588 4.076379 5.378815 5.173189
#> Track_08 16.01165 5.869619 3.814577 6.169448
#> Track_09 12.44577 21.459210 36.643358 21.460416
#> Track_13 NA 11.406049 22.371414 11.843307
#> Track_15 11.40605 NA 7.547234 1.280341
#> Track_16 22.37141 7.547234 NA 8.182026
#> Track_18 11.84331 1.280341 8.182026 NA
#>
#> $DTW_distance_metric_p_values
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 0.25 0.25 0.25 0.50 0.50 1.00
#> Track_02 NA NA 0.25 0.25 0.25 0.25 1.00
#> Track_03 NA NA NA 0.25 0.50 0.25 0.75
#> Track_04 NA NA NA NA 0.25 0.25 0.50
#> Track_07 NA NA NA NA NA 0.25 1.00
#> Track_08 NA NA NA NA NA NA 1.00
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 0.50 0.25 0.25 0.25
#> Track_02 0.75 0.50 0.25 0.25
#> Track_03 0.25 0.25 0.75 0.25
#> Track_04 0.25 0.25 0.25 0.25
#> Track_07 0.50 0.25 0.25 0.50
#> Track_08 0.25 0.25 0.25 0.25
#> Track_09 0.50 0.50 1.00 0.75
#> Track_13 NA 0.50 1.00 0.75
#> Track_15 NA NA 0.25 0.25
#> Track_16 NA NA NA 0.50
#> Track_18 NA NA NA NA
#>
#> $DTW_distance_metric_p_values_BH
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 0.4296875 0.4296875 0.4296875 0.6250000 0.6250000 1.0000000
#> Track_02 NA NA 0.4296875 0.4296875 0.4296875 0.4296875 1.0000000
#> Track_03 NA NA NA 0.4296875 0.6250000 0.4296875 0.8418367
#> Track_04 NA NA NA NA 0.4296875 0.4296875 0.6250000
#> Track_07 NA NA NA NA NA 0.4296875 1.0000000
#> Track_08 NA NA NA NA NA NA 1.0000000
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 0.6250000 0.4296875 0.4296875 0.4296875
#> Track_02 0.8418367 0.6250000 0.4296875 0.4296875
#> Track_03 0.4296875 0.4296875 0.8418367 0.4296875
#> Track_04 0.4296875 0.4296875 0.4296875 0.4296875
#> Track_07 0.6250000 0.4296875 0.4296875 0.6250000
#> Track_08 0.4296875 0.4296875 0.4296875 0.4296875
#> Track_09 0.6250000 0.6250000 1.0000000 0.8418367
#> Track_13 NA 0.6250000 1.0000000 0.8418367
#> Track_15 NA NA 0.4296875 0.4296875
#> Track_16 NA NA NA 0.6250000
#> Track_18 NA NA NA NA
#>
#> $DTW_metric_p_values_combined
#> [1] 0
#>
#> $DTW_distance_metric_simulations
#> $DTW_distance_metric_simulations[[1]]
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 34.77507 40.83665 28.97181 7.98321 39.801883 40.97504
#> Track_02 NA NA 33.15206 52.11918 20.13191 25.407164 18.04472
#> Track_03 NA NA NA 37.95267 23.53295 8.399035 19.73591
#> Track_04 NA NA NA NA 24.44538 40.943513 49.13389
#> Track_07 NA NA NA NA NA 21.926868 23.58801
#> Track_08 NA NA NA NA NA NA 11.16308
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 39.12781 26.669164 15.953157 39.613209
#> Track_02 40.79062 36.246694 21.116625 34.865976
#> Track_03 14.93785 20.189044 12.790502 5.126975
#> Track_04 25.34226 9.750902 15.373086 33.302578
#> Track_07 24.07498 16.958717 8.490390 22.822048
#> Track_08 20.46661 23.313536 12.973165 12.013410
#> Track_09 31.60224 30.167942 17.917873 23.002248
#> Track_13 NA 12.313347 12.653304 10.495460
#> Track_15 NA NA 9.390416 16.660829
#> Track_16 NA NA NA 11.736856
#> Track_18 NA NA NA NA
#>
#> $DTW_distance_metric_simulations[[2]]
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 12.45845 39.22490 66.70123 24.60561 8.035231 24.928465
#> Track_02 NA NA 35.58791 59.53738 17.91867 12.662517 16.324110
#> Track_03 NA NA NA 12.72239 18.52695 20.454961 24.738623
#> Track_04 NA NA NA NA 33.03691 42.737136 39.977566
#> Track_07 NA NA NA NA NA 14.446761 4.820698
#> Track_08 NA NA NA NA NA NA 17.287810
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 31.623826 42.359709 28.294056 23.937678
#> Track_02 22.775928 38.565890 25.353980 17.303799
#> Track_03 21.854177 4.311715 5.184952 17.583788
#> Track_04 34.014885 14.593099 19.033953 32.036203
#> Track_07 6.162870 22.039899 12.739054 1.269285
#> Track_08 20.025424 23.749981 12.542857 13.868949
#> Track_09 7.600545 27.684611 17.961896 4.639764
#> Track_13 NA 25.246732 16.600928 6.335406
#> Track_15 NA NA 7.889243 21.107408
#> Track_16 NA NA NA 12.047081
#> Track_18 NA NA NA NA
#>
#> $DTW_distance_metric_simulations[[3]]
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 19.51285 38.94657 60.13078 40.02200 36.102832 35.892140
#> Track_02 NA NA 33.11222 45.74144 33.69174 27.286147 23.896833
#> Track_03 NA NA NA 17.33850 3.67163 5.953785 15.294390
#> Track_04 NA NA NA NA 13.97407 8.426659 14.435758
#> Track_07 NA NA NA NA NA 5.192675 14.030102
#> Track_08 NA NA NA NA NA NA 8.529183
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 5.477892 11.894416 29.161582 39.077199
#> Track_02 12.051650 6.148277 25.281876 35.845563
#> Track_03 26.606883 22.020042 4.558738 7.450284
#> Track_04 46.078995 37.863686 20.287013 28.371299
#> Track_07 28.137917 23.189137 7.140877 10.253712
#> Track_08 23.745531 18.512185 6.477767 11.670501
#> Track_09 24.559188 18.657639 12.677109 21.094087
#> Track_13 NA 7.365335 18.682095 27.305054
#> Track_15 NA NA 15.022130 23.319641
#> Track_16 NA NA NA 5.848749
#> Track_18 NA NA NA NA
#>
#>
# Example 5: Simulating and comparing DTW distance in the MountTom dataset using the
# "Origin" superposition method
sbMountTom <- subset_track(MountTom, tracks = c(1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 18))
s5 <- simulate_track(sbMountTom, nsim = 3)
#> Warning: `model` is NULL. Defaulting to 'Unconstrained'.
simil_DTW_metric(sbMountTom, test = TRUE, sim = s5, superposition = "Origin")
#> 2025-12-30 00:07:30.167582 Iteration 1
#>
#> DTW metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 74.58195 78.91397 96.64398 70.79938 42.08277 35.468019
#> Track_02 NA NA 63.99860 15.82293 23.15382 63.20830 70.717161
#> Track_03 NA NA NA 71.73955 34.28973 31.76544 44.275450
#> Track_04 NA NA NA NA 18.85089 78.35770 87.732235
#> Track_07 NA NA NA NA NA 42.76890 50.107566
#> Track_08 NA NA NA NA NA NA 9.465876
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 93.33423 37.413138 52.051414 44.623207
#> Track_02 65.57425 64.313729 27.180051 64.332909
#> Track_03 14.28249 36.947737 17.754758 30.247222
#> Track_04 69.93651 80.012922 30.825070 79.119289
#> Track_07 32.87243 46.118442 9.575422 42.182148
#> Track_08 46.37700 5.752637 23.576983 2.286923
#> Track_09 58.16662 5.798243 33.668082 12.004624
#> Track_13 NA 51.806516 18.002230 44.967019
#> Track_15 NA NA 25.658456 7.668531
#> Track_16 NA NA NA 23.322257
#> Track_18 NA NA NA NA
#> ------------------------------------
#> 2025-12-30 00:07:30.213162 Iteration 2
#>
#> DTW metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 54.23687 45.21874 49.44016 71.50794 66.200748 50.16937
#> Track_02 NA NA 7.59947 12.07591 40.92355 35.146152 75.64967
#> Track_03 NA NA NA 17.30699 24.03585 19.747136 53.78668
#> Track_04 NA NA NA NA 54.70680 48.159534 81.22320
#> Track_07 NA NA NA NA NA 4.641382 57.88723
#> Track_08 NA NA NA NA NA NA 56.10677
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 70.35715 38.246414 53.14282 26.44397
#> Track_02 79.04307 59.444327 37.94523 16.21459
#> Track_03 54.32899 40.387821 20.39138 12.63175
#> Track_04 89.50376 65.038178 48.86099 14.16531
#> Track_07 52.78516 47.239386 11.03906 31.14017
#> Track_08 49.82184 42.374859 11.26962 27.12250
#> Track_09 23.79028 5.888145 35.99928 42.43423
#> Track_13 NA 22.474039 27.90860 47.60924
#> Track_15 NA NA 25.48473 30.77810
#> Track_16 NA NA NA 21.73607
#> Track_18 NA NA NA NA
#> ------------------------------------
#> 2025-12-30 00:07:30.26363 Iteration 3
#>
#> DTW metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 113.7108 75.852344 122.69581 77.46999 69.19479 43.15262
#> Track_02 NA NA 8.193052 35.02683 19.17839 44.74527 69.97435
#> Track_03 NA NA NA 24.76149 14.27356 28.90253 43.40454
#> Track_04 NA NA NA NA 47.01337 71.48320 65.05010
#> Track_07 NA NA NA NA NA 19.83662 51.52343
#> Track_08 NA NA NA NA NA NA 52.01348
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 89.950716 75.784943 37.31391 75.446867
#> Track_02 26.529842 19.578187 42.36166 23.277897
#> Track_03 20.553526 9.494872 22.86946 16.753630
#> Track_04 58.201258 11.676309 58.04265 50.430771
#> Track_07 5.254308 23.198154 21.30381 5.295366
#> Track_08 21.428358 36.170033 13.73617 15.205568
#> Track_09 62.329274 39.511223 31.33146 50.831737
#> Track_13 NA 31.105605 26.59231 3.924337
#> Track_15 NA NA 25.54778 24.486652
#> Track_16 NA NA NA 18.784094
#> Track_18 NA NA NA NA
#> ------------------------------------
#> ANALYSIS COMPLETED
#> ------------------------------------
#>
#> $DTW_distance_metric
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 8.365556 8.140001 13.288601 17.963246 14.043056 80.66455
#> Track_02 8.365556 NA 9.928547 13.870318 15.997694 11.986057 83.72110
#> Track_03 8.140001 9.928547 NA 4.242441 9.250629 6.131319 46.49768
#> Track_04 13.288601 13.870318 4.242441 NA 5.401950 3.172849 42.61717
#> Track_07 17.963246 15.997694 9.250629 5.401950 NA 5.261381 51.73148
#> Track_08 14.043056 11.986057 6.131319 3.172849 5.261381 NA 54.42241
#> Track_09 80.664547 83.721098 46.497678 42.617174 51.731485 54.422412 NA
#> Track_13 45.007884 47.197964 20.295656 19.901140 29.449500 27.328666 22.41087
#> Track_15 18.730370 17.799527 7.910351 4.165947 2.944420 4.915326 41.82503
#> Track_16 14.932046 9.836672 10.158387 8.969975 6.952380 5.349900 68.97341
#> Track_18 22.179224 20.569492 9.928217 6.396727 5.167036 6.863688 42.96907
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 45.00788 18.730370 14.932046 22.179224
#> Track_02 47.19796 17.799527 9.836672 20.569492
#> Track_03 20.29566 7.910351 10.158387 9.928217
#> Track_04 19.90114 4.165947 8.969975 6.396727
#> Track_07 29.44950 2.944420 6.952380 5.167036
#> Track_08 27.32867 4.915326 5.349900 6.863688
#> Track_09 22.41087 41.825031 68.973412 42.969072
#> Track_13 NA 20.579060 39.239001 21.761749
#> Track_15 20.57906 NA 8.572483 2.784293
#> Track_16 39.23900 8.572483 NA 10.795457
#> Track_18 21.76175 2.784293 10.795457 NA
#>
#> $DTW_distance_metric_p_values
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 0.25 0.25 0.25 0.25 0.25 1.00
#> Track_02 NA NA 0.75 0.50 0.25 0.25 1.00
#> Track_03 NA NA NA 0.25 0.25 0.25 0.75
#> Track_04 NA NA NA NA 0.25 0.25 0.25
#> Track_07 NA NA NA NA NA 0.50 0.75
#> Track_08 NA NA NA NA NA NA 0.75
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 0.25 0.25 0.25 0.25
#> Track_02 0.50 0.25 0.25 0.50
#> Track_03 0.50 0.25 0.25 0.25
#> Track_04 0.25 0.25 0.25 0.25
#> Track_07 0.50 0.25 0.25 0.25
#> Track_08 0.50 0.25 0.25 0.50
#> Track_09 0.25 1.00 1.00 0.75
#> Track_13 NA 0.25 1.00 0.50
#> Track_15 NA NA 0.25 0.25
#> Track_16 NA NA NA 0.25
#> Track_18 NA NA NA NA
#>
#> $DTW_distance_metric_p_values_BH
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 0.3819444 0.3819444 0.3819444 0.3819444 0.3819444 1.0000000
#> Track_02 NA NA 0.8250000 0.6111111 0.3819444 0.3819444 1.0000000
#> Track_03 NA NA NA 0.3819444 0.3819444 0.3819444 0.8250000
#> Track_04 NA NA NA NA 0.3819444 0.3819444 0.3819444
#> Track_07 NA NA NA NA NA 0.6111111 0.8250000
#> Track_08 NA NA NA NA NA NA 0.8250000
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 0.3819444 0.3819444 0.3819444 0.3819444
#> Track_02 0.6111111 0.3819444 0.3819444 0.6111111
#> Track_03 0.6111111 0.3819444 0.3819444 0.3819444
#> Track_04 0.3819444 0.3819444 0.3819444 0.3819444
#> Track_07 0.6111111 0.3819444 0.3819444 0.3819444
#> Track_08 0.6111111 0.3819444 0.3819444 0.6111111
#> Track_09 0.3819444 1.0000000 1.0000000 0.8250000
#> Track_13 NA 0.3819444 1.0000000 0.6111111
#> Track_15 NA NA 0.3819444 0.3819444
#> Track_16 NA NA NA 0.3819444
#> Track_18 NA NA NA NA
#>
#> $DTW_metric_p_values_combined
#> [1] 0
#>
#> $DTW_distance_metric_simulations
#> $DTW_distance_metric_simulations[[1]]
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 74.58195 78.91397 96.64398 70.79938 42.08277 35.468019
#> Track_02 NA NA 63.99860 15.82293 23.15382 63.20830 70.717161
#> Track_03 NA NA NA 71.73955 34.28973 31.76544 44.275450
#> Track_04 NA NA NA NA 18.85089 78.35770 87.732235
#> Track_07 NA NA NA NA NA 42.76890 50.107566
#> Track_08 NA NA NA NA NA NA 9.465876
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 93.33423 37.413138 52.051414 44.623207
#> Track_02 65.57425 64.313729 27.180051 64.332909
#> Track_03 14.28249 36.947737 17.754758 30.247222
#> Track_04 69.93651 80.012922 30.825070 79.119289
#> Track_07 32.87243 46.118442 9.575422 42.182148
#> Track_08 46.37700 5.752637 23.576983 2.286923
#> Track_09 58.16662 5.798243 33.668082 12.004624
#> Track_13 NA 51.806516 18.002230 44.967019
#> Track_15 NA NA 25.658456 7.668531
#> Track_16 NA NA NA 23.322257
#> Track_18 NA NA NA NA
#>
#> $DTW_distance_metric_simulations[[2]]
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 54.23687 45.21874 49.44016 71.50794 66.200748 50.16937
#> Track_02 NA NA 7.59947 12.07591 40.92355 35.146152 75.64967
#> Track_03 NA NA NA 17.30699 24.03585 19.747136 53.78668
#> Track_04 NA NA NA NA 54.70680 48.159534 81.22320
#> Track_07 NA NA NA NA NA 4.641382 57.88723
#> Track_08 NA NA NA NA NA NA 56.10677
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 70.35715 38.246414 53.14282 26.44397
#> Track_02 79.04307 59.444327 37.94523 16.21459
#> Track_03 54.32899 40.387821 20.39138 12.63175
#> Track_04 89.50376 65.038178 48.86099 14.16531
#> Track_07 52.78516 47.239386 11.03906 31.14017
#> Track_08 49.82184 42.374859 11.26962 27.12250
#> Track_09 23.79028 5.888145 35.99928 42.43423
#> Track_13 NA 22.474039 27.90860 47.60924
#> Track_15 NA NA 25.48473 30.77810
#> Track_16 NA NA NA 21.73607
#> Track_18 NA NA NA NA
#>
#> $DTW_distance_metric_simulations[[3]]
#> Track_01 Track_02 Track_03 Track_04 Track_07 Track_08 Track_09
#> Track_01 NA 113.7108 75.852344 122.69581 77.46999 69.19479 43.15262
#> Track_02 NA NA 8.193052 35.02683 19.17839 44.74527 69.97435
#> Track_03 NA NA NA 24.76149 14.27356 28.90253 43.40454
#> Track_04 NA NA NA NA 47.01337 71.48320 65.05010
#> Track_07 NA NA NA NA NA 19.83662 51.52343
#> Track_08 NA NA NA NA NA NA 52.01348
#> Track_09 NA NA NA NA NA NA NA
#> Track_13 NA NA NA NA NA NA NA
#> Track_15 NA NA NA NA NA NA NA
#> Track_16 NA NA NA NA NA NA NA
#> Track_18 NA NA NA NA NA NA NA
#> Track_13 Track_15 Track_16 Track_18
#> Track_01 89.950716 75.784943 37.31391 75.446867
#> Track_02 26.529842 19.578187 42.36166 23.277897
#> Track_03 20.553526 9.494872 22.86946 16.753630
#> Track_04 58.201258 11.676309 58.04265 50.430771
#> Track_07 5.254308 23.198154 21.30381 5.295366
#> Track_08 21.428358 36.170033 13.73617 15.205568
#> Track_09 62.329274 39.511223 31.33146 50.831737
#> Track_13 NA 31.105605 26.59231 3.924337
#> Track_15 NA NA 25.54778 24.486652
#> Track_16 NA NA NA 18.784094
#> Track_18 NA NA NA NA
#>
#>