STATISTICS Linear Statistical Models Professor Ke-Sheng Cheng Department of Bioenvironmental Systems...

STATISTICS Linear Statistical Models

Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University

The Method of Least Squares • Consider the data shown in the following

table and figure. We are interested in fitting a straight line to the points in order to obtain a simple mathematical relationship for runoff and rainfall.

112/04/10 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

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• Intuitively, we want that, for each observed value of rainfall, the corresponding value of runoff will be as close as possible to the observed value. It is equivalent to say that we want the vertical deviations to be as small as possible.


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• One method of constructing such a straight line to fit the observed data is called the method of least squares. It requires the sum of the squares of the vertical deviations of all the points from the fitted line to be a minimum.

• Let the rainfall and runoff data in the above figure be respectively represented by x and y. The fitted line is expressed by

xy 10ˆ


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Remarks

n

ini

n

inini

xx

yyxx

1

2

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)(

))((

n

ini

n

iini

xx

yxx

1

2

1

)(

)(

n

ini

n

inii

xx

yyx

1

2

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)(

)(

,0)ˆ(1

n

iii yy

n

iiii yyx

1

0)ˆ(


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Given a value of x, what dose the predicted value of y really represent?

• Given a value of x, what dose the predicted value of y really represent?– It is unlikely that the predicted value will be the

same as the observed value at all times. – It may even be possible that the predicted value is

the same as the observed value only in very few cases.

– In some cases, the predicted values are far different from observed values.

• We are sure that the linear model may overpredict or underpredict the observed values.


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Linear statistical model


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.),0(~,)|( , Given 210 Niidxxyx iiiiii

Random component

We are not able to predict y without errors due to existence of the random component. If a phenomenon is stochastic in nature, it cannot be predicted without errors.


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iii

ixY

xxYVar

x

xNxY

ii

2

10|

210

)|(

),(~)|(

model) d(Postulate

.),0(~,)|( , Given 210 Niidxxyx iiiiii

iiiii

ii

exeyy

xy

10

10

ˆˆˆ

ˆˆˆ

Coefficient of determination• How well does the least squares line explain

the variation in the data? • The coefficient of determination represents

the proportion of data variation that can be explained by the linear regression model.


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Estimating the variance of Y|x


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iii

ixY

xxYVar

xxNxYii

2

10|2

10

)|(

,),(~)|(

Note: The variance of Y|x is NOT the same as the variance of Y.

RSS (Residual sum of squares) = SSE (sum of squared errors)

n

iii

n

iii xyyyRSS

1

210

1

2 )]ˆˆ([)ˆ(


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Unbiasedness of the least squares estimators


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Confidence intervals of the regression coefficients• Pivotal quantities


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10 and

22

000 ~

1

ˆ

n

xx

n

t

s

x

ns

Q

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1 ~ˆ

n

xx

tss

Q

222

2

3 ~)2(

n

snQ

Hypothesis tests for regression coefficients


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Simple linear regression using R• Useful material– Chapter 11 of Introduction to Probability and

Statistics Using R (G. J. Kerns) is highly recommended.

– http://www.montefiore.ulg.ac.be/~kvansteen/GBIO0009-1/ac20092010/Class8/Using%20R%20for%20linear%20regression.pdf


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• Defining linear regression models


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• Conducting regressionlm(y~model)


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s

• Other useful commands


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– For prediction (x values not observed)


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Graphing the Confidence and Prediction Bands


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You may want to change it. For example, data.frame(x=seq(20,30,by=0.5))

Confidence and prediction intervals


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Line of prediction. It represents the estimated conditional expectation of y given x.


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• Multiple regression – The following slides are provided for your reference

only. Due to the time constraint, they will not be covered in this class.


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• Now let’s consider fitting a linear function of several variables. Suppose that we have the following data set:


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knnn

k

k

xxx

xxx

xxx

X

21

22212

12111

1

1

1

k

1

0

ny

y

y

Y2

1

YXB


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YXXBX TT YXXXB TT 1)(

The Linear Regression Model


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Covariance and Correlation Coefficient

• Suppose we have observed the following data. We wish to measure both the direction and the strength of the relationship between Y and X.


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The Analysis of Variance (ANOVA)


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• Given X, Y’s are independent normal random variables, i.e.,

• The residual sum of squares (or sum of squared errors, SSE) is expressed by

nI2,~ XBNY

BXYBXYSSET ˆˆ


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BXYYY

BXYXBBXYY

BXYBXYSSE

TT

TTT

T

ˆ

ˆˆˆ

ˆˆ

0ˆ BXYX T


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• The total sum of squares corrected for the mean is referred to as the total variation. This total variation is split up in two parts:– the regression part (SSRm) “explained by the model”,

and

– the residual part (SSE).


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• The ratio is known as the coefficient of determination.

• If the coefficient of determination is large then the model provides a good fit to the data. It also represents the part of the total variation which is explained by the model.

mm SSTSSRR /2


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Properties of the Estimators


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Confidence Intervals


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• The 100(1 – )% confidence interval of 2 is

2

2,

2

221,

2 )(,

)(

pnpn

spnspn


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• However, the true value of is unknown, the above equation can not be used to establish the confidence interval of .

• We then use s to substitute and it is known that has a t-distribution with

(n–p) degree of freedom.

i

i

ii

vs

ˆ


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• The 100(1 – )% confidence interval of

is

1ˆˆ

1ˆ

2,2,

2,2,

ipniiipni

pn

i

iipn

vstvstP

tvs

tP

i

ipniipni vstvst 2,2,ˆ,ˆ


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Example 1 • A scientist carries out an experiment on the

relationship between the yield Y of a crop and the amount of irrigation water X. It is believed that the relationship between expected yield and amount of irrigation water (ignore the units) can be described adequately as

xxxYE 210)|(


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• The data shown in the following table were collected in the field.


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Example 2 • Data in the following table are rainfall (x)

and runoff (y) measured during the rainy season in a study area.

• A regression model is postulated for the above data iXY X

ii 10|


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Test of Hypotheses


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