Probability before and after PCA projectionCombine 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.
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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.
$begingroup$
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
New contributor
$endgroup$
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
New contributor
$endgroup$
$begingroup$
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
2 days ago
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
New contributor
$endgroup$
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
probability probability-distributions projective-space
New contributor
New contributor
New contributor
asked Mar 11 at 10:33
MichaelMichael
11
11
New contributor
New contributor
$begingroup$
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
2 days ago
add a comment |
$begingroup$
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
2 days ago
$begingroup$
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
2 days ago
$begingroup$
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
2 days ago
add a comment |
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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
2 days ago