1 %% Decomposition of vectors
   2 
   3 1*[1 0 0] + 2*[0 1 0] + 3*[0 0 1]
   4 
   5 1*[1 1 1] + 1*[0 1 0] + 2*[0 0 1]
   6 
   7 -1*[1 -1 0] + -1*[0 1 -1] + 2*[1 1 1]
   8 
   9 %% matrix inversion
  10 % define a matrix
  11 x = [1 1; 1 -1]
  12 
  13 % multiply data with inverse matrix
  14 inv(x)*[1 1]'
  15 inv(x)
  16 
  17 %% trivial example
  18 x = [2 0; 0 3];
  19 inv(x)
  20 
  21 %% parameter estimation - forward model
  22 x = 0:0.1:8*pi; % abscissa values
  23 
  24 figure;
  25 y1 = x';         % straight line (note transpose)
  26 plot(x, y1);
  27 
  28 figure;
  29 y2 = sin(x)';    % sine curve
  30 plot(x, y2);
  31 
  32 figure;
  33 y3 = exp(-x/2)'; % exponential function
  34 plot(x, y3);
  35 
  36 % Generate data with some noise
  37 figure;
  38 y = y1/5 + y2 + 5*y3 + 0.5*randn(length(y1),1);
  39 plot(x, y);
  40 
  41 whos
  42 %% Estimating the parameters using minimum least-squares (MNLS)
  43 % creating the design matrix
  44 M = [y1 y2 y3];
  45 whos
  46 
  47 % check correlation among basis functions
  48 corr(M)
  49 
  50 % (pseudo)inverting the design matrix
  51 Minv = pinv(M);
  52 whos
  53 
  54 % getting the solution
  55 b = Minv*y
  56 
  57 % checking how solution predicts the data
  58 figure;
  59 ypred = b(1)*y1 + b(2)*y2 + b(3)*y3;
  60 plot(x, ypred)
  61 
  62 % compare with "ground truth"
  63 hold on;    % don't overwrite figure
  64 yreal = y1/5 + y2 + 5*y3;
  65 plot(x, yreal, 'r');
  66 
  67 % plot the difference between measured and predicted data
  68 figure;
  69 ydiff = y - ypred; 
  70 plot(x, ydiff);
  71 
  72 % variance of difference
  73 var(ydiff)