There are two easy ways to add noise, by scale the original data, or by mask some noise on the data.
First for a simple function $y=\sin(x)$ , the following matlab code add 10% noise to it.

N = 100;
x = linspace(-pi, pi, N);
y = sin(x);
plot(x, y, 'r');
hold on;

% add 10% noise based on gaussian
scale = 0.1;
n1 = randn(1, N); % noise with mean=0 and std=1;
y1 = y + n1.*y*scale;
plot(x, y1, 'g');

n2 = 0.1*randn(1,N)*sqrt(max(abs(y))); % noise with mean=0 and %std=max(amplitude);
y2 = y + n2;
plot(x, y2, 'b');

% Of course we can combine the two
y3 = y1 + n2;
plot(x, y3, 'm');


The final result looks like this:

We can also try to add noise to a more complicated synthetic data. For example, the famous swiss roll data[1] in manifold learning. First, we can generate the dataset by this function:
$t=\frac{3}{2}\cdot\pi\cdot(1+2r)\,where\,r\ge 0$
$x=t\cdot\cos(t)$
$y=t\cdot\sin(t)$
$z\in(z_{1}, z_2),\,where\,z_1, z_2\in\mathbb{R}$
Plot a scatter plot of $(x, y, z)$ will give us a swiss roll dataset. For example, the following matlab code will create this figure.

N = 500;
r = linspace(0,1,N);
t = (3*pi/2)*(1+2*r);
x = t.*cos(t);
y = t.*sin(t);
z = 20*rand(1,N);
scatter3(x, y, z, 12, t, 'filled');


Now after adding noise. the standard deviation of the noise is 2% of smallest dimension of the bounding box enclosing the data (as discussed in [2])

mindim = min(max(y)-min(y), max(x)-min(x));
x = x+0.02*randn(1,N)*sqrt(mindim);
y = y+0.02*randn(1,N)*sqrt(mindim);
scatter3(x, y, z, 12, t, 'filled');


1. Tenenbaum, J.B., Silva, V.D. & Langford, J.C. A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290, 2319-2323 (2000).
2. Balasubramanian, M., Schwartz, E.L., Tenenbaum, J.B., de Silva, V. & Langford, J.C. The Isomap Algorithm and Topological Stability. Science 295, 7a (2002).
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