README.md

Sequential Estimation

Online learning is very important in machine learning as it allows for the inclusion of new data samples without having to recalculate model parameters for the rest of the data. The aim of this exercise is to explore this concept.

Section 1

We will now look into online estimation of a mean vector. The objective is to apply the following formula for estimating a mean (see Bishop Section 2.3.5):

$$ \mu_{ML}^{N} = \mu_{ML}^{N-1} + \frac{1}{N}(x_n - \mu_{ML}^{N-1}) $$

Section 1.1

Let's first create a data generator. Create a function gen_data(n, k, mean, var) which returns a $n\times k$ array, $X$. This $X$ contains a sequence of $n$ vectors of dimension $k$. Each vector $x_i$ in $X$ should be $x_i \sim N_k(\mu, \sigma^2I_k)$ where:

• $N_k()$ is the k-variate normal distribution
• $\mu$ (or mean) is the mean vector of dimension $k$
• $\sigma$ (or var) is the variance.
• $I_k$ is the identity matrix

You should use np.random.multivariate_normal for this.

Example inputs and outputs (these examples use np.random.seed(1234)):

1. gen_data(2, 3, np.array([0, 1, -1]), 1.3)

[[ 0.61286571, -0.5482684 ,  0.86251906],
 [-0.40644746,  0.06323465,  0.15331182]]

2. gen_data(5, 1, np.array([0.5]), 0.5)

[[ 0.73571758],
 [-0.09548785],
 [ 1.21635348],
 [ 0.34367405],
 [ 0.13970563]]

Section 1.2

Answer this question in a raw text file and submit it as 1_2_1.txt

Lets create some data $X$. Create 300 3-dimensional data points sampled from $N_3([0, 1, -1], \sqrt{3})$

You can visualize your data using tools.scatter_3d_data to get a plot similar to the following

You can also use tools.bar_per_axis to visualize the distribution of the data per dimension:

Do you expect the batch estimate to be exactly $(0, 1, -1)$ ? Which two parameters can be used to make this estimate more accurate?

Section 1.3

Continue to section 1.4.

Section 1.4

We will now implement the sequential estimate.

We want a function that returns $N$ number of sequential estimates of the mean vector where we feed one vector from $X$ at a time into the function. We start by implementing the update equation above.

Create a function update_sequence_mean(mu, x, n) which performs the update in the equation above.

Example inputs and outputs:

mean = np.mean(X, 0)
new_x = gen_data(1, 3, np.array([0, 0, 0]), 1)
update_sequence_mean(mean, new_x, X.shape[0]+1)

Results in an array, similar to [[-0.21653761 -0.00721158 -0.15876203]] (since we're using random numbers, the values you get will probably not be exactly the same). Gradescope wil use np.random.seed(1234) before generating the data to test this function.

Section 1.5

Lets plot the estimates on all dimensions as the sequence estimate gets updated. You can use _plot_sequence_estimate() as a template. You should:

• Generate 100 3-dimensional points with mean [0, 0, 0] and variance 4.
• Set the initial estimate as $(0, 0, 0)$
• And perform update_sequence_mean for each point in the set.
• Collect the estimates as you go

For a different set of points this plot looks like the following:

Turn in your plot as 1_5_1.png

Section 1.6

Lets now plot the squared error between the estimate and the actual mean after every update.

The squared error between e.g. a ground truth $y$ and a prediction $\hat{y}$ is $(y-\hat{y})^2$.

Of course our data will be 3-dimensional so after calculating the squared error you will have a 3-dimensional error. Take the mean of those three values to get the average error across all three dimensions and plot those values.

You can use _plot_square_error and _square_error for this.

For a different distribution this plot looks like the following:

Turn in your plot as 1_6_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 so you can of course help each other out but your code should be your own.

You are not allowed to import any non-built in packages that are not already imported.

Files to turn in:

• template.py: This is your code
• 1_2_1.txt
• 1_5_1.png
• 1_6_1.png

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

Independent Section (optional)

What happens if the mean value changes (perhaps slowly) with time? What if $\mu =(0,1,-1)$ moves to $\mu=(1,-1,0)$ in 500 time ticks? How would we track the mean? Some sort of a forgetting could be added to the update equation. How would that be done?

Create this type of data and formulate a method for tracking the mean.

Plot the estimate of all dimensions and the mean squared error over all three dimensions. Turn in these plots as indep_1.png and indep_2.png.

Write a short summary how your method works.