Abstract
Skeletal representations (s-reps) have been successfully adopted to parsimoniously parametrize the shape of three-dimensional objects and have been particularly employed in analyzing hippocampus shape variation. Within this context, we provide a fully nonparametric dimension-reduction tool based on kernel smoothing for determining the main source of variability of hippocampus shapes parametrized by s-reps. The methodology introduces the so-called density ridges for data on the polysphere and involves addressing high-dimensional computational challenges. For the analyzed dataset, our model-free indexing of shape variability reveals that the spokes defining the sharpness of the elongated extremes of hippocampi concentrate the most variation among subjects.
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Acknowledgements
Both authors acknowledge support from grant PID2021-124051NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. The authors greatly acknowledge Prof. Stephen M. Pizer and Dr. Zhiyuan Liu (University of North Carolina at Chapel Hill) for kindly providing the analyzed s-reps hippocampi data. Comments by the editor and a referee are appreciated.
Proofs
Proof (Proposition 1)
For \(\bar {\mathbf {x}}=({\mathbf {x}}_1^{\prime }/\|{\mathbf {x}}_1\|,\ldots ,{\mathbf {x}}_r^{\prime }/\|{\mathbf {x}}_r\|)'=:(\bar {\mathbf {x}}_1^{\prime },\ldots ,\bar {\mathbf {x}}_r^{\prime })^{\prime }\in (\mathbb {S}^{d})^r\),
where \({\mathbf {e}}_i\) is the ith canonical vector of \(\mathbb {R}^{d+1}\), \(i=1,\ldots ,d+1\), and \(j,k=1,\ldots ,r\), \(j\neq k\). It now follows that, for \(\mathbf {x}\in (\mathbb {R}^{d+1})^r\),
Hence, for \(\mathbf {x} \in (\mathbb {S}^{d})^r\),
To obtain the Hessian of \(\bar {f}\), we first compute the entries of \(\boldsymbol {\mathcal {H}}_{jj} \bar {f}(\mathbf {x})\) for \(\mathbf {x} \in (\mathbb {R}^{d+1})^r\) and \(j=1,\ldots ,r\):
with \(p,q=1,\ldots ,d+1\). Collecting the entries in (4.20) into \(\boldsymbol {\mathcal {H}}_{jj} \bar {f}(\mathbf {x})\), it follows that, for \(\mathbf {x} \in (\mathbb {S}^{d})^r\),
where \(\mathbf {A}:=\big [{\mathbf {x}}_j\boldsymbol \nabla _j f(\mathbf {x})+({\mathbf {x}}_j\boldsymbol \nabla _j f(\mathbf {x}))'-2(\boldsymbol \nabla _j f(\mathbf {x}){\mathbf {x}}_j){\mathbf {x}}_j{\mathbf {x}}_j^{\prime }\big ]\) is a symmetric matrix that, differently from the other terms in (4.21), is non-orthogonal to \({\mathbf {x}}_j{\mathbf {x}}_j^{\prime }\):
despite being easy to check that \(({\mathbf {x}}_j{\mathbf {x}}_j^{\prime })\mathbf {A}({\mathbf {x}}_j{\mathbf {x}}_j^{\prime })=({\mathbf {I}}_{d+1}-{\mathbf {x}}_j{\mathbf {x}}_j^{\prime })\mathbf {A}({\mathbf {I}}_{d+1}-{\mathbf {x}}_j{\mathbf {x}}_j^{\prime })\)
\(=\mathbf {0}\).
In addition, for \(k,j=1,\ldots ,r\), \(k\neq j\), and \(p,q=1,\ldots ,d+1\),
By an analogous collection of terms to that in (4.21), for \(\mathbf {x} \in (\mathbb {S}^{d})^r\), \(\boldsymbol {\mathcal {H}}_{kj} \bar {f}(\mathbf {x})=({\mathbf {I}}_{d+1}-{\mathbf {x}}_k{\mathbf {x}}_k^{\prime })\boldsymbol {\mathcal {H}}_{kj} f(\mathbf {x})({\mathbf {I}}_{d+1}-{\mathbf {x}}_j{\mathbf {x}}_j^{\prime })\). □
Proof (Proposition 2)
The proof follows after recalling that the unprojected estimator \(\tilde {m}(t;h):=\sum _{j=1}^nW_{j}(t;h){\mathbf {X}}_i\) satisfies \(\tilde {m}_{-i}(t;h)=\sum _{j=1,\, j\neq i}^nW_{-i,j}(t;h){\mathbf {X}}_j\) with \(W_{-i,j}(t;h)=W_j(t;h)/(1-W_i(t;h))\) since \(\sum _{i=1}^nW_i(t;h)=1\), for all \(t\in \mathbb {R}\). □
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García-Portugués, E., Meilán-Vila, A. (2023). Hippocampus Shape Analysis via Skeletal Models and Kernel Smoothing. In: Larriba, Y. (eds) Statistical Methods at the Forefront of Biomedical Advances. Springer, Cham. https://doi.org/10.1007/978-3-031-32729-2_4
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