$1$ Generating the Data

Before we fit a model, we need data to fit it to$.$ The data set will be a set of points of the form $\{(x_{{?}^{?}}(1),y^1),(x_{{?}^{?}}(2),y^2),$dots,$(x_{{?}^{?}}(N),y^N)\}$ where each $x_{{?}^{?}}(i)in{R}^{101},y^i$inR. To generate the data points $(x_{,}y),$ the following instructions lay out how to generate $(x_0,x_1,x_2,$dots,$x_{100}),$ and then $y$ based on that $x_{?}$ vector.

Note: In the following, I am using $N(,{}^2)$ to indicate a random value sampled from a normal distribution with mean $$ and variance ${}^2.$ In order to generate random values, you need to specify a distribution $($see CS$-206),$ and normal distributions are fairly standard functionality for RNG packages. I am performing a slight abuse of notation when I write $($expression$)+N(,{}^2),$ but this should be interpreted as adding a small random bit of noise sampled from that distribution to the value of $($expression$).$ All noise terms should be taken as independent. These terms are present to represent noise or fluctuations in the measurements that you collect from the real world.

$x_0=1.$

$x_1,$dots,$x_{50}$ are i$.$i$.$d$.$ random values, with distribution $N(0,1).$

For $i=51,52,$dots,$60,x_i=x_1+0\mathrm{.}5**x_{i-50}+N(0,0\mathrm{.}1).$

Note: The way to interpret this is that, for example, $x_{55}=x_1+0\mathrm{.}5**x_5+($a little bit of noise$).$

For $i=61,62,$dots,$70,x_i=x_{i-60}-x_{i-50}+x_{i-40}+N(0,0\mathrm{.}1).$

Example: $x_{65}=x_5-x_{15}+x_{25}+($a little bit of noise$).$

For $i=71,72,$dots,$80,x_i=x_{6**(i-70)}+3**x_{i-70}+N(0,0\mathrm{.}1).$

For $i=81,82,$dots,$90,x_i=5-x_{i-10}.$

For $i=91,92,$dots,$100,x_i=0\mathrm{.}5**x_{50+(i-90)**4}+0\mathrm{.}5**x_{50+(i-90)**3}+N(0,0\mathrm{.}1).$

The output data will be generated in the following way. For any input $x_{?},$ the output is given by

$y=({\displaystyle \underset{i=1}{\overset{50}{}}}(-0\mathrm{.}88{)}^i{x}_{2i})+N(0,0\mathrm{.}01).$

Note that with this setup, none of the odd$-$indexed components of $x_{?}$ 'explicitly' show up in the value of $y.$

Note: This specifies that the value of $y$ is based on $x_{?}$ via $y=w_{*}{x}_{+}($a little noise$),$ and that the 'true' weights are given by

$w_{?}=(0,0,-0\mathrm{.}88,0,(-0\mathrm{.}88{)}^2,0,(-0\mathrm{.}88{)}^3,0,$dots,$0,(-0\mathrm{.}88{)}^{50})$

$1$

Problem $1$: Write a function that takes as input a number of data points $N,$ and gives as output a data matrix $x$ as a PyTorch tensor, with $N$ rows and $101$ columns, where each row is an $x_{?}$ vector as above, as well as a PyTorch tensor $y_{?}$ with $N$ rows and $1$ column, where each entry of $y_{?}$ is the corresponding $y$ value for that row of $x.$

Some functions that may be useful:

torch.zeros

torch.randn