[R] PCA with not non-negative definite covariance

[Stéphane Dray]

As said by Pierre Bady,
an answer to your question is NIPALS analysis.
PCA is usually obtained by the diagonalization of a variance-covariance 
matrix. But it can also be obtained by an iterative proedure which 
consists in two regressions. NIPLAS is an implementation of this 
iterative procedure and is strictly equivalent to PCA when there is no 
missing values.
The adavantage of NIPALS is that it can be used with missing values. 
However, note that the convergence is not always obtained (it depends of 
the number and distribution of missing values).
You can find a description of the method and the algorithm here:

http://biomserv.univ-lyon1.fr/~dray/articles/SD165.html

Sincerely,


Quin Wills wrote:

>My apologies (in response to the last 2 replies). I should write sensibly -
>including subject titles that make grammatical sense.
>
>(1) By analogous, I mean that using classical MDS with Euclidian distance is
>equivalent to plotting the first "k" principle components.
>(2) Agreed re. distribution assumptions.
>(3) Agreed re. the need to use some kind of imputation for calculating
>distances. I'm thinking pairwise exclusion for correlation.
>
>Re. why I want to do this is simply for graphically representing my data.
>
>Quin
>
>
>
>-----Original Message-----
>From: Berton Gunter [mailto:gunter.berton at gene.com] 
>Sent: 26 July 2006 05:10 PM
>To: 'Quin Wills'; bady at univ-lyon1.fr
>Cc: r-help at stat.math.ethz.ch
>Subject: RE: [R] PCA with not non-negative definite covariance
>
>Not sure what "completely analagous" means; mds is nonlinear, PCA is linear.
>
>In any case, the bottom line is that if you have high dimensional data with
>"many" missing values, you cannot know what the multivariate distribution
>looks like -- and you need a **lot** of data with many variables to usefully
>characterize it anyway. So you must either make some assumptions about what
>the distribution could be (including imputation methodology) or use any of
>the many exploratory techniques available to learn what you can.
>Thermodynamics holds -- you can't get something for nothing (you can't fool
>Mother Nature).
>
>-- Bert Gunter
>Genentech Non-Clinical Statistics
>South San Francisco, CA
> 
>"The business of the statistician is to catalyze the scientific learning
>process."  - George E. P. Box
> 
> 
>
>  
>
>>-----Original Message-----
>>From: r-help-bounces at stat.math.ethz.ch 
>>[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Quin Wills
>>Sent: Wednesday, July 26, 2006 8:44 AM
>>To: bady at univ-lyon1.fr
>>Cc: r-help at stat.math.ethz.ch
>>Subject: Re: [R] PCA with not non-negative definite covariance
>>
>>Thanks.
>>
>>I suppose that another option could be just to use classical
>>multi-dimensional scaling. By my understanding this is (if based on
>>Euclidian measure) completely analogous to PCA, and because it's based
>>explicitly on distances, I could easily exclude the variables 
>>with NA's on a
>>pairwise basis when calculating the distances.
>>
>>Quin
>>
>>-----Original Message-----
>>From: bady at univ-lyon1.fr [mailto:bady at univ-lyon1.fr] 
>>Sent: 25 July 2006 09:24 AM
>>To: Quin Wills
>>Cc: r-help at stat.math.ethz.ch
>>Subject: Re: [R] PCA with not non-negative definite covariance
>>
>>Hi , hi all,
>>
>>    
>>
>>>Am I correct to understand from the previous discussions on 
>>>      
>>>
>>this topic (a
>>    
>>
>>>few years back) that if I have a matrix with missing values 
>>>      
>>>
>>my PCA options
>>    
>>
>>>seem dismal if:
>>>(1)     I don’t want to impute the missing values.
>>>(2)     I don’t want to completely remove cases with missing values.
>>>(3)     I do cov() with use=”pairwise.complete.obs”, as 
>>>      
>>>
>>this produces
>>    
>>
>>>negative eigenvalues (which it has in my case!).
>>>      
>>>
>>(4) Maybe you can use the Non-linear Iterative Partial Least Squares
>>(NIPALS)
>>algorithm (intensively used in chemometry). S. Dray proposes 
>>a version of
>>this
>>procedure at http://pbil.univ-lyon1.fr/R/additifs.html.
>>
>>
>>Hope this help :)
>>
>>
>>Pierre
>>
>>
>>
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>>
>> 
>>
>>--
>>
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>>
>>    
>>
>
>  
>


-- 
Stéphane DRAY (dray at biomserv.univ-lyon1.fr )
Laboratoire BBE-CNRS-UMR-5558, Univ. C. Bernard - Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Tel: 33 4 72 43 27 57       Fax: 33 4 72 43 13 88
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