Invariants of Multidimensional Time Series Based on Their IteratedIntegral Signature
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Abstract
We introduce a novel class of features for multidimensional time series that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen’s iteratedintegral signature.
Keywords
Invariants Signature Time series Curves1 Introduction
The analysis of multidimensional time series is a standard problem in data science. Usually, as a first step, features of a time series must be extracted that are (in some sense) robust and that characterize the time series. In many applications the features should additionally be invariant to a particular group acting on the data. In Human Activity Recognition for example, the orientation of the measuring device is often unknown. This leads to the requirement of rotation invariant features [37]. In EEG analysis, invariants to the general linear group are beneficial [12]. In other applications, the labeling of coordinates is arbitrary, which leads to permutation invariant features.
As any time series in discrete time can, via linear interpolation, be thought of as a multidimensional curve, one is naturally led to the search of invariants of curves. Invariant features of curves have been treated using various approaches, mostly focussing on twodimensional curves. Among the techniques are Fourier series (of closed curves) [21, 27, 52], wavelets [6], curvature based methods [2, 36] and integral invariants [13, 35].
The aim of this work is threefold. Firstly, we adapt classical results in invariant theory regarding noncommuting polynomials (or, equivalently, multilinear maps), to our situation. These results are spread out in the literature and sometimes need a little massaging. Secondly, it lays out the usefulness of the iteratedintegral signature in the search for invariants of \(d\)dimensional curves. We show, see Sect. 7, that certain “integral invariants” found in the literature are in fact found in the signature and our approach simplifies their enumeration. Lastly, we present new geometric insights into some entries found in the signature, Sect. 3.3.^{1}
The paper is structured as follows. In the next section we introduce the iteratedintegral signature of a multidimensional curve, as well as some algebraic language to work with it. Based on this signature, we present in Sect. 3 and Sect. 4 invariants to the general linear group and the special orthogonal group. Both are based on classical results in invariant theory. For completeness, we present in Sect. 5 the invariants to permutations, which have been constructed in [1]. In Sect. 6 we show how to use all these invariants if an additional (time) coordinate is introduced. In Sect. 7 we relate our work to the integral invariants of [13] and demonstrate that the invariants presented there cannot be complete. We formulate the conjecture of completeness for our invariants and point out open algebraic questions.
For readers who want to use these invariants without having to go into the technical results, we propose the following route. The required notation is presented in the next section. The invariants are presented in Proposition 3.11, Proposition 4.4 and Proposition 5.4. Examples are given in Sect. 3.1 (in particular Remark 3.14), Example 4.7 and Example 5.6. All these invariants are also implemented in the software package [9]. For calculating the iteratedintegral signature in Python we propose using the package iisignature, as described in [40].
2 The Signature of Iterated Integrals
By a multidimensional curve\(X\) we will denote a continuous mapping \(X: [0,T] \to \mathbb{R}^{d}\) of bounded variation.^{2} The aim of this work is to find features (i.e. complex or real numbers) describing such a curve that are invariant under the general linear group, the group of rotations and the group of permutations. Note that in practical situations one is usually presented with a discrete sequence of data points in \(\mathbb{R}^{d}\), a multidimensional time series. Such a time series can be easily transformed into a (piecewise) smooth curve by linear interpolation.
to the goalFind functions \(\varPsi : \text{curves} \to \mathbb{R}\) that are invariant under the action of a group \(G\) ,
By the shuffle identity (Lemma 2.1), any polynomial function on the signature can be rewritten as a linear function on the signature. Assuming that arbitrary functions are wellapproximated by polynomial functions, we are led to the final simplification, which is the goal of this paperFind functions \(\varPsi : \text{signature of curves} \to \mathbb{R}\) that are invariant under the action of a group \(G\) .
Find linear functions \(\varPsi : \text{signature of curves} \to \mathbb{R}\) that are invariant under the action of a group \(G\) .
2.1 Algebraic Underpinning
The space \(T((\mathbb{R}^{d}))\) becomes an algebra by extending the usual product of monomials, denoted ⋅, to the whole space by bilinearity. Note that ⋅ is noncommutative.
As mentioned above, every polynomial expression in terms of the signature can be rewritten as a linear expression in (different) terms of the signature. This is the content of the following lemma, which is proven in [39] (see also [41, Corollary 3.5]).
Lemma 1
(Shuffle identity)
Remark 2
The concatenation of curves is compatible with the product on \(T((\mathbb{R}^{d}))\) in the following sense (for a proof, see for example [16, Theorem 7.11]).
Lemma 3
(Chen’s relation)
We will use the following fact repeatedly, which also explains the commonly used name tensor algebra for \(T(\mathbb{R}^{d})\).
Lemma 4
For example, with \(d=2\) and \(n=3\), we can consider the multilinear map \(\psi\) which takes \(\bigl((a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\bigr) \in \mathbb{R}^{2} \times \mathbb{R}^{2} \times \mathbb{R}^{2}\) to the number \(a_{1}a_{2}b_{3}\). It maps to \(\mathsf{poly}(\psi)=x_{1}x_{1}x_{2}\).
3 General Linear Group
Definition 1
Definition 2
Lemma 3
Proof
We can simplify the concept of \(\operatorname{GL}\) invariants further, using the next lemma. Owing to the shuffle identity, signatures of curves live in a nonlinear subset of the whole tensor algebra \(T((\mathbb{R}^{d}))\), the set of “grouplike elements” (compare [41, Sect. 3.1]). It turns out though that they linearly span all of \(T((\mathbb{R}^{d}))\).
Lemma 4
Proof
These elements span \(\pi _{n} T((\mathbb{R}^{d}))\), which finishes the proof. □
Since the action respects homogeneity, we immediately obtain that projections of invariants are invariants (take \(B = (\det A)^{w} A ^{\top }\) in the following lemma):
Lemma 5
Proof
By definition, the action of \(\operatorname{GL}\) on \(T(\mathbb{R}^{d})\) commutes with \(\pi _{n}\). □
In order to apply classical results in invariant theory, we use the bijection \(\mathsf{poly}\) between multilinear functions and noncommuting polynomials, given in Lemma 2.4.
Lemma 6
Proof
To state the following classical result, we introduce the notion of Young diagrams, which play an important role in the representation theory of the symmetric group.
The following result is classical, see for example Dieudonné [10, Sect. 2.5], [50] and [18], none of which explicitly give a basis for the invariants though. See [47, Theorem 4.1.12] for a slightly different basis.
Theorem 7
Remark 8
Identities of this type are called Plücker identities. They have a long history and are a major ingredient in the representation theory of the symmetric group. The procedure of reducing certain products of determinants to a basic set of such products is called the straightening algorithm [44, Sect. 2.6]. See also [30] and [48].
Remark 9
Remark 10
Proof of Theorem 3.7
Applying Lemma 2.4 to Theorem 3.7 we get the invariants in \(T(\mathbb{R}^{d})\).
Proposition 11
Remark 12
By Lemma 3.5, for any invariant \(\phi \in T(\mathbb{R}^{d})\) and \(n\ge 1\) we have that \(\pi _{n} \phi \) is also invariant. Hence the previous theorem characterizes all invariants we are interested in (Definition 3.1), not just homogeneous ones.
Remark 13
Note that each of these invariants \(\phi \) consists only of monomials that contain every variable \(x_{1}, \dots , x_{d}\) at least once. This implies that \(\langle S(X)_{0,T}, \phi \rangle \) consists only of iterated integrals that contain every component \(X^{1},\dots ,X^{d}\) of the curve at least once. Hence, if at least one of these components is constant, the whole expression will be zero.
Since \(\phi \) is invariant, this implies that \(\langle S(X)_{0,T}, \phi \rangle = 0\) as soon as there is some coordinate transformation under which one component is constant, that is whenever the curve \(X\) stays in a hyperplane of dimension strictly less then \(d\).
3.1 Examples
We present the invariants described in Sect. 2 for some special cases of \(d\) and \(w\).
The case \(d=2\)
Remark 14
Let us make clear that from the perspective of data analysis, the “invariant” of interest is really the action of this element in \(T(\mathbb{R}^{d})\) on the signature of a curve.
Remark 15
A similar analysis can also be carried out for the following invariants, but we refrain from doing so, since it can be easily done with a computer algebra system.
3.2 The Invariant of Weight One, in Dimension Two
Geometric Interpretation
Connection to Correlation
Remark 16
3.3 The Invariant of Weight One, in Any Dimension
This invariant is of homogeneity \(d\). The following lemma tells us that we can write \(\operatorname{Inv}_{d}\) in terms of expressions on lower homogeneities.
Lemma 17
Remark 18
Proof
The first statement follows from expressing the determinant in (4) in terms of minors with respect to the row \(r+1\) (since the \(x_{i}\) are noncommuting, this does not work with columns!).
An immediate consequence is the following lemma.
Lemma 19
Proof
In even dimension we have the phenomenon that closing a curve does not change the value of the invariant.
Lemma 20
Proof
Let \(\bar{X}\) be parametrized on \([0,2T]\) as follows: \(\bar{X} = X\) on \([0,T]\) and it is the linear path connecting \(X_{T}\) to \(X_{0}\) on \([T,2T]\). By translation invariance we can assume \(X_{0} = 0\) and by \(\operatorname{GL}(\mathbb{R}^{d})\)invariance that \(X_{T}\) lies on the \(x_{1}\) axis. Then the only component of \(\bar{X}\) that is nonconstant on \([T,2T]\) is the first one, \(\bar{X}^{1}\).
Lemma 21
Proof
Lemma 22
We now proceed to piecewise linear curves with more than \(d\) vertices.
Lemma 23
Remark 24
Remark 25
Example 26
Proof of Lemma 3.23
The case \(d=2\)
Now assume the statement is true for all dimensions strictly smaller than some \(d\). We show it is true for \(d\). \(d\)is odd
\(d\) is even
We proceed by induction on \(n\). For \(n=d\) the statement follows from Lemma 3.21.
Definition 27
Theorem 28
Proof
Lemma 29
Remark 30
Proof
Our points \(p_{i}\) lie on the moment curve. Then, by (6), any collection of points \(p_{i_{0}}, p _{i_{1}},.., p_{i_{d}}\) is in general position. This means that every facet of \(P\) must have exactly \(d\) points (and not more). Facets of \(\operatorname{ConvexHull}(P)\) with \(d\) points are characterized by Gale’s criterion ([17, Theorem 3], [53, Theorem 0.7]):
the points \(p_{i_{1}},.., p_{i_{d}}\), with distinct \(i_{j} \in \{0,..,n \}\) form a facet of \(P\) if and only if any two elements of \(\{0,..,n \} \setminus \{i_{1},.., i_{d}\}\) are separated by an even number of elements in \(\{i_{1},.., i_{d}\}\).^{11}
\(d\) odd

\(i_{\ell +1} = i_{\ell }+ 1\) for \(\ell \) odd

\(i_{d}= n\).
\(d\) even

\(i_{\ell +1} = i_{\ell }+ 1\) for \(\ell \) odd.
The statement of the lemma now follows by piecewise linear approximation of \(X\) using continuity of the convex hull, which follows from [11, Lemma 3.2], and of iterated integrals [16, Proposition 1.28, Proposition 2.7]. □
4 Rotations
Definition 1
Since \(\det (A) = 1\), any \(\operatorname{GL}\) invariant of weight \(w \ge 1\) (Sect. 3) is automatically an \(\operatorname{SO}\) invariant. But there are \(\operatorname{SO}\) invariants that are not \(\operatorname{GL}\) invariants (of any weight), for example, for \(d=2\), \(\phi := x_{1} x_{1} + x_{2} x_{2}\).
Switching to the perspective of multilinear maps, this is the map \((v_{1},v_{2}) \mapsto \langle v_{1}, v_{2} \rangle \). It is shown, see for example [50, Theorem 2.9.A], that all invariants are built from the inner product and the determinant.
Theorem 2
([28, Theorem 12.5.0.8])

\(c^{(j)} \in I(d,n)\) for each \(j=1,..,s\)

\(a^{(j)},b^{(j)} \in I(t_{j},n)\) for some \(1 \le t_{j} \le d1\) for each \(j=1,..,r\)

\(a^{(1)} \ge b^{(1)} \ge a^{(2)} \ge .. \ge b^{(r)} \ge c^{(1)} \ge .. \ge c^{(s)}\)

every number\(1,..,n\)appears in exactly one of the sequences\(a^{(1)},.., a^{(r)}, b^{(1)},.., b^{(r)}, c^{(1)},.., c^{(s)}\); (in particular\(n = 2 \cdot C_{1} + d\cdot C_{2}\)for some\(C_{1}\), \(C_{2}\)nonnegative integers)
Example 3
We give examples of these sequences for \(d=2\).
\(n=1\): There is no such set of sequences, since nonnegative integers \(C_{1}\), \(C_{2}\) with \(2\cdot C_{1} + 2 \cdot C_{2} = 1\) cannot be found.

\(c^{(1)} = (1,2)\); meaning that \(F(v_{1},v_{2}) = \langle v_{1}, v _{2} \rangle \)

\(a^{(1)} = (2)\), \(b^{(1)} = (1)\); meaning that \(F(v_{1},v_{2}) = \det [ v_{1} v_{2} ]\)
\(n=3\): There is no such set of sequences.

\(a^{(1)} = (4)\), \(b^{(1)} = (3)\), \(a^{(2)} = (2)\), \(b^{(2)} = (1)\); meaning that \(F(v_{1},v_{2},v_{3},v_{4}) = \langle v_{4}, v_{3} \rangle \langle v_{2}, v_{1} \rangle \).

\(a^{(1)} = (4)\), \(b^{(1)} = (3)\), \(c^{(1)} = (1,2)\); meaning that \(F(v_{1},v_{2},v_{3},v_{4}) = \langle v_{4}, v_{3} \rangle \det [ v _{1} v_{2} ]\).

\(a^{(1)} = (4)\), \(b^{(1)} = (2)\), \(c^{(1)} = (1,3)\)

\(a^{(1)} = (3)\), \(b^{(1)} = (2)\), \(c^{(1)} = (1,4)\)

\(c^{(1)} = (3,4)\), \(c^{(2)} = (1,2)\)

\(c^{(1)} = (2,4)\), \(c^{(2)} = (1,3)\)
In the setting of \(T(\mathbb{R}^{d})\) we have
Proposition 4
In the case \(d=2\), there is another way to arrive at a basis for the invariants. Taking inspiration from [15], which concerns rotation invariants of images, we work in the complex vector space \(T(\mathbb{C}^{2})\).^{12}
Theorem 5
Remark 6
In particular for \(d=2\) and \(n\) even, the dimension of rotation invariants on level \(n\) in \(T(\mathbb{R}^{2})\) is equal to \(\binom{n}{n/2}\).
Proof
1. The elements\(z\)are invariant
2. The elements\(z\)form a basis
Now \(x_{j_{1}} .. x_{j_{n}}: j_{\ell }\in \{1,2\}\) is a basis of \(\pi _{n} T(\mathbb{C}^{2})\) with respect to ℂ. Hence \(z_{j_{1}} .. z_{j_{n}}\) is (the map \((x_{1},x_{2}) \mapsto (z_{1},z _{2})\) is invertible). By Step 1 we have hence exhibited a basis (with respect to ℂ) for all invariants in \(\pi _{n} T(\mathbb{C} ^{2})\).
3. Real invariants
Example 7
5 Permutations
Denote by \(S_{d}\) the group of permutations of \([d] := \{1,.., d\}\).
Lemma 1
Then\(M: S_{d}\to \operatorname{GL}(\mathbb{R}^{d})\)is a group homomorphism and moreover\(M(\sigma ^{1}) = M(\sigma )^{\top }\).^{13}
Proof
Definition 2
Example 3
Proposition 4
([1, Sect. 3])
Remark 5
We are not aware of a general explicit formula for the number of partitions (i.e. the coefficients of the generating function).
Proof of Proposition 5.4
By (7), each \(M_{A}\) is permutation invariant. Moreover, since \(A \le d\), \(M_{A}\) is nonzero.
For \(A\), \(A'\) distinct set partitions of \([n]\), the monomials in \(M_{A}\) and the monomials in \(M_{A'}\) do not overlap. Hence the proposed basis is linearly independent.
Now, if \(\phi \) is permutation invariant and if for some \(i\), \(i'\), \(\nabla ( x_{i_{1}} .. x_{i_{n}} ) = \nabla ( x_{i'_{1}} .. x_{i'_{n}} )\) then the coefficient of \(x_{i}\) and \(x_{i'}\) must coincide. Hence the proposed basis spans invariants of homogeneity \(n\). □
Example 6
Consider \(d=3\)
6 An Additional (Time) Coordinate
Assume now that \(X = (X^{0},X^{1},..,X^{d}): [0,T] \to \mathbb{R}^{1+d}\). Here \(X^{0}\) plays a special role, in that we assume that it is not affected by the space transformations under consideration.
Adding an “artificial” 0th component, usually keeping track of time, \(X^{0}_{t} := t\), is a common trick to improve the expressiveness of the signature. In particular, if such an \(X^{0}\) is monotonically increasing, the enlarged curve \((X^{0},X^{1},..,X^{d})\) never has any “treelike” components (compare Sect. 7), no matter what the original \((X^{1},..,X^{d})\) was.
Consider \(\operatorname{GL}\) invariants for the moment.
Definition 1
Theorem 2
Proof
The corresponding statements for rotations and permutations are completely analogous, so we omit them.
7 Discussion and Open Problems
We have presented a novel way to extract invariant features of \(d\)dimensional curves, based on the iteratedintegral signature. We have identified all those features that can be written as a finite linear combination of terms in the signature.
We note that expressions of the form (8) are not enough to uniquely characterize a path. Indeed, the following lemma gives a counterexample to the conjecture on p. 906 in [13] that “signatures of nonequivalent curves are different” (here, the “signature” of a curve means the set of expressions of the form (8)).
Lemma 1
Then all the expressions (8) coincide on\(X^{+}\)and\(X^{}\).^{14}
Proof
Moreover, the algorithmic nature of the construction in [13] makes it difficult to proceed to invariants of higher order. In contrast, our method gives an explicit linear basis for the invariants under consideration up to any order.
Conjecture 2
In Proposition 3.11, Proposition 4.4 and Proposition 5.4 we have established a linear basis for invariants for every homogeneity. As already mentioned in Remark 3.15, owing to the shuffle identity, there are algebraic relations between elements of different homogeneity. An interesting open problem is then to find a minimal set of generators for the set of invariants, considered as a subalgebra of the shuffle algebra. This applies to all subgroups of \(\operatorname{GL}( \mathbb{R}^{d})\) and their corresponding invariants.
Lastly, a word on (computational) complexity. We have seen in Remark 3.10 the dimensions of \(\operatorname{GL}\) invariant elements (which is a lower bound on the dimensions of \(\operatorname{SO}\) invariant elements).^{15} In Remark 5.5 we have seen the dimensions for the permutation invariant elements.
Computing the signature itself up to level \(n\) has complexity \(\varOmega (d^{n})\), since \(d+ .. + d^{n}\) integrals need to be calculated. So any method that calculates the invariant features of a curve \(X\) by first calculating its signature and extracting them (see Remark 3.14) will have computational complexity dominated by the calculation of the signature. Furthermore, the calculation of the invariant elements is a computation that can be done offline (they do not depend on the curve \(X\)).

Is it possible to apply kernelization techniques similar to the ones used for the entire (noninvariant) signature in [25]? These techniques, in the noninvariant setting, allow to use information of the signature up to high levels and dimension for certain learning algorithms.

We have studied in this paper linear expressions on the signature that are invariant to a group action. This was justified by using the shuffle identity (Lemma 2.1), which tells us that any polynomial functional on the signature can in fact be linearized. One can also consider a fixed level \(n\) of the signature and look for all nonlinear expressions that are invariant under the group action. This is the classical problem of invariant theory for polynomial rings [47, Sect. 4]. On the one hand, this makes it possible to “peek ahead” in the signature, since one gets invariant information that would only be seen in linear expressions of higher levels than \(n\). On the other hand, except for special cases, there are no explicit expressions for these nonlinear invariant. One has to proceed algorithmically (for example via Derksen’s algorithm, [8]) which only works for low dimension \(d\) and low levels \(n\). Since the calculation of those nonlinear invariant elements can also be done offline it would nonetheless be nice to have a tabulation of nonlinear invariants (as far as existing algorithms can reach).

For \(\operatorname{GL}\) invariants, in Remark 3.25 we conjecture the existence of a “\(\operatorname{GL}\) invariant” signature. This could improve computation time, since no noninvariant integrals have to be computed.
Footnotes
 1.
The signature is notorious for being hard to interpret in geometric terms.
 2.
The reader might prefer to just think of a (piecewise) smooth curve.
 3.
Since \(X\) is of bounded variation the integrals are welldefined using classical RiemannStieltjes integration (see for example Chap. 6 in [43]). This generalizes the notation in the introduction above beyond smooth curves. It can be pushed much further though. In fact the following considerations are purely algebraic and hence hold for any curve for which a sensible integration theory (in particular: obeying integration by parts) exists. A relevant example is Brownian motion which, although being almost surely nowhere differentiable, nonetheless admits a stochastic (Stratonovich) integral.
 4.
Also called the “rough path signature”.
 5.
In contrast to a power series, a polynomial only has finitely many terms.
 6.One can also think of a tabloid as the following element of the vector space spanned by Young tableaux,Here the sum is over all permutations \(\pi \) that leave the elements of each row of \(t\) unchanged.$$\begin{aligned} \{ t \} = \sum_{\pi } \pi t. \end{aligned}$$
 7.
The prefactor \(1/2\) is irrelevant, so we will speak of \(\phi \) and also of \(\tfrac{1}{2} \phi \) as picking out the area.
 8.
The standard example is a time series that is discretely observed at times \(t_{i}\) and linearly interpolated in between.
 9.
Note the nomenclature used in signal analysis. A probabilist or statistician would tend to call this a “covariance” and not a “correlation”.
 10.
 11.
For example, with \(n=4\) and dimension \(d=3\), the indices \(\{0,1,2\}\), \(\{0,2,3\}\), \(\{0,3,4 \}\), \(\{0,1,4\}\), \(\{1,2,4\}\), \(\{2,3,4\}\) lead to the facets, which in this dimension are triangles.
 12.
One may think of this as the space of noncommuting polynomials in \(x_{1}\), \(x_{2}\) with complex coefficients, or, equivalently, as the complexification [42, Chap. 1] of the vector space \(T(\mathbb{R}^{2})\). An element \(A \in \operatorname{GL}(\mathbb{R}^{2})\) then acts on \(T(\mathbb{C} ^{2})\) by the prescription in Definition 3.2. More abstractly: this is the diagonal action of the complexification of \(A\) on \(T(\mathbb{C}^{2})\).
 13.
\(M\) is sometimes called the defining representation of \(S_{d}\).
 14.
Note that \(X^{+}\) and \(X^{}\) are not treelike equivalent and therefore have different (iteratedintegral) signatures. The lowest level on which they differ is level 4.
 15.
In this section we take “invariant element” to mean the elements of the space \(T(\mathbb{R}^{d})\), incarnations of which can be seen in Sect. 3.1. They do not depend on any curve \(X\) one might be interested in.
Notes
Acknowledgements
The authors thank Bernd Sturmfels for discussion on the topics of this paper. Open access funding provided by Max Planck Society.
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