[R] Taking the derivative of a quadratic B-spline

[Duncan Murdoch]

James McDermott wrote:
> Would the unique quadratic defined by the three points be the same
> curve as the curve predicted by a quadratic B-spline (fit to all of
> the data) through those same three points?

Yes, if you restrict attention to an interval between knots.  You'll 
need to re-evaluate it for each such interval (but since quadratic 
splines are continuous, you can reuse evaluations at the knots, and you 
just need one new point in each interval).

 From a practical point of view, you need to make sure that COBS really 
is giving you a quadratic spline and really is reporting all of the 
knots correctly.  Watch out for coincident knots (zero length 
intervals); you don't care about the derivative on those, but they might 
cause overflows in some calculations.

Duncan Murdoch
> 
> Jim
> 
> On 7/19/05, Duncan Murdoch <murdoch at stats.uwo.ca> wrote:
> 
>>On 7/19/2005 3:34 PM, James McDermott wrote:
>>
>>>I wish it were that simple (perhaps it is and I am just not seeing
>>>it).  The output from cobs( ) includes the B-spline coefficients and
>>>the knots.  These coefficients are not the same as the a, b, and c
>>>coefficients in a quadratic polynomial.  Rather, they are the
>>>coefficients of the quadratic B-spline representation of the fitted
>>>curve.  I need to evaluate a linear combination of basis functions and
>>>it is not clear to me how to accomplish this easily.  I was hoping to
>>>find an alternative way of getting the derivatives.
>>
>>I don't know COBS, but doesn't predict just evaluate the B-spline?  The
>>point of what I posted is that the particular basis doesn't matter if
>>you can evaluate the quadratic at 3 points.
>>
>>Duncan Murdoch
>>
>>
>>>Jim McDermott
>>>
>>>On 7/19/05, Duncan Murdoch <murdoch at stats.uwo.ca> wrote:
>>>
>>>>On 7/19/2005 2:53 PM, James McDermott wrote:
>>>>
>>>>>Hello,
>>>>>
>>>>>I have been trying to take the derivative of a quadratic B-spline
>>>>>obtained by using the COBS library.  What I would like to do is
>>>>>similar to what one can do by using
>>>>>
>>>>>fit<-smooth.spline(cdf)
>>>>>xx<-seq(-10,10,.1)
>>>>>predict(fit, xx, deriv = 1)
>>>>>
>>>>>The goal is to fit the spline to data that is approximating a
>>>>>cumulative distribution function (e.g. in my example, cdf is a
>>>>>2-column matrix with x values in column 1 and the estimate of the cdf
>>>>>evaluated at x in column 2) and then take the first derivative over a
>>>>>range of values to get density estimates.
>>>>>
>>>>>The reason I don't want to use smooth.spline is that there is no way
>>>>>to impose constraints (e.g. >=0, <=1, and monotonicity) as there is
>>>>>with COBS.  However, since COBS doesn't have the 'deriv =' option, the
>>>>>only way I can think of doing it with COBS is to evaluate the
>>>>>derivatives numerically.
>>>>
>>>>Numerical estimates of the derivatives of a quadratic should be easy to
>>>>obtain accurately.  For example, if the quadratic ax^2 + bx + c is
>>>>defined on [-1, 1], then the derivative 2ax + b, has 2a = f(1) - f(0) +
>>>>f(-1), and b = (f(1) - f(-1))/2.
>>>>
>>>>You should be able to generalize this to the case where the spline is
>>>>quadratic between knots k1 and k2 pretty easily.
>>>>
>>>>Duncan Murdoch
>>>>
>>
>>