Why is PCA sensitive to outliers?
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There are many posts on this SE that discuss robust approaches to Principal Component Analysis (PCA) but I cannot find a single good explanation of why PCA is sensitive to outliers in the first place?
machine-learning pca outliers
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Psi
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up vote
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There are many posts on this SE that discuss robust approaches to Principal Component Analysis (PCA) but I cannot find a single good explanation of why PCA is sensitive to outliers in the first place?
machine-learning pca outliers
asked 3 hours ago
Psi
1235
New contributor
Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
There are many posts on this SE that discuss robust approaches to Principal Component Analysis (PCA) but I cannot find a single good explanation of why PCA is sensitive to outliers in the first place?
machine-learning pca outliers
asked 3 hours ago
Psi
1235
New contributor
Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
There are many posts on this SE that discuss robust approaches to Principal Component Analysis (PCA) but I cannot find a single good explanation of why PCA is sensitive to outliers in the first place?
machine-learning pca outliers
machine-learning pca outliers
asked 3 hours ago
Psi
1235
New contributor
Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
Psi
1235
New contributor
Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 3 hours ago
asked 3 hours ago
Psi
1235
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Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 3 hours ago
Psi
1235
asked 3 hours ago
Psi
1235
1235
New contributor
Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Psi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
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One of the reasons is that PCA can be thought as low-rank decomposition of the data that minimizes the sum of $L_2$ norms of the residuals of the decomposition. I.e. if $Y$ is your data ($m$ vectors of $n$ dimensions), and $X$ is the PCA basis ($k$ vectors of $n$ dimensions), then the decomposition will strictly minimize
$$lVert Y-XA rVert^2_F = sum_{j=1}^{m} lVert Y_j - X A_{j.} rVert^2 $$
Here $A$ is the matrix of coefficients of PCA decomposition and $lVert
cdot rVert_F$ is a Frobenius norm of the matrix
Because the PCA minimizes the $L_2$ norms (i.e. quadratic norms) it has the same issues a least-squares or fitting a Gaussian by being sensitive to outliers.
edited 37 mins ago
dsaxton
9,39811535
answered 52 mins ago
sega_sai
43538
Thank you so much, what an awesome reply! This was exactly what I was looking for and everything makes so much sense the way you explained it!!
– Psi
43 mins ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
One of the reasons is that PCA can be thought as low-rank decomposition of the data that minimizes the sum of $L_2$ norms of the residuals of the decomposition. I.e. if $Y$ is your data ($m$ vectors of $n$ dimensions), and $X$ is the PCA basis ($k$ vectors of $n$ dimensions), then the decomposition will strictly minimize
$$lVert Y-XA rVert^2_F = sum_{j=1}^{m} lVert Y_j - X A_{j.} rVert^2 $$
Here $A$ is the matrix of coefficients of PCA decomposition and $lVert
cdot rVert_F$ is a Frobenius norm of the matrix
Because the PCA minimizes the $L_2$ norms (i.e. quadratic norms) it has the same issues a least-squares or fitting a Gaussian by being sensitive to outliers.
edited 37 mins ago
dsaxton
9,39811535
answered 52 mins ago
sega_sai
43538
Thank you so much, what an awesome reply! This was exactly what I was looking for and everything makes so much sense the way you explained it!!
– Psi
43 mins ago
add a comment |
up vote
5
down vote
accepted
One of the reasons is that PCA can be thought as low-rank decomposition of the data that minimizes the sum of $L_2$ norms of the residuals of the decomposition. I.e. if $Y$ is your data ($m$ vectors of $n$ dimensions), and $X$ is the PCA basis ($k$ vectors of $n$ dimensions), then the decomposition will strictly minimize
$$lVert Y-XA rVert^2_F = sum_{j=1}^{m} lVert Y_j - X A_{j.} rVert^2 $$
Here $A$ is the matrix of coefficients of PCA decomposition and $lVert
cdot rVert_F$ is a Frobenius norm of the matrix
Because the PCA minimizes the $L_2$ norms (i.e. quadratic norms) it has the same issues a least-squares or fitting a Gaussian by being sensitive to outliers.
edited 37 mins ago
dsaxton
9,39811535
answered 52 mins ago
sega_sai
43538
Thank you so much, what an awesome reply! This was exactly what I was looking for and everything makes so much sense the way you explained it!!
– Psi
43 mins ago
add a comment |
up vote
5
down vote
accepted
up vote
5
down vote
accepted
One of the reasons is that PCA can be thought as low-rank decomposition of the data that minimizes the sum of $L_2$ norms of the residuals of the decomposition. I.e. if $Y$ is your data ($m$ vectors of $n$ dimensions), and $X$ is the PCA basis ($k$ vectors of $n$ dimensions), then the decomposition will strictly minimize
$$lVert Y-XA rVert^2_F = sum_{j=1}^{m} lVert Y_j - X A_{j.} rVert^2 $$
Here $A$ is the matrix of coefficients of PCA decomposition and $lVert
cdot rVert_F$ is a Frobenius norm of the matrix
Because the PCA minimizes the $L_2$ norms (i.e. quadratic norms) it has the same issues a least-squares or fitting a Gaussian by being sensitive to outliers.
edited 37 mins ago
dsaxton
9,39811535
answered 52 mins ago
sega_sai
43538
One of the reasons is that PCA can be thought as low-rank decomposition of the data that minimizes the sum of $L_2$ norms of the residuals of the decomposition. I.e. if $Y$ is your data ($m$ vectors of $n$ dimensions), and $X$ is the PCA basis ($k$ vectors of $n$ dimensions), then the decomposition will strictly minimize
$$lVert Y-XA rVert^2_F = sum_{j=1}^{m} lVert Y_j - X A_{j.} rVert^2 $$
Here $A$ is the matrix of coefficients of PCA decomposition and $lVert
cdot rVert_F$ is a Frobenius norm of the matrix
Because the PCA minimizes the $L_2$ norms (i.e. quadratic norms) it has the same issues a least-squares or fitting a Gaussian by being sensitive to outliers.
edited 37 mins ago
dsaxton
9,39811535
answered 52 mins ago
sega_sai
43538
edited 37 mins ago
dsaxton
9,39811535
edited 37 mins ago
dsaxton
9,39811535
edited 37 mins ago
dsaxton
9,39811535
9,39811535
answered 52 mins ago
sega_sai
43538
answered 52 mins ago
sega_sai
43538
answered 52 mins ago
sega_sai
43538
43538
Thank you so much, what an awesome reply! This was exactly what I was looking for and everything makes so much sense the way you explained it!!
– Psi
43 mins ago
add a comment |
Thank you so much, what an awesome reply! This was exactly what I was looking for and everything makes so much sense the way you explained it!!
– Psi
43 mins ago
Thank you so much, what an awesome reply! This was exactly what I was looking for and everything makes so much sense the way you explained it!!
– Psi
43 mins ago
Thank you so much, what an awesome reply! This was exactly what I was looking for and everything makes so much sense the way you explained it!!
– Psi
43 mins ago
add a comment |
Psi is a new contributor. Be nice, and check out our Code of Conduct.
Psi is a new contributor. Be nice, and check out our Code of Conduct.
Psi is a new contributor. Be nice, and check out our Code of Conduct.
Psi is a new contributor. Be nice, and check out our Code of Conduct.
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