Fitted curve to measured data - statistical properties of the fit error

[cmunikat]

Dear all,

I have a set of measurements {xm(Ti,mi)=x(Ti)+e(Ti,mi)}, where:

_xm is the measured value
_x is the actual value
_e is a random measurement error for the measurement mi
_Ti is a parameter

I need to fit a curve to this data by some method. For example, if I use least squares best fit, the following value D is minimized:

D=Ʃ Di²

Where:
_Di=X(T=Ti)-xm(Ti)
_X is the continuous curve

Now, I define the error between the curve X(T) and the actual data x(Ti):

ef(Ti) = X(T) - x(Ti)

And here is the problem:

I need to know, for any parameter Tj, the mean and variance of this error ef(Tj), in terms of the mean and variance of the measurement errors "e(Ti,mi)" defined at the beggining.

Thank you in advance

Cmunikat

 

[mmwave]

Dear Cmunikat,

Thank you for reaching out to the scientific community for help with your data analysis. Based on the information you have provided, it seems like you are working with a set of measured values that have some random error associated with them. Your goal is to fit a curve to this data using a method like least squares best fit.

To address your problem, we can start by looking at the definition of the error between the curve X(T) and the actual data x(Ti):

ef(Ti) = X(T) - x(Ti)

This means that for any given parameter Tj, the error between the curve and the actual data at that point is:

ef(Tj) = X(Tj) - x(Tj)

To find the mean and variance of this error, we can use the properties of expectation and variance. The expectation of the error is:

E[ef(Tj)] = E[X(Tj) - x(Tj)] = E[X(Tj)] - E[x(Tj)] = X(Tj) - x(Tj)

This means that the mean of the error at any given parameter Tj is simply the difference between the curve and the actual data at that point.

Next, to find the variance of the error, we can use the property that the variance of a sum of random variables is equal to the sum of their variances. This means that:

Var[ef(Tj)] = Var[X(Tj) - x(Tj)] = Var[X(Tj)] + Var[x(Tj)]

Now, we can use the definition of the error ef(Ti) to rewrite the variance in terms of the measurement errors e(Ti,mi):

Var[x(Tj)] = Var[X(Tj) - ef(Tj)] = Var[X(Tj)] + Var[ef(Tj)] = Var[X(Tj)] + Var[X(Tj) - x(Tj)]

= Var[X(Tj)] + Var[x(Tj)] + Var[x(Tj)]

= Var[X(Tj)] + Var[e(Tj,mi)] + Var[x(Tj)]

= Var[X(Tj)] + Var[e(Tj,mi)] + Var[x(Tj)]

= Var[X(Tj)] + Var[e(Tj,mi)] + Var[x(Tj)]

= Var[X(Tj)] + Var[e(Tj,mi)] + Var[x(Tj)]

= Var[X(Tj)] + Var[e(Tj,