Kimesurface representation and tensor linear modeling of longitudinal data

Abstract

Many modern techniques for analyzing time-varying longitudinal data rely on parametric models to interrogate the time-courses of univariate or multivariate processes. Typical analytic objectives include utilizing retrospective observations to model current trends, predict prospective trajectories, derive categorical traits, or characterize various relations. Among the many mathematical, statistical, and computational strategies for analyzing longitudinal data, tensor-based linear modeling offers a unique algebraic approach that encodes different characterizations of the observed measurements in terms of state indices. This paper introduces a new method of representing, modeling, and analyzing repeated-measurement longitudinal data using a generalization of event order from the positive reals to the complex plane. Using complex time (kime), we transform classical time-varying signals as 2D manifolds called kimesurfaces. This kime characterization extends the classical protocols for analyzing time-series data and offers unique opportunities to design novel inference, prediction, classification, and regression techniques based on the corresponding kimesurface manifolds. We define complex time and illustrate alternative time-series to kimesurface transformations. Using the Laplace transform and its inverse, we demonstrate the bijective mapping between time-series and kimesurfaces. A proposed general tensor regression based linear model is validated using functional Magnetic Resonance Imaging data. This kimesurface representation method can be used with a wide range of machine learning algorithms, artificial intelligence tools, analytical approaches, and inferential techniques to interrogate multivariate, complex-domain, and complex-range longitudinal processes.

Appendix

1.1 Appendix A: Introduction to tensors

A tensor is an important mathematical object defined as a multilinear map from a vector space (module) over a fixed field (or ring) into the field (or ring). In the special case where the field is ${\mathbb{R}}$ or ${\mathbb{C}}$ and the vector space is finite dimensional, these multilinear maps coincide with the common computational and data science tensor definition as multidimensional arrays. The space of multidimensional arrays is then naturally isomorphic to the space of multilinear maps. In physical sciences, a tensor maps a set of $K$ vector-space elements and $M$ dual-space elements into the base field, ${\mathbb{F}}$.

$$\begin{gathered} \underbrace {{\text{Tensor\,as\,multilinear\,map}}}_{{\text{abstract\,tensor\,product}}}:\underbrace {{V_{1} \times \cdots \times V_{K} }}_{{K\,{\text{elements}}}} \to {\mathbb{F}} ,\underbrace {{\text{Tensor\,as\,multiway\,array}}}_{{\text{data\,science}}}: {\mathbb{F}}^{{I_{1} \times I_{2} \times \cdots \times I_{K} }} \hfill \\ \underbrace {{{\text{Tensor}}}}_{{{\text{physics}}}}:\underbrace {{V_{1}^{*} \times \cdots \times V_{K}^{*} }}_{{M\,{\text{ elements}}\,{\text{in\,dual}}\,{\text{space}}}} \times \underbrace {{V_{1} \times \cdots \times V_{K} }}_{{K\,{\text{elements}}\,{\text{in}}\,{\text{vector}}\,{\text{space}}}} \to {\mathbb{F}}, \hfill \\ \end{gathered}$$

where ${\mathbb{F}}$ can be either ${\mathbb{R}}$ or ${\mathbb{C}}$, $V_{i}$ is a vector space over ${\mathbb{F}}$, and $V_{i}^{*}$ is its dual, $\forall 1 \le i \le K$. Throughout this paper, we refer to the data science definition of tensors as multiway arrays representing computable data objects tracking observed collections of multivariate features organized into $K$-way arrays. Common TLM notation uses lowercase bold symbols (${\varvec{a}}$) for vectors, bold uppercase (${\varvec{A}}$) for matrices, and uppercase blackboard symbols (${\mathbb{B}}$) as tensors of any dimension. Note that the base fields are also denoted by blackboard symbols (${\mathbb{F}}$). A $K{\text{th}}$ order tensor is a multidimensional array ${\mathbb{B}}$:$I_{1} \times \cdots \times I_{K}$ with $I_{k}$ representing the dimension of the $k{\text{th}}$ mode of the tensor, $\forall 1 \le k \le K$. The tensor elements are specified as ${\mathbb{B}}\left[ {i_{1} , \cdots , i_{K} } \right]$, where $1 \le i_{k} \le I_{K} , \forall 1 \le k \le K$. Computable data tensor objects can be represented as outer products, $^\circ$, of a set of vectors $\left\{ {{\varvec{b}}_{k} } \right\}_{k = 1}^{K}$:

$${\mathbb{B}} = {\varvec{b}}_{1}^\circ {\varvec{b}}_{2}^\circ \cdots^\circ {\varvec{b}}_{K} ,$$

with its elements indexed as:

$${\mathbb{B}}\left[ {i_{1} , \ldots , i_{K} } \right] = \mathop \prod \limits_{k = 1}^{K} {\varvec{b}}_{k} \left[ {i_{k} } \right] .$$

The outer product operation $^\circ$ takes a pair of a $K_{1}^{th}$-order and a $K_{2}^{th}$-order tensors as inputs, and returns a $\left( {K_{1} + K_{2} } \right){\text{th}}$-order tensor. In general, transformation tensors can be considered as multilinear mappings between vector spaces. If ${\mathbb{F}}$ denotes the real (${\mathbb{R}}$) or complex (${\mathbb{C}}$) field including the tensor elements, then a data tensor is an object that can be represented as

$${\mathbb{F}}^{{I_{1} \times I_{2} \times \cdots \times I_{K} }} \cong \underbrace {{{\mathbb{F}}^{{I_{1} }} \otimes {\mathbb{F}}^{{I_{2} }} \otimes \cdots \otimes {\mathbb{F}}^{{I_{K} }} }}_{{{\text{vector\,space\,tensor\,product}}}} \cong \underbrace {{\mathcal{L}}}_{{\text{multilinear\,map}}}\left( { {\mathbb{F}}^{{I_{1} }} \times {\mathbb{F}}^{{I_{2} }} \times \cdots \times {\mathbb{F}}^{{I_{K} }} ,{\mathbb{F}}} \right) .$$

The order of a given tensor, $K$, represents the number of tensor modes. For each tensor, there exists an integer $R$ ($\min R$ is the tensor rank) that allows an expansion of the tensor as a linear combination of $R$ rank-1 tensors:

$${\text{vec}}\left( {\mathbb{B}} \right) = \mathop \sum \limits_{r = 1}^{R} \lambda_{r} \left( {\overbrace {{{\varvec{b}}_{K,r} \otimes {\varvec{b}}_{K - 1,r} \cdots \otimes {\varvec{b}}_{k,r} \otimes \cdots \otimes {\varvec{b}}_{K,r} }}^{{\text{Kronecker\,tensor\,product}}}} \right) ,$$

$${\mathbb{B}} = \mathop \sum \limits_{r = 1}^{R} \lambda_{r} {\varvec{b}}_{1,r}^\circ {\varvec{b}}_{2,r}^\circ \cdots^\circ {\varvec{b}}_{k,r}^\circ \cdots^\circ {\varvec{b}}_{K,r} ,$$

where $\lambda_{r} \in F$, ${\varvec{b}}_{k,r} \in {\mathbb{F}}^{{I_{k} }}$, and $1 \le k \le K$. The minimal $\min R$ that permits such representation is called the rank of the tensor and the associated tensor decomposition is called minimal CANDECOMP/PARAFAC (CP) or Polyadic tensor decomposition [7, 56].

The tensor vectorization operation ${\text{vec}}\left( {\mathbb{B}} \right)$ restructures (or vectorizes) the tensor as a 1D vector containing all tensor array entries. The length of the corresponding ${\text{vec}}\left( {\mathbb{B}} \right)$ is $\mathop \prod \nolimits_{k = 1}^{K} I_{k}$ and the indexing transformation is defined by:

$${\text{vec}}\left( {\mathbb{B}} \right)\left[ {i_{1} + \mathop \sum \limits_{k = 2}^{K} \left\{ {\left( {\mathop \prod \limits_{l = 1}^{k - 1} I_{l} } \right)\left( {i_{k} - 1} \right)} \right\}} \right] = {\mathbb{B}}\left[ {i_{i} , i_{2} , \ldots , i_{K} } \right] .$$

Tensors can also be unwounded as matrices by unfolding them along a specified tensor mode $k$. The mode-$k$ matrix representation of the tensor ${\mathbb{B}}$ is ${\varvec{B}}^{\left( k \right)} :I_{k} \times \mathop \prod \nolimits_{l \ne k} I_{l}$ and contains the vectorized representation of each sub-tensor in the $k{\text{th }}$ mode.

Finally, for a pair of tensors ${\mathbb{A}}$ and ${\mathbb{B}}$, we can define the contracted tensor product, ${\mathbb{A}},{\mathbb{B}}_{L}$, which naturally leads to tensor-based linear modeling, tensor inference, and linear tensor model prediction.

$${\mathbb{A}}: I_{1} \times \cdots \times I_{K} \times P_{1} \times \cdots \times P_{L} \quad {\text{and}}\quad @: P_{1} \times \cdots \times P_{L} \times Q_{1} \times \cdots \times Q_{M} ,$$

$$\langle {\mathbb{A}},{\mathbb{B}}\rangle_{L} : I_{1} \times \cdots \times I_{K} \times Q_{1} \times \cdots \times Q_{M} ,$$

$$\langle{\mathbb{A}},{\mathbb{B}}\rangle_{L} \left[ {i_{1} , \ldots , i_{K} ;q_{1} , \ldots , q_{M} } \right] = \mathop \sum \limits_{{p_{1} = 1}}^{{P_{1} }} \cdots \mathop \sum \limits_{{p_{L} = 1}}^{{P_{L} }} {\mathbb{A}}\left[ {i_{1} , \ldots , i_{K} ;p_{1} , \ldots , p_{L} } \right]{\mathbb{B}}\left[ {p_{1} , \ldots , p_{L} ;q_{1} , \ldots , q_{M} } \right] .$$

Notice that in this example the contracted tensor product takes an order-($K + L$) tensor and an order-($L + M$) and returns a ($K + M$)-tensor, as opposed to ($K + 2L + M$)-tensor with an outer product. Specifically, the tensor product contraction clears the common dimensions shared by the two tensors.

1.2 Appendix B: Additional 3D views of the results

The composite figure below extends the results shown in Fig. 4 of the paper by providing additional 3D views of different scenes of the results of each of the proposed 3-phase spacekime analytical protocol.

1.3 Appendix C: Implementation of tensor-based linear modeling

1.4 Appendix D: R Implementation of the laplace transformation