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Probability before and after PCA projection

Combine n Normal distribution Probability Sets in a limited float rangeProbability of drawing an Ace: before and afterBefore and after training running timesorhogonal projectionProbability After Sampling Without Replacement Until SuccessConditional Sample from Gaussian CopulaGeometric Probability: P(Bob comes before 1:30 and Alice comes after Bob)Expected Value where probability changes after successAverage of conditional probabilities and co-occuranceProbability of rolling first 3 with a fair die before 10th roll and after 4th roll.

0

$begingroup$

If there is a set of points generated from a multivariate normal distribution with mean and covariance matrix:

mean=[1, 2]; covariance=[5, -2; -2, 3];

Data in original space

And is thereafter projected into PCA-space using the eigenvectors of the covariance:

PCA_cov=[-5.3, 0.93; 3.28, 1.5]; data_proj=PCA_cov^(-1)*data;

Projection into PCA space

The projected data now have mean and covariance:

mean_PC=[0.034, 1.26]; sigma=[0.4, 0.75];

Why does a projected point from the original- to PCA-space not exhibit the same probability?

It is the red squares in the previous images that have been used to calculate the graph below.

Calculated probability of data and projection

In the multivariate case it is calculated with SciPy.stats.multivariate_normal package, should therefore be correct.
In the projected space it is again calculate with SciPy, not multivariate. The probability in both axis is then multiplied together.

I guess the question can be summarized as: Why is the probability of a point and its projection not the same?

Thanks in advance!

probability probability-distributions projective-space

share|cite|improve this question

asked Mar 11 at 10:33

MichaelMichael

11

New contributor

Michael is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  My question is as read at the end: Why is the probability of a point and its projection not the same? It is not projected to a lower dimension just rotated and scaled into the orthogonal eigenspace
  $endgroup$
  – Michael
  Mar 12 at 8:07
  
  
  

  
  
  
  

add a comment | 

0

$begingroup$

If there is a set of points generated from a multivariate normal distribution with mean and covariance matrix:

mean=[1, 2]; covariance=[5, -2; -2, 3];

Data in original space

And is thereafter projected into PCA-space using the eigenvectors of the covariance:

PCA_cov=[-5.3, 0.93; 3.28, 1.5]; data_proj=PCA_cov^(-1)*data;

Projection into PCA space

The projected data now have mean and covariance:

mean_PC=[0.034, 1.26]; sigma=[0.4, 0.75];

Why does a projected point from the original- to PCA-space not exhibit the same probability?

It is the red squares in the previous images that have been used to calculate the graph below.

Calculated probability of data and projection

In the multivariate case it is calculated with SciPy.stats.multivariate_normal package, should therefore be correct.
In the projected space it is again calculate with SciPy, not multivariate. The probability in both axis is then multiplied together.

I guess the question can be summarized as: Why is the probability of a point and its projection not the same?

Thanks in advance!

probability probability-distributions projective-space

share|cite|improve this question

asked Mar 11 at 10:33

MichaelMichael

11

New contributor

Michael is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  My question is as read at the end: Why is the probability of a point and its projection not the same? It is not projected to a lower dimension just rotated and scaled into the orthogonal eigenspace
  $endgroup$
  – Michael
  Mar 12 at 8:07
  
  
  

  
  
  
  

add a comment | 

$begingroup$

If there is a set of points generated from a multivariate normal distribution with mean and covariance matrix:

mean=[1, 2]; covariance=[5, -2; -2, 3];

Data in original space

And is thereafter projected into PCA-space using the eigenvectors of the covariance:

PCA_cov=[-5.3, 0.93; 3.28, 1.5]; data_proj=PCA_cov^(-1)*data;

Projection into PCA space

The projected data now have mean and covariance:

mean_PC=[0.034, 1.26]; sigma=[0.4, 0.75];

Why does a projected point from the original- to PCA-space not exhibit the same probability?

It is the red squares in the previous images that have been used to calculate the graph below.

Calculated probability of data and projection

In the multivariate case it is calculated with SciPy.stats.multivariate_normal package, should therefore be correct.
In the projected space it is again calculate with SciPy, not multivariate. The probability in both axis is then multiplied together.

I guess the question can be summarized as: Why is the probability of a point and its projection not the same?

Thanks in advance!

probability probability-distributions projective-space

share|cite|improve this question

asked Mar 11 at 10:33

MichaelMichael

11

New contributor

Michael is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

mean=[1, 2]; covariance=[5, -2; -2, 3];

PCA_cov=[-5.3, 0.93; 3.28, 1.5]; data_proj=PCA_cov^(-1)*data;

mean_PC=[0.034, 1.26]; sigma=[0.4, 0.75];

share|cite|improve this question

asked Mar 11 at 10:33

MichaelMichael

11

New contributor

Michael is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked Mar 11 at 10:33

MichaelMichael

11

New contributor

Michael is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 11 at 10:33

MichaelMichael

11

New contributor

Michael is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 11 at 10:33

MichaelMichael

11