The Vision Thing Power Thirteen

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The Vision ThingPower Thirteen

• Bivariate Normal Distribution

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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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y
x
mx 0, sx2 1 my 0, sy2 1 rx, y 0
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y
b
x
a
mx a, sx2 1 my b, sy2 1 rx, y 0
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y
x
mx 0, sx2 gt sy2 my 0 rx, y 0
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y
x
mx 0, sx2 lt sy2 my 0 rx, y 0
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y
b
x
a
mx a, sx2 gt sy2 my b rx, y gt 0
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y
b
x
a
mx a, sx2 gt sy2 my b rx, y lt 0
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Why? The Bivariate Normal Density and Circles

• f(x, y) 1/2psxsyexp(-1/2(1-r2)
  ((x-mx)/sx2 -2r ((x-mx)/sx ((y-my)/sy
  ((y-my)/sy2
• If means are zero and the variances are one and
  no correlation, then
• f(x, y) 1/2pexp(-1/2 )(x2 y2), where
  f(x,y) constant, k, for an isodensity
• ln2pk (-1/2)(x2 y2), and (x2 y2)
  -2ln2pkr2

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Ellipses

• If sx2 gt sy2, f(x,y) 1/2psxsyexp(-1/2)
  ((x-mx)/sx2 ((y-my)/sy2, and x (x-mx)
  etc.
• f(x,y) 1/2psxsyexp(-1/2) (x/sx2
  y/sy2) , where f(x,y) constant, k, and
  lnk 2psxsy (-1/2) (x/sx2 y/sy2
  )and x2/c2 y2/d2 1 is an ellipse

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Correlation and Rotation of the Axes
y
Y
X
x
mx 0, sx2 lt sy2 my 0 rx, y lt 0
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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/sy exp-1/2(1-r2)sy2
  y-my-r(x-mx)(sy/sx)
• the mean of the conditional distribution is
  my r(x - mx) )(sy/sx), i.e this is the
  expected value of y for a given value of x, xx
• E(y/xx) my r(x - mx) )(sy/sx)
• The variance of the conditional distribution is
  VAR(y/xx) sx2(1-r)2

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y
Regression line intercept my -
rmx(sy/sx) slope r(sy/sx)
my
mx
x
mx a, sx2 gt sy2 my b rx, y gt 0
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Bivariate Regression Another Perspective

• Regression line is the E(y/x) line if y and x are
  bivariate normal
• intercept my - r mx (sx/sy)
• slope r (sx/sy)

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Example Lab Six
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Example Lab Six
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Correlation Matrix

• GE INDEX GE 1.000000 0.636290 INDEX
  0.636290 1.000000

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Bivariate Regression Another Perspective

• Regression line is the E(y/x) line if y and x are
  bivariate normal
• intercept my - r mx (sx/sy)
• slope r (sx/sy)
• my 0.022218
• r 0.63629
• mx 0.014361
• (sx/sy) (0.02543/0.043669)
• intercept 0.0064
• slope 1.094

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Vs. 0.0064 Vs. 1.094
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Bivariate Normal Distribution and the Linear
probability Model
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education
Non-Players
Mean Educ Non-Players
Players
Mean educ. Players
income
mean income players
Mean income non
mx a, sx2 gt sy2 my b rx, y gt 0
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education
Non-Players
Mean Educ Non-Players
Players
Mean educ. Players
income
mean income players
Mean income Non-Players
mx a, sx2 gt sy2 my b rx, y gt 0
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education
Non-Players
Mean Educ Non-Players
Discriminating line
Players
Mean educ. Players
income
mean income players
Mean income Non-Players
mx a, sx2 gt sy2 my b rx, y gt 0
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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)23/1001/4,
  P(P)77/1003/4
• 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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education
Non-Players
Mean Educ Non-Players
Discriminating line
Players
Mean educ. players
income
mean income players
Mean income Non-Players
mx a, sx2 gt sy2 my b rx, y gt 0
Note P(P/N) is area of the non-players
distribution below (southwest) of the line
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Set Bern 3/4 1.39 -0.0216education -
0.0105income, solve for education as it depends
on income and plot
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7 non-players misclassified, as well as 14players
misclassified
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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