I am implementing M-spline in R as defined here : http://www.fon.hum.uva.nl/praat/manual/spline.html and originally in Ramsay (1988).
In short, we define a list of knots $t$ such that :
$0=t_1=…=t_k<t_{k+1}<…<t_{k+m}<t_{k+m+1}=…=t_{k+m+k}=1$
So the $k$ first knots are 0 and the $k$ last knots are 1 with $m$ inner knots.
The Mi spline of order k with knots t is define recursively as below :
$M_i(x|1,t) = 1 / (t_{i+1} – t_i), t_i ≤ x < t_{i+1},$ 0 otherwise
$M_i(x|k,t) = k [(x–t_i)M_i(x|k–1,t) + (t_{i+k}–x)M_{i+1}(x|k–1,t)] / ((k–1)(t_{i+k}–t_i))$
The problem is that if I want the M-splines of order 3 (k=3), the $M_1$ spline and the $M_{k+m}$ spline do not exist because when it calculate $M_i(x|k–1,t)$ it divides by $(t_{i+k}–t_i)$ but since now $k=3-1=2$ and $i=1$ (or $m+k$), $(t_{i+k}–t_i)$ is $(t_{1+2}–t_1)=0$ since the first three knots are 0 (same thing for the last one).
So the definition seems to have a problem. Here is my implementation in R with an example of the problem. Here k=3 and m=3.
ts = c(0,0,0,0.3,0.5,0.6,1,1,1) Mk = function(i,k,x,ts){ if(k==1){ if(ts[i]<=x && x<ts[i+1]){1/(ts[i+1]-ts[i])} else{0} } else{ #print(paste(i,k)) #print((ts[i+k]-ts[i])) k*((x-ts[i])*Mk(i,k-1,x,ts)+(ts[i+k]-x)*Mk(i+1,k-1,x,ts))/((k-1)*(ts[i+k]-ts[i])) } } Mk(1,3,.4,ts) Note that my code and the definition seems to work for all the "inner" splines.
Thanks!
Best Answer
Well, after some tweaking with my code I tried to add this line to the definition of Mk :
if(ts[i+k]-ts[i]==0){0} So that when it goes out of the list, the spline is simply zero. It worked and I could confirm that the basis was right.