Fréchet mean-based Grassmann discriminant analysis

2019 ◽  
Vol 26 (1) ◽  
pp. 63-73
Author(s):  
Hongbin Yu ◽  
Kaijian Xia ◽  
Yizhang Jiang ◽  
Pengjiang Qian
Keyword(s):  
Discriminant Analysis ◽  
Fréchet Mean ◽  
Frechet Mean

scholarly journals Epilepsy Detection in EEG Using Grassmann Discriminant Analysis Method

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Hongbin Yu ◽  
Chao Fan ◽  
Yunting Zhang
Keyword(s):  
Discriminant Analysis ◽  
Grassmann Manifold ◽  
Signal To Noise Ratio ◽  
Involuntary Movement ◽  
Experimental Scheme ◽  
Data Dimensionality Reduction ◽  
Eeg Data ◽  
Fréchet Mean ◽  
Benchmark Datasets ◽  
Frechet Mean

Epilepsy is marked by seizures stemming from abnormal electrical activity in the brain, causing involuntary movement or behavior. Many scientists have been working hard to explore the cause of epilepsy and seek the prevention and treatment. In the field of machine learning, epileptic diagnosis based on EEG signal has been a very hot research topic; many methods have been proposed, and considerable progress has been achieved. However, resorting the epileptic diagnosis techniques based on EEG to the reality applications still faces many challenges. Low signal-to-noise ratio (SNR) is one of the most important methodological challenges for EEG data collection and analysis. This paper discusses an automated diagnostic method for epileptic detection using a Fréchet Mean embedded in the Grassmann manifold analysis. Fréchet mean-based Grassmann discriminant analysis (FMGDA) algorithm to implement the EEG data dimensionality reduction and clustering task. The method is resorted to reduce Grassmann data from high-dimensional data to a relative lower-dimensional data and maximize between-class distance and minimize within-class distance simultaneously. Every EEG feature is mapped into the Grassmann manifold space first and then resort the Fréchet mean to represent the clustering center to carry out the clustering work. We designed a detailed experimental scheme to test the performance of our proposed algorithm; the test is assessed on several benchmark datasets. Experimental results have delivered that our approach leads to a significant improvement over state-of-the-art Grassmann manifold methods.

On the consistency of procrustean mean shapes

1998 ◽  
Vol 30 (1) ◽  
pp. 53-63 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Mild Condition ◽  
Shape Space ◽  
Entire Space ◽  
Fréchet Mean ◽  
Mean Shape ◽  
Frechet Mean

We discuss the uniqueness of the Fréchet mean of a class of distributions on the shape space of k labelled points in ℝ2, the supports of which could be the entire space. From this it follows that the shape of the means is the unique Fréchet mean shape of the induced distribution with respect to an appropriate metric structure, provided the distribution of k labelled points in ℝ2 is isotropic and satisfies a further mild condition. This result implies that an increasing sequence of procrustean mean shapes defined in either of the two ways used in practice will tend almost surely to the shape of the means.

On the consistency of procrustean mean shapes

1998 ◽  
Vol 30 (01) ◽  
pp. 53-63 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Mild Condition ◽  
Shape Space ◽  
Entire Space ◽  
Fréchet Mean ◽  
Mean Shape ◽  
Frechet Mean

We discuss the uniqueness of the Fréchet mean of a class of distributions on the shape space of k labelled points in ℝ2, the supports of which could be the entire space. From this it follows that the shape of the means is the unique Fréchet mean shape of the induced distribution with respect to an appropriate metric structure, provided the distribution of k labelled points in ℝ2 is isotropic and satisfies a further mild condition. This result implies that an increasing sequence of procrustean mean shapes defined in either of the two ways used in practice will tend almost surely to the shape of the means.

scholarly journals Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees

Biometrika ◽  
2017 ◽  
Vol 104 (4) ◽  
pp. 901-922 ◽  
Author(s):  
Tom M W Nye ◽  
Xiaoxian Tang ◽  
Grady Weyenberg ◽  
Ruriko Yoshida
Keyword(s):  
Principal Component Analysis ◽  
Phylogenetic Trees ◽  
Principal Component ◽  
Component Analysis ◽  
Fréchet Mean ◽  
Frechet Mean

Estimating Fréchet means in Bookstein's shape space

2000 ◽  
Vol 32 (3) ◽  
pp. 663-674 ◽  
Author(s):  
Alfred Kume ◽  
Huiling Le
Keyword(s):  
Probability Measure ◽  
Shape Space ◽  
Probability Measures ◽  
Consistent Estimate ◽  
Shape Spaces ◽  
Fréchet Mean ◽  
Frechet Mean

In [8], Le showed that procrustean mean shapes of samples are consistent estimates of Fréchet means for a class of probability measures in Kendall's shape spaces. In this paper, we investigate the analogous case in Bookstein's shape space for labelled triangles and propose an estimator that is easy to compute and is a consistent estimate of the Fréchet mean, with respect to sinh(δ/√2), of any probability measure for which such a mean exists. Furthermore, for a certain class of probability measures, this estimate also tends almost surely to the Fréchet mean calculated with respect to the Riemannian distance δ.

Sign in / Sign up

Export Citation Format

Share Document