Question

I have the following discrete data.

x=[missing, missing, missing, 758, 763, 742, 729, 721, 714, 709, 696, 680 ]

y=[87.5, 86.4, 84.5, 83.6, 83.2, 84.0 83.2, 82.6, missing, 82.0, 81.2, 80.8 ]

Use cubic spline to predict the y-value at a x-value of 735 (using carefully commented matlab code).

Please present the theory behind the mathematics of the model. I prefer if you do all this as handwritten notes

In which range could the validity of the model be and why?

Answer 1

y-value predicted = 83.6521

Matlab Code:

clf; close all; clearvars; clc

x = [758 763 742 729 721 709 696 680];
y = [83.6 83.2 84 83.2 82.6 82 81.2 80.8];

pp = spline(x,y); %Added this line in code to get coefficients of spline polynomials (read handwritten pictures)
y_pred = spline(x,y,735);

coeff (description given in attached pictures)

-6.66738690759402e-05   0.00426027105801941   -0.0260958264448695   80.8000000000000 %correspond to s₁(x)
-6.66738690759400e-05   0.00105992534237428   0.0590273159614295   81.2000000000000
0.000109045440120756   -0.00154035555158738   0.0527817232416592   82
-0.000109422851215722   0.00238528029275983   0.0629208201357286   82.6000000000000
-9.11619531040150e-05   -0.000240868136417494   0.0800761173864673   83.2000000000000
3.18145050926097e-05   -0.00379618430747408   0.0275944356158768   84
3.18145050926094e-05   -0.00226908806302882   -0.0694499223121695   83.6000000000000 % correspond to s₇(x)