1.5 McCulloch Pitts Neuron

Your first model

Recap: Six jars

What we saw in the previous chapter?

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Learning

Loss

Model

Data

Task

Evaluation

Artificial Neuron

What is the fundamental building block of Deep Learning ?

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f

Recall Biological Neuron

Where does the inspiration come from ?

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The Model

When and who proposed this model ?

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The early model of an artificial neuron is introduced by Warren McCulloch and Walter Pitts in 1943. The McCulloch-Pitts neural model is also known as linear threshold gate.

Walter Pitts was a logician who proposed the first mathematical model of a neural network. The unit of this model, a simple formalized neuron, is still the standard of reference in the field of neural networks. It is often called a McCulloch–Pitts neuron.

Warren McCulloch was a neuroscientist who created computational models based on threshold logic which split the inquiry into two distinct approaches, focused on biological processes in the brain and application of neural networks to artificial intelligence.

* Images adapted from https://www.i-programmer.info/babbages-bag/325-mcculloch-pitts-neural-networks.html

The Model

How are we going to approach this ?

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Learning

Loss

Model

Data

Task

Evaluation

The Model

What is the mathematical model ?

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One parameter, b

f

g

McCulloch and Pitts proposed a highly simplified computational model of the neuron.

The inputs can be excitatory or inhibitory

g aggregates the inputs and the function f takes a decision based on this aggregation.

\(y=0\) if any \(x_i\) is inhibitory, else

\tiny{g(x_{1},x_{2},...x_{n})=g(x)= \sum_{i=1}^{n} x_{i} }
g(x1,x2,...xn)=g(x)=∑i=1nxi\tiny{g(x_{1},x_{2},...x_{n})=g(x)= \sum_{i=1}^{n} x_{i} }
\tiny y=f(g(x))=1 \text{ if } g(x)\geq b
y=f(g(x))=1 if g(x)≥b\tiny y=f(g(x))=1 \text{ if } g(x)\geq b
\tiny=0 \text{ if } g(x) < b
=0 if g(x)&lt;b\tiny=0 \text{ if } g(x) &lt; b
\tiny \in \{0,1\}
∈{0,1}\tiny \in \{0,1\}
\tiny \in \{0,1\}
∈{0,1}\tiny \in \{0,1\}

Data and Task

What kind of data and tasks can MP neuron process ?

Pitch
in line
1
0
1
0

\(x_1\)

(pitch)

\(x_3\)

    (missing stumps)

\(x_2\)

(impact)

\(y\) (LBW)

y=\sum_{i=1}^3 x_i \geq 3
y=∑i=13xi≥3y=\sum_{i=1}^3 x_i \geq 3

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Pitch
in line		 Impact Missing stumps Is it LBW? (y)
1 0 0 0
0 1 1 0
1 1 1 1
0 1 0 0
Pitch
in line		 Impact Missing stumps
1 0 0
0 1 1
1 1 1
0 1 0
Pitch
in line		 Impact
1 0
0 1
1 1
0 1

\(x_1\)

\(x_2\)

\(x_3\)

\(y\)

Data and Task

What kind of data and tasks can MP neuron process ?

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Boolean inputs

Boolean output

\(x_1\)

b

y=\sum_{i=1}^n x_i \geq b
y=∑i=1nxi≥by=\sum_{i=1}^n x_i \geq b
Launch (within 6 months) 0 1 1 0 0 1 0 1 1
Weight (<160g) 1 0 1 0 0 0 1 0 0
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1
dual sim 1 1 0 0 0 1 0 1 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1
NFC 0 1 1 0 1 0 1 1 1
Radio 1 0 0 1 1 1 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0
Price > 20k 0 1 1 0 0 0 1 1 1
Like (y) 1 0 1 0 1 1 0 1 0
Launch (within 6 months) 0 1 1 0 0 1 0 1 1
Weight (<160g) 1 0 1 0 0 0 1 0 0
Launch (within 6 months) 0 1 1 0 0 1 0 1 1

?

Launch (within 6 months) 0 1 1 0 0 1 0 1 1
Weight (<160g) 1 0 1 0 0 0 1 0 0
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1
dual sim 1 1 0 0 0 1 0 1 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1
NFC 0 1 1 0 1 0 1 1 1
Radio 1 0 0 1 1 1 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0
Price > 20k 0 1 1 0 0 0 1 1 1

\(x_2\)

\(x_n\)

\(y\)

Loss Function

How do we compute the loss ?

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Launch (within 6 months) 0 1 1 0 0 1 0 1 1
Weight (<160g) 1 0 1 0 0 0 1 0 0
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1
dual sim 1 1 0 0 0 1 0 1 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1
NFC 0 1 1 0 1 0 1 1 1
Radio 1 0 0 1 1 1 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0
Price > 20k 0 1 1 0 0 0 1 1 1
Like (y) 1 0 1 0 1 1 0 1 0
loss / error = y-\hat{y}
loss/error=y−y^loss / error = y-\hat{y}
\hat{y}
y^\hat{y}
x_1
x1x_1
x_2
x2x_2
x_n
xnx_n
b
bb
\hat{y}=\sum_{i=1}^n x_i \geq b
y^=∑i=1nxi≥b\hat{y}=\sum_{i=1}^n x_i \geq b
Launch (within 6 months) 0 1 1 0 0 1 0 1 1
Weight (<160g) 1 0 1 0 0 0 1 0 0
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1
dual sim 1 1 0 0 0 1 0 1 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1
NFC 0 1 1 0 1 0 1 1 1
Radio 1 0 0 1 1 1 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0
Price > 20k 0 1 1 0 0 0 1 1 1
Like (y) 1 0 1 0 1 1 0 1 0
Prediction    0
\hat{y}
y^\hat{y}

Loss Function

How do we compute the loss ?

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loss = \sum_i y_i-\hat{y_i}
loss=∑iyi−yi^loss = \sum_i y_i-\hat{y_i}
\hat{y}
y^\hat{y}
x_1
x1x_1
x_2
x2x_2
x_n
xnx_n
b
bb
\hat{y}=\sum_{i=1}^n x_i \geq b
y^=∑i=1nxi≥b\hat{y}=\sum_{i=1}^n x_i \geq b
Launch (within 6 months) 0 1 1 0 0 1 0 1 1
Weight (<160g) 1 0 1 0 0 0 1 0 0
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1
dual sim 1 1 0 0 0 1 0 1 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1
NFC 0 1 1 0 1 0 1 1 1
Radio 1 0 0 1 1 1 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0
Price > 20k 0 1 1 0 0 0 1 1 1
Like (y) 1 0 1 0 1 1 0 1 0
\hat{y}
y^\hat{y}
\hat{y}
y^\hat{y}
Launch (within 6 months) 0 1 1 0 0 1 0 1 1
Weight (<160g) 1 0 1 0 0 0 1 0 0
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1
dual sim 1 1 0 0 0 1 0 1 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1
NFC 0 1 1 0 1 0 1 1 1
Radio 1 0 0 1 1 1 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0
Price > 20k 0 1 1 0 0 0 1 1 1
Like (y) 1 0 1 0 1 1 0 1 0
Prediction 0 0 1 0 0 1 1 1 0

Loss Function

How do we compute the loss ?

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loss = \sum_i y_i-\hat{y_i} = 0
loss=∑iyi−yi^=0loss = \sum_i y_i-\hat{y_i} = 0
\hat{y}
y^\hat{y}
x_1
x1x_1
x_2
x2x_2
x_n
xnx_n
b
bb
\hat{y}=\sum_{i=1}^n x_i \geq b
y^=∑i=1nxi≥b\hat{y}=\sum_{i=1}^n x_i \geq b
Launch (within 6 months) 0 1 1 0 0 1 0 1 1 0
Weight (<160g) 1 0 1 0 0 0 1 0 0 1
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1 0
dual sim 1 1 0 0 0 1 0 1 0 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1 0
NFC 0 1 1 0 1 0 1 1 1 0
Radio 1 0 0 1 1 1 0 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0 0
Price > 20k 0 1 1 0 0 0 1 1 1 0
Like?      (y) 1 0 1 0 1 1 0 1 0 0
prediction          1 0 0 1 1 1 1 0 0 0
loss 0 0 1 -1 0 0 -1 1 0 0
\hat{y}
y^\hat{y}
loss = \sum_i y_i-\hat{y_i}
loss=∑iyi−yi^loss = \sum_i y_i-\hat{y_i}
loss = \sum_i (y_i-\hat{y_i})^2
loss=∑i(yi−yi^)2loss = \sum_i (y_i-\hat{y_i})^2

Learning Algorithm

How do we train our model?

(c) One Fourth Labs

Only one parameter, can afford Brute Force search

b = 1 ?
b=1?b = 1 ?
\hat{y}=\sum_{i=1}^n x_i \geq b
y^=∑i=1nxi≥b\hat{y}=\sum_{i=1}^n x_i \geq b
loss = \sum_i (y_i-\hat{y_i})^2
loss=∑i(yi−yi^)2loss = \sum_i (y_i-\hat{y_i})^2
b = 2 ?
b=2?b = 2 ?
b = ?
b=?b = ?
Launch (within 6 months) 0 1 1 0 0 1 0 1 1 0
Weight (<160g) 1 0 1 0 0 0 1 0 0 1
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1 0
dual sim 1 1 0 0 0 1 0 1 0 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1 0
NFC 0 1 1 0 1 0 1 1 1 0
Radio 1 0 0 1 1 1 0 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0 0
Price > 20k 0 1 1 0 0 0 1 1 1 0
Like?      (y) 1 0 1 0 1 1 0 1 0 0
prediction          1 1 1 1 1 1 1 1 1 1
Launch (within 6 months) 0 1 1 0 0 1 0 1 1 0
Weight (<160g) 1 0 1 0 0 0 1 0 0 1
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1 0
dual sim 1 1 0 0 0 1 0 1 0 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1 0
NFC 0 1 1 0 1 0 1 1 1 0
Radio 1 0 0 1 1 1 0 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0 0
Price > 20k 0 1 1 0 0 0 1 1 1 0
Like?      (y) 1 0 1 0 1 1 0 1 0 0
prediction          1 1 1 1 1 1 1 1 1 0
Launch (within 6 months) 0 1 1 0 0 1 0 1 1 0
Weight (<160g) 1 0 1 0 0 0 1 0 0 1
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1 0
dual sim 1 1 0 0 0 1 0 1 0 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1 0
NFC 0 1 1 0 1 0 1 1 1 0
Radio 1 0 0 1 1 1 0 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0 0
Price > 20k 0 1 1 0 0 0 1 1 1 0
Like?      (y) 1 1 1 0 0 1 1 1 0 0
prediction          - - - - - - - - - -

Evaluation

How does MP perform?

(c) One Fourth Labs

Launch (within 6 months) 0 1 1 0 0 1 0 1 1 0
Weight (<160g) 1 0 1 0 0 0 1 0 0 1
Screen size (<5.9 in) 1 0 1 0 1 0 1 0 1 0
dual sim 1 1 0 0 0 1 0 1 0 0
Internal memory (>= 64 GB, 4GB RAM) 1 1 1 1 1 1 1 1 1 0
NFC 0 1 1 0 1 0 1 1 1 0
Radio 1 0 0 1 1 1 0 0 0 0
Battery(>3500mAh) 0 0 0 1 0 1 0 1 0 0
Price > 20k 0 1 1 0 0 0 1 1 1 0
Like?      (y) 1 1 1 0 0 1 1 1 0 0
predicted        1 1 0 1 1 1 1 0 0 0
loss = \sum_i (y_i-\hat{y_i})^2
loss=∑i(yi−yi^)2loss = \sum_i (y_i-\hat{y_i})^2
1 0 0 1
0 1 1 1
0 1 1 1
0 1 0 0
1 0 0 0
0 0 1 0
1 1 1 0
1 1 1 0
0 0 1 0
0 1 0 0
0 1 1 0

Training data

Test data

Accuracy=\frac{\text{Number of correct predictions}}{\text{Total number of predictions}}
Accuracy=Number of correct predictionsTotal number of predictionsAccuracy=\frac{\text{Number of correct predictions}}{\text{Total number of predictions}}
= \frac{3}{4} = 75\%
=34=75%= \frac{3}{4} = 75\%
\hat{y} = 1 \text{ if } \sum_{i=1}^n w_i x_i \geq b
y^=1 if ∑i=1nwixi≥b\hat{y} = 1 \text{ if } \sum_{i=1}^n w_i x_i \geq b
\hat{y} = 0 \text{ otherwise }
y^=0 otherwise \hat{y} = 0 \text{ otherwise }

\(w_1\)

\(w_2\)

\(w_n\)

Geometric Interpretation

How to interpret the model of MP neuron gemoetrically?

(c) One Fourth Labs

Screen size (>5 in) 1 0 1 0 1 0 1 0 1 0
Battery (>2000 mAh) 0 0 0 1 0 1 0 1 0 0
Like 1 0 1 0 1 1 0 1 0 0
\hat{y}
y^\hat{y}
x_1
x1x_1
x_2
x2x_2
b
bb
\hat{y}=\sum_{i=1}^n x_i \gt 1
y^=∑i=1nxi&gt;1\hat{y}=\sum_{i=1}^n x_i \gt 1

Geometric Interpretation

How to interpret the model of MP neuron gemoetrically?

(c) One Fourth Labs

Screen size (>5 in) 1 0 1 1 1 0 1 0 1 0
Battery (>2000mAh) 0 0 0 1 0 1 1 1 1 0
Like 1 0 1 0 1 1 0 1 0 0
\hat{y}
y^\hat{y}
x_1
x1x_1
x_2
x2x_2
b
bb
\hat{y}=\sum_{i=1}^n x_i \gt ?
y^=∑i=1nxi&gt;?\hat{y}=\sum_{i=1}^n x_i \gt ?

Linear

Fixed Slope

Few possible intercepts (b's)

Take-aways

So will you use MP neuron?

(c) One Fourth Labs

\( \{0, 1\} \)

Classification

loss = \sum_i (y_i-\hat{y_i})^2
loss=∑i(yi−yi^)2loss = \sum_i (y_i-\hat{y_i})^2
Accuracy=\frac{\text{Number of correct predictions}}{\text{Total number of predictions}}
Accuracy=Number of correct predictionsTotal number of predictionsAccuracy=\frac{\text{Number of correct predictions}}{\text{Total number of predictions}}

Loss

Model

Data

Task

Evaluation

Learning

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