1 Linear Regression With One Variable

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    Linearregress

    withonevaria

    Model

    representaMachineLearning

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    0

    100

    200

    300

    400

    500

    0 500 1000 1500 2000

    0

    100

    200

    300

    400

    500

    0 500 1000 1500 2000

    HousingPrices

    (Portland,OR)

    Price(in1000s

    ofdollars)

    Size(feet2)

    SupervisedLearning

    Giventherightanswerfor

    eachexampleinthedata.

    RegressionProblem

    Predictreal-valuedoutp

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    Nota6on:

    m=Numberoftrainingexamplesxs=inputvariable/features

    ys=outputvariable/targetvariable

    Sizeinfeet2(x) Price($)in1

    2104 40

    141 232

    1534 315852 178

    Trainingsetof

    housingprices

    (Portland,OR)

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    TrainingSet

    LearningAlgorithm

    h

    Sizeof

    house

    Es6mated

    price

    Howdowerep

    Linearregressionwith

    Univariatelinearregre

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    Costfunc6

    MachineLearning

    Linearregress

    withonevaria

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    Howtochooses?

    TrainingSet

    Hypothesis:

    s:Parameters

    Sizeinfeet2(x) Price($)in1

    2104 40

    141 232

    1534 315852 178

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    0

    1

    2

    3

    0 1 2 3

    0

    1

    2

    3

    0 1 2 3

    0

    1

    2

    3

    0

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    y

    x

    Idea:Choosesothat

    isclosetoforour

    trainingexamples

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    Costfunc6

    intui6onIMachineLearning

    Linearregress

    withonevaria

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    Hypothesis:

    Parameters:

    CostFunc6on:

    Goal:

    Simplifie

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    0

    1

    2

    3

    0 1 2 3

    y

    x

    (forfixed,thisisafunc6onofx) (func6onoftheparam

    0

    1

    2

    3

    -0.5 0 0.5 1

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    0

    1

    2

    3

    0 1 2 3

    y

    x

    (forfixed,thisisafunc6onofx) (func6onoftheparam

    0

    1

    2

    3

    -0.5 0 0.5 1

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    0

    1

    2

    3

    -0.5 0 0.5 1

    y

    x

    (forfixed,thisisafunc6onofx) (func6onoftheparam

    0

    1

    2

    3

    0 1 2 3

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    Costfunc6

    intui6onIMachineLearning

    Linearregress

    withonevaria

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    Hypothesis:

    Parameters:

    CostFunc6on:

    Goal:

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    Gradient

    descentMachineLearning

    Linearregress

    withonevaria

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    Havesomefunc6on

    Want

    Outline:

    Startwithsome Keepchangingtoreduce

    un6lwehopefullyendupataminimum

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    1

    0

    J(0,1)

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    0

    1

    J(0,1)

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    Gradientdescentalgorithm

    Correct:Simultaneousupdate Incorrect:

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    Gradientdes

    intui6onMachineLearning

    Linearregress

    withonevaria

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    Gradientdescentalgorithm

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    Ifistoosmall,gradientdescentcanbeslow.

    Ifistoolarge,gradientdescent

    canovershoottheminimum.Itmay

    failtoconverge,orevendiverge.

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    atlocalop6ma

    Currentvalueof

    Gradient descent can converge to a local

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    Gradientdescentcanconvergetoalocal

    minimum,evenwiththelearningratefixed.

    Asweapproachalocal

    minimum,gradient

    descentwillautoma6callytakesmallersteps.So,no

    needtodecreaseover

    6me.

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    Gradientdesc

    linearregress

    MachineLearning

    Linearregress

    withonevaria

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    Gradientdescentalgorithm LinearRegressionM

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    Gradientdescentalgorithm

    u

    simu

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    1

    0

    J(0,1)

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    (forfixed,thisisafunc6onofx) (func6onofthepara

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    BatchGradientDescent

    Batch:Eachstepofgradientdescusesallthetrainingexamples.


Chapter 5 Regression BPS - 5th Ed. Chapter 51. Linear Regression Objective: To quantify the linear relationship between an explanatory variable (x)

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Linear Regression - Wharton Finance - Finance …finance.wharton.upenn.edu/.../Linear-Regression-Slides.pdf · 2009-10-07 · regression model, 2-variable linear regression model)

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Online Instrumental Variable Regression with Applications to …bboots3/files/OIVR_AAAI.pdf · 2015-12-07 · Online Instrumental Variable Regression with Applications to Online Linear

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