Return to Answer

Mahalanobis distance is equivalent to the Euclidean distance on the PCA-transformed data (not just PCA-rotated!), where by "PCA-transformed" I mean (i) first rotated to become uncorrelated, and (ii) then scaled to become standardized. This is what @ttnphns said in the comments above and what both @DmitryLaptev and @whuber meant and explicitly wrote in their answers that you linked to (one and two), so I encourage you to re-read their answers and make sure this point becomes clear.

This means that you can make your code work simply by replacing pc$x with scale(pc$x) in the fourth line from the bottom.

Regarding your second question, with $n<p$, covariance matrix is singular and hence Mahalanobis distance is undefined. Indeed, think about Euclidean distance in the PCA-transformed data; when $n<p$, some of the eigenvalues of covariance matrix are zero and the corresponding PCs have zero variance (all the data points are projected to zero). It is therefore impossible to standardize these PCs, as it is impossible to divide by zero. Mahalanobis distance cannot be defined "in these directions".

What one can do, is to focus exclusively on the subspace where the data actually lie, and define Mahalanobis distance in this subspace. This is equivalent to doing PCA and keeping only non-zero components, which is I think what you suggested in your question #2. So the answer to this is yes. I am not sure though how useful this can be in practice, as this distance is likely to be very unstable (near-zero eigenvalues are known with very bad precision, but are going to be inverted in the Mahalanobis formula, possible yielding gross errors).

Mahalanobis distance is equivalent to the Euclidean distance on the PCA-transformed data (not just PCA-rotated!), where by "PCA-transformed" I mean (i) first rotated to become uncorrelated, and (ii) then scaled to become standardized. This is what @ttnphns said in the comments above and what both @DmitryLaptev and @whuber meant and explicitly wrote in their answers that you linked to (one and two), so I encourage you to re-read their answers and make sure this point becomes clear.

This means that you can make your code work simply by replacing pc$x with scale(pc$x) in the fourth line from the bottom.

Regarding your second question, with $n<p$, covariance matrix is singular and hence Mahalanobis distance is undefined. Indeed, think about Euclidean distance in the PCA-transformed data; when $n<p$, some of the eigenvalues of covariance matrix are zero and the corresponding PCs have zero variance (all the data points are projected to zero). It is therefore impossible to standardize these PCs, as it is impossible to divide by zero. Mahalanobis distance cannot be defined "in these directions".

What one can do, is to focus exclusively on the subspace where the data actually lie, and define Mahalanobis distance in this subspace. This is equivalent to doing PCA and keeping only non-zero components, which is I think what you suggested in your question #2. So the answer to this is yes. I am not sure though how useful this can be in practice, as this distance is likely to be very unstable (near-zero eigenvalues are known with very bad precision, but are going to be inverted in the Mahalanobis formula, possible yielding gross errors).

Mahalanobis distance is equivalent to the Euclidean distance on the PCA-transformed data (not just PCA-rotated!), where by "PCA-transformed" I mean (i) first rotated to become uncorrelated, and (ii) then scaled to become standardized. This is what @ttnphns said in the comments above and what both @DmitryLaptev and @whuber meant and explicitly wrote in their answers that you linked to (one and two), so I encourage you to re-read their answers and make sure this point becomes clear.

This means that you can make your code work simply by replacing pc$x with scale(pc$x) in the fourth line from the bottom.

Regarding your second question, with $n<p$, covariance matrix is singular and hence Mahalanobis distance is undefined. However, one can still define it in the subspace where the data actually lie -- this is equivalent to doing PCA and keeping only non-zero components. So the answer is yes. I am not sure though how useful this can be.

Mahalanobis distance is equivalent to the Euclidean distance on the PCA-transformed data (not just PCA-rotated!), where by "PCA-transformed" I mean (i) first rotated to become uncorrelated, and (ii) then scaled to become standardized. This is what @ttnphns said in the comments above and what both @DmitryLaptev and @whuber meant and explicitly wrote in their answers that you linked to (one and two), so I encourage you to re-read their answers and make sure this point becomes clear.

This means that you can make your code work simply by replacing pc$x with scale(pc$x) in the fourth line from the bottom.

Regarding your second question, with $n<p$, covariance matrix is singular and hence Mahalanobis distance is undefined. Indeed, think about Euclidean distance in the PCA-transformed data; when $n<p$, some of the eigenvalues of covariance matrix are zero and the corresponding PCs have zero variance (all the data points are projected to zero). It is therefore impossible to standardize these PCs, as it is impossible to divide by zero. Mahalanobis distance cannot be defined "in these directions".

What one can do, is to focus exclusively on the subspace where the data actually lie, and define Mahalanobis distance in this subspace. This is equivalent to doing PCA and keeping only non-zero components, which is I think what you suggested in your question #2. So the answer to this is yes. I am not sure though how useful this can be in practice, as this distance is likely to be very unstable (near-zero eigenvalues are known with very bad precision, but are going to be inverted in the Mahalanobis formula, possible yielding gross errors).

This is just expanding slightly on @ttnphns remarks in the comments above.

Mahalanobis distance is equivalent to the Euclidean distance on the PCA-transformed data (not just PCA-rotated!), where by "PCA-transformed" I mean (i) first rotated to become uncorrelated, and (ii) then scaled to become standardized. This is what both @DmitryLaptev and @whuber meant and explicitly wrote in their answers that you linked to, so I encourage you to re-read their answers and make sure this point becomes clear.

This means that you can make your code work simply by replacing pc$x with scale(pc$x) in the fourth line from the bottom.

Regarding your second question, with $n<p$, covariance matrix is singular and hence Mahalanobis distance is undefined. However, one can still define it by using the subspace where the data actually lie -- this is equivalent to doing PCA and keeping only non-zero components. So the answer is yes. I am not sure though how useful this can be.

Mahalanobis distance is equivalent to the Euclidean distance on the PCA-transformed data (not just PCA-rotated!), where by "PCA-transformed" I mean (i) first rotated to become uncorrelated, and (ii) then scaled to become standardized. This is what @ttnphns said in the comments above and what both @DmitryLaptev and @whuber meant and explicitly wrote in their answers that you linked to (one and two), so I encourage you to re-read their answers and make sure this point becomes clear.

This means that you can make your code work simply by replacing pc$x with scale(pc$x) in the fourth line from the bottom.

Regarding your second question, with $n<p$, covariance matrix is singular and hence Mahalanobis distance is undefined. However, one can still define it in the subspace where the data actually lie -- this is equivalent to doing PCA and keeping only non-zero components. So the answer is yes. I am not sure though how useful this can be.