Statistical - Cubic Regression (Cubic Spline Fit) ?

[Wes Loewer]

(06-01-2022 03:20 PM)Thomas Klemm Wrote:  Use a cubic spline if you want a function that goes smoothly through the given data points.
Disadvantage: The function is defined piecewise.

This is consistent with the Prime's spline command. However, some cubic spline implementations don't limit themselves to y=f(x), but instead treat x and y parametrically, finding x(t) and y(t) where t is the parametric variable. The GeoGebra help page says:

Quote:The result of the spline command is a curve. Spline algorithm is used for x and y coordinates separately: first we determine values of t that correspond to the points (based on Euclidian distances between the points), then we find cubic splines as functions t->x and t->y.

In GeoGebra, if the first and last points are the same, forming a loop, the algorithm also smooths out where they join.

[jonmoore]

One of my favourite resources for curve fitting per se is the one that's included with Graphpad Prism. Luckily, their help pages are available to non-customers too.

https://www.graphpad.com/guides/prism/la.../index.htm

Their general statistics guide is great for both beginners and those looking for a refresher on the principles of statistical analysis.

https://www.graphpad.com/guides/prism/la.../index.htm

[Albert Chan]

(06-03-2022 09:42 AM)Wes Loewer Wrote:  This is consistent with the Prime's spline command. However, some cubic spline implementations don't limit themselves to y=f(x), but instead treat x and y parametrically, finding x(t) and y(t) where t is the parametric variable.

This is not a "limit". We can get same kind of results with CAS spline command.

XCAS> T,X,Y := range(5), [0,2,3,4,5], [1,0,3,5,4]
XCAS> tx := spline(T,X) :;
XCAS> ty := spline(T,Y) :;

To interpolate for x=1, we solve for t first.

XCAS> t1 := fsolve(tx[0]=1,x=.5)      → 0.451840260174
XCAS> [tx[0], ty[0]] (x=t1)               → [1.0, 0.156460522575]

We could also use complex numbers, combined 2 splines into 1

XCAS> txy := spline(T, X+i*Y) :;
XCSS> txy[0](x=t1)                          → 1.0 + 0.156460522575*i

Note: curve shape is *very* different than cubic-spline of X,Y

XCAS> spline(X,Y)[0](x=1.)               → -0.433139534884

[Wes Loewer]

(06-06-2022 01:03 PM)Albert Chan Wrote:  XCAS> T,X,Y := range(5), [0,2,3,4,5], [1,0,3,5,4]
XCAS> tx := spline(T,X) :;
XCAS> ty := spline(T,Y) :;

I think this is how I would have handled it as well. GeoGebra adds a twist by having T be based on the distance between the points instead of the point number. The result is a more intuitive curve when two points are close together.

(06-06-2022 01:03 PM)Albert Chan Wrote:  We could also use complex numbers, combined 2 splines into 1

Very interesting. I'm always fascinated by the use of complex numbers (or even quaternions) in applications that are originally intended for real numbers.

[jonmoore]

I took a look through my 48 book collection and found a group of programs that can be used in 48/49/50 series calculators (the book in question is Calculus on the HP-48G/GX - Grapevine Publications). The focus in the book is on calculating integrals, but the techniques/programs can be used for more general statistical curve fitting workflows too.

It's available in digital form via:
https://literature.hpcalc.org/items/1500

But here's so examples from the book.

[rprosperi]

I'm not sure if there is interest in the 71B User Library Cubic Spline interpolation program I mentioned, however it's taken weeks to find it, and since I have it now, here's a scan of the user library program description, including an overview, a little theory review, sample problem, sample program run output and BASIC program listing.

https://1drv.ms/b/s!AiI5Ei2M2Ja2mRDm0BOo...n?e=LdBPIH

Note: The HP-71B MATH PAC ROM (or JFG's enhanced 2B version) must be installed for this to work.