[FDA] 2. Derivatives of Functional Data

When we want to estimate data using FDA, it is useful to use derivative of function since derivative represents the shape of function.

And calculating the derivative is straightforward like following equation

$$X_n(t) = \sum_{k=1}^K c_{nk} B_k(t) \longrightarrow X_n^{'}(t) = \sum_k c_{nk} B_k^{'}(t)$$

 

However, there exists some case that function is not differentiable. Thus, we can use following approximation technique to derive the first derivative of the function.

$$X^{'}(t_i) \approx \frac{X(t_i) - X(t_{i-1})}{t_i - t_{i-1}}$$

 

After calculating its derivative, we plot this result. Then we can know that the function varies a lot if plot fluctuates many times.

 

These are plots of the results of differentiating the same function using different methods.

I used function below

$$ f(x) = sin(x) + log(x)$$

And the shape of this function is like this

function f(x) = sin(x) + log(x)

 

And the derivative of this function is this

$$ f^{'}(x) = cos(x) + \frac{1}{x}$$

 

exact equation and approximation results

Using this equation, I made two plots. Left one is the result of using true derivative and inputs the values between 0.1 to 15 by 0.05. Right one is the result of using above derivative approximation equation.

In conclusion, I think the shape of these two plots is quite similar. 

 

In FDA, there is some cases that the function is not differentiable. In that case, we can use approximation derivative formula effectively.