Evaluating Total Error for Continuous Functions f and g

[RaduAndrei]

Consider two functions f, g that take on values at t=0, t=1, t=2.

Then the total error between them is:

total error = mod(f(0)-g(0)) + mod(f(1)-g(1)) + mod(f(2)-g(2))

where mod is short for module.

This seems reasonable enough.

Now, consider the two functions to be continuous on [0,2].
What is the total error now?

My guess is that it is the integral of the absolute value of their difference divided by the length of the interval:
total error =1/2 * integral from 0 to 2 of mod(f(x)-g(x)) dx

Is this right?

Or is the error evaluation done in a different way?

 

[RaduAndrei]

Thinking about it, measuring the error between the curves using the area between them is good enough. And dividing by the length of the interval is not necessary. It would be necessary if we would want the error to be measured in the units of the oy axis.

Other measure could be using the distance from linear algebra.
http://math.stackexchange.com/questions/760819/is-there-any-statistical-method-to-compare-two-curves

But not necessary.The first option is also nice. I guess.

 

[pasmith]

Thinking about it, measuring the error between the curves using the area between them is good enough.

If $f(x) = 10^7 \sin \pi x$ and $g(x) = 0$ then $$\int_0^2 f(x) - g(x)\,dx = 0.$$ Does it seem reasonable to say that the error between $f$ and $g$ is zero?

Errors should be defined in terms of norms. There are many norms one can place on the space of continuous functions on $[0,2]$, for example $$\|f\|_{\infty} = \max \{|f(x)| : x \in [0,2]\}$$ or $$\|f\|_p = \left( \int_0^2 |f(x)|^p\,dx \right)^{1/p},\quad p \geq 1.$$

 

[RaduAndrei]

No. I take the module. I integrate the module. I specified this in the first post.

It is not zero if integrating the absolute value.

 

[pasmith]

No. I take the module. I integrate the module. I specified this in the first post.

You then say "measuring the error between the curves using the area between them" - ie $\int_0^2 f(x) - g(x)\,dx$ - "is good enough", which of course it isn't.

 

[RaduAndrei]

What if I do the following thing.

So, consider the function f(x) that takes on values at x=0, x=1, x=2. So the space between two points on the x-axis is 1.
And I want to sum the values:
sum = f(0)+f(1)+f(2)

Now, I want to make the space between the points infinitesimally small. For this I write like this:
sum = f(0) + f(0+Dx) + f(0+2Dx) where Dx = 1
And I make Dx to go to 0. Thus:

sum = f(0) + f(0+dx) + f(0+2dx)
And I sum like this from x = 0 to x = 2.

Then I do this for mod(f(x)-g(x)). So I sum the absolute value of their differences at each point dx on the x axis. Would this work? Something similar is made when going from Fourier series to Fourier transform.

I mean, the sum would have to be infinite. But, in theory, I think it is correct.
Or not. If it is infinite, who decides that it should stop at x=2 or x=10. For both it is sum from k=0 to inf of f(k*dx). I don't know.

 

[RaduAndrei]

You then say "measuring the error between the curves using the area between them" - ie $\int_0^2 f(x) - g(x)\,dx$ - "is good enough", which of course it isn't.

The absolute value of the area. It was self-implied from the first post.