simplex uses time delay embedding on a single time series to generate an attractor reconstruction, and then applies the simplex projection algorithm to make forecasts.

s_map is similar to simplex, but uses the S-map algorithm to make forecasts.

simplex(time_series, lib = c(1, NROW(time_series)), pred = lib,
norm = 2, E = 1:10, tau = 1, tp = 1, num_neighbors = "e+1",
stats_only = TRUE, exclusion_radius = NULL, epsilon = NULL,
silent = FALSE)

s_map(time_series, lib = c(1, NROW(time_series)), pred = lib,
norm = 2, E = 1, tau = 1, tp = 1, num_neighbors = 0,
theta = c(0, 1e-04, 3e-04, 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 0.5,
0.75, 1, 1.5, 2, 3, 4, 6, 8), stats_only = TRUE,
exclusion_radius = NULL, epsilon = NULL, silent = FALSE,
save_smap_coefficients = FALSE)

## Arguments

time_series either a vector to be used as the time series, or a data.frame or matrix with at least 2 columns (in which case the first column will be used as the time index, and the second column as the time series) a 2-column matrix (or 2-element vector) where each row specifies the first and last *rows* of the time series to use for attractor reconstruction (same format as lib), but specifying the sections of the time series to forecast. the distance measure to use. see 'Details' the embedding dimensions to use for time delay embedding the lag to use for time delay embedding the prediction horizon (how far ahead to forecast) the number of nearest neighbors to use. Note that the default value will change depending on the method selected. (any of "e+1", "E+1", "e + 1", "E + 1" will peg this parameter to E+1 for each run, any value < 1 will use all possible neighbors.) specify whether to output just the forecast statistics or the raw predictions for each run excludes vectors from the search space of nearest neighbors if their *time index* is within exclusion_radius (NULL turns this option off) excludes vectors from the search space of nearest neighbors if their *distance* is farther away than epsilon (NULL turns this option off) prevents warning messages from being printed to the R console the nonlinear tuning parameter (theta is only relevant if method == "s-map") specifies whether to include the s_map coefficients with the output (and forces stats_only = FALSE, as well)

## Value

For simplex, a data.frame with components for the parameters and forecast statistics:

 E embedding dimension tau time lag tp prediction horizon nn number of neighbors num_pred number of predictions rho correlation coefficient between observations and predictions mae mean absolute error rmse root mean square error perc percent correct sign p_val p-value that rho is significantly greater than 0 using Fisher's z-transformation const_pred_rho same as rho, but for the constant predictor const_pred_mae same as mae, but for the constant predictor const_pred_rmse same as rmse, but for the constant predictor const_pred_perc same as perc, but for the constant predictor const_p_val same as p_val, but for the constant predictor model_output data.frame with columns for the time index, observations, predictions, and estimated prediction variance (if stats_only == FALSE)

For s_map, the same as for simplex, but with additional columns:

 theta the nonlinear tuning parameter smap_coefficients data.frame with columns for the s-map coefficients (if save_smap_coefficients == TRUE) smap_coefficient_covariances list of covariance matrices for the s-map coefficients (if save_smap_coefficients == TRUE)

## Details

simplex is typically applied, and the embedding dimension varied, to find an optimal embedding dimension for the data. Thus, the default parameters are set so that passing a time series as the only argument will run over E = 1:10 (embedding dimension), using leave-one-out cross-validation over the whole time series, and returning just the forecast statistics.

s_map is typically applied, with fixed embedding dimension, and theta varied, to test for nonlinear dynamics in the data. Thus, the default parameters are set so that passing a time series as the only argument will run over a default list of thetas (0, 0.0001, 0.0003, 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 0.5, 0.75, 1.0, 1.5, 2, 3, 4, 6, and 8), using E = 1, leave-one-out cross-validation over the whole time series, and returning just the forecast statistics.

norm = 2 (default) uses the "L2 norm", Euclidean distance: $$distance(a,b) := \sqrt{\sum_i{(a_i - b_i)^2}}$$ norm = 1 uses the "L1 norm", Manhattan distance: $$distance(a,b) := \sum_i{|a_i - b_i|}$$ Other values generalize the L1 and L2 norm to use the given argument as the exponent, P, as: $$distance(a,b) := \sum_i{(a_i - b_i)^P}^{1/P}$$

## Examples

data("two_species_model")
ts <- two_species_model$x[1:200] simplex(ts, lib = c(1, 100), pred = c(101, 200))#> E tau tp nn num_pred rho mae rmse perc p_val #> 1 1 1 1 2 99 0.9977065 0.008357515 0.013002536 1 1.744958e-241 #> 2 2 1 1 3 98 0.9989009 0.005136110 0.009018716 1 3.100170e-293 #> 3 3 1 1 4 97 0.9990626 0.004970217 0.008331537 1 1.637664e-302 #> 4 4 1 1 5 96 0.9971511 0.008104056 0.014523326 1 2.152115e-219 #> 5 5 1 1 6 95 0.9965447 0.009534573 0.015990520 1 1.387630e-204 #> 6 6 1 1 7 94 0.9948537 0.012735600 0.019566559 1 4.606555e-178 #> 7 7 1 1 8 93 0.9943411 0.013903466 0.020459728 1 1.264351e-170 #> 8 8 1 1 9 92 0.9928574 0.016085801 0.023026414 1 9.197431e-156 #> 9 9 1 1 10 91 0.9927218 0.016557143 0.023240397 1 5.043687e-153 #> 10 10 1 1 11 90 0.9921240 0.017697481 0.024493244 1 3.913787e-147 #> const_pred_num_pred const_pred_rho const_pred_mae const_pred_rmse #> 1 99 -0.9696726 0.3771995 0.3811690 #> 2 98 -0.9694899 0.3776359 0.3816164 #> 3 97 -0.9693895 0.3779297 0.3819369 #> 4 96 -0.9692005 0.3781315 0.3821729 #> 5 95 -0.9690482 0.3782716 0.3823513 #> 6 94 -0.9687106 0.3779132 0.3820240 #> 7 93 -0.9684948 0.3777218 0.3818742 #> 8 92 -0.9682642 0.3780649 0.3822440 #> 9 91 -0.9680677 0.3783219 0.3825359 #> 10 90 -0.9679411 0.3786576 0.3829010 #> const_pred_perc const_p_val #> 1 1 1 #> 2 1 1 #> 3 1 1 #> 4 1 1 #> 5 1 1 #> 6 1 1 #> 7 1 1 #> 8 1 1 #> 9 1 1 #> 10 1 1 data("two_species_model") ts <- two_species_model$x[1:200]
simplex(ts, stats_only = FALSE)#> Warning: Found overlap between lib and pred. Enabling cross-validation with exclusion radius = 0.#>     E tau tp nn num_pred       rho         mae       rmse perc         p_val
#> 1   1   1  1  2      199 0.9935132 0.010898960 0.02190263    1  0.000000e+00
#> 2   2   1  1  3      198 0.9941356 0.006718955 0.02076716    1  0.000000e+00
#> 3   3   1  1  4      197 0.9920315 0.007295780 0.02431598    1  0.000000e+00
#> 4   4   1  1  5      196 0.9906450 0.008829094 0.02631030    1 9.682911e-304
#> 5   5   1  1  6      195 0.9962264 0.008826986 0.01670992    1  0.000000e+00
#> 6   6   1  1  7      194 0.9958433 0.011211758 0.01750450    1  0.000000e+00
#> 7   7   1  1  8      193 0.9955036 0.011909085 0.01821520    1  0.000000e+00
#> 8   8   1  1  9      192 0.9937719 0.014373767 0.02148182    1  0.000000e+00
#> 9   9   1  1 10      191 0.9934040 0.015176019 0.02210354    1  0.000000e+00
#> 10 10   1  1 11      190 0.9923382 0.016699364 0.02398609    1  0.000000e+00
#>    const_pred_num_pred const_pred_rho const_pred_mae const_pred_rmse
#> 1                  199     -0.9503299      0.3763936       0.3805352
#> 2                  198     -0.9684038      0.3765168       0.3806739
#> 3                  197     -0.9688821      0.3770916       0.3811776
#> 4                  196     -0.9688946      0.3774204       0.3814957
#> 5                  195     -0.9690341      0.3776509       0.3817309
#> 6                  194     -0.9688659      0.3775119       0.3816094
#> 7                  193     -0.9687451      0.3774507       0.3815690
#> 8                  192     -0.9686880      0.3776819       0.3818055
#> 9                  191     -0.9686819      0.3778536       0.3819894
#> 10                 190     -0.9685374      0.3778696       0.3820268
#>    const_pred_perc const_p_val model_output
#> 1                1           1 c(2, 3, ....
#> 2                1           1 c(2, 3, ....
#> 3                1           1 c(2, 3, ....
#> 4                1           1 c(2, 3, ....
#> 5                1           1 c(2, 3, ....
#> 6                1           1 c(2, 3, ....
#> 7                1           1 c(2, 3, ....
#> 8                1           1 c(2, 3, ....
#> 9                1           1 c(2, 3, ....
#> 10               1           1 c(2, 3, ....
data("two_species_model")
ts <- two_species_model$x[1:200] s_map(ts, E = 2)#> Warning: Found overlap between lib and pred. Enabling cross-validation with exclusion radius = 0.#> E tau tp nn theta num_pred rho mae rmse perc #> 1 2 1 1 0 0.0000 198 0.9657371 0.039812393 0.04982476 1 #> 2 2 1 1 0 0.0001 198 0.9657415 0.039810163 0.04982157 1 #> 3 2 1 1 0 0.0003 198 0.9657504 0.039805703 0.04981520 1 #> 4 2 1 1 0 0.0010 198 0.9657817 0.039790090 0.04979291 1 #> 5 2 1 1 0 0.0030 198 0.9658708 0.039745464 0.04972923 1 #> 6 2 1 1 0 0.0100 198 0.9661814 0.039589064 0.04950655 1 #> 7 2 1 1 0 0.0300 198 0.9670588 0.039142497 0.04887196 1 #> 8 2 1 1 0 0.1000 198 0.9700165 0.037573331 0.04666727 1 #> 9 2 1 1 0 0.3000 198 0.9775100 0.033059289 0.04052216 1 #> 10 2 1 1 0 0.5000 198 0.9833757 0.028643565 0.03492574 1 #> 11 2 1 1 0 0.7500 198 0.9880897 0.023991575 0.02962951 1 #> 12 2 1 1 0 1.0000 198 0.9904746 0.020860058 0.02651966 1 #> 13 2 1 1 0 1.5000 198 0.9925683 0.018068919 0.02342138 1 #> 14 2 1 1 0 2.0000 198 0.9946577 0.015367977 0.01987157 1 #> 15 2 1 1 0 3.0000 198 0.9974683 0.010502762 0.01370791 1 #> 16 2 1 1 0 4.0000 198 0.9973697 0.009755731 0.01395530 1 #> 17 2 1 1 0 6.0000 198 0.9951959 0.009410954 0.01894897 1 #> 18 2 1 1 0 8.0000 198 0.9907575 0.009328022 0.02649087 1 #> p_val const_pred_num_pred const_pred_rho const_pred_mae #> 1 3.557737e-176 198 -0.9684038 0.3765168 #> 2 3.465763e-176 198 -0.9684038 0.3765168 #> 3 3.288867e-176 198 -0.9684038 0.3765168 #> 4 2.737698e-176 198 -0.9684038 0.3765168 #> 5 1.620137e-176 198 -0.9684038 0.3765168 #> 6 2.566921e-177 198 -0.9684038 0.3765168 #> 7 1.257304e-179 198 -0.9684038 0.3765168 #> 8 5.214022e-188 198 -0.9684038 0.3765168 #> 9 9.363677e-215 198 -0.9684038 0.3765168 #> 10 1.081861e-244 198 -0.9684038 0.3765168 #> 11 6.481307e-280 198 -0.9684038 0.3765168 #> 12 8.307358e-305 198 -0.9684038 0.3765168 #> 13 0.000000e+00 198 -0.9684038 0.3765168 #> 14 0.000000e+00 198 -0.9684038 0.3765168 #> 15 0.000000e+00 198 -0.9684038 0.3765168 #> 16 0.000000e+00 198 -0.9684038 0.3765168 #> 17 0.000000e+00 198 -0.9684038 0.3765168 #> 18 3.033391e-308 198 -0.9684038 0.3765168 #> const_pred_rmse const_pred_perc const_p_val #> 1 0.3806739 1 1 #> 2 0.3806739 1 1 #> 3 0.3806739 1 1 #> 4 0.3806739 1 1 #> 5 0.3806739 1 1 #> 6 0.3806739 1 1 #> 7 0.3806739 1 1 #> 8 0.3806739 1 1 #> 9 0.3806739 1 1 #> 10 0.3806739 1 1 #> 11 0.3806739 1 1 #> 12 0.3806739 1 1 #> 13 0.3806739 1 1 #> 14 0.3806739 1 1 #> 15 0.3806739 1 1 #> 16 0.3806739 1 1 #> 17 0.3806739 1 1 #> 18 0.3806739 1 1 data("two_species_model") ts <- two_species_model$x[1:200]
s_map(ts, E = 2, theta = 1, save_smap_coefficients = TRUE)#> Warning: Found overlap between lib and pred. Enabling cross-validation with exclusion radius = 0.#>   E tau tp nn theta num_pred       rho        mae       rmse perc         p_val
#> 1 2   1  1  0     1      198 0.9904746 0.02086006 0.02651966    1 8.307358e-305
#>   const_pred_num_pred const_pred_rho const_pred_mae const_pred_rmse
#> 1                 198     -0.9684038      0.3765168       0.3806739
#>   const_pred_perc const_p_val model_output smap_coefficients
#> 1               1           1 c(2, 3, ....      c(NaN, -....
#>   smap_coefficient_covariances
#> 1                 NULL, c(....