Geometric mean, harmonic mean and loss functions
$begingroup$
Consider a sequence $(x_i)_{i in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss
$$ text{mode}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^0 $$
The median is the minimizing argument for the $L_1$ loss
$$ text{median}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^1 $$
The arithmetic mean is the minimizing argument for the $L_2$ loss
$$ text{arithmetic mean}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^2 $$
Can we find similar results for the harmonic mean or the geometric mean?
Thanks!
statistics average means
edited Mar 5 at 20:02
mikado
1217
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
$endgroup$
add a comment |
$begingroup$
Consider a sequence $(x_i)_{i in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss
$$ text{mode}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^0 $$
The median is the minimizing argument for the $L_1$ loss
$$ text{median}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^1 $$
The arithmetic mean is the minimizing argument for the $L_2$ loss
$$ text{arithmetic mean}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^2 $$
Can we find similar results for the harmonic mean or the geometric mean?
Thanks!
statistics average means
edited Mar 5 at 20:02
mikado
1217
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
$endgroup$
add a comment |
$begingroup$
Consider a sequence $(x_i)_{i in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss
$$ text{mode}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^0 $$
The median is the minimizing argument for the $L_1$ loss
$$ text{median}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^1 $$
The arithmetic mean is the minimizing argument for the $L_2$ loss
$$ text{arithmetic mean}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^2 $$
Can we find similar results for the harmonic mean or the geometric mean?
Thanks!
statistics average means
edited Mar 5 at 20:02
mikado
1217
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
$endgroup$
Consider a sequence $(x_i)_{i in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss
$$ text{mode}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^0 $$
The median is the minimizing argument for the $L_1$ loss
$$ text{median}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^1 $$
The arithmetic mean is the minimizing argument for the $L_2$ loss
$$ text{arithmetic mean}[ ; (x_i)_{i in I} ; ] = argmin_{u in mathbb{R}} ; ; sum_{i in I} ; | x_i - u|^2 $$
Can we find similar results for the harmonic mean or the geometric mean?
Thanks!
statistics average means
statistics average means
edited Mar 5 at 20:02
mikado
1217
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
edited Mar 5 at 20:02
mikado
1217
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
edited Mar 5 at 20:02
mikado
1217
edited Mar 5 at 20:02
mikado
1217
edited Mar 5 at 20:02
mikado
1217
1217
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
asked Jul 27 '16 at 16:48
Estienne GranetEstienne Granet
1213
1213
add a comment |
add a comment |
1 Answer
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$begingroup$
A simple manipulation of the relationship between arithmetic and geometric mean gives the following expression for the geometric mean
$$arg min_{u in mathbb{R}^+} sum_{i in I}|log frac{x_i}{u}|^2$$
Interestingly, this might be generalised to complex values.
answered Mar 4 at 22:56
mikadomikado
1217
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add a comment |
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1 Answer
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$begingroup$
A simple manipulation of the relationship between arithmetic and geometric mean gives the following expression for the geometric mean
$$arg min_{u in mathbb{R}^+} sum_{i in I}|log frac{x_i}{u}|^2$$
Interestingly, this might be generalised to complex values.
answered Mar 4 at 22:56
mikadomikado
1217
$endgroup$
add a comment |
$begingroup$
A simple manipulation of the relationship between arithmetic and geometric mean gives the following expression for the geometric mean
$$arg min_{u in mathbb{R}^+} sum_{i in I}|log frac{x_i}{u}|^2$$
Interestingly, this might be generalised to complex values.
answered Mar 4 at 22:56
mikadomikado
1217
$endgroup$
add a comment |
$begingroup$
A simple manipulation of the relationship between arithmetic and geometric mean gives the following expression for the geometric mean
$$arg min_{u in mathbb{R}^+} sum_{i in I}|log frac{x_i}{u}|^2$$
Interestingly, this might be generalised to complex values.
answered Mar 4 at 22:56
mikadomikado
1217
$endgroup$
A simple manipulation of the relationship between arithmetic and geometric mean gives the following expression for the geometric mean
$$arg min_{u in mathbb{R}^+} sum_{i in I}|log frac{x_i}{u}|^2$$
Interestingly, this might be generalised to complex values.
answered Mar 4 at 22:56
mikadomikado
1217
edited Mar 4 at 23:09
answered Mar 4 at 22:56
mikadomikado
1217
answered Mar 4 at 22:56
mikadomikado
1217
answered Mar 4 at 22:56
mikadomikado
1217
1217
add a comment |
add a comment |
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