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Optimizing vector values for maximum correlation
2019 Community Moderator ElectionHow to test for Overall Significance?If my data looks like this, is linear regression just never gonna be useful?Converting feature coefficients to probabilities scores for linear modelsDeriving backpropagation equations “natively” in tensor formRemoving Categorial Features in Linear RegressionMy Neural network in Tensorflow does a bad job in comparison to the same Neural network in KerasDealing with diverse groups in regressioncomparison of linear Q-learning and DQNCan we think of neurons as maps between matrices?Can I use Linear Regression to model a nonlinear function?
$begingroup$
I'm new to ML, linear algebra, statistics, etc. so bear with me on the terminology...
I’m looking to find a vector that produces the maximum correlation for the relationship between 1) all relationships among dimensions of the vector (determined by subtraction) and 2) some output value produced by said relationships. I'm specifically using this to create a sports ranking system that takes a number of matches and the resulting scores and attempts to assign a value to the teams that can be used to predict future scores. In other words, the difference between any two team's ratings should be predictive of the next match between the two.
So for example, if I have 3 teams, A, B, and C, each with unknown ratings:
$$
beginarrayc
A&?\
B&?\
C&?\
endarray
$$
If each team played the other once, the left table would be used to calculate their rating differences (column team's rating minus row team's rating). The right table would be the difference in scores in the respective matchups.
$$
beginequation
beginarrayc
&A&B&C \
hline
A&*& B - A& C - A\
B&A - B& *& C - B\
C&A - C& B - C& *\
endarray
Rightarrow
beginarrayc
&A&B&C \
hline
A&0&3&6\
B&-3&0&3\
C&-6&-3&0\
endarray
endequation
$$
Here is a possible solution that, for this example, would result in a perfect correlation between team rating differentials and score differentials.
$$
beginarrayc
A&1\
B&2\
C&3\
endarray
$$
This would be the regression line where x2 is the column team's rating and x1 is the row team's rating.
$$y = 3 * (x_2 - x_1)$$
It’s worth noting that what matters is the relationship between the various values (not their nominal values) since this would be another possible solution:
$$
beginarrayc
A&2\
B&4\
C&6\
endarray
$$
Which would result in a linear equation that looks like this, which would also have a correlation of 1:
$$y = 3over 2 * (x_2 - x_1)$$
What I want to do is find a method to determine values for A, B, and C that maximizes the correlation between the pairwise differences and resulting output values. The one additional catch for the teams example is that not every team will play every other team so any resulting matrices will be asymmetrical (assuming that matters).
Are there any existing techniques to address this problem?
linear-regression linear-algebra
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I'm new to ML, linear algebra, statistics, etc. so bear with me on the terminology...
I’m looking to find a vector that produces the maximum correlation for the relationship between 1) all relationships among dimensions of the vector (determined by subtraction) and 2) some output value produced by said relationships. I'm specifically using this to create a sports ranking system that takes a number of matches and the resulting scores and attempts to assign a value to the teams that can be used to predict future scores. In other words, the difference between any two team's ratings should be predictive of the next match between the two.
So for example, if I have 3 teams, A, B, and C, each with unknown ratings:
$$
beginarrayc
A&?\
B&?\
C&?\
endarray
$$
If each team played the other once, the left table would be used to calculate their rating differences (column team's rating minus row team's rating). The right table would be the difference in scores in the respective matchups.
$$
beginequation
beginarrayc
&A&B&C \
hline
A&*& B - A& C - A\
B&A - B& *& C - B\
C&A - C& B - C& *\
endarray
Rightarrow
beginarrayc
&A&B&C \
hline
A&0&3&6\
B&-3&0&3\
C&-6&-3&0\
endarray
endequation
$$
Here is a possible solution that, for this example, would result in a perfect correlation between team rating differentials and score differentials.
$$
beginarrayc
A&1\
B&2\
C&3\
endarray
$$
This would be the regression line where x2 is the column team's rating and x1 is the row team's rating.
$$y = 3 * (x_2 - x_1)$$
It’s worth noting that what matters is the relationship between the various values (not their nominal values) since this would be another possible solution:
$$
beginarrayc
A&2\
B&4\
C&6\
endarray
$$
Which would result in a linear equation that looks like this, which would also have a correlation of 1:
$$y = 3over 2 * (x_2 - x_1)$$
What I want to do is find a method to determine values for A, B, and C that maximizes the correlation between the pairwise differences and resulting output values. The one additional catch for the teams example is that not every team will play every other team so any resulting matrices will be asymmetrical (assuming that matters).
Are there any existing techniques to address this problem?
linear-regression linear-algebra
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I'm new to ML, linear algebra, statistics, etc. so bear with me on the terminology...
I’m looking to find a vector that produces the maximum correlation for the relationship between 1) all relationships among dimensions of the vector (determined by subtraction) and 2) some output value produced by said relationships. I'm specifically using this to create a sports ranking system that takes a number of matches and the resulting scores and attempts to assign a value to the teams that can be used to predict future scores. In other words, the difference between any two team's ratings should be predictive of the next match between the two.
So for example, if I have 3 teams, A, B, and C, each with unknown ratings:
$$
beginarrayc
A&?\
B&?\
C&?\
endarray
$$
If each team played the other once, the left table would be used to calculate their rating differences (column team's rating minus row team's rating). The right table would be the difference in scores in the respective matchups.
$$
beginequation
beginarrayc
&A&B&C \
hline
A&*& B - A& C - A\
B&A - B& *& C - B\
C&A - C& B - C& *\
endarray
Rightarrow
beginarrayc
&A&B&C \
hline
A&0&3&6\
B&-3&0&3\
C&-6&-3&0\
endarray
endequation
$$
Here is a possible solution that, for this example, would result in a perfect correlation between team rating differentials and score differentials.
$$
beginarrayc
A&1\
B&2\
C&3\
endarray
$$
This would be the regression line where x2 is the column team's rating and x1 is the row team's rating.
$$y = 3 * (x_2 - x_1)$$
It’s worth noting that what matters is the relationship between the various values (not their nominal values) since this would be another possible solution:
$$
beginarrayc
A&2\
B&4\
C&6\
endarray
$$
Which would result in a linear equation that looks like this, which would also have a correlation of 1:
$$y = 3over 2 * (x_2 - x_1)$$
What I want to do is find a method to determine values for A, B, and C that maximizes the correlation between the pairwise differences and resulting output values. The one additional catch for the teams example is that not every team will play every other team so any resulting matrices will be asymmetrical (assuming that matters).
Are there any existing techniques to address this problem?
linear-regression linear-algebra
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I'm new to ML, linear algebra, statistics, etc. so bear with me on the terminology...
I’m looking to find a vector that produces the maximum correlation for the relationship between 1) all relationships among dimensions of the vector (determined by subtraction) and 2) some output value produced by said relationships. I'm specifically using this to create a sports ranking system that takes a number of matches and the resulting scores and attempts to assign a value to the teams that can be used to predict future scores. In other words, the difference between any two team's ratings should be predictive of the next match between the two.
So for example, if I have 3 teams, A, B, and C, each with unknown ratings:
$$
beginarrayc
A&?\
B&?\
C&?\
endarray
$$
If each team played the other once, the left table would be used to calculate their rating differences (column team's rating minus row team's rating). The right table would be the difference in scores in the respective matchups.
$$
beginequation
beginarrayc
&A&B&C \
hline
A&*& B - A& C - A\
B&A - B& *& C - B\
C&A - C& B - C& *\
endarray
Rightarrow
beginarrayc
&A&B&C \
hline
A&0&3&6\
B&-3&0&3\
C&-6&-3&0\
endarray
endequation
$$
Here is a possible solution that, for this example, would result in a perfect correlation between team rating differentials and score differentials.
$$
beginarrayc
A&1\
B&2\
C&3\
endarray
$$
This would be the regression line where x2 is the column team's rating and x1 is the row team's rating.
$$y = 3 * (x_2 - x_1)$$
It’s worth noting that what matters is the relationship between the various values (not their nominal values) since this would be another possible solution:
$$
beginarrayc
A&2\
B&4\
C&6\
endarray
$$
Which would result in a linear equation that looks like this, which would also have a correlation of 1:
$$y = 3over 2 * (x_2 - x_1)$$
What I want to do is find a method to determine values for A, B, and C that maximizes the correlation between the pairwise differences and resulting output values. The one additional catch for the teams example is that not every team will play every other team so any resulting matrices will be asymmetrical (assuming that matters).
Are there any existing techniques to address this problem?
linear-regression linear-algebra
linear-regression linear-algebra
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 37 mins ago
SuperCodeBrah
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
asked 4 hours ago
SuperCodeBrahSuperCodeBrah
1011
1011
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
SuperCodeBrah is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
SuperCodeBrah 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 |
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