A research paper presents an equation for 'correlation distance between between two vectors'. But I cannot...
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Hallo Mathematics StackExchange,
I currently trying to pick apart a research paper titled 'Accounting for Label Unscertainty in Machine Learning for Detection of Acute Respiratory Distress Syndrome' (https://ieeexplore.ieee.org/document/8304750).
The problem that I am having is understanding where the first equation in this paper comes from, and its not cited so I haven't had any luck looking through the sources. But its presented as follows:
To implement this sampling strategy, we first calculated pairwise correlation distance matrices to represent dependency over the span of each patient’s time-series data. Given an m-by-n matrix for each patient’s data, where m is the number of times the patient was observed, and each observation is treated as 1-by-n row vectors, the correlation distance between vectors $X_a$ and $X_b$ for a single pair of observations is defined as:
$$d_{ab} = 1 - frac{(X_a-tilde{X_a})(X_b-tilde{X_b})}{sqrt{(X_a-tilde{X_a})(X_a-tilde{X_a})^prime} sqrt{(X_b-tilde{X_b})(X_b-tilde{X_b})^prime}}$$ where $tilde{X_a}=frac{1}{n}sum_{j}X_{aj}$ and $tilde{X_b}=frac{1}{n}sum_{j}X_{bj}$.
Using this correlation distance formula, an m-by-m correlation distance matrix can be derived for all observations on the patient, taken pairwise.
I have done some reading into correlation distance and have seen quite a few formulas that take the same form as above, but those formulas do not subtract from one. Here is an example (although not quite the same): https://stats.stackexchange.com/questions/269834/how-to-calculate-one-number-from-pearson-correlation-distance-of-more-than-two-v
Also, I do not understand what the primes signify in the denominator. I'm at a loss of where to even begin.
So here are my main questions:
- Why does this equation subtract from one?
- What do the primes signify?
- It seems $tilde{X_a}$ and $tilde{X_b}$ are scalars. I don't know how to interpret the subraction of a scalar from a 1xn vector.
machine-learning correlation
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
$endgroup$
add a comment |
$begingroup$
Hallo Mathematics StackExchange,
I currently trying to pick apart a research paper titled 'Accounting for Label Unscertainty in Machine Learning for Detection of Acute Respiratory Distress Syndrome' (https://ieeexplore.ieee.org/document/8304750).
The problem that I am having is understanding where the first equation in this paper comes from, and its not cited so I haven't had any luck looking through the sources. But its presented as follows:
To implement this sampling strategy, we first calculated pairwise correlation distance matrices to represent dependency over the span of each patient’s time-series data. Given an m-by-n matrix for each patient’s data, where m is the number of times the patient was observed, and each observation is treated as 1-by-n row vectors, the correlation distance between vectors $X_a$ and $X_b$ for a single pair of observations is defined as:
$$d_{ab} = 1 - frac{(X_a-tilde{X_a})(X_b-tilde{X_b})}{sqrt{(X_a-tilde{X_a})(X_a-tilde{X_a})^prime} sqrt{(X_b-tilde{X_b})(X_b-tilde{X_b})^prime}}$$ where $tilde{X_a}=frac{1}{n}sum_{j}X_{aj}$ and $tilde{X_b}=frac{1}{n}sum_{j}X_{bj}$.
Using this correlation distance formula, an m-by-m correlation distance matrix can be derived for all observations on the patient, taken pairwise.
I have done some reading into correlation distance and have seen quite a few formulas that take the same form as above, but those formulas do not subtract from one. Here is an example (although not quite the same): https://stats.stackexchange.com/questions/269834/how-to-calculate-one-number-from-pearson-correlation-distance-of-more-than-two-v
Also, I do not understand what the primes signify in the denominator. I'm at a loss of where to even begin.
So here are my main questions:
- Why does this equation subtract from one?
- What do the primes signify?
- It seems $tilde{X_a}$ and $tilde{X_b}$ are scalars. I don't know how to interpret the subraction of a scalar from a 1xn vector.
machine-learning correlation
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
$endgroup$
$begingroup$
Here a quick start: Primes denote matrix transpose
$endgroup$
– dbx
Dec 3 '18 at 12:37
$begingroup$
The part on the right is the normalised correlation, between 0 and 1
$endgroup$
– Damien
Dec 3 '18 at 13:38
add a comment |
$begingroup$
Hallo Mathematics StackExchange,
I currently trying to pick apart a research paper titled 'Accounting for Label Unscertainty in Machine Learning for Detection of Acute Respiratory Distress Syndrome' (https://ieeexplore.ieee.org/document/8304750).
The problem that I am having is understanding where the first equation in this paper comes from, and its not cited so I haven't had any luck looking through the sources. But its presented as follows:
To implement this sampling strategy, we first calculated pairwise correlation distance matrices to represent dependency over the span of each patient’s time-series data. Given an m-by-n matrix for each patient’s data, where m is the number of times the patient was observed, and each observation is treated as 1-by-n row vectors, the correlation distance between vectors $X_a$ and $X_b$ for a single pair of observations is defined as:
$$d_{ab} = 1 - frac{(X_a-tilde{X_a})(X_b-tilde{X_b})}{sqrt{(X_a-tilde{X_a})(X_a-tilde{X_a})^prime} sqrt{(X_b-tilde{X_b})(X_b-tilde{X_b})^prime}}$$ where $tilde{X_a}=frac{1}{n}sum_{j}X_{aj}$ and $tilde{X_b}=frac{1}{n}sum_{j}X_{bj}$.
Using this correlation distance formula, an m-by-m correlation distance matrix can be derived for all observations on the patient, taken pairwise.
I have done some reading into correlation distance and have seen quite a few formulas that take the same form as above, but those formulas do not subtract from one. Here is an example (although not quite the same): https://stats.stackexchange.com/questions/269834/how-to-calculate-one-number-from-pearson-correlation-distance-of-more-than-two-v
Also, I do not understand what the primes signify in the denominator. I'm at a loss of where to even begin.
So here are my main questions:
- Why does this equation subtract from one?
- What do the primes signify?
- It seems $tilde{X_a}$ and $tilde{X_b}$ are scalars. I don't know how to interpret the subraction of a scalar from a 1xn vector.
machine-learning correlation
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
$endgroup$
Hallo Mathematics StackExchange,
I currently trying to pick apart a research paper titled 'Accounting for Label Unscertainty in Machine Learning for Detection of Acute Respiratory Distress Syndrome' (https://ieeexplore.ieee.org/document/8304750).
The problem that I am having is understanding where the first equation in this paper comes from, and its not cited so I haven't had any luck looking through the sources. But its presented as follows:
To implement this sampling strategy, we first calculated pairwise correlation distance matrices to represent dependency over the span of each patient’s time-series data. Given an m-by-n matrix for each patient’s data, where m is the number of times the patient was observed, and each observation is treated as 1-by-n row vectors, the correlation distance between vectors $X_a$ and $X_b$ for a single pair of observations is defined as:
$$d_{ab} = 1 - frac{(X_a-tilde{X_a})(X_b-tilde{X_b})}{sqrt{(X_a-tilde{X_a})(X_a-tilde{X_a})^prime} sqrt{(X_b-tilde{X_b})(X_b-tilde{X_b})^prime}}$$ where $tilde{X_a}=frac{1}{n}sum_{j}X_{aj}$ and $tilde{X_b}=frac{1}{n}sum_{j}X_{bj}$.
Using this correlation distance formula, an m-by-m correlation distance matrix can be derived for all observations on the patient, taken pairwise.
I have done some reading into correlation distance and have seen quite a few formulas that take the same form as above, but those formulas do not subtract from one. Here is an example (although not quite the same): https://stats.stackexchange.com/questions/269834/how-to-calculate-one-number-from-pearson-correlation-distance-of-more-than-two-v
Also, I do not understand what the primes signify in the denominator. I'm at a loss of where to even begin.
So here are my main questions:
- Why does this equation subtract from one?
- What do the primes signify?
- It seems $tilde{X_a}$ and $tilde{X_b}$ are scalars. I don't know how to interpret the subraction of a scalar from a 1xn vector.
machine-learning correlation
machine-learning correlation
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
edited Dec 3 '18 at 12:26
Aaron Elliott
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
asked Dec 3 '18 at 12:18
Aaron ElliottAaron Elliott
11
11
$begingroup$
Here a quick start: Primes denote matrix transpose
$endgroup$
– dbx
Dec 3 '18 at 12:37
$begingroup$
The part on the right is the normalised correlation, between 0 and 1
$endgroup$
– Damien
Dec 3 '18 at 13:38
add a comment |
$begingroup$
Here a quick start: Primes denote matrix transpose
$endgroup$
– dbx
Dec 3 '18 at 12:37
$begingroup$
The part on the right is the normalised correlation, between 0 and 1
$endgroup$
– Damien
Dec 3 '18 at 13:38
$begingroup$
Here a quick start: Primes denote matrix transpose
$endgroup$
– dbx
Dec 3 '18 at 12:37
$begingroup$
Here a quick start: Primes denote matrix transpose
$endgroup$
– dbx
Dec 3 '18 at 12:37
$begingroup$
The part on the right is the normalised correlation, between 0 and 1
$endgroup$
– Damien
Dec 3 '18 at 13:38
$begingroup$
The part on the right is the normalised correlation, between 0 and 1
$endgroup$
– Damien
Dec 3 '18 at 13:38
add a comment |
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$begingroup$
Here a quick start: Primes denote matrix transpose
$endgroup$
– dbx
Dec 3 '18 at 12:37
$begingroup$
The part on the right is the normalised correlation, between 0 and 1
$endgroup$
– Damien
Dec 3 '18 at 13:38