how to generate this dataset using matlab?

I'm here asking a question about a postgraduate Bayesian coursework I'm doing. It's about data generation. this is the data generation process the paper asks me to follow

It's not clear and my lecturer at my university illustrated further that, the 45 betas will be evenly allocated to the 15 groups, which mean each group would have 3 betas. Also, it will be in the order of : [1(Z1=0) 1(Z1=1) 1(Z1=2)]' [1(Z2=0) 1(Z2=1) 1(Z2=2)]' ....... [1(Z15=0) 1(Z15=1) 1(Z15=2)]' where the '1(...)' is the indicator function, according to what the teacher says.

the teacher also says after getting the Z vector, which is supposed to be a 15*1 vector, we should be able to use the Z vector to generate the X matrix (n*45, where n is number of observations and 45 is number of betas), which will then be used to calculate y and hence generate the dataset.

My question here is how exactly can a 15*1 vector (Z) be used to generate a n*45 matrix (the X matrix)? (because this is what the teacher told me explicitly in email and he's getting impatient to say anything more than that...)

And more importantly, how to use matlab to get this X matrix?

Best regards

Answer 1

[Kartik Saxena]

Hi,

Based on your description, it seems like you are trying to generate a design matrix `X` for a categorical variable `Z` that has 15 groups, each with 3 levels. The indicator function `1(Zi=j)` is a binary indicator that is 1 if the i-th group is at level j, and 0 otherwise. This is a common way to encode categorical variables for regression analysis.

To generate the matrix `X` in MATLAB, you would need to follow these steps for each observation:

1. For each group, create a vector of length 3 with all zeros except for a 1 at the position corresponding to the level of the group.

2. Concatenate these vectors for all 15 groups to form a row of the design matrix `X`.

3. Repeat this process for all observations `n` to form the full `n x 45` design matrix.

I hope this resolves your issue.