NN : Forward and Back Propagation
Q1. Sigmoid and
softmax functions
Which of the following
statements is true for a neural network having more than one output neuron ?
Choose the correct answer
from below:
A.
In a neural network where the output
neurons have the sigmoid activation, the sum of all the outputs from the
neurons is always 1.
B.
In a neural network where the output
neurons have the sigmoid activation, the sum of all the outputs from the
neurons is 1 if and only if we have just two output neurons.
C.
In a neural network where the output
neurons have the softmax activation, the sum of all the outputs from the
neurons is always 1.
D.
The softmax function is a special
case of the sigmoid function
Ans: C
Correct option : In
a neural network where the output neurons have the softmax activation, the sum
of all the outputs from the neurons is always 1.
Explanation :
- For the sigmoid activation, when we
have more than one neuron, it is possible to have the sum of outputs from
the neurons to have any value.
- The softmax classifier outputs the
probability distribution for each class, and the sum of the probabilities
is always 1.
- The Sigmoid function is
the special case of the Softmax function where the number
of classes is 2.
Q2. Forward
propagation
Given the independent and
dependent variables in X and y, complete the code
to calculate the results of the forward propagation for
a single neuron on each observation of the dataset.
The code should print the
calculated labels for each observation of the given dataset i.e. X.
Input Format:
Two lists are taken as
the inputs. First list should be the independent variable(X) and the second
list should be the dependent variable(y)
Output Format:
A numpy array consisting
of labels for each observation.
Sample Input:
X
= [[100, 129, 157, 133], [168, 150, 30, 19], [4, 148, 106, 74], [123, 195, 60,
93], [169, 40, 188, 179], [40, 59, 29, 94], [165, 126, 16, 99], [167, 157, 65,
23], [128, 87, 37, 111], [191, 154, 89, 134], [101, 41, 145, 112], [43, 110,
197, 118], [147, 22, 109, 139], [11, 161, 135, 119], [26, 48, 199, 182], [96,
100, 82, 87], [149, 2, 8, 10], [5, 38, 166, 100], [193, 117, 59, 164], [133, 5,
38, 163], [88, 177, 84, 114], [9, 132, 177, 24], [94, 130, 83, 131], [77, 11,
141, 81], [154, 198, 175, 98], [21, 148, 170, 122], [185, 145, 101, 183], [100,
196, 111, 11], [97, 147, 112, 11], [25, 97, 95, 45], [6, 89, 88, 38], [51, 16,
151, 3], [90, 174, 122, 157], [2, 133, 121, 199], [15, 78, 163, 180], [103,
118, 7, 179], [102, 179, 157, 183], [113, 139, 195, 122], [55, 88, 68, 117],
[115, 185, 93, 102], [139, 82, 3, 165], [135, 29, 78, 11], [11, 16, 60, 123],
[103, 191, 187, 129], [146, 181, 28, 192], [85, 73, 136, 139], [117, 179, 81,
183], [15, 131, 106, 28], [58, 78, 111, 65], [76, 11, 25, 103], [11, 90, 162,
129], [144, 1, 16, 33], [33, 172, 40, 72], [106, 83, 160, 151], [68, 159, 150,
64], [31, 79, 83, 15], [51, 140, 173, 10], [105, 80, 70, 21], [195, 80, 64,
129], [50, 96, 107, 82], [185, 150, 15, 143], [28, 71, 27, 57], [58, 13, 146,
78], [20, 71, 183, 44], [91, 44, 15, 87], [77, 157, 95, 110], [132, 28, 193,
49], [177, 87, 57, 41], [194, 175, 17, 20], [166, 64, 134, 150], [79, 74, 162,
168], [166, 149, 34, 117], [160, 170, 127, 44], [99, 41, 103, 155], [48, 127,
138, 68], [17, 3, 101, 94], [29, 102, 123, 158], [194, 60, 135, 179], [73, 192,
145, 168], [21, 94, 154, 143], [17, 10, 145, 131], [73, 29, 195, 199], [132,
189, 90, 100], [134, 32, 81, 119], [118, 37, 119, 27], [51, 78, 187, 86], [95,
8, 56, 29], [156, 162, 186, 127], [126, 111, 144, 59], [7, 140, 32, 75], [40,
0, 109, 92], [165, 175, 61, 103], [178, 68, 185, 119], [132, 105, 36, 80],
[165, 117, 35, 176], [128, 49, 185, 9], [50, 176, 12, 198], [124, 164, 99,
102], [36, 30, 114, 147], [166, 172, 35, 14]]
y = [1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1,
1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1,
0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1,
1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0]
Sample Output:
[1 1 1 1 1 1 1 1 1 1 1 1
0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1
1 1 1 1 0 1]
Fill the Missing Code
import
numpy as np
np.random.seed(2)
#independent
variables
X
= np.array(eval(input()))
#dependent
variable
y
= np.array(eval(input()))
m
= X.shape[__] #no. of samples
n
= X.shape[__] #no. of features
c
= #no. of classes in the data and therefore no. of neurons in the
layer
#weight
vector of dimension (number of features, number of neurons in the layer)
w
= np.random.randn(___, ___)
#bias
vector of dimension (1, number of neurons in the layer)
b
= np.zeros((___, ___))
#(weighted
sum + bias) of dimension (number of samples, number of classes)
z
= ____
#exponential
transformation of z
a
= np.exp(z)
#Perform
the softmax on a
a
= ____
#calculate
the label for each observation
y_hat
= ____
print(y_hat)
Final Code:
import
numpy as np
np.random.seed(2)
#independent
variables
X
= np.array(eval(input()))
#dependent
variable
y
= np.array(eval(input()))
#‘m’
and ‘n’ refers to the no. of rows and columns in the dataset respectively.’c’
refers to the number of classes in y.
m
= X.shape[0] #no. of samples
n
= X.shape[1] #no. of features
c
= len(np.unique(y)) #no. of classes in the data and therefore no. of
neurons in the layer
#Initializing
weights randomly
#weight
vector of dimension (number of features, number of neurons in the layer)
w
= np.random.randn(n, c)
#Initializing
biases as zero
#bias
vector of dimension (1, number of neurons in the layer)
b
= np.zeros((1, c))
#Finding
the output ‘z’
#(weighted
sum + bias) of dimension (number of samples, number of classes)
z
= np.dot(X, w) + b
#Applying
the softmax activation function on the output
#exponential
transformation of z
a
= np.exp(z)
a
= a/np.sum(a, axis = 1, keepdims = True)
#Calculating
the label for each observation
y_hat
= np.argmax(a, axis = 1)
print(y_hat)
Q3. Same layer still
different output
Why do two neurons in the
same layer produce different outputs even after using the same kind of function
(i.e. wT.x + b)?
Choose the correct answer
from below:
A.
Because the weights are not the same
for the neurons.
B.
Because the input for each neuron is
different.
C.
Because weights of all neurons are
updated using different learning rates.
D.
Because only biases (b) of all
neurons are different, not the weights.
Ans: A
Correct option: Because
the weights are not the same for the neurons.
Explanation :
- The weights for each neuron in a
layer are different. Thus the output of each neuron ( wT.x + b ) will be
different
- The input for each neuron in a layer
is the same. In a fully connected network, each neuron in a particular
layer gets inputs from each neuron in the previous layer.
- There may be different learning rates
for each model weight depending on the type of optimizer used, but that is
not the reason for the neurons to give different outputs.
- In a fully connected network, each
neuron has two trainable parameters : a bias and a weight. The values of
bias and weight for any two neurons in a layer need not be the same, since
they keep changing during the model training.
Q4. Will he watch
the movie?
We want to predict
whether a user would watch a movie or not. Each movie has a certain number of
features, each of which is explained in the image.
Now take the case of the
movie Avatar having the features vector as [9,1,0,5].
According to an algorithm, these features are assigned the weights [0.8,0.2,0.5,0.4] and bias=-10.
For a user X, predict whether he will watch the movie or not if the threshold
value(θ) is 10?
Note:
If the output of the neuron is greater than θ then the user will watch the
movie otherwise not.
Choose the correct answer
from below:
A.
Yes, the user will watch the movie
with neuron output = -0.6
B.
No, the user will not watch the movie
with neuron output = -0.6
C.
No, the user will watch the movie
with neuron output = 2.5
D.
Yes, the user will watch the movie
with neuron output = 2.5
Ans: B
Correct option :No,
the user will not watch the movie with neuron output with -0.6
Explanation :
The output of a neuron is
obtained by taking the weighted sum of inputs and adding the bias term to it.
The output of neuron is :
(0.8).(9)+(0.2).(1)+(0).(0.5)+(5).(0.4)−10 =−0.6<10,
therefore he won’t watch the movie.
Q5. Dance festival
The dance festival is in
this coming weekend and you like dancing as everyone does. You want to decide
whether to go to the festival or not. And the decision depends on the stated
three factors: ( x1, x2, x3 are boolean variables )
- x1: Is the
weather good? (x1=1 means good)
- x2: Will your
friend accompany you? (x2=1 means she will accompany)
- x3: Is the
festival near public transit? (x3=1 means ‘Yes’)
All of the above variables/factors are binary. You are using the following
rules for deciding whether to go or not (1 means you will go):
Output:
0:
if the ∑inwixi<threshold
1: if the ∑inwixi≥threshold}
Which of the given
options will be the values of the correct weights for deciding with threshold=5
using the above rules, given the condition that you won't go unless the weather
is good, and will definitely go if the weather is good?
Choose the correct answer
from below:
A.
[6,2,2]
B.
[4,2,4]
C.
[3,2,1]
D.
[4,2,2]
Ans: A
Correct option : [6,2,2]
Explanation :
We can try substituting
the values of weights for different values of variables:
a. x1=1,x2=0,x3=0:Weightedsum=6x1+2x0+2x0=6>5 , therefore Yes, you will go.
b. x1=0,x2=1,x3=1:Weightedsum=6x0+2x1+2x1=4<5 therefore No, you won’t go.
c. x1=1,x2=0,x3=1:Weightedsum=6x1+2x0+2x1=8>5, therefore Yes
According to the above
calculations weights [6,2,2] give the required outcomes for the necessary
condition of good weather (x1=1).
It gives a value greater than the threshold if x1=1, otherwise a smaller value
irrespective of the values of x2 and x3.
Other options don’t give
these desired outputs
Q6. Value of weight
While training a simple
neural network, this is what we get with the given input:
If the neural network is
defined as:
then what value of W will
satisfy the actual output values given that bias equals
to 1.
Choose the correct answer
from below:
A.
2
B.
5
C.
10
D.
7
Ans: A
Correct option : 2
Explanation:
- This problem can be solved by simple
substitution in equations. It is given that bias is equals to 1. And W is
same for all of them.
- We want to find the value W which
gives the results as the actual output given in the table.
Hence, for input = 1,
- w×1+1=3, which
gives the value of W = 2
Similarly for input
as 2, we will get
- w×2+1=5, again
the value of W = 2
And finally when input
= 4,
- w×4+1=9, again
the value of W = 2
Therefore 2 is the
correct answer
Q7. Vectorize
Consider the following
code snippet:
How do you vectorize
this?
Note:
All x, y and z are NumPy arrays.
Choose the correct answer
from below:
A.
z = x + y
B.
z= x * y.T
C.
z = x + y.T
D.
z = x.T + y.T
Ans: C
Correct option
: z=x+y.T
Explanation :
The shape of x is (10,6)
The shape of y is (6,1)
We can observe from the
question that all elements of y are added element-wise to z, while y[j] is
added to each element in the jth column in z.
For this to happen, we will have to convert the y array to shape (1,6) and then
add it to x, which will result in broadcasting of y into shape (10,6) .
Thus the answer is
z=x+y.T
Q8. Identify the
Function
Mark the correct option
for the below-mentioned statements:
(a) It
is possible for a perceptron that it adds up all the weighted inputs it
receives, and if the sum exceeds a specific value, it outputs a 1. Otherwise,
it just outputs a 0.
(b) Both
artificial and biological neural networks learn from past experiences.
Choose the correct answer
from below:
A.
Both the mentioned statements are
true.
B.
Both the mentioned statements are
false.
C.
Only statement (a) is true.
D.
Only statement (b) is true.
Ans: A
Correct option:
Both the statements are true.
Explanation :
Implementation of
statement (a) is called step function and yes it is possible.
Both of artificial and
biological neural networks learn from past experiences.
The artificial networks are trained on data to make predictions. The weights
assigned to each neuron continuously changes during the training process to reduce the error.
Q9. Find the Value
of 'a'
Given below is a neural
network with one neuron that takes two float numbers as inputs.
If the model uses the
sigmoid activation function, What will be the value of 'a' for the given
x1 and x2 _____(rounded off to 2 decimal places)?
Choose the correct answer
from below:
A.
0.57
B.
0.22
C.
0.94
D.
0.75
Ans: A
Correct option :
Explanation :
The value of z will
be :
- z= w1.x1+w2.x2+b
- z = (0.5×0.55) + (−0.35×0.45) + 0.15 = 0.2675
The value of a will
be :
- a= f(z) = σ(0.2675) = 1/(1+e(−z))=1.7652901=0.5664=0.57
