Extrapolation error of linear regression lines for a two-cluster data set
I am studying on some linear regression problems using the least-squares method and stumbled upon a problem regarding the error when extrapolating far-away datapoints.
About the problem: For a point $y$ given by best fitting line, described by
$y = ax + b$,
where $a$ and $b$ are the coefficients achieved by applying the least-squares-method to a given data set,
we know that generally, the error or the variance for $y$, will increase, the further we move away from our actual data, while the variance will be minimal at the mean position of $x$ of our given data set. This is so far totally clear for me.
But now I wondered:
Say, I have a given data set that consists of two clusters, with many datapoints around a very small negative position $j$ and another cluster at a very big positive position $k$.
Wouldn't the overall error of the best fitting line be decreased now? Will our line, for example for interpolating points in between, be a better fit with this kind of measurement, than a line that comes from only one data cluster int the 'middle' of the two-cluster version?
I hope I explained my problem sufficiently and I am excited for your suggestions!
regression least-squares data-analysis error-propagation
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I am studying on some linear regression problems using the least-squares method and stumbled upon a problem regarding the error when extrapolating far-away datapoints.
About the problem: For a point $y$ given by best fitting line, described by
$y = ax + b$,
where $a$ and $b$ are the coefficients achieved by applying the least-squares-method to a given data set,
we know that generally, the error or the variance for $y$, will increase, the further we move away from our actual data, while the variance will be minimal at the mean position of $x$ of our given data set. This is so far totally clear for me.
But now I wondered:
Say, I have a given data set that consists of two clusters, with many datapoints around a very small negative position $j$ and another cluster at a very big positive position $k$.
Wouldn't the overall error of the best fitting line be decreased now? Will our line, for example for interpolating points in between, be a better fit with this kind of measurement, than a line that comes from only one data cluster int the 'middle' of the two-cluster version?
I hope I explained my problem sufficiently and I am excited for your suggestions!
regression least-squares data-analysis error-propagation
edited yesterday
t.ysn
1257
asked yesterday
felixei
11
New contributor
felixei 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 |
I am studying on some linear regression problems using the least-squares method and stumbled upon a problem regarding the error when extrapolating far-away datapoints.
About the problem: For a point $y$ given by best fitting line, described by
$y = ax + b$,
where $a$ and $b$ are the coefficients achieved by applying the least-squares-method to a given data set,
we know that generally, the error or the variance for $y$, will increase, the further we move away from our actual data, while the variance will be minimal at the mean position of $x$ of our given data set. This is so far totally clear for me.
But now I wondered:
Say, I have a given data set that consists of two clusters, with many datapoints around a very small negative position $j$ and another cluster at a very big positive position $k$.
Wouldn't the overall error of the best fitting line be decreased now? Will our line, for example for interpolating points in between, be a better fit with this kind of measurement, than a line that comes from only one data cluster int the 'middle' of the two-cluster version?
I hope I explained my problem sufficiently and I am excited for your suggestions!
regression least-squares data-analysis error-propagation
edited yesterday
t.ysn
1257
asked yesterday
felixei
11
New contributor
felixei is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I am studying on some linear regression problems using the least-squares method and stumbled upon a problem regarding the error when extrapolating far-away datapoints.
About the problem: For a point $y$ given by best fitting line, described by
$y = ax + b$,
where $a$ and $b$ are the coefficients achieved by applying the least-squares-method to a given data set,
we know that generally, the error or the variance for $y$, will increase, the further we move away from our actual data, while the variance will be minimal at the mean position of $x$ of our given data set. This is so far totally clear for me.
But now I wondered:
Say, I have a given data set that consists of two clusters, with many datapoints around a very small negative position $j$ and another cluster at a very big positive position $k$.
Wouldn't the overall error of the best fitting line be decreased now? Will our line, for example for interpolating points in between, be a better fit with this kind of measurement, than a line that comes from only one data cluster int the 'middle' of the two-cluster version?
I hope I explained my problem sufficiently and I am excited for your suggestions!
regression least-squares data-analysis error-propagation
regression least-squares data-analysis error-propagation
edited yesterday
t.ysn
1257
asked yesterday
felixei
11
New contributor
felixei is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
t.ysn
1257
asked yesterday
felixei
11
New contributor
felixei is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited yesterday
t.ysn
1257
edited yesterday
t.ysn
1257
edited yesterday
t.ysn
1257
1257
asked yesterday
felixei
11
New contributor
felixei is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
felixei
11
asked yesterday
felixei
11
11
New contributor
felixei is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
felixei is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Check out our Code of Conduct.
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