Can different weights help determine the difference between two functions?

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For others, a different measure altogether may be more appropriate.In summary, Josh is asking for help in determining a measure for how different two functions are from each other. He suggests using an integral of the squared difference between the functions as a possible method. Another person suggests using an inner product for this purpose, which can be defined by integrating the product of the functions with a weight function. They also mention that the choice of weight function may depend on the specific problem at hand.
  • #1
member 428835
Hi PF!

Can any of you help me determine a good measure for how "different" two functions are from each other?

I've thought of using something like ##\int_\Omega (f-g)^2 \, dx##. Can anyone recommend a good technique and direct me to the theory so I can understand it well?

Thanks so much!

Josh
 
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  • #2
Your proposal is an option and is a special case of using an inner product for the purpose. You can view the function space as a linear vector space and generally define an inner product by integrating the product of the functions, possibly together with a positive weight function. Your proposal would be equivalent to taking the inner product of the difference function with itself, much like you could determine the distance between two points in any vector space with an inner product.
 
  • #3
So does this difference function (with weight 1) seem like a good representation for distance, or is there a better method in your opinion?
 
  • #4
This would depend on exactly what you are looking to do. For some problems, it is natural to use different weights.
 

Related to Can different weights help determine the difference between two functions?

1. What is the purpose of minimizing functions in scientific research?

Minimizing functions are used in scientific research to find the optimal solution or value for a given problem. This helps to reduce errors and improve the accuracy of scientific experiments and models.

2. How do you minimize a function mathematically?

To minimize a function, you can use calculus to find the critical points, where the derivative of the function is equal to 0. Then, you can use the second derivative test to determine if the critical point is a minimum or maximum.

3. Can minimizing functions be applied to real-world problems?

Yes, minimizing functions can be applied to a variety of real-world problems, such as optimizing production processes, minimizing costs, or maximizing profits. It is a powerful tool in decision-making and problem-solving.

4. Are there any limitations to using minimizing functions?

Minimizing functions can only provide an optimal solution if the function is differentiable and continuous. In some cases, there may be multiple local minima, making it challenging to find the global minimum.

5. How does minimizing functions relate to gradient descent?

Gradient descent is an iterative optimization algorithm that uses minimizing functions to find the minimum value of a function. It is commonly used in machine learning and artificial intelligence to adjust the parameters of a model to minimize the error between predicted and actual values.

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