Can I Scale Down Basis Functions Without Losing Zero Force?

In summary, the conversation discusses using basis functions ##\phi_i(x)## which become very large, approximately ##O(\sinh(12 j))##, for the ##jth## function. To address this issue, the functions are being forced to zero at 3 and 3.27, although these values are still far from zero. The speaker is seeking advice on how to significantly scale down the functions to be closer to zero. They also mention that for some functions, although they are close to zero in size, their values at x=3 can be as small as 10^16, which is small compared to their average height of 10^30 with sharp gradients. However, they later mention that the issue has
  • #1
member 428835
Hi PF!

I'm working with some basis functions ##\phi_i(x)##, and they get out of control big, approximately ##O(\sinh(12 j))## for the ##jth## function. What I am doing is forcing the functions to zero at approximately 3 and 3.27. I've attached a graph so you can see. Looks good, but in fact these values are very far from zero. Any ideas on what I can do to get these scaled down significantly so they are forced to zero?

I should say, each function individually get very close to zero regarding it's size, but for some cases that means ##\phi(x=3) = 10^{16}##, which is very small considering it's average height is about ##10^{30}## with sharp gradients.
 

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  • #2
Nevermind, I have it working now!
 

Related to Can I Scale Down Basis Functions Without Losing Zero Force?

1. What are basis functions and why are they important?

Basis functions are mathematical functions used to represent complex data in a simpler form. They are important in data analysis and machine learning because they help to reduce the dimensionality of data and make it easier to interpret and analyze.

2. What does it mean for basis functions to be "too huge"?

When we say that basis functions are "too huge", it means that there are a large number of basis functions being used to represent the data. This can lead to overfitting, where the model fits the training data too closely and does not generalize well to new data.

3. How does having too many basis functions affect the performance of a model?

Having too many basis functions can lead to overfitting, where the model becomes too complex and is not able to generalize well to new data. This can result in poor performance and inaccurate predictions.

4. How can we prevent basis functions from becoming too large?

One way to prevent basis functions from becoming too large is by using regularization techniques, such as Lasso or Ridge regression, which penalize complex models and encourage simpler solutions. Another approach is to carefully select the appropriate number and type of basis functions for the data.

5. Are there any advantages to using a large number of basis functions?

While having too many basis functions can lead to overfitting, it can also allow for more complex and accurate representations of the data. In some cases, this may be necessary for achieving high performance in certain types of models. However, it is important to carefully balance the trade-off between model complexity and performance.

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