How to normalize vector to unit length

how to normalize vector of features to unit length to generate a probability density function (pdf) also what the normalization can do for the vector?

Answer 1

[John D'Errico]

Vector norms are linear, in the sense that for constant k and vector V,

norm(k*V) = k*norm(V)

So all you need do is

V = V/norm(V);

Which will force the norm(V) to now be 1.

Your other question, "what can a norm do for a vector" makes no sense. Sorry. If you can clarify what you mean, I might be able to answer.

[Jan]

Slightly faster:

V = V / sqrt(V' * V);  % for column vectors, for rows: V * V'

[John D'Errico]

At the same time, norm is more robust. The computation that Jan shows will fail on some vectors where norm will not.

A very simple example where that is true is:

V = [1e200, 1e190];
norm(V)
ans =
                    1e+200

sqrt(V*V')
ans =
   Inf

[Jan]

Thanks, John. I was not aware that norm() uses the stable hypot approach.

Answer 2

[vahid rowghanian]

For a 2-D feature vector R that variables are along columns and samples are along rows, use the following code to normalize the feature to unity range (0-1) with respect to min and max values of each column (feature vector).

    Rmax = repmat(max(R), size(R,1), 1);

    Rmin = repmat(min(R), size(R,1), 1);

    R_unity = (R - Rmin)./(Rmax - Rmin);

For normalizing gray or 3-D or more (any number of channel matrices) that contain negative or positive values that need to be confined in unity range (0-1), the code below will help:

im = double(im);

immin = repmat(min(min(im)), size(im,1), size(im,2));

immax = repmat(max(max(im)), size(im,1), size(im,2));

imu = (im - immin)./(immax - immin);

The Matlab function normalize(A), normalizes vector or matrix A to the center 0 and standard deviation 1. The result will be in range (-1,1).

In case by normalization you mean to make the sum of each column to be equal to one, one possible way for matrix D which can even be a multidimensional is:

Dnorm = bsxfun(@rdivide, D, sum(D));

Now, each column summation will be one (see sum(Dnorm) ).

Answer 3

[Steven Lord]

The normalize and vecnorm functions may also be of use to you.

[Oleksii Doronin]

Function normalize does not give the needed answer. Instead, it treats vector as a set of points, then centers it around 0 and then normalizes.