Prediction with Regression

An Introduction to Linear Regression and Shrinkage Methods

Ehsan Khoddam Mohammadi

•Prediction•Estimation•Bias Variance Trade-Off•Regression•Ordinary Least square•Ridge regression•Lasso

Outline

Predictiondefinitionset of inputs: X1, X2, …, Xp

the output: YWe want to analyze the relationship between

these variables (interpretation)We want to estimate output based on inputs

(prediction)

Predictionsame concept in different literatures

Machine learning: supervised learningFinance: forecastingPolitics: predictionEstimation theory: function approximation

Statistics ML Economy

X1, X2, …, Xn

Predictors Features Independent variables

Y Response Class Dependent variables

RegressionWhy?Well-performed and accurate in both Interpretation

and PredictionStrong fundamental in math, statistics and

computationMany modern and advanced methods are based on

Regression, even they are variant of regressionNew methods are still invented for regression:

Nobel prize are still given to investigations in regression, Hot topic

Could be formulated as optimization problem: that’s the reason I choose it for this class, it’s more

related to subject of class than any other methods I’ve known for prediction

RegressionclassificationLinear Regression

Least square Best sub-sut Selection, Regression with feature selectionStepwise RegressionShrinkage regularization for Regression:

Ridge Regression Lasso Regression

Non-Linear RegressionNumerical Data fittingANNDiscrete regression

Logistic Regression

Before proceeding with regressionLet’s investigate on some statistical property of

ESTIMATION

Estimating the parameterassume that we have iid (identically

independent distributed) samples X1, . . . ,Xn with unknown distribution.

Estimating p.d.f of them is too hard in many situations, Instead of that, We want to estimate a parameter θ .

is estimation of θ, it is function of X1, . . . ,Xn .

Bias-Variance dilemma Definition 1 : The bias of an estimator is . If it is 0, the estimator is

said to be unbiased.

Definition 2 : The mean squared error (MSE) of an estimator is .

An interesting equation:

ˆ ˆ( ) ( )Bias E

2ˆ(( ) )E

2 2ˆ ˆ ˆ(( ) ) ( ) ( )E Var Bias

What does it really mean?

[Image from “More on Regularization and (Generalized) Ridge Operators”, Takane,(2007)]

Test and training error as a function of model complexity.

[ Image from “The Elements of Statistical Learning”,Second Edition, Hastie et al. (2008)]

Linear RegressionModel

Set of training data :

Linear Regression model:

Real-valued coefficients β need to be estimated

1 1

1 2

, )...( , )

Each ( , ,..., )

( n n i

Ti i i ip

y x y y

x x x x

x

01

( )p

j jj

f X X

Linear RegressionLeast square

Most popular estimation methodMinimize the Residual Sum of Squares:

2

1

20

1 1

( ) ( ( ))

( )

N

i ii

pN

i ij ji j

RSS y f x

y x

How do we minimize it?

Linear RegressionLeast square

Let’s rewrite last formula in this form:

Quadratic function (not a point here but we shall use this property later)

Differentiating respect to β and set it to zero:

Unique Solution: ;

( ) ( ) ( )TRSS y X y X

2 ( ) 0TRSSX y X

1ˆ ( )T TX X X y

Under which assumptions we could obtain unique solution?

ˆy X

Linear RegressionLeast square, AssumptionsX should be full-rank, hence is p.d and

invertible, unique solution could be obtainedIn another word, features vectors should be

linearly independent or uncorrelated What will be happened to β if X would be non-

full-rank matrix or some features would be highly correlated?

TX X

Linear RegressionLeast square, flaws

Low bias but High variance:

and one could estimate Var(y) by:

It’s hard to find meaning-full relation if we have too many features.

What would you recommend to solve these problems?

1 2ˆ( ) ( )TVar X X

2 1ˆ ( )

1 iyN p

Linear RegressionImprovementsModel Selection (Feature Selection):

Best-Subset Selection (Branch and Leap , Furnival (1974))

Step-wise Selection (Greedy approach, sub-optimal but preferred)

mRMR (using mutual information criterion for selection)

Shrinkage Methods: impose constraint on βRidge RegressionLasso Regression

Ridge RegressionWhen you have a problem want to be solved

in statistics, There is always a Russian statistician waiting for you to solve it. (Be careful! just in statistics I guarantee , they will betray you in any other situations)

Andrey Nikolayevich Tychonoff provides a Tikhonov (!!!) regularization for ill-posed problems , Also known as Ridge Regression in statistics.

Ridge Regressionfirst attempt

Remember this?:Tychonoff added a term to avoid singularity and changed above formula to this:

Now, the inverse could be computed even if Is not of full-rank, Also β is still linear function of y.Every thing start from above formula but now we

have better point of view than Tychonoff, let’s take a look!

1ˆ ( )T TX X X y

1ˆ ( )ridge T TX X I X y TX X

Ridge Regressionbetter motivationTo avoid high variance of β we just impose a

constraint on it, our problem is now an optimization problem with constraints.

2

01 1

2

1

ˆ arg min ,

subject to

pNridge

i ij ji j

p

jj

y x

t

Even better representation: using lagrangian form

Or again even better! in matrix representation form, we could differentiate this formula and set it to zero

Could you guess the solution?

2

20

1 1 1

ˆ arg minp pN

ridgei ij j j

i j j

y x

( ) ( ) ( )T TRSS y X y X

Could you find a relation between β and βridge when inputs are orthonormal?

LASSO

Least Absolute Selection and Shrinkage Operator

LASSOWe impose L1-norm constraint on our

regression

No close form exists, it’s non-linear function of y

2

01 1

1

ˆ arg min ,

subject to | |

pNlasso

i ij ji j

p

jj

y x

t

How could you solve above problem? (hint: ask Mr.Iranmehr!)

LASSOWhy?First attempt for usage of L1-norm, show

significant results in signal processing, denoising [Chen et al. (1998)]

Base method for LAR (new and novel method for regression, not covered here)

[Efron et al. (2004)]Good for Sparse model selection where p>N

[Donoho (2006b)]

REFERENCES“The Elements of Statistical Learning”,

Second Edition, Hastie et al. , 2008“More on Regularization and (Generalized)

Ridge Operators”, Takane, 2007“Bias, Variance and MSE of Estimators”, Guy

Lebanon, 2004“Least Squares Optimization with L1-Norm

Regularization”, Mark Schmidt, 2005“Regularization: Ridge Regression and the

LASSO”, Tibshirani, 2006