几个数理统计相关的问题
几个数理统计相关的问题
为什么样本方差的分母是 ?
如果已知随机变量
那么采样后用
来近似
一般说来
我们要证明可以用
来近似
首先我们知道
根据中心极限定理,
易知
如果我们令
则有
而
故
故
得到的就是无偏估计.
只考虑一阶和二阶统计量的高斯过程
设有两个高斯过程
其中
这与二元正态分布是很类似的.
低通滤波
函数的低通滤波(?)对于函数
可以认为是
的变形.
统计量的公式
对于一个 Markov chain
PCA, ICA, CCA, PLS
PCA, ICA
PCA 和 ICA 都是降维算法. 不同于 PCA 得到的主成分在时间和空间上都不相关(左右奇异向量都是正交的),ICA 得到的主成分只在一个部分上具有最大的统计独立性.
PCA 目标是找一组轴,使得数据在这组轴上的投影方差最大. ICA 的想法是观察到的信号是由一些独立的信号混合而成的. By the central limit theorem, any linear mixture of independent variables will be more "Gaussian" than the original variables. Thus, ICA seeks to create a new set of axes. The axes are oriented such that the projection of data points onto the axes is maximally non-Gaussian. We can use kurtois, negentropy, or mutual information to measure the non-Gaussianity.
Independent components can be interpreted as the dominant functional networks or modes of activity that contribute to the observed neuroimaging data[1].
cvPCA
cross-validated PCA is a method to derive unbiased estimation of the (major part) of the eigenspectrum of the (sampled) covariance matrix.
Assume we have several trials of neural activity recordings
Theorem 1 and 2 in [SI, Stringer et al.,
2019[2]]
tell us that how the cvPCA method is implemented. Note that the
requirement in theorem 2 that one source of noise has dimensions
orthogonal to the signal dimensions cannot be satisfied in some cases,
e.g., the sample correlation matrix is full rank. However, we often have
Let's now describe how cvPCA is implemented. Suppose we have a
training recording
First, we can do PCA on the training recording
where
Next, we project the test recording
The estimated
Canonical Correlation Analysis (CCA)
The goal of CCA is to relate two sets of data,
CVA (Friston et al. 1996, Strother et al. 2002), linear discriminant analysis (LDA), and multivariate analysis of variance are special case of CCA.
We want to create pairs of new variable that are linear combinations
of the original variables in
Reference
- McIntosh, Anthony R., & Bratislav Mišić. Multivariate Statistical Analyses for Neuroimaging Data. Annual Review of Psychology 64, 1 (2013): 499–525. https://doi.org/10.1146/annurev-psych-113011-143804. ↩︎
- Stringer, Carsen, Marius Pachitariu, Nicholas Steinmetz, Matteo Carandini & Kenneth D. Harris. High-Dimensional Geometry of Population Responses in Visual Cortex. Nature 571, 7765 (2019): 361–65. https://doi.org/10.1038/s41586-019-1346-5. ↩︎