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VB .NET Helper Newsletter  Rod Stephens
 Nov 24, 2010 10:38 PST 

I wanted to get a quick newsletter out to wish those here in the U.S. a
Happy Thanksgiving.
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This newsletter includes 2 examples in both VB 6 and .NET versions. The
examples don't include any code (see the previous examples referenced in
the descriptions and download the code for details) so the descriptions
of the VB 6 and .NET versions are the same, so they're only described
once here.
-----
Have a great weekend and thanks for subscribing!

Rod
RodSte-@vb-helper.com

Twitter feeds:
    VBHelper
    CSharpHelper
----------
==========

    Both Contents:
1. New HowTo: Find an exponential least squares fit for a set of points
in Visual Basic 6
2. New HowTo: Find an exponential least squares fit for a set of points
in Visual Basic .NET
3. New HowTo: Find a least squares Weibull curve fit for a set of points
in Visual Basic 6
4. New HowTo: Find a least squares Weibull curve fit for a set of points
in Visual Basic .NET
==========
++++++++++
      <Both>
++++++++++
==========
1. New HowTo: Find an exponential least squares fit for a set of points
in Visual Basic 6
http://www.vb-helper.com/howto_curve_fit_exponential.html
http://www.vb-helper.com/HowTo/howto_curve_fit_exponential.zip
==========
2. New HowTo: Find an exponential least squares fit for a set of points
in Visual Basic .NET
http://www.vb-helper.com/howto_net_curve_fit_exponential.html
http://www.vb-helper.com/HowTo/howto_net_curve_fit_exponential.zip

(Picture at
http://www.vb-helper.com/howto_net_curve_fit_exponential.png.)

The following examples show how to fit a set of points that lie along a
line or polynomial equation:

    - Find a linear least squares fit for a set of points in Visual
Basic .NET
      http://www.vb-helper.com/howto_net_linear_least_squares.html

    - Find a polynomial least squares fit for a set of points in Visual
Basic .NET
      http://www.vb-helper.com/howto_net_polynomial_least_squares.html

    - Find a linear least squares fit for a set of points in Visual
Basic 6
      http://www.vb-helper.com/howto_linear_least_squares.html

    - Find a polynomial least squares fit for a set of points in Visual
Basic 6
      http://www.vb-helper.com/howto_polynomial_least_squares.html

The basic idea behind these algorithms is to:

    - Calculate an error function containing unknown coefficients

    - Find the partial derivatives of the error function with respect to
the coefficients

    - To find the coefficients that give the smallest error, set the
partial derivatives equal to zero and solve for the coefficients

For linear and polynomial least squares, the partial derivatives happen
to have a linear form so you can solve them relatively easily by using
Gaussian elimination.

But what if the points don't lie along a polynomial? What if the partial
derivatives of the error function are not easy to solve? This entry
examines such an example.

Consider the points in the picture and suppose you want to find values
a, b, and c to make the equation F(x) = a + b * e^(c * x) fit the curve.
You can begin as before, finding the error function and taking its
partial derivatives. The following equation shows the error function.

(See the web pages for the equation in a nicer format.)

    Sum[(yi - F(x,a,b,c))^2] = Sum[(yi - (a+b*e^(c*x)))^2]

Now you can take the partial derivatives of this function with respect
to a, b, and c. For example, the following equation shows the partial
derivative with respect to c.

(See the web pages for the equation in a nicer format.)

    Sum[2*(yi - (a+b*e^(c*x))*(-b*e^(c*x)*x)]

After finding the partial derivatives, you would set them equal to 0 and
solve for a, b, and c. Unfortunately the previous equation is a mess.
The partial derivatives with respect to a and b are a bit simpler but,
frankly, I don't know how to solve the system of these three equations.
(If you can solve them, please email me at RodSte-@csharphelper.com
and tell me how.)

The point of setting the partial derivatives of E equal to 0 and solving
was to minimize the error function E. Although you can't directly solve
the equations, you can use them to minimize E.

This program starts with a guess (a, b, c). It then calculates the
partial derivatives at that point to get a gradient vector. The gradient
vector <va, vb, vc> gives the error function's direction of maximum
increase.

To think of this another way, pretend the error function is a function
E(x, y) that defines a surface in three-dimensional space. The gradient
<vx, vy> tells the direction to move that increases the function the
most. In other words, it tells you how to move directly uphill on the
surface.

Conversely the negative of the gradient <-vx, -vy> tells you how to move
directly downhill on the surface.

That's what the program does. It calculates the gradient at the guess
point (a, b, c) and moves a small distance in the opposite direction. If
the value at the new point is greater than the value at the current
point, then the program tries moving a smaller distance in the same
direction. It continues trying smaller distances until it finds one that
works. It then updates (a, b, c) to the new point and starts over.

If you consider the three-dimensional analogy again, the program moves
downhill until it can't find any way to more farther downhill or it
exceeds a maximum number of moves.

See the code for the details about partial derivatives and how the
program controls its search.

There is one piece of the process that's a bit fuzzy. The program picks
test points (a, b, c) within an area where I suspect the solution may
lie. It then follows the gradients from each of those points and picks
the best solution. Where you pick those points depends on the equation
F.

The number of points that you pick and the maximum number of iterations
that you should use for each point will depend on the shape of the
surface defined by the error function E. If the function is relatively
smooth, then any initial point should lead eventually to a good
solution, so you may only need to pick a few points and follow them for
many iterations.

In contrast, if E defines a wrinkly surface with many local minima, then
from a given starting point the program may become stuck and not reach a
global minimum. In that case, you should pick many initial points and
iterate them as long as you can. In the three-dimensional analogy, the
difference is between a single smooth hillside and terrain full of
bumps, valleys, and sinkholes.
==========
3. New HowTo: Find a least squares Weibull curve fit for a set of points
in Visual Basic 6
http://www.vb-helper.com/howto_curve_fit_weibull.html
http://www.vb-helper.com/HowTo/howto_curve_fit_weibull.zip
==========
4. New HowTo: Find a least squares Weibull curve fit for a set of points
in Visual Basic .NET
http://www.vb-helper.com/howto_net_curve_fit_weibull.html
http://www.vb-helper.com/HowTo/howto_net_curve_fit_weibull.zip

(Picture at http://www.vb-helper.com/howto_net_curve_fit_weibull.png.)

Not long ago, a client wanted to fit sets of data points to Weibull
curves of the form:

(See the web pages for the equation in a nicer format.)

    F(x) = a - b * e^(-c*x^d)

The earlier example "Find an exponential least squares fit for a set of
points in Visual Basic .NET"
(http://www.vb-helper.com/howto_net_curve_fit_exponential.html) explains
how to find a least squares fit for exponential data. This example uses
a similar technique with the following differences:

    - This example uses a different function F, error function E, and
partial derivatives
    - This example's functions have 4 parameters a, b, c, and d instead
of just 3
    - The ranges for test values of a, b, c, and d are different so they
fit the data better

See the earlier example for an explanation of how this technique works.

For more information on classical Weibull functions, see:

    - Weibull Distribution at Wolfram MathWorld
      http://mathworld.wolfram.com/WeibullDistribution.html

    - Weibull distribution at Wikipedia
      http://en.wikipedia.org/wiki/Weibull_distribution
==========
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