polynomial interpoltaion & least squares

0 184360.634246042 -211658.511711780 -6853898.76885147

5 219214.256803219 -226785.422707026 -6852351.69754470

10 254061.148627043 -241905.367897281 -6850593.66616310

15 288900.239807604 -257017.882918880 -6848624.72472167

20 323730.460592783 -272122.503660514 -6846444.92962074

25 358550.741525596 -287218.766139992 -6844054.34362999

30 393360.013332653 -302306.206701186 -6841453.03589562

35 428157.207073006 -317384.361885925 -6838641.08193523

40 462941.254118769 -332452.768510410 -6835618.56363502

45 497711.086198094 -347510.963664615 -6832385.56924254

50 532465.635437569 -362558.484708884 -6828942.19336887

55 567203.834312849 -377594.869407128 -6825288.53697975

60 601924.615866041 -392619.655658098 -6821424.70739058

In a general form, the polynomials of degree n can be represented as follows: y = a0t 0 + a1t 1 + a2t 2 + ... + ant n (1)

where y contains the measurements (orbit components x, y or z here) and t is the time vector corresponding to each measurement.

Here the polynomial modeling of the given data y, is the estimation of the unknown coefficients a0, a1,..., an. Assuming that

values: y = [y1, y2, y3, ..., ym]m×1 (2)

are given at time points: t = [t1, t2, t3, ..., tm]m×1. (3)

Then n + 1 coefficients x = [a0, a1, ..., an]1×(n+1) of the polynomials degree n can be simply estimated using least-squares

adjustment: x = (ATA) −1AT y, (4)

i have this data of satelite positions(x,y,z) for 1 minute with 5 sec sampling. 1st colmn is time with 5 sec spacing.

how could i interpolate the positions to 1 second spacing using polynomial. is there a built in fnctn ?

Answer 1

[Jon]

I'm not totally clear on what you are trying to do from your description, but here are few things that I think will help you.

You can fit a polynomial to known data using MATLAB's polyfit function. This will give you a vector of polynomial coefficients in descending order.

Once you have the polynomial coefficients you can evaluate them for any value of x using MATLAB's polyval function.

For help on polyfit type doc polyfit on the command line, for help on polyval type doc polyval on the command line

[Trae]

coordinate positions xyz(column2,3,4) at time spacing of 5ecs for 1min(1st column is the data of time).

i need todo an upsampling of this coordinate postns by interpolating the positions to a 1 seconds spacing. for this i need to use a polynomial function with a reasonable degree. and also estimate the polynomial coefficients using least squares adjustment.

[Jon]

Did you read the documentation for polyfit (does the least squares fit) and polyval evaluates at any values of t, so you can use it to interpolate. Do you have a question on using those functions? Is there some reason those can't be used to solve your problem? Please try to write some code, attach and explain what problems or errors you have

[Trae]

load('Orbit.mat')

[x] = Orbit(:,2);

[y] = Orbit(:,3);

[z] = Orbit(:,4);

plot3(x,y,z,'*-')

title('original positions of satellite(1min @5sec samplingn)','fontsize',14);

xlabel('X', 'FontSize', 16);

ylabel('Y', 'FontSize', 16);

zlabel('Z', 'FontSize', 16);

grid on;

legend('pos of satelite @ 5sec intervals for 1 min');

[T] = Orbit(:,1);

t = 0:60;

a = interp1(T,x,t);

b = interp1(T,y,t);

c = interp1(T,z,t);

figure

plot3(x,y,z,'*-')

title('original positions of satellite(1min @5sec samplingn)','fontsize',14);

xlabel('X', 'FontSize', 16);

ylabel('Y', 'FontSize', 16);

zlabel('Z', 'FontSize', 16);

grid on;

legend('pos of satelite @ 5sec intervals for 1 min');

figure

plot3(a,b,c,'*-')

title('upsampled positions of satellite(1min @1sec samplingn)','fontsize',14);

xlabel('X', 'FontSize', 16);

ylabel('Y', 'FontSize', 16);

zlabel('Z', 'FontSize', 16);

grid on;

legend('pos of satelite @ 1sec intervals for 1 min');

[Trae]

[Trae]

[Trae]

the 1st figure is the plot of original data

the 2nd figure is the plot of interpolated data to 1 sec spacing using interp1.

now i need to get the same results as in the 2nd figure using polynomial interpolation.

[Jon]

Hi, I don't have time at the moment to go through your code in detail, but from a quick look, maybe you want to use the method argument for interp1. This allows you to choose from various methods for performing the interpolation including 'pchip', 'cubic', 'spline' etc.

y = interp1(t,x,tq,'spline')

You can find out more by typing doc interp1 on the command line