--- title: "CanonicalWarpingCorrespondence" language: "en" type: "Symbol" summary: "CanonicalWarpingCorrespondence[s1, s2] gives the canonical time warping (CTW) correspondence between sequences s1 and s2. CanonicalWarpingCorrespondence[s1, s2, warp] uses warp as initial warping correspondence. CanonicalWarpingCorrespondence[s1, s2, warp, win] uses a window win for local search." keywords: - CTW - canonical time warping - DTW - canonical correlation analysis - CCA - local canonical time warping - LCTW canonical_url: "https://reference.wolfram.com/language/ref/CanonicalWarpingCorrespondence.html" source: "Wolfram Language Documentation" related_guides: - title: "Sequence Alignment & Comparison" link: "https://reference.wolfram.com/language/guide/SequenceAlignmentAndComparison.en.md" related_functions: - title: "CanonicalWarpingDistance" link: "https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.en.md" - title: "WarpingDistance" link: "https://reference.wolfram.com/language/ref/WarpingDistance.en.md" - title: "WarpingCorrespondence" link: "https://reference.wolfram.com/language/ref/WarpingCorrespondence.en.md" - title: "SmithWatermanSimilarity" link: "https://reference.wolfram.com/language/ref/SmithWatermanSimilarity.en.md" - title: "NeedlemanWunschSimilarity" link: "https://reference.wolfram.com/language/ref/NeedlemanWunschSimilarity.en.md" --- # CanonicalWarpingCorrespondence CanonicalWarpingCorrespondence[s1, s2] gives the canonical time warping (CTW) correspondence between sequences s1 and s2. CanonicalWarpingCorrespondence[s1, s2, warp] uses warp as initial warping correspondence. CanonicalWarpingCorrespondence[s1, s2, warp, win] uses a window win for local search. ## Details and Options * Canonical time warping (CTW) iteratively performs spatial transformations and dynamic time warping on the reference sequence ``s1`` and the query sequence ``s2`` to find the alignment with minimal distance between sequences. * The sequences ``si`` can be lists of numeric scalars or vectors. In contrast to dynamic time warping, elements of ``s1`` and ``s2`` can be of different dimensions. * ``CanonicalWarpingCorrespondence`` returns ``{{n1, …, nk}, {m1, …, mk}}`` of non-decreasing positions such that ``s1[[ni]]`` correspond to ``s2[[mi]]``. * Corresponding positions attempt to minimize the distance $\sum _{i=1}^k \left(\alpha _i\cdot s_1\left[\left[n_i\right]\right]-\beta _i\cdot s_2\left[\left[m_i\right]\right]\right){}^2$ over all possible such positions and with the constraint that all elements of ``s1`` and ``s2`` are represented as some ``s1[[ni]]`` and ``s2[[mj]]``, respectively. * Spatial transformation matrices ``α`` and ``β`` are computed in each iteration using canonical correlation analysis. * Compute the effective distance using ``CanonicalWarpingDistance``. * Possible settings for the search window ``win`` are: | | | | | --- | --- | --- | | [image] | Automatic | a full search | | [image] | r | a slanted band window of radius $r$ | | [image] | {"SlantedBand", r} | a slanted band window of radius $r$ | | [image] | {"Band", r} | band window of radius $r$ (Sakoe–Chiba) | | [image] | {"Parallelogram", a} | parallelogram window placed at origin with slopes $a$ and $1/a$ (Itakura) | * The following options are supported: | | | | | ---------------- | --------- | ---------------------------------------------- | | DistanceFunction | Automatic | distance function used in dynamic time warping | | MaxIterations | Automatic | maximal number of iterations | | Method | Automatic | additional parameters | * The following options are available through ``Method -> opts`` : | | | | | ------------------- | --------- | ------------------------------------------------------------------------- | | "DimensionsToKeep" | Automatic | dimensions after projection | | "EnergyThreshold" | Automatic | fraction of "energy" to be kept | | "Lambdas" | Automatic | regularization values | | "MatchingIntervals" | Automatic | whether to match the query to the full reference or part of the reference | * Possible settings for the ``"MatchingIntervals"`` option include: | | | | ------------- | ----------------------------------------- | | Automatic | a full match | | "Flexible" | flexible at both ends | | "FlexibleEnd" | flexible only at the end of the interval | ## Examples (13) ### Basic Examples (2) Find the canonical time warping correspondence between two scalar sequences: ```wl In[1]:= CanonicalWarpingCorrespondence[{1, 2, 3, 1}, {8, 8, 7, 7}] Out[1]= {{1, 1, 2, 3, 4}, {1, 2, 3, 3, 4}} ``` --- Find the canonical time warping correspondence between a 3D path and a 2D line segment: ```wl In[1]:= CanonicalWarpingCorrespondence[{{0, 0, 0}, {1, 1, 2}, {3, 5, 5}, {1, 3, 7}}, {{0, 0}, {4, 4}}] Out[1]= {{1, 2, 3, 4}, {1, 1, 2, 2}} ``` ### Scope (9) #### Data (6) Find the correspondence between two sequences of numerical scalars with different length: ```wl In[1]:= CanonicalWarpingCorrespondence[{1, 1, 1, 2, 2, 3, 4}, {11, 13, 17, 19}] Out[1]= {{1, 2, 3, 4, 5, 6, 7}, {1, 1, 1, 2, 2, 3, 4}} ``` --- Find the correspondence between two sequences of vectors: ```wl In[1]:= CanonicalWarpingCorrespondence[{{0, 0}, {-1, 0}, {-1, 1}}, {{-1, 2}, {1, -2}, {0, 7}}] Out[1]= {{1, 2, 3}, {1, 2, 3}} ``` --- Find the correspondence between a sequence of scalars and a sequence of 3D points: ```wl In[1]:= CanonicalWarpingCorrespondence[{1, 1, 1, 2, 2, 3, 4}, {{-1, -2, -3}, {1, 2, 3}, {0, 0, 0}}]//TextGrid Out[1]= | | | | | | | | | - | - | - | - | - | - | - | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | 1 | 1 | 1 | 2 | 3 | 3 | 3 | ``` --- Find the correspondence between two sequences of quantities with compatible units: ```wl In[1]:= s1 = QuantityArray[{1, 2.5, 4}, "Decimeters"]; s2 = {Quantity[0.1, "Kilometers"], Quantity[200, "Inches"]}; In[2]:= CanonicalWarpingCorrespondence[s1, s2] Out[2]= {{1, 2, 3}, {1, 1, 2}} ``` All units are converted to base SI units: ```wl In[3]:= CanonicalWarpingCorrespondence[s1, s2] == CanonicalWarpingCorrespondence[UnitConvert[s1], UnitConvert[s2]] Out[3]= True ``` --- Find the correspondence between two sequences of quantity vectors with compatible units: ```wl In[1]:= s1 = QuantityArray[{{1, 2.5}, {4, 5}, {5, 4}}, {"Seconds", "Minutes"}] Out[1]= QuantityArray[StructuredArray`StructuredData[{3, 2}, {{{1, 2.5}, {4, 5}, {5, 4}}, {"Seconds", "Minutes"}, {{1}, {2}}}]] In[2]:= s2 = Table[Quantity[(i + j) / 60, "Hours"], {i, 5}, {j, 2}] Out[2]= {{Quantity[1/30, "Hours"], Quantity[1/20, "Hours"]}, {Quantity[1/20, "Hours"], Quantity[1/15, "Hours"]}, {Quantity[1/15, "Hours"], Quantity[1/12, "Hours"]}, {Quantity[1/12, "Hours"], Quantity[1/10, "Hours"]}, {Quantity[1/10, "Hours"], Quantity[7/60, "Hours"]}} In[3]:= CanonicalWarpingCorrespondence[s1, s2]//TextGrid Out[3]= | | | | | | | - | - | - | - | - | | 1 | 2 | 2 | 2 | 3 | | 1 | 2 | 3 | 4 | 5 | ``` --- Find the correspondence between a sequence of quantities and a sequence of scalars: ```wl In[1]:= squant = QuantityArray[{1, 2.5, 4, 5}, "Kilometers"]; sscal = {500, 400, 300}; In[2]:= CanonicalWarpingCorrespondence[squant, sscal]//TextGrid Out[2]= | | | | | | - | - | - | - | | 1 | 2 | 3 | 4 | | 1 | 2 | 2 | 3 | ``` The scalars are interpreted as if they had a compatible unit from SI base units: ```wl In[3]:= CanonicalWarpingCorrespondence[squant, QuantityArray[sscal, "meters"]]//TextGrid Out[3]= | | | | | | - | - | - | - | | 1 | 2 | 3 | 4 | | 1 | 2 | 2 | 3 | ``` #### Initial Warping (1) By default, the shorter sequence is first warped uniformly to match the length of the longer sequence: ```wl In[1]:= s1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}; s2 = {1, 1, 1, 1, 1, 2, 3, 7, 8, 9, 10}; In[2]:= CanonicalWarpingCorrespondence[s1, s2] Out[2]= {{1, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11}} ``` Construct the initial uniform warping manually: ```wl In[3]:= CanonicalWarpingCorrespondence[s1, s2, {Range[Length[s1]], ArrayResample[Range[Length[s2]], Length[s1], Resampling -> "NearestLeft"]}] Out[3]= {{1, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11}} ``` #### Search Window (2) By default, an unconstrained search is performed: ```wl In[1]:= s1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}; s2 = {1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; In[2]:= CanonicalWarpingCorrespondence[s1, s2] Out[2]= {{1, 2, 2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 11, 12, 13, 14}} ``` --- Use a ``"SlantedBand"`` search window: ```wl In[1]:= s1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}; s2 = {1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; In[2]:= CanonicalWarpingCorrespondence[s1, s2, Automatic, {"SlantedBand", 1}] Out[2]= {{1, 2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14}} ``` ### Options (1) #### MaxIterations (1) By default, the maximal number of iterations is 50: ```wl In[1]:= s1 = Table[{7t * Cos[1.8t], 3t * Sin[1.8t]}, {t, Range[0, 4Pi, 4Pi / 650]}]; s2 = Table[{5t * Cos[t], -3t * Sin[t]}, {t, Range[0, 6Pi, 6Pi / 500]}]; Graphics[{{Red, Point[s1]}, {Blue, Point[s2]}}] Out[1]= [image] In[2]:= AbsoluteTiming[CanonicalWarpingCorrespondence[s1, s2];] Out[2]= {1.67922, Null} ``` Decreasing the maximal number of iterations may reduce the timing: ```wl In[3]:= AbsoluteTiming[CanonicalWarpingCorrespondence[s1, s2, MaxIterations -> 20];] Out[3]= {0.675765, Null} ``` ### Properties & Relations (1) Canonical time warping is robust to spatial transformations. Translation invariance: ```wl In[1]:= s1 = Table[{Cos[0.6t] * Cos[t], Cos[0.6t] * Sin[t]}, {t, 0, 6Pi, 2Pi / 100}]; s2 = TranslationTransform[{2, 2}] /@ s1; In[2]:= ListLinePlot[{s1, s2}] Out[2]= [image] In[3]:= CanonicalWarpingCorrespondence[s1, s2]//ListLinePlot Out[3]= [image] ``` Scale invariance: ```wl In[4]:= s1 = Table[{2Cos[t], Sin[t]}, {t, 0, 2Pi, 2Pi / 20}]; s2 = 3s1; ListLinePlot[{s1, s2}] Out[4]= [image] In[5]:= CanonicalWarpingCorrespondence[s1, s2]//ListLinePlot Out[5]= [image] ``` Rotation invariance: ```wl In[6]:= s1 = Table[{Cos[3t] * Cos[t], Cos[3t] * Sin[t]}, {t, 0, 2Pi, 2Pi / 100}]; s2 = RotationTransform[Pi] /@ s1; In[7]:= ListLinePlot[{s1, s2}] Out[7]= [image] In[8]:= CanonicalWarpingCorrespondence[s1, s2]//ListLinePlot Out[8]= [image] ``` ## See Also * [`CanonicalWarpingDistance`](https://reference.wolfram.com/language/ref/CanonicalWarpingDistance.en.md) * [`WarpingDistance`](https://reference.wolfram.com/language/ref/WarpingDistance.en.md) * [`WarpingCorrespondence`](https://reference.wolfram.com/language/ref/WarpingCorrespondence.en.md) * [`SmithWatermanSimilarity`](https://reference.wolfram.com/language/ref/SmithWatermanSimilarity.en.md) * [`NeedlemanWunschSimilarity`](https://reference.wolfram.com/language/ref/NeedlemanWunschSimilarity.en.md) ## Related Guides * [Sequence Alignment & Comparison](https://reference.wolfram.com/language/guide/SequenceAlignmentAndComparison.en.md) ## History * [Introduced in 2016 (11.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn110.en.md)