On the explanatory power of principal components
On the explanatory power of principal components
复制标题
论主成分的解释力
DOI：
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发表时间：
2014
期刊：
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通讯作者：
J. Dazard
中科院分区：
文献类型：
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作者：
D. A. Díaz;J. Rao;J. Dazard
我们证明了，如果在$p$维向量空间中有一个正交基($u_1，\ldots，u_p$)，并且选择$p+1$向量$v_1，\ldots，v_p$和$w$使得这些向量遍历原点，则$w$比$v_1，\ldots，v_p$更接近于基中的所有向量的概率至少是1/2，并且随着$p$在区间[-1，1]上增加到无限正态分布而收敛，即$\Phi(1)-\Phi(-1)\约0.6826$。如果我们以正交基作为主成分的方向，这一结果对于回归和其他学习环境中的主成分分析具有相关的结果。
We show that if we have an orthogonal base ($u_1,\ldots,u_p$) in a $p$-dimensional vector space, and select $p+1$ vectors $v_1,\ldots, v_p$ and $w$ such that the vectors traverse the origin, then the probability of $w$ being to closer to all the vectors in the base than to $v_1,\ldots, v_p$ is at least 1/2 and converges as $p$ increases to infinity to a normal distribution on the interval [-1,1]; i.e., $\Phi(1)-\Phi(-1)\approx0.6826$. This result has relevant consequences for Principal Components Analysis in the context of regression and other learning settings, if we take the orthogonal base as the direction of the principal components.