Principal component analysis (PCA) with small number of observations

In summary, the conversation discusses the use of principal component analysis (PCA) on hyperspectral data with 200 observations and ~1000 bands. It is mentioned that the estimated variance-covariance matrix is singular due to the smaller number of observations compared to variables. The questions raised are whether PCA can still be performed with this limitation and if the maximum number of meaningful principal components is equal to 199. The suggestion is made to search for "PCA in cluster analysis" and references are requested.
  • #1
miguelcc
1
0
Dear all,
I'd like to apply principal component analysis (PCA) to hyperspectral data (~1000 bands). The number of observations is 200.
The estimated variance covarance matrix is singular because the number of observations is smaller than the number of variables.

My questions are,

Can I still perform PCA (number of variables is < number of observations)?

Is the maximum number of meaninful principal components equal to 199?

Could you also provide me with references, please?

Thanks a lot in advance.

MiguelCC
 
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  • #2
I am not sure but I would search for: "PCA in cluster analysis" since this is a method for dimension reduction of the phase space. Wikipedia has a good overview on PCA.
 

1. What is Principal Component Analysis (PCA)?

Principal Component Analysis (PCA) is a statistical technique used to reduce the dimensionality of a large dataset while retaining as much of its original information as possible. It does this by creating new variables, called principal components, that are linear combinations of the original variables. These principal components are ordered by their ability to explain the variance in the data, with the first component explaining the most variance.

2. How does PCA work with a small number of observations?

PCA can still be applied to datasets with a small number of observations, but the results may not be as reliable due to the limited amount of data. In this case, it is important to use cross-validation techniques to assess the stability of the results and to avoid overfitting the data.

3. What are the benefits of using PCA with a small number of observations?

PCA can still provide useful insights and help to identify patterns and relationships in the data, even with a small number of observations. It can also help to reduce the dimensionality of the data, making it easier to visualize and interpret. Additionally, PCA can be a useful tool for data exploration and feature selection in small datasets.

4. What are the limitations of using PCA with a small number of observations?

One major limitation of using PCA with a small number of observations is that the results may not be as reliable or generalizable as with a larger dataset. This is because there is a higher risk of overfitting the data, which can lead to misleading or spurious results. Additionally, the interpretation of the principal components may be less meaningful due to the limited amount of data.

5. Are there any alternatives to using PCA with a small number of observations?

Yes, there are several alternative techniques that can be used in place of PCA when working with small datasets. These include factor analysis, multidimensional scaling, and non-linear dimensionality reduction methods. It is important to carefully consider the specific goals and characteristics of the dataset when selecting the most appropriate technique.

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