Approximating measurement data with $ae^{bx}+c$
$begingroup$
I have measurement data which looks like the sum of an exponential and a constant function. It is enough to see it as a continuous function, say $f(x)$.
I am looking for the $a$, $b$, $c$ values for which
$$intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx$$
minimal.
I think the problem is well-known, although typically such functions are approximated with polynoms.
What I did:
First I've tried to find the local minimums, so I created an equation system using
$$frac{d}{d{a,b,c}}intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx=0$$
The result was a practically not calculable equation system, after filling some sheet of paper with formulas. Using the Lambert W wouldn't be a problem for me, but the situation was too complex to apply even that.
However, such a well-known and probably useful problem probably has already some results.
Actually, even a local minimum would be already useful, I don't suspect multiple local minima in the $(a,b,c) in mathbb{R}^3$ space.
Or, maybe an entirely different direction should be used?
exponential-function approximation lambert-w
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
$endgroup$
add a comment |
$begingroup$
I have measurement data which looks like the sum of an exponential and a constant function. It is enough to see it as a continuous function, say $f(x)$.
I am looking for the $a$, $b$, $c$ values for which
$$intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx$$
minimal.
I think the problem is well-known, although typically such functions are approximated with polynoms.
What I did:
First I've tried to find the local minimums, so I created an equation system using
$$frac{d}{d{a,b,c}}intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx=0$$
The result was a practically not calculable equation system, after filling some sheet of paper with formulas. Using the Lambert W wouldn't be a problem for me, but the situation was too complex to apply even that.
However, such a well-known and probably useful problem probably has already some results.
Actually, even a local minimum would be already useful, I don't suspect multiple local minima in the $(a,b,c) in mathbb{R}^3$ space.
Or, maybe an entirely different direction should be used?
exponential-function approximation lambert-w
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
$endgroup$
1
$begingroup$
are you sure to want a minimizing of the linear error, and not of a log error ?
$endgroup$
– G Cab
Dec 19 '18 at 23:52
$begingroup$
@GCab In the current case, yes. In a broader sense, the data looks like the sum of an exponential and a constant function, thus I think trying to minimize the logarithmic error wouldn't help too much.
$endgroup$
– peterh
Dec 20 '18 at 9:42
add a comment |
$begingroup$
I have measurement data which looks like the sum of an exponential and a constant function. It is enough to see it as a continuous function, say $f(x)$.
I am looking for the $a$, $b$, $c$ values for which
$$intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx$$
minimal.
I think the problem is well-known, although typically such functions are approximated with polynoms.
What I did:
First I've tried to find the local minimums, so I created an equation system using
$$frac{d}{d{a,b,c}}intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx=0$$
The result was a practically not calculable equation system, after filling some sheet of paper with formulas. Using the Lambert W wouldn't be a problem for me, but the situation was too complex to apply even that.
However, such a well-known and probably useful problem probably has already some results.
Actually, even a local minimum would be already useful, I don't suspect multiple local minima in the $(a,b,c) in mathbb{R}^3$ space.
Or, maybe an entirely different direction should be used?
exponential-function approximation lambert-w
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
$endgroup$
I have measurement data which looks like the sum of an exponential and a constant function. It is enough to see it as a continuous function, say $f(x)$.
I am looking for the $a$, $b$, $c$ values for which
$$intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx$$
minimal.
I think the problem is well-known, although typically such functions are approximated with polynoms.
What I did:
First I've tried to find the local minimums, so I created an equation system using
$$frac{d}{d{a,b,c}}intlimits_0^1big(ae^{bx}+c-f(x)big)^2dx=0$$
The result was a practically not calculable equation system, after filling some sheet of paper with formulas. Using the Lambert W wouldn't be a problem for me, but the situation was too complex to apply even that.
However, such a well-known and probably useful problem probably has already some results.
Actually, even a local minimum would be already useful, I don't suspect multiple local minima in the $(a,b,c) in mathbb{R}^3$ space.
Or, maybe an entirely different direction should be used?
exponential-function approximation lambert-w
exponential-function approximation lambert-w
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
asked Dec 19 '18 at 21:22
peterhpeterh
2,17351731
2,17351731
1
$begingroup$
are you sure to want a minimizing of the linear error, and not of a log error ?
$endgroup$
– G Cab
Dec 19 '18 at 23:52
$begingroup$
@GCab In the current case, yes. In a broader sense, the data looks like the sum of an exponential and a constant function, thus I think trying to minimize the logarithmic error wouldn't help too much.
$endgroup$
– peterh
Dec 20 '18 at 9:42
add a comment |
1
$begingroup$
are you sure to want a minimizing of the linear error, and not of a log error ?
$endgroup$
– G Cab
Dec 19 '18 at 23:52
$begingroup$
@GCab In the current case, yes. In a broader sense, the data looks like the sum of an exponential and a constant function, thus I think trying to minimize the logarithmic error wouldn't help too much.
$endgroup$
– peterh
Dec 20 '18 at 9:42
1
1
$begingroup$
are you sure to want a minimizing of the linear error, and not of a log error ?
$endgroup$
– G Cab
Dec 19 '18 at 23:52
$begingroup$
are you sure to want a minimizing of the linear error, and not of a log error ?
$endgroup$
– G Cab
Dec 19 '18 at 23:52
$begingroup$
@GCab In the current case, yes. In a broader sense, the data looks like the sum of an exponential and a constant function, thus I think trying to minimize the logarithmic error wouldn't help too much.
$endgroup$
– peterh
Dec 20 '18 at 9:42
$begingroup$
@GCab In the current case, yes. In a broader sense, the data looks like the sum of an exponential and a constant function, thus I think trying to minimize the logarithmic error wouldn't help too much.
$endgroup$
– peterh
Dec 20 '18 at 9:42
add a comment |
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$begingroup$
are you sure to want a minimizing of the linear error, and not of a log error ?
$endgroup$
– G Cab
Dec 19 '18 at 23:52
$begingroup$
@GCab In the current case, yes. In a broader sense, the data looks like the sum of an exponential and a constant function, thus I think trying to minimize the logarithmic error wouldn't help too much.
$endgroup$
– peterh
Dec 20 '18 at 9:42