$4.$ Programming a perceptron: Consider a classification problem of two continuous input variables and a binary class variable. Apply perceptron weight training algorithm to find a line that seperates the two classes.

Choose a learning rate

Stat with initial weights for each variable and a bias term which are small numbers: uniformly distributed between $-0\mathrm{.}1$ and $+0\mathrm{.}1$

Iterate until accuracy is $100\%$ for P$1$ or accuracy reachs a reasonable value for P$2$

At each step

Update weights and bias

Monitor accuracy and the weights and bias

When stopped: monitor the final weights and accuracy

Solve the problem for the two datasets P$1$ and P$2.$ P$1$ isa linearly seperable data whereas P$2$ cannot be perectly separated by a single perceptron.

Each problem is solved by a different program in Python

Input Generation

Generate $100$ data points of two continuous inputs uniformly distributed between $0$ and $10$

Make s scatter plot of your data

These are the input features of two continuous variable used in both P$1$ and P$2$

P$1$: A Linearly Seperable Dataset

Labesl the data with two lnearly seperable classes

Choose a line in $2-$dimensional input space specified by w$0,$w$1$ and w$3$ the three we$4$igths of the perceptron

Such as L: w$1*$x$1+$ w$2*$x$2+$ w$0=0$

Such as L: $1*$x$1+1*$x$210=0$

Where x$1$ and x$2$ are input features and w$0,$w$1$ and w$2$ are weigths to be learned by the perceptron

Here as you set the line such that

Above lhis line if L $>=0$ the output class is yes or $1$

$$f L $<0$ the class is not buy or $0$

For all $100$ input of two features between $0$ and $10$

Apply the above formula to generate a label

So at the end of labeling approximately $50\%$ of your date is labeled as class $1$ and the other $50\%$ is of class $2$

Then implement a percvepron training alorithm in Python

Start with initial weights

Choose a learning rate

In each eopch

Pass over all data point

and make a littel update of the weights

Aster several epochs

plot the labeled daa together with the learned line

Monitor the accuracy rate as well

Note Since the problem is linearly seperable by the line whose equation you know

The final accuracy shoud be $100\%$

You know the line as the generator of he datga.

Let the perceptr$$n learn it: learn the weigths w$0,$w$1$ and w$2$

P$2$ A dataset that is not linearly seperable

In P$1$ ht eline you set say x$1$ x$210=10$ perfectlyseperates the two classes

Say above it is class $1$ and below it as class $0$

Now lst us generatre classes assign labels to the say $100$ inputs so as to generate a data set that is not linearly seperable.

Think of two additional lines just $2$ units above and below the seperating line.

Say LU: x$1+$ x$2-12=0$ and

LL: x$1+$ x$28=0$

How to label an non$-$seperable class

Pass over all $100$ data

If LU is satisfied x$1+$ x$2-12>=0$ Label as class $1$

els if LL is satisfied x$1+$ x$2-8<=0$ Label as class $0$

else $//$ data is in beween LU and LU

generate a ranom number between $0$ and $1$ say r

if r $<0\mathrm{.}5$ label as class$1$ otherwise label as class $0$

This is how the inputs are labeled as a not linearly seperable data

Tht is a line $$ a single perceptron cannot seperarte with $100\%$ accuracy

Make a sctter plot of the data includeing the labeled classes

Appy the perceptron algorithm you depelop to this data

After several $10$s or $100$s of eqochs

Monitor the accuracy rate or the error rate

Make scatter plot of the data as well as the lerend line

Make a comperison with P$1$