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Introduction to Artificial Neural Networks (ANNs) - Step-by-Step Training & Testing Example

The document provides a step-by-step guide on training and testing artificial neural networks (ANNs) at Menoufia University, focusing on classification techniques. It covers key concepts including linear and nonlinear classifiers, the structure of neural networks with input, hidden, and output layers, and the use of various activation functions. Additionally, the document discusses the importance of weights and biases in neural network training.

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Artificial Neural Networks (ANNs) Step-By-Step Training & Testing Example MENOUFIA UNIVERSITY FACULTY OF COMPUTERS AND INFORMATION ALL DEPARTMENTS ARTIFICIAL INTELLIGENCE ‫المنوفية‬ ‫جامعة‬ ‫والمعلومات‬ ‫الحاسبات‬ ‫كلية‬ ‫األقسام‬ ‫جميع‬ ‫الذكاء‬‫اإلصطناعي‬ ‫المنوفية‬ ‫جامعة‬ Ahmed Fawzy Gad ahmed.fawzy@ci.menofia.edu.eg
Neural Networks & Classification
Linear Classifiers
Linear Classifiers
Linear Classifiers
Linear Classifiers Complex Data
Linear Classifiers Complex Data
Linear Classifiers Complex Data
Not Solved Linearly
Not Solved Linearly
Nonlinear Classifiers
Nonlinear Classifiers Training
Nonlinear Classifiers Training
Nonlinear Classifiers Training
Classification Example B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567
Neural Networks B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Hidden Output
Neural Networks B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567
Input Layer B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B
Output Layer B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE 𝒀𝒋
Weights B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 Weights=𝑾𝒊 𝒀𝒋
Activation Function B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
Activation Function B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
Activation Function B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
Activation Function Components B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
Activation Function Inputs B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 s 𝒀𝒋
Activation Function Inputs B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝒀𝒋
Activation Function Inputs B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝒀𝒋
Activation Function Inputs B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 𝒀𝒋
Activation Function Inputs B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 𝒀𝒋
Activation Function Inputs B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 𝒀𝒋
Activation Function Outputs B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 Class Label 𝒀𝒋
Activation Functions Piecewise Linear Sigmoid Signum
Activation Functions Which activation function to use? B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Outputs Class Labels Activation Function TWO Class Labels TWO Outputs One that gives two outputs. B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Which activation function to use? 𝑪𝒋𝒀𝒋
Activation Functions Piecewise Linear Sigmoid SignumSignum
Activation Function B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋
Bias B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
Bias B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
Bias B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝑿 𝟎 = +𝟏
Bias B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(+𝟏 ∗ 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
Bias B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=0
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=0
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=+v
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=+v
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=-v
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=+v
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) Same Concept Applies to Bias S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
Bias Importance B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
Learning Rate R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏
Summary of Parameters Inputs 𝑿 𝒎 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑)
Summary of Parameters Weights 𝑾 𝒎 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑) 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑)
Summary of Parameters Bias 𝒃 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
Summary of Parameters Sum Of Products (SOP) 𝒔 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
Summary of Parameters Activation Function 𝒔𝒈𝒏 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
Summary of Parameters Outputs 𝒀𝒋 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
Summary of Parameters Learning Rate η R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
Other Parameters Step n R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏𝒏 = 𝟎, 𝟏, 𝟐, … 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
Other Parameters Desired Output 𝒅𝒋 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏𝒏 = 𝟎, 𝟏, 𝟐, … B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 𝒅 𝒏 = −𝟏, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟏 (𝑹𝑬𝑫) +𝟏, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟐 (𝑩𝑳𝑼𝑬) 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
Neural Networks Training Steps Weights Initialization Inputs Application Sum of Inputs-Weights Products Activation Function Response Calculation Weights Adaptation Back to Step 2 1 2 3 4 5 6
Regarding 5th Step: Weights Adaptation • If the predicted output Y is not the same as the desired output d, then weights are to be adapted according to the following equation: 𝑾 𝒏 + 𝟏 = 𝑾 𝒏 + η 𝒅 𝒏 − 𝒀 𝒏 𝑿(𝒏) Where 𝑾 𝒏 = [𝒃 𝒏 , 𝑾 𝟏(𝒏), 𝑾 𝟐(𝒏), 𝑾 𝟑(𝒏), … , 𝑾 𝒎(𝒏)]
Neural Networks Training Example Step n=0 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=0: η = .001 𝑋 𝑛 = 𝑋 0 = +1, 255, 0, 0 𝑊 𝑛 = 𝑊 0 = −1, −2, 1, 6.2 𝑑 𝑛 = 𝑑 0 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=0 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=0 - SOP 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1+255*-2+0*1+0*6.2 =-511 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=0 - Output 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 -511 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟎 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −𝟓𝟏𝟏 = −𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
Neural Networks Training Example Step n=0 - Output 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 -511 −𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟎 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −𝟓𝟏𝟏 = −𝟏 -1 RED B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=0 Predicted Vs. Desired 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 -511 −𝟏 =+1𝑿 𝟎 -1 -1 𝒀 𝒏 = 𝒀 𝟎 = −𝟏 𝐝 𝒏 = 𝒅 𝟎 = −𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=1 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=1: η = .001 𝑋 𝑛 = 𝑋 1 = +1, 248, 80, 68 𝑊 𝑛 = 𝑊 1 = 𝑊 0 = −1, −2, 1, 6.2 𝑑 𝑛 = 𝑑 1 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=1 248 80 68 RED/BLUE -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=1 - SOP 248 80 68 RED/BLUE -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1+248*-2+80*1+68*6.2 =4.6 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=1 - Output 248 80 68 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 4.6 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟏 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 𝟒. 𝟔 = +𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
Neural Networks Training Example Step n=1 - Output 248 80 68 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 4.6 +𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟏 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 𝟒. 𝟔 = +𝟏 +1 BLUE B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=1 Predicted Vs. Desired 248 80 68 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 4.6 +𝟏 =+1𝑿 𝟎 -1 +1 𝒀 𝒏 = 𝒀 𝟏 = +𝟏 𝐝 𝒏 = 𝒅 𝟏 = −𝟏 ∵ 𝒀 𝒏 ≠ 𝒅 𝒏 ∴ Weights are Incorrect. Adaptation Required B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Weights Adaptation • According to 𝑾 𝒏 + 𝟏 = 𝑾 𝒏 + η 𝒅 𝒏 − 𝒀 𝒏 𝑿(𝒏) • Where n = 1 𝑾 𝟏 + 𝟏 = 𝑾 𝟏 + η 𝒅 𝟏 − 𝒀 𝟏 𝑿(𝟏) 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + .001 −𝟏 − (+𝟏) +𝟏, 𝟐𝟒𝟖, 𝟖𝟎, 𝟔𝟖 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + .001 −𝟐 +𝟏, 𝟐𝟒𝟖, 𝟖𝟎, 𝟔𝟖 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + −. 𝟎𝟎𝟐 +𝟏, 𝟐𝟒𝟖, 𝟖𝟎, 𝟔𝟖 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + −. 𝟎𝟎𝟐, −. 𝟒𝟗𝟔, −. 𝟏𝟔, −. 𝟏𝟑𝟔 𝑾 𝟐 = −𝟏. 𝟎𝟎𝟐, −𝟐. 𝟒𝟗𝟔, . 𝟖𝟒, 𝟔. 𝟎𝟔𝟒
Neural Networks Training Example Step n=2 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=2: η = .001 𝑋 𝑛 = 𝑋 2 = +1, 0, 0, 255 𝑊 𝑛 = 𝑊 2 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 2 = +1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=2 0 0 255 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=2 - SOP 0 0 255 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+0*- 2.496+0*.84+255*6.064 =1545.32 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=2 - Output 0 0 255 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1545 .32 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟐 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1545.32 = +𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
Neural Networks Training Example Step n=2 - Output 0 0 255 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1545 .32 +𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟐 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1545.32 = +𝟏 +1 BLUE B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=2 Predicted Vs. Desired 0 0 255 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1545 .32 +𝟏 =+1𝑿 𝟎 +1 𝒀 𝒏 = 𝒀 𝟐 = +𝟏 𝐝 𝒏 = 𝒅 𝟐 = +𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=3 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=3: η = .001 𝑋 𝑛 = 𝑋 3 = +1, 67, 15, 210 𝑊 𝑛 = 𝑊 3 = 𝑊 2 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 3 = +1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=3 67 15 210 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=3 - SOP 67 15 210 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+67*- 2.496+15*.84+210*6.064 =1349.542 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=3 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1349 .542 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟑 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1349.542 = +𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 67 15 210 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
Neural Networks Training Example Step n=3 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1349 .542 +𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟑 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1349.542 = +𝟏 +1 BLUE B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 67 15 210
Neural Networks Training Example Step n=3 Predicted Vs. Desired 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1349 .542 +𝟏 =+1𝑿 𝟎 +1 𝒀 𝒏 = 𝒀 𝟑 = +𝟏 𝐝 𝒏 = 𝒅 𝟑 = +𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 67 15 210
Neural Networks Training Example Step n=4 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=4: η = .001 𝑋 𝑛 = 𝑋 4 = +1, 255, 0, 0 𝑊 𝑛 = 𝑊 4 = 𝑊 3 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 4 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=4 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=4 - SOP 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+255*- 2.496+0*.84+0*6.064 =-637.482 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=4 - Output 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟒 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −637.482 = −𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
Neural Networks Training Example Step n=4 - Output 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 −𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟒 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −637.482 = −𝟏 -1 RED B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=4 Predicted Vs. Desired 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 −𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟒 = −𝟏 𝐝 𝒏 = 𝒅 𝟒 = −𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=5 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=5: η = .001 𝑋 𝑛 = 𝑋 5 = +1, 248, 80, 68 𝑊 𝑛 = 𝑊 5 = 𝑊 4 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 5 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
Neural Networks Training Example Step n=5 248 80 68 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=5 - SOP 248 80 68 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+248*- 2.496+80*.84+68*6.064 =-31.306 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
Neural Networks Training Example Step n=5 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 31.3 06 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟓 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −31.306 = −𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 248 80 68 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
Neural Networks Training Example Step n=5 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 31.3 06 −𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟓 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −31.306 = −𝟏 -1 RED B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 248 80 68
Neural Networks Training Example Step n=5 Predicted Vs. Desired 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 −𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟓 = −𝟏 𝐝 𝒏 = 𝒅 𝟓 = −𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 248 80 68
Correct Weights • After testing the weights across all samples and results were correct then we can conclude that current weights are correct ones for training the neural network. • After training phase we come to testing the neural network. • What is the class of the unknown color of values of R=150, G=100, B=180?
Testing Trained Neural Network (R, G, B) = (150, 100, 180) Trained Neural Networks Parameters η = .001 𝑊 = −1.002, −2.496, .84, 6.064
Testing Trained Neural Network (R, G, B) = (150, 100, 180) SOP 150 100 180 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+150*- 2.496+100*.84+180*6.064 =800.118 RED/BLUE 𝒀 6.064 .84 -2.496 -1.002
Testing Trained Neural Network (R, G, B) = (150, 100, 180) Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 800. 118 =+1𝑿 𝟎 𝒀 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 800.118 = +𝟏 sgn RED/BLUE 𝒀 6.064 .84 -2.496 -1.002 150 100 180 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
Testing Trained Neural Network (R, G, B) = (150, 100, 180) Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 800. 118 +𝟏 =+1𝑿 𝟎 𝒀 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 800.118 = +𝟏 +1 BLUE 6.064 .84 -2.496 -1.002 150 100 180

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Introduction to Artificial Neural Networks (ANNs) - Step-by-Step Training & Testing Example

  • 1.
    Artificial Neural Networks(ANNs) Step-By-Step Training & Testing Example MENOUFIA UNIVERSITY FACULTY OF COMPUTERS AND INFORMATION ALL DEPARTMENTS ARTIFICIAL INTELLIGENCE ‫المنوفية‬ ‫جامعة‬ ‫والمعلومات‬ ‫الحاسبات‬ ‫كلية‬ ‫األقسام‬ ‫جميع‬ ‫الذكاء‬‫اإلصطناعي‬ ‫المنوفية‬ ‫جامعة‬ Ahmed Fawzy Gad ahmed.fawzy@ci.menofia.edu.eg
  • 2.
    Neural Networks &Classification
  • 3.
  • 4.
  • 5.
  • 6.
  • 7.
  • 8.
  • 9.
  • 10.
  • 11.
  • 12.
  • 13.
  • 14.
  • 15.
    Classification Example B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567
  • 16.
    Neural Networks B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Hidden Output
  • 17.
    Neural Networks B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567
  • 18.
    Input Layer B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B
  • 19.
    Output Layer B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE 𝒀𝒋
  • 20.
    Weights B (BLUE)G (GREEN)R(RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE 𝑾 𝟏 𝑾 𝟐 𝑾 𝟑 Weights=𝑾𝒊 𝒀𝒋
  • 21.
    Activation Function B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
  • 22.
    Activation Function B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
  • 23.
    Activation Function B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
  • 24.
    Activation Function Components B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 𝒀𝒋
  • 25.
    Activation Function Inputs B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 s 𝒀𝒋
  • 26.
    Activation Function Inputs B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝒀𝒋
  • 27.
    Activation Function Inputs B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝒀𝒋
  • 28.
    Activation Function Inputs B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 𝒀𝒋
  • 29.
    Activation Function Inputs B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 𝒀𝒋
  • 30.
    Activation Function Inputs B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 ss=SOP(𝑿𝒊, 𝑾𝒊) 𝑿𝒊=Inputs 𝑾𝒊=Weights 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊 𝒀𝒋
  • 31.
    Activation Function Outputs B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 Class Label 𝒀𝒋
  • 32.
  • 33.
    Activation Functions Which activationfunction to use? B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Outputs Class Labels Activation Function TWO Class Labels TWO Outputs One that gives two outputs. B (BLUE)G (GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Which activation function to use? 𝑪𝒋𝒀𝒋
  • 34.
  • 35.
    Activation Function B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋
  • 36.
    Bias B (BLUE)G (GREEN)R(RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
  • 37.
    Bias B (BLUE)G (GREEN)R(RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
  • 38.
    Bias B (BLUE)G (GREEN)R(RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝑿 𝟎 = +𝟏
  • 39.
    Bias B (BLUE)G (GREEN)R(RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(+𝟏 ∗ 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
  • 40.
    Bias B (BLUE)G (GREEN)R(RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
  • 41.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑)
  • 42.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y
  • 43.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b
  • 44.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b
  • 45.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept
  • 46.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=0
  • 47.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=0
  • 48.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=+v
  • 49.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=+v
  • 50.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=-v
  • 51.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) X Y y=x+b Y-Intercept b=+v
  • 52.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) Same Concept Applies to Bias S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
  • 53.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
  • 54.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
  • 55.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
  • 56.
    Bias Importance B (BLUE)G(GREEN)R (RED) 00255 RED 6880248 25500 BLUE 2101567 Input Output R G B RED/BLUE W1 W2 W3 F(s)s 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) s=(𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) S= 𝟏 𝒎 𝑿𝒊 𝑾𝒊+BIAS
  • 57.
    Learning Rate R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏
  • 58.
    Summary of Parameters Inputs𝑿 𝒎 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑)
  • 59.
    Summary of Parameters Weights𝑾 𝒎 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑) 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑)
  • 60.
    Summary of Parameters Bias𝒃 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
  • 61.
    Summary of Parameters SumOf Products (SOP) 𝒔 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
  • 62.
    Summary of Parameters ActivationFunction 𝒔𝒈𝒏 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
  • 63.
    Summary of Parameters Outputs𝒀𝒋 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
  • 64.
    Summary of Parameters LearningRate η R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
  • 65.
    Other Parameters Step n R G B RED/BLUE W1 W2 W3 𝑿𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏𝒏 = 𝟎, 𝟏, 𝟐, … 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
  • 66.
    Other Parameters Desired Output𝒅𝒋 R G B RED/BLUE W1 W2 W3 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn 𝒀𝒋 =+1𝑿 𝟎 W0 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) 𝟎 ≤ η ≤ 𝟏𝒏 = 𝟎, 𝟏, 𝟐, … B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 𝒅 𝒏 = −𝟏, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟏 (𝑹𝑬𝑫) +𝟏, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟐 (𝑩𝑳𝑼𝑬) 𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, 𝑿 𝟑) W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, 𝑾 𝟑)
  • 67.
    Neural Networks TrainingSteps Weights Initialization Inputs Application Sum of Inputs-Weights Products Activation Function Response Calculation Weights Adaptation Back to Step 2 1 2 3 4 5 6
  • 68.
    Regarding 5th Step:Weights Adaptation • If the predicted output Y is not the same as the desired output d, then weights are to be adapted according to the following equation: 𝑾 𝒏 + 𝟏 = 𝑾 𝒏 + η 𝒅 𝒏 − 𝒀 𝒏 𝑿(𝒏) Where 𝑾 𝒏 = [𝒃 𝒏 , 𝑾 𝟏(𝒏), 𝑾 𝟐(𝒏), 𝑾 𝟑(𝒏), … , 𝑾 𝒎(𝒏)]
  • 69.
    Neural Networks Training Example Stepn=0 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=0: η = .001 𝑋 𝑛 = 𝑋 0 = +1, 255, 0, 0 𝑊 𝑛 = 𝑊 0 = −1, −2, 1, 6.2 𝑑 𝑛 = 𝑑 0 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 70.
    Neural Networks Training Example Stepn=0 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 71.
    Neural Networks Training Example Stepn=0 - SOP 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1+255*-2+0*1+0*6.2 =-511 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 72.
    Neural Networks Training Example Stepn=0 - Output 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 -511 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟎 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −𝟓𝟏𝟏 = −𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
  • 73.
    Neural Networks Training Example Stepn=0 - Output 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 -511 −𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟎 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −𝟓𝟏𝟏 = −𝟏 -1 RED B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 74.
    Neural Networks Training Example Stepn=0 Predicted Vs. Desired 255 0 0 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 -511 −𝟏 =+1𝑿 𝟎 -1 -1 𝒀 𝒏 = 𝒀 𝟎 = −𝟏 𝐝 𝒏 = 𝒅 𝟎 = −𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 75.
    Neural Networks Training Example Stepn=1 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=1: η = .001 𝑋 𝑛 = 𝑋 1 = +1, 248, 80, 68 𝑊 𝑛 = 𝑊 1 = 𝑊 0 = −1, −2, 1, 6.2 𝑑 𝑛 = 𝑑 1 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 76.
    Neural Networks Training Example Stepn=1 248 80 68 RED/BLUE -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 77.
    Neural Networks Training Example Stepn=1 - SOP 248 80 68 RED/BLUE -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 -1 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1+248*-2+80*1+68*6.2 =4.6 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 78.
    Neural Networks Training Example Stepn=1 - Output 248 80 68 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 4.6 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟏 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 𝟒. 𝟔 = +𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
  • 79.
    Neural Networks Training Example Stepn=1 - Output 248 80 68 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 4.6 +𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟏 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 𝟒. 𝟔 = +𝟏 +1 BLUE B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 80.
    Neural Networks Training Example Stepn=1 Predicted Vs. Desired 248 80 68 -2 1 6.2 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 4.6 +𝟏 =+1𝑿 𝟎 -1 +1 𝒀 𝒏 = 𝒀 𝟏 = +𝟏 𝐝 𝒏 = 𝒅 𝟏 = −𝟏 ∵ 𝒀 𝒏 ≠ 𝒅 𝒏 ∴ Weights are Incorrect. Adaptation Required B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 81.
    Weights Adaptation • Accordingto 𝑾 𝒏 + 𝟏 = 𝑾 𝒏 + η 𝒅 𝒏 − 𝒀 𝒏 𝑿(𝒏) • Where n = 1 𝑾 𝟏 + 𝟏 = 𝑾 𝟏 + η 𝒅 𝟏 − 𝒀 𝟏 𝑿(𝟏) 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + .001 −𝟏 − (+𝟏) +𝟏, 𝟐𝟒𝟖, 𝟖𝟎, 𝟔𝟖 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + .001 −𝟐 +𝟏, 𝟐𝟒𝟖, 𝟖𝟎, 𝟔𝟖 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + −. 𝟎𝟎𝟐 +𝟏, 𝟐𝟒𝟖, 𝟖𝟎, 𝟔𝟖 𝑾 𝟐 = −𝟏, −𝟐, 𝟏, 𝟔. 𝟐 + −. 𝟎𝟎𝟐, −. 𝟒𝟗𝟔, −. 𝟏𝟔, −. 𝟏𝟑𝟔 𝑾 𝟐 = −𝟏. 𝟎𝟎𝟐, −𝟐. 𝟒𝟗𝟔, . 𝟖𝟒, 𝟔. 𝟎𝟔𝟒
  • 82.
    Neural Networks Training Example Stepn=2 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=2: η = .001 𝑋 𝑛 = 𝑋 2 = +1, 0, 0, 255 𝑊 𝑛 = 𝑊 2 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 2 = +1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 83.
    Neural Networks Training Example Stepn=2 0 0 255 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 84.
    Neural Networks Training Example Stepn=2 - SOP 0 0 255 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+0*- 2.496+0*.84+255*6.064 =1545.32 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 85.
    Neural Networks Training Example Stepn=2 - Output 0 0 255 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1545 .32 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟐 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1545.32 = +𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
  • 86.
    Neural Networks Training Example Stepn=2 - Output 0 0 255 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1545 .32 +𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟐 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1545.32 = +𝟏 +1 BLUE B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 87.
    Neural Networks Training Example Stepn=2 Predicted Vs. Desired 0 0 255 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1545 .32 +𝟏 =+1𝑿 𝟎 +1 𝒀 𝒏 = 𝒀 𝟐 = +𝟏 𝐝 𝒏 = 𝒅 𝟐 = +𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 88.
    Neural Networks Training Example Stepn=3 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=3: η = .001 𝑋 𝑛 = 𝑋 3 = +1, 67, 15, 210 𝑊 𝑛 = 𝑊 3 = 𝑊 2 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 3 = +1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 89.
    Neural Networks Training Example Stepn=3 67 15 210 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 90.
    Neural Networks Training Example Stepn=3 - SOP 67 15 210 RED/BLUE 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+67*- 2.496+15*.84+210*6.064 =1349.542 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 91.
    Neural Networks Training Example Stepn=3 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1349 .542 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟑 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1349.542 = +𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 67 15 210 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
  • 92.
    Neural Networks Training Example Stepn=3 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1349 .542 +𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟑 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 1349.542 = +𝟏 +1 BLUE B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 67 15 210
  • 93.
    Neural Networks Training Example Stepn=3 Predicted Vs. Desired 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 1349 .542 +𝟏 =+1𝑿 𝟎 +1 𝒀 𝒏 = 𝒀 𝟑 = +𝟏 𝐝 𝒏 = 𝒅 𝟑 = +𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 67 15 210
  • 94.
    Neural Networks Training Example Stepn=4 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=4: η = .001 𝑋 𝑛 = 𝑋 4 = +1, 255, 0, 0 𝑊 𝑛 = 𝑊 4 = 𝑊 3 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 4 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 95.
    Neural Networks Training Example Stepn=4 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 96.
    Neural Networks Training Example Stepn=4 - SOP 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+255*- 2.496+0*.84+0*6.064 =-637.482 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 97.
    Neural Networks Training Example Stepn=4 - Output 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟒 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −637.482 = −𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
  • 98.
    Neural Networks Training Example Stepn=4 - Output 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 −𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟒 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −637.482 = −𝟏 -1 RED B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 99.
    Neural Networks Training Example Stepn=4 Predicted Vs. Desired 255 0 0 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 −𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟒 = −𝟏 𝐝 𝒏 = 𝒅 𝟒 = −𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 100.
    Neural Networks Training Example Stepn=5 • In each step in the solution, the parameters of the neural network must be known. • Parameters of step n=5: η = .001 𝑋 𝑛 = 𝑋 5 = +1, 248, 80, 68 𝑊 𝑛 = 𝑊 5 = 𝑊 4 = −1.002, −2.496, .84, 6.064 𝑑 𝑛 = 𝑑 5 = −1 B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567
  • 101.
    Neural Networks Training Example Stepn=5 248 80 68 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 102.
    Neural Networks Training Example Stepn=5 - SOP 248 80 68 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+248*- 2.496+80*.84+68*6.064 =-31.306 RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002
  • 103.
    Neural Networks Training Example Stepn=5 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 31.3 06 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟓 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −31.306 = −𝟏 sgn RED/BLUE 𝒀(𝒏) B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 248 80 68 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
  • 104.
    Neural Networks Training Example Stepn=5 - Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 31.3 06 −𝟏 =+1𝑿 𝟎 𝒀 𝒏 = 𝒀 𝟓 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 −31.306 = −𝟏 -1 RED B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 248 80 68
  • 105.
    Neural Networks Training Example Stepn=5 Predicted Vs. Desired 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 - 637. 482 −𝟏 =+1𝑿 𝟎 -1 𝒀 𝒏 = 𝒀 𝟓 = −𝟏 𝐝 𝒏 = 𝒅 𝟓 = −𝟏 ∵ 𝒀 𝒏 = 𝒅 𝒏 ∴ Weights are Correct. No Adaptation B (BLUE)G (GREEN)R (RED) 00255 RED = -1 6880248 25500 BLUE = +1 2101567 6.064 .84 -2.496 -1.002 248 80 68
  • 106.
    Correct Weights • Aftertesting the weights across all samples and results were correct then we can conclude that current weights are correct ones for training the neural network. • After training phase we come to testing the neural network. • What is the class of the unknown color of values of R=150, G=100, B=180?
  • 107.
    Testing Trained NeuralNetwork (R, G, B) = (150, 100, 180) Trained Neural Networks Parameters η = .001 𝑊 = −1.002, −2.496, .84, 6.064
  • 108.
    Testing Trained NeuralNetwork (R, G, B) = (150, 100, 180) SOP 150 100 180 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 F(s)s sgn =+1𝑿 𝟎 s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+𝑿 𝟑 𝑾 𝟑) =+1*-1.002+150*- 2.496+100*.84+180*6.064 =800.118 RED/BLUE 𝒀 6.064 .84 -2.496 -1.002
  • 109.
    Testing Trained NeuralNetwork (R, G, B) = (150, 100, 180) Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 800. 118 =+1𝑿 𝟎 𝒀 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 800.118 = +𝟏 sgn RED/BLUE 𝒀 6.064 .84 -2.496 -1.002 150 100 180 𝒔𝒈𝒏 𝒔 = +𝟏, 𝒔 ≥ 𝟎 −𝟏, 𝒔 < 𝟎
  • 110.
    Testing Trained NeuralNetwork (R, G, B) = (150, 100, 180) Output 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 800. 118 +𝟏 =+1𝑿 𝟎 𝒀 = 𝑺𝑮𝑵 𝒔 = 𝑺𝑮𝑵 800.118 = +𝟏 +1 BLUE 6.064 .84 -2.496 -1.002 150 100 180