First a little terminology:

  • A vector has dimensions. E.g. the vector $$(1.3, 94.2, 821.84)$$ is three dimensional. (Thus multidimensional).
  • A vector also has a grade, which is (by definition) 1.
  • A matrix is a multi-grade vector, it has a grade of 2.
  • A tensor has also grades, which might be larger than 2.
  • A time series is a sequence of scalars where each scalar value has an index. In a way it is also a vector, but I in the following I say that a usual time series (with scalar as it's values) is 1 dimensional.
  • A time series is multi-dimensional if each index has multiple values - thus with each index there is a vector associated with it. So the dimensionality of the time series is determined by the dimensionality of the variable associated with each index. E.g. the vector may contain values for temperature and pressure.
  • A series is the general concept of a time series where each index is not necessarily a time stamp, but just any index.
  • The grade of a series is the number of indexes the series has. For a time series, usually, multiple indexes make little sense, because it would mean there are totally independent time stamps. This might only make sense in some weird time travelling/multiverse szenario I guess. For space however this does make sense.
  • An example for a multi-grade would be an image and it's pixels as scalar values.
  • A series can be uni-dimensional, uni-grade (the standard case), multi-dimensional, uni-grade (temperature and pressure example), it can be uni-dimensional, multi-grade (e.g. a gray-scale image), or it could be multi-dimensional, multi-grade (e.g. an image with multiple colour channels).

I am aware that there are DTW algorithms for the multi-dimensional, uni-grade case of DTW. There are two ways of doing this: Independent case is to calculate a DTW for each dimension and then add the DTW-distances. Dependent case is to have one distance matrix and make the multi-dimensional comparission simultationsly. http://www.cs.ucr.edu/~eamonn/Multi-Dimensional_DTW_Journal.pdf describes this. From my perspective, the dependent case is a "real" mutli-dimensional DTW. This has been discussed in Multidimensional dynamic time warping

Finally my question: How is it possible to build a DTW not for multi-dimensional data, but for multi-grade data?

Make no mistake: What I am not looking for a way of transforming the multi-grade data into multi-dimensional data.

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