README.md

Introduction Assignment

This is an introductory assignment for the course. By solving this assignment you will get more familiar with the tools we will be using in the course.

Solving this assignment is not a requirement and will not be a part of your final grade, but is helpful to get you started with the tools you need to be able to use during future assignments in this course.

Setting up

Before starting, make sure that you have installed Python requirements as explained in the main README.

All your code should be turned in as a python script, i.e.

• A single file with a .py ending. Although you can use the template.ipynb notebook in your development and then port your code over to template.py if that suits you.
• The file should contain all the functions with exactly the same function names as listed in the assignment description below
• Use the supplied template.py file to fill in your code
• Plots should be turned in as well. The naming convention for submitted plots is as follows: The Z-th plot under Part X.Y should be turned in as X_Y_Z.png

Requirements

You will need the following requirements to finish this requirement.

• numpy: https://numpy.org/
• matplotlib: https://matplotlib.org/

You can run pip install -r requirements.txt to install all the requirements.

Part 1

1.1

Define the Gaussian probability density function normal_prob(x, sigma, mu) where: * $\sigma$ is the standard deviation * $\mu$ is the mean of the distribution * $x$ is any real value

The Gaussian PDF is given by

$$p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} }$$

$p(x)$ will give you the probability of the value $x$ according to the Gaussian PDF.

Make sure to use numpy math operators, for example:

• for $x^2$ do: np.power(x, 2)
• for $\pi$ do: np.pi
• for $e^2$ do: np.exp(2)
• For multiplication/division you can simply use * and / respectively. If done correctly, you can even supply $x$ as an $n$-dimensional numpy array and you will get $n$ probabilities as a result.

Example inputs and outputs:

• normal(0, 1, 0) = 0.3989422804014327
• normal(3, 1, 5) = 0.05399096651318806
• normal(np.array([-1,0,1]), 1, 0) = array([0.24197072, 0.39894228, 0.24197072])

1.2

Define a function plot_normal(sigma, mu, x_start, x_end) that plots a normal distribution on the given range, [x_start, x_end]. This function should work similarily to examples.plot_multiple in that it should only plot onto a figure without actually showing it. Inside the function, create 500 linear spaced values using np.linspace.

Calling plot_normal(0.5, 0, -2, 2) should result in the following plot:

Now plot the three following normal distributions onto the same figure on the range $[-5, 5]$:

• $\mu = 0, \sigma=0.5$
• $\mu = 1, \sigma=0.25$
• $\mu = 1.5, \sigma=1$

Save the figure as 1_2_1.png

Part 2

2.1

Create a function normal_mixture(x, sigmas, mus, weights) which defines a Gaussian mixture function. A Gaussian mixture density function combines multiple gaussian densities with weights that sum to 1. It is given by:

$$p(x) = \sum_{i=1}\frac{\phi_i}{\sqrt{ 2 \pi \sigma_i^2 }}\text{exp}\left({ - \frac{ (x - \mu_i)^2 } {2 \sigma_i^2} }\right)$$

where $\phi_i$ is the $i$-th mixture weight, corresponding to the $i$-th gaussian.

Assume that the x argument is always a np.ndarray of some sort.

Example inputs and outputs:

Example 1:

normal_mixture(np.linspace(-5, 5, 5), [0.5, 0.25, 1], [0, 1, 1.5], [1/3, 1/3, 1/3])

Output: [8.89852205e-11 4.56012216e-05 3.09312492e-01 8.06579074e-02 2.90894232e-04]

Example 2:

normal_mixture(np.linspace(-2, 2, 4), [0.5], [0], [1])

Output: [2.67660452e-04 3.28020149e-01 3.28020149e-01 2.67660452e-04]

2.2

Lets compare three individual normal components with the evenly-weighted mixture of the three. You should use your plot_normal function for the individual components but a different method must be used for the mixture.

Try the following values:

• $\mu = 0, \sigma=0.5, \phi=1/3$
• $\mu = -0.5, \sigma=1.5, \phi=1/3$
• $\mu = 1.5, \sigma=0.25, \phi=1/3$

Plot each individual normal distribution as well as the mixture onto the same figure. Save this figure as 2_2_1.png

Part 3

3.1

Define a function sample_gaussian_mixture(sigmas, mus, weights, n_samples) that samples n_samples from a given gaussian mixture. To sample a Gaussian mixture it is best to do the following:

1. Use np.random.multinomial to determine how many times each normal component is selected to sample from. For example, to throw a dice 10 times using this function you would do: np.random.multinomial(10, [1/6.]*6) since each side of the dice is equally likely, i.e. $1/6$. In your case, these probabilities are determined by the weights in your mixture
2. Use np.random.normal to sample the selected normal component
3. Repeat step 2 until you have n_samples

This function should return a numpy.ndarray

Example inputs and outputs (Use np.random.seed(0) to get similar values):

Example 1:

sample_gaussian_mixture([0.1, 1], [-1, 1], [0.9, 0.1], 3)
= [-0.92584083 -0.84470863 -1.22683282]

Example 2:

[0.1, 1, 1.5], [1, -1, 5], [0.1, 0.1, 0.8], 10)
= [1.0978738  1.2408932  7.80133699 3.53408318 6.42513263 4.77296419
 4.84517172 5.61589775 5.21606536 7.18141026]

3.2

Lets now see how well samples follow a specific mixture of normal probability distributions.

Our mixture will have the following parameters:

• $\mu = 0, \sigma=0.3, \phi=0.2$
• $\mu = -1, \sigma=0.5, \phi=0.3$
• $\mu = 1.5, \sigma=1, \phi=0.5$

After drawing:

1. 10 samples
2. 100 samples
3. 500 samples
4. 1000 samples

Plot a histogram of the samples, using e.g plt.hist(samples, 100, density=True), as well as the gaussian mixture.

You should use plt.subplot beteween each experiment, i.e. just before you draw the first plot, do: plt.subplot(141), and just before the second do: plt.subplot(142)

For a different mixture, the plot looks like this after 100 draws:

Save this plot as 3_2_1.png

What to turn in to Gradescope

Read this carefully before you submit your solution.

You should edit template.py to include your own code.

This is an individual project, you can of course help each other out but your code should be your own.

You can use built-in python modules as you wish, however you are not allowed to install and import packages than are not already imported.

Files to turn in:

• template.py: This is your code
• 1_2_1.png
• 2_2_1.png
• 3_2_1.png

Make sure the file names are exact. Submission that do not pass the first two tests in Gradescope will not be graded.