# DLFNLP Exercise 1

## 1
### a)
11

### b)
$$
\begin{aligned}
\begin{bmatrix}
14 \\
10 \\
\end{bmatrix}
\end{aligned}
$$

### c)
$$
\begin{aligned}
\begin{bmatrix}
8 & 20 \\
5 & 13 \\
\end{bmatrix}
\end{aligned}
$$

### d)

$det(X) = 2*3 - 1*4 = 2 \implies \text{invertible}$

$$
\begin{aligned}
\begin{bmatrix}
1.5 & -2 \\
-0.5 & 1 \\
\end{bmatrix}
\end{aligned}
$$

### e)
2

$$
\begin{aligned}
\begin{bmatrix}
1.5 & -2 \\
-0.5 & 1 \\
\end{bmatrix}
\end{aligned}
$$

## 2
### a)

$3 * x^2 + 1$

### b)
$40x^3 - 8x$

### c)
$\frac{16x^2+4x-24}{64x^2+16x+1}$

### d)
$24*(3x-2)^7$

### e)
$\frac{2x + 1}{x^2 + x}$

## 3.

## 4.
### b)
Fully connected layer => every beuron has weights to the next layer + every neuron (besides input neurons) have a bias

From this follows there are:
$D_X * D_H + D_H + D_H * D_Y + D_Y = (D_X+1)D_H + (D_H+1)D_Y$
neurons.

## 5.

We had some problems with mismatched dimensionality in our backpropagation method. Because of this we sadly didn't manage to calculate the accuracy before the deadline.

import numpy as np
import random
from sklearn.datasets import load_iris

class nn:
    W_1 = np.random.rand(8,4)
    W_2 = np.random.rand(3,8)

    B_1 = np.random.rand(8)
    B_2 = np.random.rand(3)
    
    h_1 = None
    h_2 = None
    a_2 = None
    
    @staticmethod
    def cross_entropy(y,y_hat):
        return -np.sum([y[i] * np.log(y_hat[i]) for i in range(len(y))])
    
    @staticmethod
    def relu(x):
        return np.array([xi if xi > 0 else 0 for xi in x])
    
    
    
    @staticmethod
    def softmax(x):
        denom = np.sum([np.exp(xi) for xi in x])
        return np.array([np.exp(xi)/denom for xi in x])
    
    def softmax_cross_d(self,x,y):
        return self.softmax(x) - y
    
    def forward(self,X):
        a_1 = self.W_1 @ X - self.B_1
        self.h_1 = self.relu(a_1)
        
        self.a_2 = self.W_2 @ self.h_1 - self.B_2
        self.h_2 = self.softmax(self.a_2)
        return self.h_2
    
    def fit(self,train,target,epochs,learning_rate):
        for epoch in range(epochs):
            for i,X in enumerate(train):
                y_hat = self.forward(X)
                err = self.cross_entropy(target[i],y_hat)
                l2_err = self.softmax_cross_d(y_hat,target[i])
                l1_err = np.transpose(self.W_2) @ l2_err
                
                 
                self.W_2 = self.W_2 - (learning_rate * l2_err.dot(np.transpose(self.h_2)))
                self.W_1 = self.W_1 - (learning_rate * l1_err.dot(np.transpose(self.h_1)))
                
                self.B_2 = self.B_2 - learning_rate * l2_err 
                self.B_1 = self.B_1 - learning_rate * l1_err 
            print(err)
            
data = load_iris()
data_len = len(data["target"])
train_num = int(0.8 * data_len)
test_num = data_len - train_num

test_data = []
test_targets = []

test_rows = random.sample(range(data_len),test_num)
for i in test_rows:
    test_data.append(data["data"][i])
    test_targets.append(data["target"][i])

train_data = []
train_targets = []

for i in range(data_len):
    if i not in test_rows:
        train_data.append(data["data"][i])
        train_targets.append(data["target"][i])

for i in range(len(train_targets)):
    train_targets[i] = np.eye(3)[train_targets[i]]
    
for i in range(len(test_targets)):
    train_targets[i] = np.eye(3)[test_targets[i]]

model = nn()

model.fit(np.array(train_data),np.array(train_targets),200,10 ** -4)

## 6.
### a)
1) holy

hearken
serpent
perished
faithfulness
appear
sayingspeak
baruch
common
solemn
grain

2) woman

olivetree
threshingfloor
jacob
sinoffering
nathan
pot
philip
angel
fountain
nebuzaradan

3) light

light
delight
delighteth
lightnings
lightning
delighted
lights
twilight
flight