主成分分析与正则化估计：低秩因子模型研究

需积分: 5 109 浏览量更新于2024-07-15 收藏 380KB PDF 举报

身份认证购VIP最低享 7 折!

领优惠券(最高得80元）

资源详情

资源推荐

If we write the SVD of X as X = U

′

, the singular values in

are of O

(

√

NT ) magnitude.

To facilitate asymptotic analysis, we consider the scaled data

Z =

√

, svd(Z) = UDV

′

, D =

√

Note that the left and right singular vectors of Z are the same as th ose for X. However, while the

ﬁrst r singular values of X (i.e.

) diverge and the remaining N − r are boun ded as N, T → ∞,

the r singular values of Z (i.e. D

) are bounded and the r emaining N − r singular values tend to

zero.

The model for the scaled data Z is

Z = F

∗

′

+ e

∗

(5)

where F

∗

√

, Λ

∗

√

, and e

∗

√

. Based on th e svd of Z = UDV

′

, we deﬁn e the PC

estimates as:

= U

1/2

(6a)

= V

1/2

. (6b)

Notably, the PC and the APC estimates are equivalent up to a scale transformation. In particu-

lar,

1/2

√

and

−1/2

√

. One can constru ct the APC factor estimates directly from a svd

of XX

′

and rescale the eigenvalues, or one can construct the PC factor estimates from a svd of the

rescaled data Z.

While (

) emerges from the optimization problem of min

F,Λ

kZ −F Λ

′

is an estimate for F

∗

√

, not for F

. Similarly,

is an estimate for Λ

∗

√

. For estimation

of F

and Λ

, we deﬁne

F =

√

T U

1/2

(7a)

Λ =

√

N V

1/2

. (7b)

That is to say,

F =

√

and

Λ =

√

. It follow s that

′

= D

F =

F D

1/2

Λ =

ΛD

−1/2

In contrast, APC uses the normalization

′

= I

, and Λ

′

Λ is diagonal. The unit length normal-

ization makes it inconvenient to impose restrictions on the APC estimates of F , a limitation that

is important for the analysis to follow.

There may be numerical advantages to using PC over APC. The d ocumentation of prcomp in R notes ’the

calculation is done by a singular value decomposition of the (centered and possibly scaled) data matrix, not by using

eigenvalues on the covariance matrix. This is generally the preferred method for numerical accuracy.

To establish the large sample properties of the PC estimates, we need to compare the estimates

with a rotation of the true factors. Given the relation between

F and

F , it is natural to deﬁ ne the

new rotation m atrix as

1/2

This leads to the identities

√

−

′

) =

√

1/2

(

−

′

√

T (

−

−1

) =

√

T D

−1/2

(

−

−1

From the limiting distributions of

and

, we obtain:

Proposition 1 Suppose that the data are generated by (1) and Assu mption A holds and both N

and T go to inﬁnity. Then the PC-estimates

F and

Λ satisfy:

(i)

√

−

′

)

−→N



0, D

1/2

Avar(

) D

1/2



≡ N



0, Avar(

)



;

(ii)

√

T (

−

−1

)

−→N



0, D

−1/2

Avar(

) D

−1/2



≡ N



0, Avar(

)



;

(iii) Let A

(

) =

′

Avar(

)

′

Avar(

)

. Then (A

(

))

−1/2

(

−C

)

−→N(0, 1).

The PC estimate of F

and Λ

are

√

N and

√

T consistent respectively, just like the APC estimates.

It follows that the estimated common component

is m in(

√

T ) consistent. Analogous to

Lemma 1, we can deﬁne asymptotically equivalent rotation matrices. Let

1,NT

= (Λ

0′

)(

′

)

−1

and

2,NT

= (F

0′

)

−1

0′

F ). Proposition 1 holds with

replaced by either

1,NT

2,NT

3 Rank and Nuclear-Norm Minimization

In large dimensional factor analysis, it is customary to choose an r ∈ [0, rmax] that provides a good

ﬁt while taking model complexity into account. Bai and Ng (2002) suggests a class of consistent

factor selection criteria, that is, rules that yield br such that prob(br = r) → 1 as N, T → ∞.

While criteria in this class have little diﬃculty singling out the dominant factors, they tend to

be too liberal. Over-estimating the number of factors is possible when validity of Assumption A

is questionable. The two leading causes are weak loadings, and idiosyncratic errors with large

variances. The ﬁrst has the implication that the smaller eigenvalues do not increase suﬃciently f ast

with N , an issue emphasized in Onatski (2011) and others. The second has the implication that

r+1-th eigenvalue (which should not increase with N) is not well separated from the r-th eigenvalue

(which should increase with N ). This is because outliers can increase the variance in an otherwise

剩余36页未读，继续阅读

Quant0xff

粉丝: 1w+
资源: 459

主成分分析与正则化估计：低秩因子模型研究

A tutorial on Principal Components Analysis.pdf

深度学习神经网络(英文版PDF教程）

01-camera-models.pdf

stata factor

explained_variance_ratio = pca.explained_variance_ratio_

pcl 3d segmentation setups

pca = PCA(n_components=5) pca.fit(Data) Std_pca=pca.transform(Data)

给我一段代码 主成分分析

principal components analysis

group lasso PCA求解算法参考文献

What is PCA

flask web开发实战.pdf

用matlab写一段PCA滤波函数

function [train_pca,test_pca,dataset_cumsum,percent_explained] = pcaForRF(train,test,threshold)

one-hot encoding pca

最新资源

给我一段代码主成分分析