Andrews plot function for multivariate control chart limits UCL and LCL

I am working on Andrews plot for quality control chart. So I need to define UCL and LCL for that. I have attached the images of the paper for the reference and better clarity.

Please tell me how can I develop a vector for A(theta) in the attached image. I also need to use this function for the monitoring the out-of-control points at the later stage.

Answer 1

[Paul]

Brute force approach ...

Use symbolic for illustration

syms theta
p = 5;
A = reshape([sin((1:p)*theta) ; cos((1:p)*theta)],1,[])

A = 

A = [1/sqrt(sym(2)) A];
A = A(1:p)

A = 

Also, you may be interested in andrewsplot

[Nainsi Gupta]

Thank you for your answer Paul.

How can I define theta for a limit -pi to pi? And then how to plot it.

[Paul]

x = 1:6; % an observation

% non-vectorized approach, but easy(?) to follow
syms theta
p = size(x,2);
A = reshape([sin((1:p)*theta) ; cos((1:p)*theta)],1,[]);
A = [1/sqrt(sym(2))+0*theta A];
A = A(1:p)

A = 

A = matlabFunction(A)

A = function_handle with value:
    @(theta)[sqrt(2.0)./2.0,sin(theta),cos(theta),sin(theta.*2.0),cos(theta.*2.0),sin(theta.*3.0)]

theta = linspace(-pi,pi,100);
aplot1 = 0*theta;
for ii = 1:numel(aplot1)
    aplot1(ii) = sum(x.*A(theta(ii)),2);
end
figure
plot(theta,aplot1)

% non-symbolic, vectorized approach
neven = ceil((p-1)/2);
nodd  = floor((p-1)/2);
aplot2 = x(1)/sqrt(2) + sum(x(2:2:end).*sin(theta(:).*(1:neven)),2) + sum(x(3:2:end).*cos(theta(:).*(1:nodd)),2);
figure
plot(theta,aplot2)

norm(aplot1(:)-aplot2(:))

ans = 6.3429e-15

Answer 2

[KALYAN ACHARJYA]

syms theta 
% P variate vector 
%Cross check here, just random data, it can be variable
p=10; 
data_sin=sin((1:round(p/2)+1)*theta);
data_cos=cos((1:round(p/2))*theta);
data(1:2:2*numel(data_sin))=data_sin;
data(2:2:end)=data_cos;
data=[1/sqrt(2),data]

data = 

[Nainsi Gupta]

How can I define a limit for theta (-pi<theta<pi)

[Nainsi Gupta]

Also there is some issue with the code. if p=3, then data should contain 3 elements. similarly for other values of p