Family: EchoTime native component
Kind: similarity
s_shape = GM(clip(r(x,y), 0, 1), clip(r(Delta x, Delta y), 0, 1))Combines level correlation with first-difference correlation so the headline match does not ignore local dynamics.
Documentation
Similarity Methods Atlas
A full audit of EchoTime's current comparison math plus the 127 extracted time-series similarity methods from ts_similarity_package_v2_pkg, each with an explicit mathematical expression.
Audit summary
This page imports the full method registry from `ts_similarity_package_v2_pkg` and audits which parts fit EchoTime's report-first product direction.
What EchoTime now exposes
- max_ncc / best_shift / sbd now cover explicit lead-lag and shift-aware similarity.
- independent_max_ncc / independent_sbd now cover the channel-independent multivariate sliding variants highlighted by the v2 package.
- periodogram_distance and trend_distance now add direct frequency-domain and trend-feature distances.
- ordinal_pattern_js_distance and linear_trend_model_distance now cover symbolic rhythm comparison and lightweight trend-model comparison.
- twed_distance now covers timestamp-aware elastic comparison for irregular series.
- erp_distance, lcss_similarity, lcss_distance, and edr_distance now cover robust partial-match and gap-aware elastic matching.
- acf_distance now adds a rhythm-focused structural distance beside the existing report-level spectral view.
Current EchoTime comparison layer
shape_similarity dtw_similarity trend_similarity derivative_similarity spectral_similarity pearson_r spearman_rho kendall_tau best_lag_pearson_r normalized_mutual_information
These are the similarity characterizations EchoTime already uses in raw-series comparison reports. Every entry below includes the mathematical expression used or approximated by the current code path.
Family: EchoTime native component
Kind: similarity
s_dtw = GM(exp(-DTW_w(x,y)/0.45), exp(-DTW_w(Delta x, Delta y)/0.35))Turns constrained DTW distances into a similarity score and blends level-space and derivative-space warping.
Family: EchoTime native component
Kind: similarity
s_trend = GM(clip(r(T(x), T(y)), 0, 1), clip(r(Delta T(x), Delta T(y)), 0, 1))Smooths both series first, then compares both the trend level and the trend slope.
Family: EchoTime native component
Kind: similarity
s_diff = clip(r(Delta x, Delta y), 0, 1)Focuses on whether local changes move together even when the levels differ.
Family: EchoTime native component
Kind: similarity
s_spec = GM(1 - JSD(P_x, P_y), clip(r(P_x - U, P_y - U), 0, 1))Compares normalized spectra of differenced series, mixing JS overlap with structural similarity relative to a uniform spectrum U.
Family: EchoTime reference metric
Kind: similarity
r(x,y) = sum_t (x_t - x_bar)(y_t - y_bar) / sqrt(sum_t (x_t - x_bar)^2 sum_t (y_t - y_bar)^2)Familiar linear correlation exposed directly in every raw-series similarity report.
Family: EchoTime reference metric
Kind: similarity
rho(x,y) = corr(rank(x), rank(y))Rank-order correlation used as a robustness check against nonlinear monotone relationships.
Family: EchoTime reference metric
Kind: similarity
tau(x,y) = (C - D) / choose(n, 2)Pairwise concordance measure that stays interpretable when rank ordering matters more than amplitude.
Family: EchoTime reference metric
Kind: similarity
max_lag_r(x,y) = max_ell r(x_t, y_{t-ell})Searches over a bounded lag window to show whether a shift-aware linear relationship is stronger than the aligned one.
Family: EchoTime reference metric
Kind: similarity
NMI(X,Y) = I(X;Y) / sqrt(H(X) H(Y))Histogram-based nonlinear dependence score used as a familiar cross-check against linear metrics.
Operating modes for elastic methods
- `mode="exact"` keeps the full elastic score path and stays the default when you need a final value for a report, notebook, or paper.
- `mode="fast"` is the screening path for large candidate pools and interactive shortlist work; it favors speed over exact-path fidelity.
- `window=...` is available on `lcss_similarity`, `lcss_distance`, `edr_distance`, `erp_distance`, and `twed_distance` so you can keep the dynamic-programming path local when large warps are not plausible.
- Recommended workflow: score all candidates with `mode="fast"` first, then rerun the shortlisted pairs with `mode="exact"` before you present the result as a final analog claim.
Implemented and high-fit additions from ts_similarity_package_v2_pkg
This table stays selective: it shows only the methods already added to EchoTime and the strongest candidates from the 127-method v2 registry. In the current build, the full high-fit shortlist is already implemented.
| Method | EchoTime fit | EchoTime API | Formula | Why it fits |
|---|---|---|---|---|
| independent_max_ncc sliding | Implemented in EchoTime | independent_max_ncc | $s(X,Y)=\frac{1}{d}\sum_c \max_k NCC_k(X^{(c)},Y^{(c)})$ | This method is now available as a public low-level EchoTime similarity function. |
| independent_sbd sliding | Implemented in EchoTime | independent_sbd | $d(X,Y)=\frac{1}{d}\sum_c SBD(X^{(c)},Y^{(c)})$ | This method is now available as a public low-level EchoTime similarity function. |
| max_normalized_cross_correlation sliding | Implemented in EchoTime | max_ncc | $s(x,y)=\max_k NCC_k(x,y)$ | This method is now available as a public low-level EchoTime similarity function. |
| sbd sliding | Implemented in EchoTime | sbd | $d(x,y)=1-\max_k NCC_k(x,y)$ | This method is now available as a public low-level EchoTime similarity function. |
| edr elastic | Implemented in EchoTime | edr_distance | $EDR_{\epsilon}(x,y)$ | This method is now available as a public low-level EchoTime similarity function. |
| erp elastic | Implemented in EchoTime | erp_distance | $ERP(x,y;g)$ | This method is now available as a public low-level EchoTime similarity function. |
| lcss_distance elastic | Implemented in EchoTime | lcss_distance | $d(x,y)=1-s_{LCSS}(x,y)$ | This method is now available as a public low-level EchoTime similarity function. |
| lcss_similarity elastic | Implemented in EchoTime | lcss_similarity | $s(x,y)=LCSS_{\epsilon}(x,y)/\min(n,m)$ | This method is now available as a public low-level EchoTime similarity function. |
| twed elastic | Implemented in EchoTime | twed_distance | $TWED_{\lambda,\nu}(x,y)$ | This method is now available as a public low-level EchoTime similarity function. |
| acf_distance feature-based | Implemented in EchoTime | acf_distance | $d(x,y)=\|ACF(x)-ACF(y)\|_2$ | This method is now available as a public low-level EchoTime similarity function. |
| ordinal_pattern_js_distance feature-based | Implemented in EchoTime | ordinal_pattern_js_distance | $d(x,y)=JS(\Pi(x),\Pi(y))$ | This method is now available as a public low-level EchoTime similarity function. |
| periodogram_distance feature-based | Implemented in EchoTime | periodogram_distance | $d(x,y)=\|P_x-P_y\|_2$ | This method is now available as a public low-level EchoTime similarity function. |
| trend_distance feature-based | Implemented in EchoTime | trend_distance | $d(x,y)=\|\phi_{\mathrm{trend}}(x)-\phi_{\mathrm{trend}}(y)\|_2$ | This method is now available as a public low-level EchoTime similarity function. |
| linear_trend_model_distance model-based | Implemented in EchoTime | linear_trend_model_distance | $d(x,y)=\|\theta_{\mathrm{trend}}(x)-\theta_{\mathrm{trend}}(y)\|_2$ | This method is now available as a public low-level EchoTime similarity function. |
Pointwise comparison on an already aligned timeline.
| Method | Output | Metric | Complexity | Unequal length | Multivariate | Implementation | EchoTime fit | EchoTime API | Formula | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| euclidean | distance | yes | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sqrt{\sum_t \|x_t-y_t\|_2^2}$ | Equal-length pointwise Euclidean distance. |
| squared_euclidean | distance | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sum_t \|x_t-y_t\|_2^2$ | Squared Euclidean loss. |
| rmse_distance | distance | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sqrt{\frac{1}{nd}\sum_t \|x_t-y_t\|_2^2}$ | Root-mean-square error over aligned points. |
| manhattan | distance | yes | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sum_t \|x_t-y_t\|_1$ | L1 distance. |
| mae_distance | distance | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\frac{1}{nd}\sum_t \|x_t-y_t\|_1$ | Mean absolute error. |
| minkowski | distance | yes (p≥1) | O(n·d) | no | yes | exact | Low-priority addition | - | $d_p(x,y)=\left(\sum_t \|x_t-y_t\|_p^p\right)^{1/p}$ | Generalized Lp distance. |
| chebyshev | distance | yes | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\max_t \|x_t-y_t\|_{\infty}$ | Maximum deviation distance. |
| canberra | distance | yes | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sum_i \frac{|x_i-y_i|}{|x_i|+|y_i|}$ | Relative L1 distance sensitive near zero. |
| bray_curtis | distance-like | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\frac{\sum_i |x_i-y_i|}{\sum_i |x_i+y_i|}$ | Bray-Curtis dissimilarity. |
| sorensen | distance-like | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\frac{\sum_i |x_i-y_i|}{\sum_i (|x_i|+|y_i|)}$ | Sorensen dissimilarity. |
| soergel | distance-like | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\frac{\sum_i |x_i-y_i|}{\sum_i \max(|x_i|,|y_i|)}$ | Relative distance normalized by max magnitudes. |
| kulczynski | distance-like | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\frac{\sum_i |x_i-y_i|}{\sum_i \min(|x_i|,|y_i|)}$ | Kulczynski-style relative dissimilarity. |
| clark | distance-like | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sqrt{\sum_i\left(\frac{|x_i-y_i|}{|x_i|+|y_i|}\right)^2}$ | Clark distance. |
| wave_hedges | distance-like | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sum_i \frac{|x_i-y_i|}{\max(|x_i|,|y_i|)}$ | Wave-Hedges distance. |
| lorentzian | distance-like | not guaranteed | O(n·d) | no | yes | reference | Low-priority addition | - | $d(x,y)=\sum_i \log(1+|x_i-y_i|)$ | Log-compressed deviation. |
| cosine_similarity | similarity | no | O(n·d) | no | yes | exact | Low-priority addition | - | $s(x,y)=\frac{\langle x,y\rangle}{\|x\|\|y\|}$ | Directional similarity. |
| cosine_distance | distance-like | no | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=1-s_{\cos}(x,y)$ | Monotone transform of cosine similarity. |
| angular_distance | distance | yes on normalized vectors | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\arccos(s_{\cos}(x,y))/\pi$ | Angle-based distance. |
| pearson_similarity | similarity | no | O(n·d) | no | yes | exact | Conceptually covered | pearson_r | $r(x,y)=\frac{\sum_i(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum_i(x_i-\bar x)^2}\sqrt{\sum_i(y_i-\bar y)^2}}$ | Linear correlation. |
| pearson_distance | distance-like | no | O(n·d) | no | yes | exact | Conceptually covered | pearson_r | $d(x,y)=1-r(x,y)$ | Correlation distance. |
| spearman_similarity | similarity | no | O(n \log n) | no | yes | reference | Conceptually covered | spearman_rho | $s(x,y)=r(\mathrm{rank}(x),\mathrm{rank}(y))$ | Rank correlation similarity. |
| spearman_distance | distance-like | no | O(n \log n) | no | yes | reference | Conceptually covered | spearman_rho | $d(x,y)=1-\rho(x,y)$ | Distance-like rank dissimilarity. |
| kendall_similarity | similarity | no | O(n^2) | no | yes | reference | Conceptually covered | kendall_tau | $s(x,y)=\tau_b(x,y)$ | Kendall tau-b rank agreement. |
| kendall_distance | distance-like | no | O(n^2) | no | yes | reference | Conceptually covered | kendall_tau | $d(x,y)=1-\tau_b(x,y)$ | Distance-like form of Kendall agreement. |
| jensen_shannon | distance | yes | O(n) | no | yes | exact | Conceptually covered | spectral_similarity | $d(x,y)=\sqrt{\frac{1}{2}KL(x\|m)+\frac{1}{2}KL(y\|m)}$ | Symmetric divergence for nonnegative normalized sequences. |
| kl_divergence | divergence | no | O(n) | no | yes | exact | Low-priority addition | - | $D_{KL}(x\|y)=\sum_i p_i\log\frac{p_i}{q_i}$ | Asymmetric divergence. |
| jeffreys_divergence | divergence | no | O(n) | no | yes | exact | Low-priority addition | - | $J(x,y)=KL(x\|y)+KL(y\|x)$ | Symmetrized KL divergence. |
| bhattacharyya_distance | distance-like | no | O(n) | no | yes | exact | Low-priority addition | - | $d(x,y)=-\log\sum_i \sqrt{p_i q_i}$ | Distribution overlap distance. |
| hellinger_distance | distance | yes | O(n) | no | yes | exact | Low-priority addition | - | $d(x,y)=\sqrt{\frac{1}{2}\sum_i(\sqrt{p_i}-\sqrt{q_i})^2}$ | Probability-distribution distance. |
| chi_square_distance | distance-like | no | O(n) | no | yes | exact | Low-priority addition | - | $d(x,y)=\frac{1}{2}\sum_i\frac{(x_i-y_i)^2}{x_i+y_i}$ | Chi-square distance for histograms. |
| seuclidean_distance | distance | yes | O(n) | no | yes | reference | Low-priority addition | - | $d(x,y)=\sqrt{\sum_i \frac{(x_i-y_i)^2}{v_i}}$ | Variance-standardized Euclidean distance. |
| znorm_euclidean | distance | yes | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\|\hat x-\hat y\|_2$ | Euclidean after per-channel z-normalization. |
| znorm_manhattan | distance | yes | O(n·d) | no | yes | exact | Low-priority addition | - | $d(x,y)=\|\hat x-\hat y\|_1$ | Manhattan after z-normalization. |
| cid_distance | distance-like | no | O(n·d) | no | yes | reference | Low-priority addition | - | $CID(x,y)=ED(x,y)\cdot \frac{\max(CE(x),CE(y))}{\min(CE(x),CE(y))}$ | Complexity-invariant distance. |
Shift-aware comparison under pure translation or lead-lag offsets.
| Method | Output | Metric | Complexity | Unequal length | Multivariate | Implementation | EchoTime fit | EchoTime API | Formula | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| max_normalized_cross_correlation | similarity | no | O(n \log n) or O(n^2) | yes | yes | exact | Implemented in EchoTime | max_ncc | $s(x,y)=\max_k NCC_k(x,y)$ | Maximum normalized cross-correlation over shifts. |
| sbd | distance-like | no | O(n \log n) or O(n^2) | yes | yes | exact | Implemented in EchoTime | sbd | $d(x,y)=1-\max_k NCC_k(x,y)$ | Shape-based distance. |
| shift_euclidean_distance | distance-like | no | O(n^2·d) | yes | yes | reference | Possible addition | - | $d(x,y)=\min_k \|x_{\Omega_k}-y_{\Omega_k}\|_2$ | Best-overlap Euclidean distance over integer shifts. |
| shift_rmse_distance | distance-like | no | O(n^2·d) | yes | yes | reference | Possible addition | - | $d(x,y)=\min_k \sqrt{\frac{1}{|\Omega_k|}\sum_{t\in\Omega_k}(x_t-y_{t-k})^2}$ | Best-overlap RMSE. |
| shift_manhattan_distance | distance-like | no | O(n^2·d) | yes | yes | reference | Possible addition | - | $d(x,y)=\min_k \|x_{\Omega_k}-y_{\Omega_k}\|_1$ | Best-overlap L1 distance. |
| shift_mae_distance | distance-like | no | O(n^2·d) | yes | yes | reference | Possible addition | - | $d(x,y)=\min_k \frac{1}{|\Omega_k|}\|x_{\Omega_k}-y_{\Omega_k}\|_1$ | Best-overlap MAE. |
| shift_cosine_similarity | similarity | no | O(n^2·d) | yes | yes | reference | Possible addition | - | $s(x,y)=\max_k s_{\cos}(x_{\Omega_k},y_{\Omega_k})$ | Best-overlap cosine similarity. |
| shift_cosine_distance | distance-like | no | O(n^2·d) | yes | yes | reference | Possible addition | - | $d(x,y)=1-\max_k s_{\cos}(x_{\Omega_k},y_{\Omega_k})$ | Distance-like shift-invariant cosine measure. |
| shift_pearson_similarity | similarity | no | O(n^2·d) | yes | yes | reference | Conceptually covered | best_lag_pearson_r | $s(x,y)=\max_k r(x_{\Omega_k},y_{\Omega_k})$ | Best-overlap Pearson correlation. |
| shift_pearson_distance | distance-like | no | O(n^2·d) | yes | yes | reference | Conceptually covered | best_lag_pearson_r | $d(x,y)=1-\max_k r(x_{\Omega_k},y_{\Omega_k})$ | Distance-like shift-invariant correlation measure. |
| independent_sbd | distance-like | no | O(dn\log n) | yes | multivariate only | exact | Implemented in EchoTime | independent_sbd | $d(X,Y)=\frac{1}{d}\sum_c SBD(X^{(c)},Y^{(c)})$ | Channel-independent SBD. |
| independent_max_ncc | similarity | no | O(dn\log n) | yes | multivariate only | exact | Implemented in EchoTime | independent_max_ncc | $s(X,Y)=\frac{1}{d}\sum_c \max_k NCC_k(X^{(c)},Y^{(c)})$ | Channel-independent maximum NCC. |
Dynamic-programming alignments that allow local stretching or compression.
| Method | Output | Metric | Complexity | Unequal length | Multivariate | Implementation | EchoTime fit | EchoTime API | Formula | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| dtw | distance | no | O(nm) | yes | yes | exact | Conceptually covered | dtw_similarity | $DTW(x,y)=\min_{\pi}\sqrt{\sum_{(i,j)\in\pi} \|x_i-y_j\|_2^2}$ | Dynamic Time Warping. |
| cdtw | distance | no | O(nw) | yes | yes | exact | Conceptually covered | dtw_similarity | $DTW_w(x,y)$ with Sakoe-Chiba window | Constrained DTW. |
| itakura_dtw | distance | no | O(nm) | yes | yes | reference | Possible addition | - | $DTW_{\mathrm{Itakura}}(x,y)$ | DTW under Itakura parallelogram constraint. |
| wdtw | distance | no | O(nm) | yes | yes | exact | Possible addition | - | $WDTW(x,y)=\min_{\pi}\sqrt{\sum_{(i,j)\in\pi} w_{|i-j|}\|x_i-y_j\|_2^2}$ | Weighted DTW. |
| ddtw | distance | no | O(nm) | yes | yes | exact | Conceptually covered | dtw_similarity | $DTW(\Delta x,\Delta y)$ | Derivative DTW. |
| wddtw | distance | no | O(nm) | yes | yes | exact | Possible addition | - | $WDTW(\Delta x,\Delta y)$ | Weighted derivative DTW. |
| soft_dtw | distance-like | no | O(nm) | yes | yes | exact | Possible addition | - | $sDTW_{\gamma}(x,y)$ | Soft minimum relaxation of DTW. |
| soft_dtw_divergence | distance-like | no | O(nm) | yes | yes | exact | Possible addition | - | $D^{\gamma}(x,y)=sDTW(x,y)-\frac{1}{2}sDTW(x,x)-\frac{1}{2}sDTW(y,y)$ | Soft-DTW divergence. |
| lcss_similarity | similarity | no | O(nm) | yes | yes | exact | Implemented in EchoTime | lcss_similarity | $s(x,y)=LCSS_{\epsilon}(x,y)/\min(n,m)$ | Longest Common Subsequence similarity. |
| lcss_distance | distance-like | no | O(nm) | yes | yes | exact | Implemented in EchoTime | lcss_distance | $d(x,y)=1-s_{LCSS}(x,y)$ | Distance-like LCSS form. |
| edr | distance | no | O(nm) | yes | yes | exact | Implemented in EchoTime | edr_distance | $EDR_{\epsilon}(x,y)$ | Edit distance on real sequence. |
| swale_similarity | similarity | no | O(nm) | yes | yes | reference | Possible addition | - | $SWALE_{\epsilon}(x,y)$ | Similarity with reward and penalty. |
| erp | distance | yes | O(nm) | yes | yes | exact | Implemented in EchoTime | erp_distance | $ERP(x,y;g)$ | Edit distance with real penalty. |
| msm | distance | yes | O(nm) | yes | univariate only | exact | Possible addition | - | $MSM(x,y)$ | Move-Split-Merge distance. |
| twed | distance | yes | O(nm) | yes | yes | exact | Implemented in EchoTime | twed_distance | $TWED_{\lambda,\nu}(x,y)$ | Time Warp Edit distance. |
| discrete_frechet | distance | yes | O(nm) | yes | yes | reference | Possible addition | - | $\delta_F(x,y)$ | Discrete Fréchet distance. |
| needleman_wunsch_similarity | similarity | no | O(nm) | yes | yes | reference | Possible addition | - | $NW(x,y)$ | Global alignment score. |
| smith_waterman_similarity | similarity | no | O(nm) | yes | yes | reference | Possible addition | - | $SW(x,y)$ | Local alignment score. |
| shape_dtw | distance | no | O(nmr) | yes | yes | reference | Possible addition | - | $DTW(\phi_{\mathrm{shape}}(x),\phi_{\mathrm{shape}}(y))$ | DTW on local shape descriptors. |
| independent_dtw | distance | no | O(dnm) | yes | multivariate only | exact | Possible addition | - | $d(X,Y)=\frac{1}{d}\sum_c DTW(X^{(c)},Y^{(c)})$ | Channel-independent DTW average. |
| independent_cdtw | distance | no | O(dnw) | yes | multivariate only | exact | Possible addition | - | $d(X,Y)=\frac{1}{d}\sum_c cDTW(X^{(c)},Y^{(c)})$ | Channel-independent constrained DTW. |
| independent_wdtw | distance | no | O(dnm) | yes | multivariate only | exact | Possible addition | - | $d(X,Y)=\frac{1}{d}\sum_c WDTW(X^{(c)},Y^{(c)})$ | Channel-independent weighted DTW. |
| independent_ddtw | distance | no | O(dnm) | yes | multivariate only | exact | Possible addition | - | $d(X,Y)=\frac{1}{d}\sum_c DDTW(X^{(c)},Y^{(c)})$ | Channel-independent derivative DTW. |
| independent_erp | distance | yes | O(dnm) | yes | multivariate only | exact | Possible addition | - | $d(X,Y)=\frac{1}{d}\sum_c ERP(X^{(c)},Y^{(c)})$ | Channel-independent ERP. |
Similarity through kernel scores rather than direct distances.
| Method | Output | Metric | Complexity | Unequal length | Multivariate | Implementation | EchoTime fit | EchoTime API | Formula | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| linear_kernel | similarity | PSD if input space valid | O(n·d) | no | yes | exact | Low-priority addition | - | $K(x,y)=\langle x,y \rangle$ | Linear kernel on flattened vectors. |
| polynomial_kernel | similarity | depends on parameters | O(n·d) | no | yes | exact | Low-priority addition | - | $K(x,y)=(\gamma \langle x,y \rangle + c)^p$ | Polynomial kernel. |
| sigmoid_kernel | similarity | not always PSD | O(n·d) | no | yes | exact | Low-priority addition | - | $K(x,y)=\tanh(\gamma \langle x,y \rangle + c)$ | Sigmoid kernel. |
| rbf_kernel | similarity | yes | O(n·d) | no | yes | exact | Low-priority addition | - | $K(x,y)=\exp(-\gamma \|x-y\|_2^2)$ | Gaussian kernel over aligned vectors. |
| dtw_rbf_kernel | similarity | not guaranteed | O(nm) | yes | yes | reference | Low-priority addition | - | $K(x,y)=\exp(-\gamma DTW(x,y)^2)$ | RBF on DTW distance. |
| sbd_kernel | similarity | not guaranteed | O(n \log n) or O(n^2) | yes | yes | reference | Low-priority addition | - | $K(x,y)=\exp(-\gamma SBD(x,y))$ | RBF-like kernel on SBD. |
| soft_dtw_kernel | similarity | not guaranteed | O(nm) | yes | yes | reference | Low-priority addition | - | $K(x,y)=\exp(-D^\gamma(x,y)/\gamma)$ | Kernelized soft-DTW divergence. |
| gak | similarity | yes after normalization in practice | O(nm) | yes | yes | exact | Low-priority addition | - | $K_G(x,y)=\sum_{\pi} \prod_{(i,j)\in\pi} k(x_i,y_j)$ | Global Alignment Kernel. |
| lgak | similarity | not a metric | O(nm) | yes | yes | exact | Low-priority addition | - | $\log(K_G(x,y)+\epsilon)$ | Log-transformed GAK. |
| weighted_ncc_kernel | similarity | not guaranteed | O(n \log n) or O(n^2) | yes | yes | reference | Low-priority addition | - | $K(x,y)=\sum_k w_k \cdot NCC_k(x,y)$ | Shift-aware weighted cross-correlation kernel. |
Compare derived features instead of raw waveforms.
| Method | Output | Metric | Complexity | Unequal length | Multivariate | Implementation | EchoTime fit | EchoTime API | Formula | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| stats_distance | distance | yes in feature space | O(n·d) | yes | yes | exact | Possible addition | - | $d(x,y)=\|\phi_{\mathrm{stats}}(x)-\phi_{\mathrm{stats}}(y)\|_2$ | Distance on basic summary statistics. |
| robust_stats_distance | distance | yes in feature space | O(n·d) | yes | yes | exact | Possible addition | - | $d(x,y)=\|\phi_{\mathrm{robust}}(x)-\phi_{\mathrm{robust}}(y)\|_2$ | Distance on robust statistics. |
| acf_distance | distance | yes in feature space | O(nL) | yes | univariate only | exact | Implemented in EchoTime | acf_distance | $d(x,y)=\|ACF(x)-ACF(y)\|_2$ | Autocorrelation-feature distance. |
| spectral_cosine_similarity | similarity | no | O(n \log n) | yes | yes | exact | Conceptually covered | spectral_similarity | $s(x,y)=s_{\cos}(|FFT(x)|,|FFT(y)|)$ | Cosine similarity in the magnitude spectrum. |
| spectral_distance | distance-like | no | O(n \log n) | yes | yes | exact | Possible addition | - | $d(x,y)=1-s_{\cos}(|FFT(x)|,|FFT(y)|)$ | Distance-like spectral measure. |
| periodogram_distance | distance | yes in feature space | O(n \log n) | yes | yes | exact | Implemented in EchoTime | periodogram_distance | $d(x,y)=\|P_x-P_y\|_2$ | Distance between normalized periodograms. |
| paa_distance | distance | yes in feature space | O(n·d) | yes | yes | exact | Possible addition | - | $d(x,y)=\|PAA(x)-PAA(y)\|_2$ | Piecewise Aggregate Approximation distance. |
| dwt_distance | distance | yes in feature space | O(n·d) | yes | yes | reference | Possible addition | - | $d(x,y)=\|W(x)-W(y)\|_2$ | Distance on Haar wavelet coefficients. |
| histogram_distance | distance | yes in feature space | O(n·d) | yes | yes | exact | Possible addition | - | $d(x,y)=\|h(x)-h(y)\|_2$ | Value-histogram distance. |
| permutation_entropy_distance | distance-like | no | O(n) | yes | univariate only | exact | Possible addition | - | $d(x,y)=|H_{\pi}(x)-H_{\pi}(y)|$ | Absolute difference in permutation entropy. |
| ordinal_pattern_js_distance | distance | yes | O(n) | yes | univariate only | exact | Implemented in EchoTime | ordinal_pattern_js_distance | $d(x,y)=JS(\Pi(x),\Pi(y))$ | Jensen-Shannon distance between ordinal-pattern distributions. |
| trend_distance | distance | yes in feature space | O(n·d) | yes | yes | exact | Implemented in EchoTime | trend_distance | $d(x,y)=\|\phi_{\mathrm{trend}}(x)-\phi_{\mathrm{trend}}(y)\|_2$ | Distance on trend/intercept/residual features. |
| sax_hamming_distance | distance | yes on symbolic strings | O(n) | yes | univariate only | exact | Possible addition | - | $d(x,y)=\frac{1}{w}\sum_{i=1}^w [SAX_i(x)\neq SAX_i(y)]$ | Hamming distance on SAX words. |
| sax_mindist | distance | lower-bounding | O(w) | yes | univariate only | exact | Possible addition | - | $MINDIST(\hat x,\hat y)=\sqrt{\frac{n}{w}}\sqrt{\sum_i dist(\hat x_i,\hat y_i)^2}$ | Lower-bounding symbolic distance in SAX space. |
| bag_of_patterns_cosine_similarity | similarity | no | O(nw) | yes | univariate only | reference | Possible addition | - | $s(x,y)=s_{\cos}(BoP(x),BoP(y))$ | Cosine similarity on bag-of-patterns histograms. |
| bag_of_patterns_distance | distance-like | no | O(nw) | yes | univariate only | reference | Possible addition | - | $d(x,y)=1-s_{\cos}(BoP(x),BoP(y))$ | Distance-like bag-of-patterns measure. |
| independent_stats_distance | distance | yes in feature space | O(dn) | yes | multivariate only | exact | Possible addition | - | $d(X,Y)=\frac{1}{d}\sum_c \|\phi(X^{(c)})-\phi(Y^{(c)})\|_2$ | Channel-independent stats distance. |
| independent_paa_distance | distance | yes in feature space | O(dn) | yes | multivariate only | exact | Possible addition | - | $d(X,Y)=\frac{1}{d}\sum_c \|PAA(X^{(c)})-PAA(Y^{(c)})\|_2$ | Channel-independent PAA distance. |
Compare fitted dynamics or transition structure.
| Method | Output | Metric | Complexity | Unequal length | Multivariate | Implementation | EchoTime fit | EchoTime API | Formula | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| ar_distance | distance | yes in parameter space | O(np^2) | yes | univariate only | exact | Possible addition | - | $d(x,y)=\|\theta_{AR}(x)-\theta_{AR}(y)\|_2$ | Distance between autoregressive coefficients. |
| ar_spectrum_distance | distance | yes in spectral space | O(np^2 + Fp) | yes | univariate only | reference | Possible addition | - | $d(x,y)=\|\log S_{AR}(x)-\log S_{AR}(y)\|_2$ | Distance between AR-implied log spectra. |
| ar_prediction_error_distance | distance-like | no | O(np^2) | yes | univariate only | reference | Possible addition | - | $d(x,y)=\frac{1}{2}(MSE_{x\to y}+MSE_{y\to x})$ | Symmetric cross-prediction error. |
| markov_distance | distance | yes in transition space | O(n) | yes | univariate only | exact | Possible addition | - | $d(x,y)=\|P_x-P_y\|_F$ | Distance between discretized transition matrices. |
| linear_trend_model_distance | distance | yes in parameter space | O(n) | yes | univariate only | exact | Implemented in EchoTime | linear_trend_model_distance | $d(x,y)=\|\theta_{\mathrm{trend}}(x)-\theta_{\mathrm{trend}}(y)\|_2$ | Distance between linear trend models. |
Encode first, then compare in a latent or compressed space.
| Method | Output | Metric | Complexity | Unequal length | Multivariate | Implementation | EchoTime fit | EchoTime API | Formula | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=d_{\mathcal Z}(\phi(x),\phi(y))$ | Generic embedding-space distance. |
| embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\mathcal Z}(\phi(x),\phi(y))$ | Generic embedding-space similarity. |
| paa_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the PAA encoder. |
| paa_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the PAA encoder. |
| dft_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the DFT-magnitude encoder. |
| dft_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the DFT-magnitude encoder. |
| stats_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the basic-statistics encoder. |
| stats_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the basic-statistics encoder. |
| robust_stats_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the robust-statistics encoder. |
| robust_stats_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the robust-statistics encoder. |
| acf_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the ACF encoder. |
| acf_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the ACF encoder. |
| wavelet_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the wavelet encoder. |
| wavelet_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the wavelet encoder. |
| histogram_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the histogram encoder. |
| histogram_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the histogram encoder. |
| periodogram_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the periodogram encoder. |
| periodogram_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the periodogram encoder. |
| trend_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the trend-feature encoder. |
| trend_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the trend-feature encoder. |
| random_projection_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the random projection encoder. |
| random_projection_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the random projection encoder. |
| identity_embedding_distance | distance | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $d(x,y)=\|\phi(x)-\phi(y)\|_2$ | Euclidean distance using the identity-resample encoder. |
| identity_embedding_similarity | similarity | depends on latent metric | depends on encoder | yes | yes | adapter | Low-priority addition | - | $s(x,y)=s_{\cos}(\phi(x),\phi(y))$ | Cosine similarity using the identity-resample encoder. |