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Can we apply PCA on linear dataset?
Hierarchical linear regression can help answer this question. If your data is complex (i.e. you have many variables) you can apply PCA to reduce the number of variables/find the “latent variables”. These latent variables can then be used in the hierarchical linear regression.
Which is a latent variable model in nonlinear dimensionality reduction?
The self-organizing map (SOM, also called Kohonen map) and its probabilistic variant generative topographic mapping (GTM) use a point representation in the embedded space to form a latent variable model based on a non-linear mapping from the embedded space to the high-dimensional space.
How is principal component analysis used in nonlinear dimensionality reduction?
By comparison, if Principal component analysis, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter ‘A’) that sample this manifold vary in a non-linear manner.
How does nonlinear dimensionality reduction help machine learning?
Algorithms that operate on high-dimensional data tend to have a very high time complexity. Many machine learning algorithms, for example, struggle with high-dimensional data. Reducing data into fewer dimensions often makes analysis algorithms more efficient, and can help machine learning algorithms make more accurate predictions.
How to do dimensionality reduction in categorical data?
Amelia includes some limited capacity to deal with ordinal and nominal variables. As for dimensionality reduction for categorical data (i.e. a way to arrange variables into homogeneous clusters), I would suggest the method of Multiple Correspondence Analysis which will give you the latent variables that maximize the homogeneity of the clusters.