I have a non linear model to fit my data.

I have to evaluate the goodness of fit so I calculated the coefficient
of determination R and the standard error.
I though that coefficient of determination R and coefficient of lienar
correlation r were the same thing but this is not the case. In
particular I have some cases where R2 is negative.

I use tthe following formulae:

y = true data
f = fitted data

[ML notation]

rmse = sqrt( ( sum( ( f - y ).^2 ) ) /n );

r = ( n * sum( f .* y ) - sum( f ) * sum( y ) ) / sqrt( ( n *
sum( y .^ 2 )-sum( y ) ^ 2 ) * ( n * sum( f .^ 2 ) - sum( f ) ^
2 ) );

ss_tot = sum( ( y - mean( y ) ) .^2 );
ss_res = sum( ( f - y ) .^2 )
R2 = ( ss_tot - ss_res) / ss_tot
R = sqrt( R2 )

Could you explain me in simple words the reason of the differencies
between r and R and the reason of negative R2?

And, in case of negative R2, what I should use, apart the standard
error rmse, to evaluate the goodness of fit?

Thanks

On Jul 16, 6:11 am, Beorne <matteo... at (no spam) gmail.com> wrote:



I have a non linear model to fit my data.

I have to evaluate the goodness of fit so I calculated the coefficient
of determination R and the standard error.
I though that coefficient of determination R and coefficient of lienar
correlation r were the same thing but this is not the case. In
particular I have some cases where R2 is negative.

I use tthe following formulae:

y = true data
f = fitted data

[ML notation]

rmse = sqrt( ( sum( ( f - y ).^2 ) ) /n );

r = ( n * sum( f .* y ) - sum( f ) * sum( y ) ) / sqrt( ( n *
sum( y .^ 2 )-sum( y ) ^ 2 ) * ( n * sum( f .^ 2 ) - sum( f ) ^
2 ) );

ss_tot = sum( ( y - mean( y ) ) .^2 );
ss_res = sum( ( f - y ) .^2 )
R2 = ( ss_tot - ss_res) / ss_tot
R = sqrt( R2 )

Could you explain me in simple words the reason of the differencies
between r and R and the reason of negative R2?

And, in case of negative R2, what I should use, apart the standard
error rmse, to evaluate the goodness of fit?

Thanks

That can happen if your model lacks additive and multiplicative
constants that are free to be estimated, or if it has such constants
but they were estimated by a procedure that does not minimize the sum
of squares of the residuals.

On 16 Lug, 16:56, Ray Koopman <koop... at (no spam) sfu.ca> wrote:
On Jul 16, 6:11 am, Beorne <matteo... at (no spam) gmail.com> wrote:

I have a non linear model to fit my data.

I have to evaluate the goodness of fit so I calculated the coefficient
of determination R and the standard error.
I though that coefficient of determination R and coefficient of lienar
correlation r were the same thing but this is not the case. In
particular I have some cases where R2 is negative.

I use tthe following formulae:

y = true data
f = fitted data

[ML notation]

rmse = sqrt( ( sum( ( f - y ).^2 ) ) /n );

r = ( n * sum( f .* y ) - sum( f ) * sum( y ) ) / sqrt( ( n *
sum( y .^ 2 )-sum( y ) ^ 2 ) * ( n * sum( f .^ 2 ) - sum( f ) ^
2 ) );

ss_tot = sum( ( y - mean( y ) ) .^2 );
ss_res = sum( ( f - y ) .^2 )
R2 = ( ss_tot - ss_res) / ss_tot
R = sqrt( R2 )

Could you explain me in simple words the reason of the differencies
between r and R and the reason of negative R2?

And, in case of negative R2, what I should use, apart the standard
error rmse, to evaluate the goodness of fit?

Thanks

That can happen if your model lacks additive and multiplicative
constants that are free to be estimated, or if it has such constants
but they were estimated by a procedure that does not minimize the
sum of squares of the residuals.

My model is a power curve of the type
y = a * x ^ b + c (a,b,c parameters)
estimated minimizing sum of suqares.

On Jul 16, 8:41 am, Beorne <matteo... at (no spam) gmail.com> wrote:





On 16 Lug, 16:56, Ray Koopman <koop... at (no spam) sfu.ca> wrote:
On Jul 16, 6:11 am, Beorne <matteo... at (no spam) gmail.com> wrote:

I have a non linear model to fit my data.

I have to evaluate the goodness of fit so I calculated the coefficient
of determination R and the standard error.
I though that coefficient of determination R and coefficient of lienar
correlation r were the same thing but this is not the case. In
particular I have some cases where R2 is negative.

I use tthe following formulae:

y = true data
f = fitted data

[ML notation]

rmse = sqrt( ( sum( ( f - y ).^2 ) ) /n );

r = ( n * sum( f .* y ) - sum( f ) * sum( y ) ) / sqrt(  ( n *
sum( y .^ 2 )-sum( y ) ^ 2 ) * ( n * sum( f .^ 2 ) - sum( f ) ^
2 )   );

ss_tot = sum( ( y - mean( y ) ) .^2 );
ss_res = sum( ( f - y ) .^2 )
R2 = ( ss_tot - ss_res) / ss_tot
R = sqrt( R2 )

Could you explain me in simple words the reason of the differencies
between r and R and the reason of negative R2?

And, in case of negative R2, what I should use, apart the standard
error rmse, to evaluate the goodness of fit?

Thanks

That can happen if your model lacks additive and multiplicative
constants that are free to be estimated, or if it has such constants
but they were estimated by a procedure that does not minimize the
sum of squares of the residuals.

My model is a power curve of the type
y = a * x ^ b + c  (a,b,c parameters)
estimated minimizing sum of suqares.

Then the parameters must have been poorly estimated. This is really
a one-parameter problem, because there are closed-form expressions
for the least-squares a and c for any given b.  Let w = x^b.  Then

a = (n*sum(w*y)-(sum w)(sum y)) / (n*sum(w^2)-(sum w)^2),

c = ((sum y) - a*(sum w)) / n.

Mimize  n*sum((y-f)^2) = (n*sum(y^2)-(sum y)^2) -
        (n*sum(w*y)-(sum w)(sum y))^2 / (n*sum(w^2)-(sum w)^2)

as a function of b, then get the corresponding a and c.- Nascondi testo citato

On 16 Lug, 17:37, Ray Koopman <koop... at (no spam) sfu.ca> wrote:
On Jul 16, 8:41 am, Beorne <matteo... at (no spam) gmail.com> wrote:
On 16 Lug, 16:56, Ray Koopman <koop... at (no spam) sfu.ca> wrote:
On Jul 16, 6:11 am, Beorne <matteo... at (no spam) gmail.com> wrote:

I have a non linear model to fit my data.

I have to evaluate the goodness of fit so I calculated the coefficient
of determination R and the standard error.
I though that coefficient of determination R and coefficient of lienar
correlation r were the same thing but this is not the case. In
particular I have some cases where R2 is negative.

I use tthe following formulae:

y = true data
f = fitted data

[ML notation]

rmse = sqrt( ( sum( ( f - y ).^2 ) ) /n );

r = ( n * sum( f .* y ) - sum( f ) * sum( y ) ) / sqrt( ( n *
sum( y .^ 2 )-sum( y ) ^ 2 ) * ( n * sum( f .^ 2 ) - sum( f ) ^
2 ) );

ss_tot = sum( ( y - mean( y ) ) .^2 );
ss_res = sum( ( f - y ) .^2 )
R2 = ( ss_tot - ss_res) / ss_tot
R = sqrt( R2 )

Could you explain me in simple words the reason of the differencies
between r and R and the reason of negative R2?

And, in case of negative R2, what I should use, apart the standard
error rmse, to evaluate the goodness of fit?

Thanks

That can happen if your model lacks additive and multiplicative
constants that are free to be estimated, or if it has such constants
but they were estimated by a procedure that does not minimize the
sum of squares of the residuals.

My model is a power curve of the type
y = a * x ^ b + c (a,b,c parameters)
estimated minimizing sum of suqares.

Then the parameters must have been poorly estimated. This is really
a one-parameter problem, because there are closed-form expressions
for the least-squares a and c for any given b. Let w = x^b. Then

a = (n*sum(w*y)-(sum w)(sum y)) / (n*sum(w^2)-(sum w)^2),

c = ((sum y) - a*(sum w)) / n.

Mimize n*sum((y-f)^2) = (n*sum(y^2)-(sum y)^2) -
(n*sum(w*y)-(sum w)(sum y))^2 / (n*sum(w^2)-(sum w)^2)

as a function of b, then get the corresponding a and c.

Good to know. I found the model using the matlab curve fitting toolbox.
I'm studying how much the model does generalizes on different sets,
the model has not been taken on the same points I'm trying to match.
Clearly the points giving me negative R2 fit very bad.

But, in general, in which cases r2 and R2 are different?
And when R2 can be negative?
Is more correct r2 or R2 to evaluate the goodness of match to a model?