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1 The Vision Thing Power Thirteen Bivariate Normal Distribution

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2 Outline Circles around the origin Circles translated from the origin Horizontal ellipses around the (translated) origin Vertical ellipses around the (translated) origin Sloping ellipses

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3 x y x = 0, x 2 =1 y = 0, y 2 =1 x, y = 0

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4 x y x = a, x 2 =1 y = b, y 2 =1 x, y = 0 a b

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5 x y x = 0, x 2 > y 2 y = 0 x, y = 0

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6 x y x = 0, x 2 < y 2 y = 0 x, y = 0

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7 x y x = a, x 2 > y 2 y = b x, y > 0 a b

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8 x y x = a, x 2 > y 2 y = b x, y < 0 a b

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9 Why? The Bivariate Normal Density and Circles f(x, y) = {1/[2 x y ]}*exp{(-1/[2(1- )]* ([(x- x )/ x ] 2 -2 ([(x- x )/ x ] ([(y- y )/ y ] + ([(y- y )/ y ] 2 } If means are zero and the variances are one and no correlation, then f(x, y) = {1/2 }exp{(-1/2 )*(x 2 + y 2 ), where f(x,y) = constant, k, for an isodensity ln2 k =(-1/2)*(x 2 + y 2 ), and (x 2 + y 2 )= -2ln2 k=r 2

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10 Ellipses If x 2 > y 2, f(x,y) = {1/[2 x y ]}*exp{(-1/2)* ([(x- x )/ x ] 2 + ([(y- y )/ y ] 2 }, and x* = (x- x ) etc. f(x,y) = {1/[2 x y ]}exp{(-1/2)* ([x*/ x ] 2 + [y*/ y ] 2 ), where f(x,y) =constant, k, and ln{k [2 x y ]} = (-1/2) ([x*/ x ] 2 + [y*/ y ] 2 ) and x 2 /c 2 + y 2 /d 2 = 1 is an ellipse

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11 x y x = 0, x 2 < y 2 y = 0 x, y < 0 Correlation and Rotation of the Axes Y’ X’

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12 Bivariate Normal: marginal & conditional If x and y are independent, then f(x,y) = f(x) f(y), i.e. the product of the marginal distributions, f(x) and f(y) The conditional density function, the density of y conditional on x, f(y/x) is the joint density function divided by the marginal density function of x: f(y/x) = f(x, y)/f(x)

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Conditional Distribution f(y/x)= 1/[ y ]exp{[-1/2(1- y 2 ]* [y- y - x- x )( y / x )]} the mean of the conditional distribution is: y + (x - x ) )( y / x ), i.e this is the expected value of y for a given value of x, x=x*: E(y/x=x*) = y + (x* - x ) )( y / x ) The variance of the conditional distribution is: VAR(y/x=x*) = x 2 (1- ) 2

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14 x y x = a, x 2 > y 2 y = b x, y > 0 xx yy Regression line intercept: y - x ( y / x ) slope: ( y / x )

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15 Bivariate Regression: Another Perspective Regression line is the E(y/x) line if y and x are bivariate normal –intercept: y - x x / y ) –slope: x / y )

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16 Example: Lab Six

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17 Example: Lab Six

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18 Correlation Matrix GEINDEX GE 1.000000 0.636290 INDEX 0.636290 1.000000

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19 Bivariate Regression: Another Perspective Regression line is the E(y/x) line if y and x are bivariate normal –intercept: y - x x / y ) –slope: x / y ) – y = 0.022218 – – x = 0.014361 – x / y ) = (0.02543/0.043669) = –intercept = 0.0064 –slope = 1.094

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21 Vs. 0.0064 Vs. 1.094

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22 Bivariate Normal Distribution and the Linear probability Model

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23 income education x = a, x 2 > y 2 y = b x, y > 0 mean income players Mean educ. Players Mean Educ Non-Players Mean income non Non-Players Players

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24 income education x = a, x 2 > y 2 y = b x, y > 0 mean income players Mean educ. Players Mean Educ Non-Players Mean income Non-Players Non-Players Players

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25 income education x = a, x 2 > y 2 y = b x, y > 0 mean income players Mean educ. Players Mean Educ Non-Players Mean income Non-Players Non-Players Players Discriminating line

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26 Discriminant Function, Linear Probability Function, and Decision Theory, Lab 6 Expected Costs of Misclassification –E(C) = C(P/N)P(P/N)P(N)+C(N/P)P(N/P)P(P) Assume C(P/N) = C(N/P) Relative Frequencies P(N)=23/100~1/4, P(P)=77/100~3/4 Equalize two costs of misclassification by setting fitted value of P(P/N), i.e.Bern to 3/4 –E(C) = C(P/N)(3/4)(1/4)+C(N/P)(1/4)(3/4)

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27 income education x = a, x 2 > y 2 y = b x, y > 0 mean income players Mean educ. players Mean Educ Non-Players Mean income Non-Players Non-Players Players Discriminating line Note: P(P/N) is area of the non-players distribution below (southwest) of the line

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28 Set Bern = 3/4 = 1.39 -0.0216*education - 0.0105*income, solve for education as it depends on income and plot

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29 7 non-players misclassified, as well as 14players misclassified

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31 Decision Theory Moving the discriminant line, I.e. changing the cutoff value from 0.75 to 0.5, changes the numbers of those misclassified, favoring one population at the expense of another you need an implicit or explicit notion of the costs of misclassification, such as C(P/N) and C(N/P) to make the necessary judgement of where to draw the line

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