Kernel Density Estimation - Find the pdf and perform integration and derivative on this function

I have performed KDE on data:

 [f,xi] = ksdensity(data,'bandwidth',0.5,'Function','pdf');

I have the following code which aims to solve for theta - roots of first derivative of the pdf. What I did was plot the pdf from kernel estimation, using Basic Fitting to obtain the function, then solve for theta.

syms theta %create symbolic variable theta
assume(theta,'real') %theta is real
pf = poly2sym(fit.coeff,theta); %calling our fitted polynomials

g = diff(pf,theta);
g0=solve(g,theta);
theta = double(g0);

However, Basic Fitting doesn't fit very well with the pdf. How can I obtain pdf and take derivative without producing too much residuals? (see pic)

Additionally, theta has to follow three conditions: -smaller than the highest pdf value -pdf evaluation of theta must be smaller than 0.8 times of that of the highest pdf value -integral from min x value to theta of pdf must be larger than 0.05

 θ < μ, f(θ)<0.8·f(μ) and  ∫ f (x)dx > 0.05 from min x to θ.

Is there a way to incorporate loop and boolean to automate solving for theta.

[Sarah Mohamed]

Have you tried fitting a spline to the data?

[Tam Ho]

Fitting a spline would get very close to the real pdf function but won't produce a polynomials with coefficients which can be differentiated.

Is there a way to differentiate the spline function?

[Christoph]

ksdensity is a non-parametric fitting method. Basically, you seem to fit a (10th degree) polynomial to your data, convert the polynomial to a sym object, determine a zero of its derivative and then again convert to double. You do not need the symbolic detour here, you can just use polyder + roots to obtain all zeros. Tam Ho´s comment recommends fitting a spline, and indeed, you could do that and use fnder() for computing derivatives. Another approach would be to use traditional signal processing approach, smooth the data and use a functions such as findpeaks on a numerical derivative of the signal.

[Tam Ho]

How to " use traditional signal processing approach, smooth the data and use a functions such as findpeaks on a numerical derivative of the signal." ? Can you give an example please?

Essentially the analysis procedure requires finding the pdf of the data and perform derivation on that.