Ordinal Pattern Kernel for Brain Connectivity Network Classification

Graph kernels are a class of kernel functions measuring the similarities between graphs. There is a map $\phi$, which could implicitly embed the original graph data set $\mathcal{G}$ into a Hilbert space $\mathcal{H}$, $\phi$: $\mathcal{G}$ $\rightarrow$ $\mathcal{H}$. In $\mathcal{G}$, graph kernel $\mathcal{K}$: $\mathcal{G}$ $\times$ $\mathcal{G}$ $\rightarrow$ $\mathbb{R}$ is a function associated with $\mathcal{H}$, given two graphs G₁ and G₂, G₁, G₂ $\in$ $\mathcal{G}$, graph kernel $\mathcal{K}$ is interpreted as a dot product in the high dimensional space $\mathcal{H}$, $\mathcal{K}\left(G_{1},G_{2}\right)=\langle\phi\left(G_{1}\right),\phi\left(G_{2}\right)\rangle_{\mathcal{H}}$. If $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS), then $\mathcal{K}$ is a positive definite kernel.

A and B are two nonempty sets, $\varphi$ is a map from A to B. $\circ$ and $\bar{\circ}$ are respectively the algebraic operation on A and B. If $\forall\emph{a},\emph{b}\in\emph{A}$, then $\varphi\left(\emph{a}\circ\emph{b}\right)=\varphi\left(\emph{a}\right)\bar{\circ}~{}\varphi\left(\emph{b}\right)$, $\varphi$ is called the homomorphic mapping from A to B. If $\varphi$ is a homomorphic and onto mapping from A to B, we call A and B are homomorphic. If $\varphi$ is a homomorphic and bijective mapping from A to B, then $\varphi$ is called the isomorphic mapping from A to B. We call A and B are isomorphic, $\emph{A}\cong\emph{B}$.

A weighted network or graph G consists of a set of nodes V, edges E and weight vectors W, $\emph{G}=\left(\emph{V},\emph{E},\emph{W}\right)$. W is the weight vector for those edges with the i-th element $\emph{W}\left(e_{i}\right)$ representing the connection strength of the edge e_i, $\emph{e}_{i}\in\emph{E}$. The ordinal pattern (OP) defined in graph G is a set including ordinal nodes and ordinal edges $\emph{OP}=\left(\emph{V}_{\emph{op}},\emph{E}_{\emph{op}}\right)$. $\emph{E}_{\emph{op}}$ is a ordinal edge set, $\emph{E}_{\emph{op}}=\{\emph{e}_{1},\emph{e}_{2},\cdots,\emph{e}_{i},\emph{e}_{j},\cdots,\emph{e}_{M}\}\subseteq\emph{E}$, all $0<\emph{i}<\emph{j}\leq\emph{M}$, $\emph{W}\left(\emph{e}_{i}\right)>\emph{W}\left(\emph{e}_{j}\right)$, $\emph{e}_{i}$ and $\emph{e}_{j}$ are called ordinal edges. $\emph{V}_{\emph{op}}$ is a vertex set where vertexes are in ordinal edges included in $\emph{E}_{\emph{op}}$. The illustration of ordinal patterns could be seen in Figure 1. OP₁, OP₂ and OP₃ are ordinal patterns from a weighted network.

A graph or network could be decomposed into multiple ordinal patterns. The set consisting of all ordinal patterns is called ordinal pattern set $\left(\emph{OPs}\right)$. OPs₁ and OPs₂ are two ordinal pattern sets of graph G₁ and G₂, OP₁=$\left(\emph{V}_{\emph{op${}_{1}$}},\emph{E}_{\emph{op${}_{1}$}}\right)$ and OP₂=$\left(\emph{V}_{\emph{op${}_{2}$}},\emph{E}_{\emph{op${}_{2}$}}\right)$ are two ordinal patterns, OP₁$\in$OPs₁, OP₂$\in$OPs₂. An ordinal pattern isomorphism between two ordinal patterns OP₁ and OP₂ is a bijective mapping $\varphi:\emph{V}_{\emph{op${}_{1}$}}\to\emph{V}_{\emph{op${}_{2}$}}$, i.e. $\forall\emph{v}_{\emph{op${}_{1}$}},\emph{v'}_{\emph{op${}_{1}$}}\in\emph{V}_{\emph{op${}_{1}$}}$: $(\emph{v}_{\emph{op${}_{1}$}},\emph{v'}_{\emph{op${}_{1}$}})\in\emph{E}_{\emph{op${}_{1}$}}$ $\Leftrightarrow$ $\varphi\left(\emph{v}_{\emph{op${}_{1}$}}\right),\varphi\left(\emph{v'}_{\emph{op${}_{1}$}}\right)\in\emph{V}_{\emph{op${}_{2}$}}$, $(\varphi\left(\emph{v}_{\emph{op${}_{1}$}}\right),\varphi\left(\emph{v'}_{\emph{op${}_{1}$}}\right))\in\emph{E}_{\emph{op${}_{2}$}}$. OP₁ and OP₂ are isomorphic, written $\emph{OP${}_{1}$}\cong\emph{OP${}_{2}$}$. V${}_{1}^{{}^{\prime}}$ $\subseteq$ $\emph{V}_{\emph{op${}_{1}$}}$ and V${}_{2}^{{}^{\prime}}$ $\subseteq$ $\emph{V}_{\emph{op${}_{2}$}}$ are subsets of ordinal pattern vertices. An ordinal pattern isomorphism $\tau$ of OP₁[V${}_{1}^{{}^{\prime}}$] and OP₂[V${}_{2}^{{}^{\prime}}$] is called sub-ordinal pattern isomorphism (SOPI) of OP₁ and OP₂.

We suppose that a graph G has N ordinal patterns, hence ordinal pattern set of graph G is $\emph{OPs}=\{\emph{OP${}_{1}$},\cdots,\emph{OP${}_{i}$},\cdots,\emph{OP${}_{N}$}\}$, OP_i is the i-th ordinal pattern, $1\leq\emph{i}\leq\emph{N}$. Two graphs G₁ and G₂ have their own OPs respectively, they are OPs₁ and OPs₂. $\varphi_{ij}$ is the isomorphism mapping from ordinal pattern OP_i to ordinal pattern OP_j,OP_i$\in$OPs₁, OP_j$\in$OPs₂. Let $\Psi\left(\emph{OP${}_{i}$},\emph{OP${}_{j}$}\right)$ refer to the set which includes all sub-ordinal pattern isomorphisms (SOPIs) of OP_iand OP_j and $\Upsilon$: $\Psi\left(\emph{OP${}_{i}$},\emph{OP${}_{j}$}\right)\rightarrow\mathbb{R}^{+}$ a weight function. The sub-ordinal pattern isomorphism kernel is defined as:

Theorem 1. $k_{sopi}$ is positive semidefinite (p.s.d) kernel

Proof. Kernel $k_{sopi}$ counts the number of isomorphisms between sub-ordinal patterns in ordinal pattern OP_i and OP_j. We have known that the kernel counting the number of isomorphisms between graph sub-structures is $p.s.d$ [19]. In $k_{sopi}$, sub-ordinal patterns could be treated as the special sub-structures where the nodes and edges are from ordinal patterns. Hence, $k_{sopi}$ is $p.s.d$. $\Box$

The sub-ordinal pattern isomorphism kernel measures the similarity between two ordinal patterns by counting the number of sub-ordinal pattern isomorphisms, $\bigtriangleup:=\emph{k}_{sopi}\left(\emph{OP${}_{i}$},\emph{OP${}_{j}$}\right)$. Then, we get the ordinal pattern kernel between two graphs G₁ and G₂, defined as:

where OPs₁ and OPs₂ are the ordinal pattern sets of G₁ and G₂, iso-count($\cdot$) is a function calculating isomorphism:

where a weight function $\lambda_{o}$: $OPs\to\mathbb{R}^{+}$, which count the node number when two ordinal patterns are isomorphic.

If the ordinal patterns have the node and edge attributes, the above formula is not appropriate to defining a mapping to preserve their attributes. We need to generalize the formula (2) to the common ordinal patterns with attributes. The new kernel is called ordinal pattern attribute (OPA) kernel.

where K_V and K_E are two positive semidefinite kernel functions defined on ordinal pattern node and edge attribute features.

Theorem 2. The ordinal pattern attribute kernel is p.s.d

Before we prove theorem 2, we need to know R relation in ordinal pattern (seeing supplement) and R-ordinal pattern convolution. Suppose OPs₁ and OPs₁ are two ordinal pattern set $\left(OPs\right)$ in graph G₁ and G₂. Ordinal pattern $OP\in\emph{OPs${}_{1}$}$ and $OP^{\prime}\in\emph{OPs${}_{2}$}$, $OP=\left(V_{op},E_{op}\right)$, $OP^{\prime}=\left(V_{op^{\prime}},E_{op^{\prime}}\right)$. The decompositions of $OP$ and $OP^{\prime}$: $\vec{OP}=OP_{1},\cdots,OP_{M}$ and $\vec{OP^{\prime}}=OP^{\prime}_{1},\cdots,OP^{\prime}_{M}$ are two parts of $OP$ and $OP^{\prime}$. For $1\leq i\leq M$, we define a function iso-count_i($OP_{i}$,$OP^{\prime}_{i}$) on $OP_{i}$ and $OP^{\prime}_{i}$ that could be used to measure the similarity of ordinal patterns $OP_{i}$ and $OP^{\prime}_{i}$. For $OP_{i}=\left(V_{op_{i}},E_{op_{i}}\right)$, $OP^{\prime}_{i}=\left(V_{op^{\prime}_{i}},E_{op^{\prime}_{i}}\right)$, we define two positive semidefinite kernels $K_{V_{i}}\left(v_{op_{i}},v_{op^{\prime}_{i}}\right)$ and $K_{E_{i}}\left(e_{op_{i}},e_{op^{\prime}_{i}}\right)$ that could be used to measure the similarity of the ordinal nodes and edges, $v_{op_{i}}\in V_{op_{i}}$, $v_{op^{\prime}_{i}}\in V_{op^{\prime}_{i}}$, $e_{op_{i}}\in E_{op_{i}}$, $e_{op^{\prime}_{i}}\in E_{op^{\prime}_{i}}$. Then we define the kernel $K\left(OPs_{1},OPs_{2}\right)$ measuring the similarity between ordinal pattern set OPs₁ and OPs₂ as the following ordinal pattern convolution.

where $K\left(OPs_{1},OPs_{2}\right)$ is a symmetric function on $\mathbb{Q}\times\mathbb{Q}$, $\mathbb{Q}=\{OP:R^{-1}\left(OP\right)is~{}nonempty\}$. K is kernel defined on iso-count_i($OP_{i}$,$OP^{\prime}_{i}$), $K_{V_{i}}\left(v_{op_{i}},v_{op^{\prime}_{i}}\right)$ and $K_{E_{i}}\left(e_{op_{i}},e_{op^{\prime}_{i}}\right)$ by R relation. Hence K is called R-ordinal pattern convolution which is the zero extension of K to $OPs\times OPs$. If R is finite, then K is a finite convolution.

Theorem 3. R-ordinal pattern convolution is a kernel on $OPs\times OPs$

Proof. Let U denote $SOPs_{1}\times\cdots\times SOPs_{M}$, $SOPs_{i}$ is ordinal pattern subset, $1\leq i\leq M$, since iso-count_i($OP_{i}$,$OP^{\prime}_{i}$), $K_{V_{i}}\left(v_{op_{i}},v_{op^{\prime}_{i}}\right)$ and $K_{E_{i}}\left(e_{op_{i}},e_{op^{\prime}_{i}}\right)$ are kernels. We know that $\bar{K}\left(\vec{OP},\vec{OP^{\prime}}\right)$ is a kernel on $U\times U$, obviously it is the kernel closure under tensor product

Since R is finite, according to Lemma 1, $\bar{K^{\prime}}\left(R^{-1}\left(OP\right),R^{-1}\left(OP^{\prime}\right)\right)$ is a kernel on the product of the set of all not empty $R^{-1}\left(OP\right)$ such that $OP\in OPs$ with itself.

Since R-ordinal pattern convolution is the zero extension of $K\left(OPs_{1},OPs_{2}\right)=\bar{K^{\prime}}\left(R^{-1}\left(OP\right),R^{-1}\left(OP^{\prime}\right)\right)$, it follows that it is a kernel on $OPs\times OPs$. The OPA kernel is a R-ordinal pattern convolution kernel , where each kernel is p.s.d, hence OPA kernel is p.s.d.$\Box$

Lemma 1 If K is a kernel on a set $U\times U$, for all nonempty and finite set $X,Y\subseteq U$. We define a function $K^{\prime}\left(X,Y\right)=\sum_{x\in X,y\in Y}K\left(x,y\right)$. Then $K^{\prime}$ is a kernel on the product of the set of all nonempty and finite subsets of U with itself.

Proof. For any finite nonempty subset $X\subseteq U$, let $f_{X}=\sum_{u\in X}K_{u}\in H_{0}$, where $H_{0}$ is the pre-Hilbert space associated with $K^{\prime}$. If for nonempty finite $X,Y\subseteq U$, we define $K^{\prime}\left(X,Y\right)=\sum_{u\in X,v\in Y}K\left(u,v\right)$, then by Equation (1) in supplement, $K^{\prime}\left(X,Y\right)=\langle f_{X},f_{Y}\rangle$. Because an inner product is a kernel, it follows that $K^{\prime}$ is a kernel on the product of the set of all nonempty finite subsets of U with itself.$\Box$

Ordinal pattern kernel in formula (2) is a special case of ordinal pattern attribute kernel in formula (4). When we do not take node and edge attributes into the computation of ordinal pattern kernel, the kernel on node and edge attributes is equal 1, $\emph{K${}_{{V_{op}},{E_{op}}}$}=1$, hence formula (4) degenerate to formula (2).

These kernels guarantee that exactly the conditions of ordinal pattern isomorphism are fulfilled. Hence, the $OP$ kernel is a special case of OPA kernel, we could obtain the following corollary.

Corollary 1. The ordinal pattern kernel is p.s.d

Although we have known how to calculate ordinal pattern kernel and ordinal pattern attribute kernel, there is another problem in them. The problem is that computing the kernels including ordinal pattern kernel and ordinal pattern attribute kernel based on ordinal pattern isomorphism are NP-hard.

Theorem 4. Computing the kernel based on ordinal pattern isomorphism is NP-hard.

Proof. Let OP_n$\in$OPs be the ordinal pattern with n edges and let e_op be a vector in the ordinal pattern feature space which is defined by the mapping $\phi$: OPs $\rightarrow$ $\mathcal{H}$ into Hilbert space $\mathcal{H}$ with one feature $\phi_{OP}$ for each ordinal pattern $OP\in OPs$, $OP^{\prime}\in OPs$, $\phi_{OP}=\lambda_{|\varepsilon\left(\emph{OP}\right)|}|\{OP^{\prime}\in OPs:OP^{\prime}\cong OP\}|$, where $\lambda$ is a sequence $\lambda_{1}$, $\lambda_{2}$,$\cdots$, $\lambda_{N}$ of weights $\left(\lambda_{i}\in\mathbb{R};\lambda_{i}>0~{}for~{}all~{}i\in\emph{N}\right)$. In e_op, the features corresponding to OP equal 1 and others equal 0. Let $OP^{\prime}$ be any ordinal pattern with m vertices. According to [11] for linear independent {$\phi$$\left(\emph{OP${}_{n}$}\right)$ }${}_{n\in\emph{N}}$, we could always find $\alpha_{1}$,$\cdots$,$\alpha_{m}$ in polynomial time, make that $\alpha_{1}\phi\left(OP_{1}\right)$+,$\cdots$,+$\alpha_{m}\phi\left(OP_{m}\right)$=e${}_{OP_{m}}$. Then $\alpha_{1}\langle\phi\left(OP_{1}\right)$,$\phi\left(OP^{\prime}\right)\rangle$+,$\cdots$,+$\alpha_{m}\langle\phi\left(OP_{m}\right)$,$\phi\left(OP^{\prime}\right)\rangle$ >0 if and only if $OP^{\prime}$ has a Hamiltonian path. We all known that finding a Hamiltonian path is NP-complete. Hence, computing the kernel based on ordinal pattern isomorphism is NP-hard. $\Box$

Here, we detect the establishment process of depth-first-based ordinal pattern in detail. A weighted network or graph G consists of a set of nodes V, edges E and weight vectors W, $\emph{G}=\left(\emph{V},\emph{E},\emph{W}\right)$. $\forall u\in\emph{V}$, the neighborhood vertex set of a vertex u: $\delta\left(\emph{u}\right)=\{v:\left(u,v\right)\in E,v\in V\}$, the edge weight set between vertex u and its neighborhood vertexes: $\mathbb{W}\left(u\right)=\{\emph{W}\left(u,v\right):v\in\delta\left(\emph{u}\right),\left(u,v\right)\in E\}$. In graph G, we arbitrarily choose a node as the start node v₀ and use the depth-first search algorithm to seek the deepest ordinal pattern of node v₀. The detailed process of constructing depth-first-based ordinal pattern for node v₀ could be seen in Algorithm 1.

We could use Algorithm 1 to calculate the DOP for each node in graph G. Subsequently, we could utilize these ordinal patterns to construct the depth-first-based ordinal pattern kernel k_DOP between graph G₁ and G₂ with attributes. Because ordinal pattern kernel is a special case of ordinal pattern attribute kernel. Here, we only detect the depth-first-based ordinal pattern attribute kernel.

where DOP$\left(v_{i}\right)$ and DOP$\left(v_{j}\right)$ are the depth-first-based ordinal patterns or the deepest ordinal patterns of node $v_{i}$ and $v_{j}$. $\emph{K${}_{V}$}\left(v_{DOP},v^{\prime}_{DOP}\right)$ and $\emph{K${}_{E}$}\left(e_{DOP},e^{\prime}_{DOP}\right)$ are the kernels defined on node and edge attributes in DOP$\left(v_{i}\right)$ and DOP$\left(v_{j}\right)$.

where $\lambda$ is a sequence $\lambda_{1}$,$\lambda_{2}$,$\cdots$,$\lambda_{N}$ of weights $\left(\lambda_{i}\in\mathbb{R};\lambda_{i}>0~{}for~{}all~{}i\in N\right)$. Obviously, formula $\left(10\right)$ is a special case of formula $\left(4\right)$. Here, $match\left(DOP\left(v_{i}\right),DOP\left(v_{j}\right)\right)$ could also be calculated by the matched node numbers between $DOP\left(v_{i}\right)$ and $DOP\left(v_{j}\right)$.

Theorem 5. Depth-first-based ordinal patternn attribute kernel is p.s.d.

Proof. According to the definitions of the sub-ordinal pattern isomorphisms and the corresponding sub-ordinal pattern isomorphism kernel, we know that $match\left(DOP\left(v_{i}\right),DOP\left(v_{j}\right)\right)$ is p.s.d. K${}_{{V_{DOP}},{E_{DOP}}}$ is a p.s.d kernel defined in node and edge attributes. $\emph{k${}_{DOP}$}\left(\emph{G${}_{1}$},\emph{G${}_{2}$}\right)$ is a R-ordinal pattern convolution kernel, hence depth-first-based ordinal pattern attribute kernel is p.s.d. $\Box$

Kernel selection. In real applications, such as brain neuroimaging, each brain structure is abstracted into a network (or graph) where each node and edge may have multi-dimensional attributes. These attributes are from Euclidean spaces, we could utilize the Gaussian RBF kernel or linear kernel to compute the kernel K_V and K_E in node and edge attributes. If the attributes are discrete, we use Delta kernel $k_{attribute}\left(a,b\right)=\mathcal{T}_{\{a=b\}}$ [12].

The brain network data used in the experiments are based on brain disease patients and normal controls, which are constructed from the resting state functional magnetic resonance imaging (fMRI) data deriving from the online database ADNI¹¹1http://adni.loni.usc.edu/. Brain diseases are Alzheimer’s disease (AD), early mild cognitive impairment (EMCI), late mild cognitive impairment (LMCI). MCI consists of EMCI and LMCI. The normal controls (NC) matched to brain disease patients are used to classify brain diseases and seek the discriminative sub-structures from brain disease networks. The fMRI data are preprocessed with statistical parametric mapping (SPM) ²²2http://www.fil.ion.ucl.ac.uk/spm and resting-state fMRI analysis toolkit (REST) ³³3http://www.restfmri.net. The detailed fMRI data preprocessing steps could be seen in supplement.

After processing the fMRI data of brain disease patients and normal controls, we need to transform the preprocessed fMRI data into brain networks. The whole-brain cortical and subcortical structures are subdivided into 90 brain regions for each subject based on the AAL atlas. The linear correlation between mean time series of a pair of brain regions is then calculated to measure the functional connectivity. At last, a $90\times 90$ fully-connected weighted functional network is constructed for each subject. The detailed contents of constructing brain network could be seen in supplement.

In the experiments, uniform weight $\lambda$ is chosen from $\{10^{-2},10^{-1},\cdots,10^{2}\}$. We compare our kernel with state-of-the-art graph kernels including shortest path kernel (SP) [9], Weisfeiler-Lehman subtree kernel (WL-ST) [17], Weisfeiler-Lehman shortest path kernel (WL-SP) [17], random walk kernel (RW) [10], pyramid match kernel (PM) [21], Wasserstein Weisfeiler-Lehman graph kernel (WWL) [18], GraphHopper kernel (GH) [23], Truncated Tree Based Graph Kernels (Tree++) [24]. In robustness experiment, we randomly discard partial data in each brain network by 25% missing rate.

Support Vector Machine (SVM) [25] as our final classifier is exploited to conduct the classification experiment. We perform leave-one-out cross-validation for all the classifications, using one sample for testing and the others for training. The tradeoff parameter C in the SVM is selected from $\{10^{-3},10^{-2},\cdots,10^{3}\}$.

The classification accuracy, method robustness and discriminative sub-structures are very important in brain network analysis. We report the classification accuracies in Table 1 and investigate the robustness of our depth-first-based ordinal pattern kernel in the special missing rate in Figure 2 and the discriminative ordinal patterns in Figure 3.