Linearizing two variable function

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I have another linearization question similar to the one in here. This time, I have got two variables in my equation and I am in search of an "$A+Brho$" or possibly "$A+Brho+Ctheta$" approximation. Here is my equation:

$$W = fractheta2(1-rho)$$

where $theta,rhoin mathbbR^+$ and $rhoin[0,1)$ i.e., $0leq rho <1$.

I tried to come up with "$A+Brho$", although I feel like the correct form of the linearization should be "$A+Brho+Ctheta$". I followed Leibovici's linear regression method with Taylor series.

I minimized the norm:

$$F = int_a^b left(A + B rho - fractheta2(1-rho)right)^2$$

After integration, I came up with the following two equations:

$fracpartial Wpartial A = - 2 A a + 2 A b - B a^2 + B b^2 - theta logleft (a - 1 right ) + theta logleft (b - 1 right )$
$fracpartial Wpartial B = - A a^2 + A b^2 - frac2 B a^33 + frac2 B b^33 - a theta + b theta - theta logleft (a - 1 right ) + theta logleft (b - 1 right )$

Setting $a=0.0$ and $b=0.1$, I came up with the following approximation (which is still nonlinear):

$Wapprox 0.499055theta + 0.554939thetarho$

I do not know if this makes life easier or not, but, we have the following relationship between $rho$ and $theta$:

$$rho = sum_jin J fraclambda_jmu_j$$
and
$$theta = sum_jin J fraclambda_jmu_j^2$$

Additionally, I am not really in search of a Newton/Newton-Raphson linearization, as I believe a linear line with a single point approximation does not satisfactorily represent the curve in this case. Considering $thetain mathbbR^+$, I do not think, Newton derived methods would help me.

Any recommendation is appreciated.

asked Jul 25 at 18:01

user8028576

277

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up vote
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$$W = fractheta2(1-rho)$$

where $theta,rhoin mathbbR^+$ and $rhoin[0,1)$ i.e., $0leq rho <1$.

I tried to come up with "$A+Brho$", although I feel like the correct form of the linearization should be "$A+Brho+Ctheta$". I followed Leibovici's linear regression method with Taylor series.

I minimized the norm:

$$F = int_a^b left(A + B rho - fractheta2(1-rho)right)^2$$

After integration, I came up with the following two equations:

Setting $a=0.0$ and $b=0.1$, I came up with the following approximation (which is still nonlinear):

$Wapprox 0.499055theta + 0.554939thetarho$

I do not know if this makes life easier or not, but, we have the following relationship between $rho$ and $theta$:

$$rho = sum_jin J fraclambda_jmu_j$$
and
$$theta = sum_jin J fraclambda_jmu_j^2$$

Any recommendation is appreciated.

asked Jul 25 at 18:01

user8028576

277

add a commentÂ |Â

up vote
0
down vote

favorite

$$W = fractheta2(1-rho)$$

where $theta,rhoin mathbbR^+$ and $rhoin[0,1)$ i.e., $0leq rho <1$.

I tried to come up with "$A+Brho$", although I feel like the correct form of the linearization should be "$A+Brho+Ctheta$". I followed Leibovici's linear regression method with Taylor series.

I minimized the norm:

$$F = int_a^b left(A + B rho - fractheta2(1-rho)right)^2$$

After integration, I came up with the following two equations:

Setting $a=0.0$ and $b=0.1$, I came up with the following approximation (which is still nonlinear):

$Wapprox 0.499055theta + 0.554939thetarho$

I do not know if this makes life easier or not, but, we have the following relationship between $rho$ and $theta$:

$$rho = sum_jin J fraclambda_jmu_j$$
and
$$theta = sum_jin J fraclambda_jmu_j^2$$

Any recommendation is appreciated.

asked Jul 25 at 18:01

user8028576

277

$$W = fractheta2(1-rho)$$

where $theta,rhoin mathbbR^+$ and $rhoin[0,1)$ i.e., $0leq rho <1$.

I tried to come up with "$A+Brho$", although I feel like the correct form of the linearization should be "$A+Brho+Ctheta$". I followed Leibovici's linear regression method with Taylor series.

I minimized the norm:

$$F = int_a^b left(A + B rho - fractheta2(1-rho)right)^2$$

After integration, I came up with the following two equations:

Setting $a=0.0$ and $b=0.1$, I came up with the following approximation (which is still nonlinear):

$Wapprox 0.499055theta + 0.554939thetarho$

I do not know if this makes life easier or not, but, we have the following relationship between $rho$ and $theta$:

$$rho = sum_jin J fraclambda_jmu_j$$
and
$$theta = sum_jin J fraclambda_jmu_j^2$$

Any recommendation is appreciated.

asked Jul 25 at 18:01

user8028576

277

asked Jul 25 at 18:01

user8028576

277

asked Jul 25 at 18:01

user8028576

277

asked Jul 25 at 18:01

user8028576

277

add a commentÂ |Â

1 Answer
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Since nos you face a surface, we can consider that we need to minimize $$G = int_a^b int_c^dleft(A + B rho+C theta - fractheta2(1-rho)right)^2,drho,dtheta$$ with respect to $A,B,C$.

I shall not reproduce here the analytical expression of neither $G$ or the partial derivatives $fracpartial Gpartial A$, $fracpartial Gpartial B$, $fracpartial Gpartial C$ (they are really messy) but the solutions are quite simple (I did not finish the simplifications).

$$4(a-b)^3 A=-3 (a+b) (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b)^3 B=3 (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b) C=log (b-1)-log (a-1)$$

Using $a=0$, $b=frac 110$, $c=frac 910$, $d=frac 1110$, this would lead to
$$A=30-285 log left(frac109right)qquad B=300 left(19 log left(frac109right)-2right)qquad C=5 log left(frac109right)$$ that is to say
$$Aapprox -0.027747 qquad B approx 0.554939 qquad C approx 0.526803$$

In order to check the validity of the results, I generated a data set of $fractheta2(1-rho)$ with steps $Delta rho=Delta theta=frac1100$ between the selected bounds.

A linear regression gave the following results
$$beginarrayclclclclc
text & textEstimate & textStandard Error & textConfidence Interval \
A & -0.027756 & 0.00130 & -0.030326,-0.025186 \
B & +0.555113 & 0.00248 & +0.550223,+0.560002 \
C & +0.526900 & 0.00130 & +0.524346,+0.529454 \
endarray$$ which seems to confirm.

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

Dr. Leibovici, this was exactly what in my mind was. I just couldn't figure out to integrate twice, over $a-b$ and $c-d$. I really highly appreciate for your very well-written answer. Now, I exactly know what to do for 2+ variable linearization.
â€“Â user8028576
Jul 26 at 13:24

add a commentÂ |Â

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1 Answer
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1 Answer
1

active

oldest

votes

up vote
1
down vote

accepted

Since nos you face a surface, we can consider that we need to minimize $$G = int_a^b int_c^dleft(A + B rho+C theta - fractheta2(1-rho)right)^2,drho,dtheta$$ with respect to $A,B,C$.

$$4(a-b)^3 A=-3 (a+b) (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b)^3 B=3 (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b) C=log (b-1)-log (a-1)$$

In order to check the validity of the results, I generated a data set of $fractheta2(1-rho)$ with steps $Delta rho=Delta theta=frac1100$ between the selected bounds.

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

Dr. Leibovici, this was exactly what in my mind was. I just couldn't figure out to integrate twice, over $a-b$ and $c-d$. I really highly appreciate for your very well-written answer. Now, I exactly know what to do for 2+ variable linearization.
â€“Â user8028576
Jul 26 at 13:24

add a commentÂ |Â

up vote
1
down vote

accepted

Since nos you face a surface, we can consider that we need to minimize $$G = int_a^b int_c^dleft(A + B rho+C theta - fractheta2(1-rho)right)^2,drho,dtheta$$ with respect to $A,B,C$.

$$4(a-b)^3 A=-3 (a+b) (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b)^3 B=3 (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b) C=log (b-1)-log (a-1)$$

In order to check the validity of the results, I generated a data set of $fractheta2(1-rho)$ with steps $Delta rho=Delta theta=frac1100$ between the selected bounds.

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

Dr. Leibovici, this was exactly what in my mind was. I just couldn't figure out to integrate twice, over $a-b$ and $c-d$. I really highly appreciate for your very well-written answer. Now, I exactly know what to do for 2+ variable linearization.
â€“Â user8028576
Jul 26 at 13:24

add a commentÂ |Â

up vote
1
down vote

accepted

Since nos you face a surface, we can consider that we need to minimize $$G = int_a^b int_c^dleft(A + B rho+C theta - fractheta2(1-rho)right)^2,drho,dtheta$$ with respect to $A,B,C$.

$$4(a-b)^3 A=-3 (a+b) (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b)^3 B=3 (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b) C=log (b-1)-log (a-1)$$

In order to check the validity of the results, I generated a data set of $fractheta2(1-rho)$ with steps $Delta rho=Delta theta=frac1100$ between the selected bounds.

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

Since nos you face a surface, we can consider that we need to minimize $$G = int_a^b int_c^dleft(A + B rho+C theta - fractheta2(1-rho)right)^2,drho,dtheta$$ with respect to $A,B,C$.

$$4(a-b)^3 A=-3 (a+b) (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b)^3 B=3 (c+d) ((a+b-2) log (a-1)-(a+b-2) log (b-1)-2 a+2 b)$$
$$2(a-b) C=log (b-1)-log (a-1)$$

In order to check the validity of the results, I generated a data set of $fractheta2(1-rho)$ with steps $Delta rho=Delta theta=frac1100$ between the selected bounds.

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

answered Jul 26 at 8:03

Claude Leibovici

111k1055126

Dr. Leibovici, this was exactly what in my mind was. I just couldn't figure out to integrate twice, over $a-b$ and $c-d$. I really highly appreciate for your very well-written answer. Now, I exactly know what to do for 2+ variable linearization.
â€“Â user8028576
Jul 26 at 13:24

add a commentÂ |Â

Dr. Leibovici, this was exactly what in my mind was. I just couldn't figure out to integrate twice, over $a-b$ and $c-d$. I really highly appreciate for your very well-written answer. Now, I exactly know what to do for 2+ variable linearization.
â€“Â user8028576
Jul 26 at 13:24

Dr. Leibovici, this was exactly what in my mind was. I just couldn't figure out to integrate twice, over $a-b$ and $c-d$. I really highly appreciate for your very well-written answer. Now, I exactly know what to do for 2+ variable linearization.
â€“Â user8028576
Jul 26 at 13:24

add a commentÂ |Â

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