syoukera commented 4 years ago

@article{PARENTE2013340, title = "Principal component analysis of turbulent combustion data: Data pre-processing and manifold sensitivity", journal = "Combustion and Flame", volume = "160", number = "2", pages = "340 - 350", year = "2013", issn = "0010-2180", doi = "https://doi.org/10.1016/j.combustflame.2012.09.016", url = "http://www.sciencedirect.com/science/article/pii/S0010218012002775", author = "Alessandro Parente and James C. Sutherland", keywords = "Combustion, Low-dimensional manifolds, Outliers, Pre-processing, Principal component analysis, Scaling", abstract = "Principal component analysis has demonstrated promise in its ability to identify low-dimensional chemical manifolds in turbulent reacting systems by providing a basis for the a priori parameterization of such systems based on a reduced number of parameterizing variables. Previous studies on PCA have only mentioned the importance of data pre-processing and scaling on the PCA analysis, without detailed consideration. This paper assesses the influence of data-preprocessing techniques on the size-reduction process accomplished through PCA. In particular, a methodology is proposed to identify and remove outlier observations from the datasets on which PCA is performed. Moreover, the effect of centering and scaling techniques on the PCA manifold is assessed and discussed in detail, to investigate how different scalings affect the size of the manifold and the accuracy in the reconstruction of the state-space. Finally, the sensitivity of the chemical manifold to flow characteristics is considered, to investigate its invariance with respect to the Reynolds number. Several high-fidelity experimental datasets from the TNF workshop database are considered in the present work to demonstrate the effectiveness of the proposed methodologies." }

syoukera commented 4 years ago

Abstruct

PCAの前処理って重要と言われていたけど，ちゃんと検討されていないよね
外れ値（Outlier）の検出と除去の方法を提案する
centeringとscaling手法がマニホールドの大きさと精度にどう影響するか
化学反応のマニホールドの流れの特性への感度を検討する

syoukera commented 4 years ago

2. Principal component analysis

PCAで元の値を再現するときには，問題の非線形性を再現することができない．この問題はLocal PCAを使えば部分的に解決されている．
2.1. Outlier detection and removal with PCA
ひとつの変数を閾値として外れ値を検出することはできない
PCAそれ自体が，多変数での外れ値検出に有効である
マハラノビス距離D_Mを使う．確か分散を考慮した平均値からの距離
データの中でD_Mが大きいものを切り捨てる
D_Mは主成分PCからも計算できる
第1主成分で最も大きく，番号が増えるほど分散の小さい主成分になるので考慮する必要がなくなる
以下の手法が提案されている
1. Multivariate trimming.

D_Mを計算→外れ値を除外→D_Mを再計算のループを繰り返す．

2. Principal components classifier.

メジャーとマイナーの二つの関数を用意．それらが全体の分散の99%の時の値と比較される．
PC1とPC2の結果では外れ値がはっきりわかる
外れ値を取り除くとPC1とPC2の値も変わっている