Minimizing e2: A Refresher in Calculus

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Minimizing e2 A Refresher in Calculus

• Minimizing error
• The derivative
• Slope at a point
• Differentiation
• Rules of Derivation
• Power rule
• Constants Sums
• Products Quotients
• The Chain-rule
• Critical points
• Factoring equations

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Whats in e?

• Measurement error
• Imperfect operationalizations
• Imperfect measure application
• Bias and noise
• Modeling error/misspecification
• Missing model explanation
• Incorrect assumptions about associations
• Incorrect assumptions about distributions
• More bias and more noise
• Sum of errors affecting measurement and model?
• Ideally, lots of small and independent influence
• Results in random quality of error
• When error is systematic, may bias estimates

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Measuring Error Residuals2
Objective to estimate model such that
we minimize ?e. For computational reasons,
we minimize ?e2.
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The Derivative

• Minimization requires specifying an estimator of
  b0 and b1 such that ?e2 is at the lowest possible
  value.
• Easy in the linear case, but ?e2 is curvilinear
  (quadratic)
• Need calculus of derivatives
• Allow identification of slope at a point

The function is f(X) The derivative
is f(X) Said as f-prime X or
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More on Derivatives

• If we knew the formula for f(X), we could plug
  in the value of X to find the slope at that
  point.
• That means differential calculus is preoccupied
  with the rules for defining the derivative, given
  the various possible functional forms of f(X)
• Now take a deep breath

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Power Rule Constant Rule

• If f(X) xn, then f(X) nxn-1
• Example if f(X) x6, then f(X) 6x5
• If f(X) c, then f(X) 0
• Example if f(X) 5, then f(X) 0

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Calculating Slope for (x,y) Pairs
Y f(X) x2
f(X) 2x
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More Rules

• A derivative of a constant times a function
• If f(X) c u(X), then f(X) c u(x)
• Example f(X)5x2 f(x) 5 2x 10x
• Example plot Y f(x) x2 - 6x 5
• Find f(x).

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Plot and Derivative for Y f(x) x2-6x5
Y f(X)
Y f(X)
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More Rules

• Differentiating a sum
• If f(x) u(x) v(x) then
• f(x) u(x) v(x)
• Example f(x) 32x 4x2
• f(x) 32 8x
• If f(x) 4x2 - 3x, what is f(x)?

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Product Rule

• If f(x) u(x) v(x) then
• f(x) u(x) v(x) u(x) v(x)
• Example f(x) x3(x - 5) find f(x)
• f(x) x3 1 3x2 (x - 5)
• x3 3x3 - 15x2 4x3 - 15x2
• You get the same result using only the power rule
• But the product rule is easier when f(x) is
  complex
• Example f(x) (x4 3)(3x3 1) find f(x)
• f(x) (x4 3) 9x2 4x3 (3x3 1)

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Quotient Rule
Example
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Chain Rule

• If f(x)u(x)n then
• f(x) nu(x)n-1 u(x)
• Example if f(x) (7x2 - 2x 13)5 then
• f(x) 5(7x2 - 2x 13)4 (14x - 2)
• Try this if f(x) (3x 1)10 then f(x) ?

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Critical Points

• Finding minima and maxima
• Key where f(x) 0, slope is zero
• Example if f(x) x2 - 4x 5
• then f(x) 2x - 4
• Set (2x - 4) 0, then x - 2 0
• so x 2 when f(x) 0
• Y coordinate when x 2 is 1 (calculate it!!)
• So f(x) has a critical point at x 2, y 1

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Critical Points, Continued

• How do you determine if its a minima or a
  maxima?
• How many humps in the functional form?
• At least exponent minus one
• Identify x,y coordinates and plot
• Identify the derivative f(x) on either side of
• the critical point

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Identification of Critical Point for f(x) x2 -
4x 5
If f(x) 2x - 4, then the value of f(x)
is negative (slope is negative) up to 2,
and positive (slope is positive) above
2. Therefore, the critical point at (2,0) is
a minimum
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Factoring Ugly Functions
Sometimes a quadratic functional form can be
simplified by factoring. When the equation can
be written as
ax2 bx c The factors can be derived as
follows
For example f(x) x3 - 6x2 9x f(x) 3x2 -
12x 9 To find the critical points, wed need
to find the values at which f(x) 0.
Factoring reduces f(x) to f(x) (x -
3)(3x - 3) so both x 3 and x 1 are CPs
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Partial Derivation
When an equation includes two variables, one can
take a partial derivative with respect to only
one variable. The other variable is simply
treated as a constant
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Some Puzzles

• 1. Differentiate f(x) x3 - 4x 7
• 2. Differentiate f(x) (x - 3)(x2 7x 10)
• 3. Differentiate f(x) (x2 - 5)/(x 7)
• 4. Differentiate f(x) (x2 - 3x 5)12
• 5. Find all critical points for
• Y f(x) x3 3x2 1
• Plot the function, identify the CPs.
• 6. Find all critical points for
• Y f(x) x4 - 8x3 18x2 - 27
• Plot, and identify CPs.
• 7. Y f(x,z) x4 15z2 2xz2 - 456z12 Find
  f(x)