Modelleerimine ja Juhtimine Tehisnärvivõrgudega
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Modelleerimine ja Juhtimine Tehisnärvivõrgudega Identification and Control with artificial neural networks (2nd part). Eduard Petlenkov, Automaatikainstituut. Eduard Petlenkov, Automaatikainstituut, 2013. Supervised learning.

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Modelleerimine ja Juhtimine Tehisnärvivõrgudega

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Modelleerimine ja juhtimine tehisn rviv rgudega

Modelleerimine ja Juhtimine Tehisnärvivõrgudega

Identification and Control with artificial neural networks

(2nd part)

Eduard Petlenkov,

Automaatikainstituut

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Supervised learning

Supervised learning is a training algorithm based on known input-output data.

X is an input vector;

Ypis a vector of reference outputs corresponding to input vector X

Y is a vector of NN’s outputs corresponding to input vector X

NNis a mathematica function of the network (Y=NN(X))

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Õpetamine (2)

  • Computation of NN’s outputs corresponding to the current NN’s paraneters;

  • Computation of NN’s error;

  • Updating NN’s parameters by using selected training algorithm in order to minimize the error.

Network training procedure consists of the following 3 steps:

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Gradient descent error backpropagation (1)

Error function:

- NN’s output

-inputs used for training;

- NN’s function;

- etalons (references) for outputs=desired outputs.

function to be minimized:

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Gradient descent error backpropagation (2)

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Global minimum problem

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Training of a double-layer perceptron using Gradient descent error backpropagation (1)

i - number of input;

j – number of neuron in the hidden layer;

k – number of output neuron;

NN’s input at time instance t;

;

- Values oh the hidden layer weighting coefficients at time instance t;

- Values oh the hidden layer biases at time instance t;

- Activation function of the hidden layer neurons

- Outputs of the output layer neurons at time instancet;

- Values oh the output layer weighting coefficients at time instance t;

- Values oh the output layer biases at time instance t;

- Activation function of the output layer neurons;

- Outputs of the network at time instance t;

Eduard Petlenkov, Automaatikainstituut, 2012


Modelleerimine ja juhtimine tehisn rviv rgudega

Training of a double-layer perceptron using Gradient descent error backpropagation (2)

error:

gradients:

Signals distributing information about error from output layer to previous layers (that is why the method is called error backpropagation):

Are derivatives of the corresponding layers activation functions.

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Training of a double-layer perceptron using Gradient descent error backpropagation (3)

Updated weighting coefficients:

learning rate

Eduard Petlenkov, Automaatikainstituut, 2013


Modelleerimine ja juhtimine tehisn rviv rgudega

Gradient Descent (GD) method vs Levenberg-Marquardt (LM) method

- Gradient

- Hessian

GD:

LM:

Eduard Petlenkov, Automaatikainstituut, 2012


Modelleerimine ja juhtimine tehisn rviv rgudega

Näide MATLABis (1)

P1=(rand(1,100)-0.5)*20% Juhuslikud sisendid. 100 väärtust vahemikus [-10 +10]

P2=(rand(1,100)-0.5)*20

P=[P1;P2]% Närvivõrgu sisendite moodustamine

T=0.3*P1+0.9*P2 % Vastavate väljundite arvutamine

net=newff([-10 10;-10 10],[5 1],{'purelin' 'purelin'},'traingd') % Närvivõrgu genereerimine

teinesisend

[min max]

neuronite arvud igas kihis

Viimase kihi neuronite arv = väljundite arv

esimene sisend

[min max]

Teise kihi neuronite aktiveerimisfunktsioon

Esimese kihi neuronite aktiveerimisfunktsioon

õppimisalgoritm

net.trainParam.show=1;

Treenimise parameetrid

net.trainParam.epochs=100;


Modelleerimine ja juhtimine tehisn rviv rgudega

Näide MATLABis (2)

net=train(net,P,T) % Närvivõrgu treenimine

out=sim(net,[3 -2]) % Närvivõrgu simuleerimine

Teise sisendi väärtus

Esimese sisendi väärtus

W1=net.IW{1,1} % esimese kihi kaalukoefitsientide maatriks

W2=net.LW{2,1} % teise kihi kaalukoefitsientide maatriks

B1=net.b{1} % esimese kihi neuronite nihete vektor

B2=net.b{2} % teise kihi neuronite nihete vektor

Eduard Petlenkov, Automaatikainstituut, 2011


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