斯坦福大学机器学习课程讲义—— 多变量的线性回归模型表达.pptx

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1、Size(feet2)Price($1000)210446014162321534315852178Multiple features(variables).第1页/共40页Size(feet2)Number of bedroomsNumber of floorsAge of home(years)Price($1000)2104514546014163240232153432303158522136178Multiple features(variables).Notation:=number of features=input(features)of training example.=v

2、alue of feature in training example.第3页/共40页Hypothesis:Previously:第4页/共40页For convenience of notation,define .Multivariate linear regression.第5页/共40页第6页/共40页Linear Regression with multiple variablesGradient descent for multiple variablesMachine Learning第7页/共40页Hypothesis:Cost function:Parameters:(si

3、multaneously update for every )RepeatGradient descent:第8页/共40页(simultaneously update )Gradient DescentRepeatPreviously(n=1):New algorithm :Repeat(simultaneously update for )第9页/共40页第10页/共40页Linear Regression with multiple variablesGradient descent in practice I:Feature ScalingMachine Learning第11页/共4

4、0页E.g.=size(0-2000 feet2)=number of bedrooms(1-5)Feature ScalingIdea:Make sure features are on a similar scale.size(feet2)number of bedrooms第12页/共40页Feature ScalingGet every feature into approximately a range.第13页/共40页Replace with to make features have approximately zero mean(Do not apply to ).Mean

5、normalizationE.g.第14页/共40页第15页/共40页Linear Regression with multiple variablesGradient descent in practice II:Learning rateMachine Learning第16页/共40页Gradient descent-“Debugging”:How to make sure gradient descent is working correctly.-How to choose learning rate .第17页/共40页Example automatic convergence t

6、est:Declare convergence if decreases by less than in one iteration.0100200300400No.of iterationsMaking sure gradient descent is working correctly.第18页/共40页Making sure gradient descent is working correctly.Gradient descent not working.Use smaller .No.of iterationsNo.of iterationsNo.of iterations-For

7、sufficiently small ,should decrease on every iteration.-But if is too small,gradient descent can be slow to converge.第19页/共40页Summary:-If is too small:slow convergence.-If is too large:may not decrease on every iteration;may not converge.To choose ,try第20页/共40页第21页/共40页Linear Regression with multipl

8、e variablesFeatures and polynomial regressionMachine Learning第22页/共40页Housing prices prediction第23页/共40页Polynomial regressionPrice(y)Size(x)第24页/共40页Choice of featuresPrice(y)Size(x)第25页/共40页第26页/共40页Linear Regression with multiple variablesNormal equationMachine Learning第27页/共40页Gradient DescentNor

9、mal equation:Method to solve for analytically.第28页/共40页Intuition:If 1DSolve for(for every )第29页/共40页Size(feet2)Number of bedroomsNumber of floorsAge of home(years)Price($1000)12104514546011416324023211534323031518522136178Size(feet2)Number of bedroomsNumber of floorsAge of home(years)Price($1000)210

10、4514546014163240232153432303158522136178Examples:第30页/共40页 examples ;features.E.g.If第32页/共40页is inverse of matrix .Octave:pinv(X*X)*X*y第33页/共40页 training examples,features.Gradient DescentNormal EquationNo need to choose .Dont need to iterate.Need to choose .Needs many iterations.Works well even whe

11、n is large.Need to computeSlow if is very large.第34页/共40页第35页/共40页Linear Regression with multiple variablesNormal equation and non-invertibility(optional)Machine Learning第36页/共40页Normal equation-What if is non-invertible?(singular/degenerate)-Octave:pinv(X*X)*X*y第37页/共40页What if is non-invertible?Redundant features(linearly dependent).E.g.size in feet2 size in m2Too many features(e.g.).-Delete some features,or use regularization.第38页/共40页第39页/共40页