function?[pdfgrid]=akde1d（Xgridgam）
%%?fast?adaptive?kernel?density?estimation?in?one-dimension;
%??provides?optimal?accuracy/speed?tradeoff?controlled?with?parameter?“gam“;
%?INPUTS:???X??-?data?as?a?‘n‘?by?‘1‘?vector;
%
%?????????grid?-?（optional）?mesh?over?which?density?is?to?be?computed;
%????????????????default?mesh?uses?2^12?points?over?range?of?data;
%
%??????????gam?-?（optional）?cost/accuracy?tradeoff?parameter?where?gam%????????????????default?value?is?gam=ceil（n^（1/3））+20;?larger?values
%????????????????may?result?in?better?accuracy?but?always?reduce?speed;
%????????????????to?speedup?the?code?reduce?the?value?of?“gam“;?
%
%?OUTPUT:?pdf?-?the?value?of?the?estimated?density?at?‘grid‘
%
%%??EXAMPLE:
%???data=[exp（randn（10^31））];?%?log-normal?sample
%???[pdfgrid]=akde1d（data）;?plot（gridpdf）
%
%?Note:?If?you?need?a?very?fast?estimator?use?my?“kde.m“?function.
%?This?routine?is?more?adaptive?at?the?expense?of?speed.?Use?“gam“
%?to?control?a?speed/accuracy?tradeoff.?
%
%%??Reference:
%??Kernel?density?estimation?via?diffusion
%??Z.?I.?Botev?J.?F.?Grotowski?and?D.?P.?Kroese?（2010）
%??Annals?of?Statistics?Volume?38?Number?5?pages?2916-2957.
[nd]=size（X）;
%?begin?scaling?preprocessing
MAX=max（X[]1）;MIN=min（X[]1）;scaling=MAX-MIN;
MAX=MAX+scaling/10;MIN=MIN-scaling/10;scaling=MAX-MIN;
X=bsxfun（@minusXMIN）;X=bsxfun（@rdivideXscaling）;
if?（nargin<2）|isempty（grid）?%?failing?to?provide?grid
????grid=（MIN:scaling/（2^12-1）:MAX）‘;
end
mesh=bsxfun（@minusgridMIN）;mesh=bsxfun（@rdividemeshscaling）;
if?nargin<3?%?failing?to?provide?speed/accuracy?tradeoff
????gam=ceil（n^（1/3））+20;
end
%?end?preprocessing
%?algorithm?initialization
del=.2/n^（d/（d+4））;perm=randperm（n）;mu=X（perm（1:gam）:）;
w=rand（1gam）;w=w/sum（w）;Sig=del^2*rand（gamd）;
ent=-Inf;
for?iter=1:1500?%?begin?algorithm
????Eold=ent;
????[wmuSigdelen