A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
A taxonomy on the basis of graph shapes
• assume functions f:R  R
• focus on behavior for x ‘in the long run’ (x 
 and x  - , or as far as they get)
• heuristic: no guarantee for ‘correctness’
• may need a bit of tuning to get the right
parameterization
• start with searching a match with few as
possible parameters
• experiment!
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Linear
Suggested
parameterisation
y = ax + b
Parameters
a: slope with the +x axis; b: intercept with the y-axis
How to fit
linear least squares
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(http://en.wikipedia.org/wiki/Linear_least_squares_(mathematics) );
Example
Remarks
The world record time on 100 m sprint as a function of time (this example
shows the limitations of extrapolating simple models; see Edwards &
Hamson, page 10 and further)
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Piecewise linear (tent- or V-shape, sharp bend)
Suggested
parameterisation
y = a abs(x - x0) + b
Parameters
a: slope with the +x axis; b: height of the apex; x0 :
location of the (+ or -) apex
How to fit
first estimate x0; next linear least squares to find a and b
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(http://en.wikipedia.org/wiki/Linear_least_squares_(mathematics) );
Example
Accurate measurements using compensation method (e.g., Wheatstone
bridge for measuring resistance, capacity, inductance)
Remarks
Possibly the slopes of left- and right segments are too different; then: treat
as two separate lines
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Suggested
parameterisation
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Horizontal asymptote left (right); unbounded increase/decrease right (left)
or vice versa
y = exp (b (x-x0)) + a
Parameters
a determines the height of the asymptote; the sign of b determines which
side (left or right) the asymptote; x0 determines the rate of
increase/decrease
How to fit
first estimate a; next take log(y-a) =b(x-x0) and estimate b and x0 using
linear least squares.
Example
Proportional growth, e.g. financial assets (compound interest), populations;
absorption in a medium (x=thickness), radioactive decay (x=time), …
Remarks
Alternative parameterisation: y = y0 exp bx + a
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Suggested
parameterisation
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Vertical asymptote left (right); unbounded increase/decrease right (left) or
vice versa
y = a log (b (x-x0)), b(x-x0) >0
Parameters
x0 determines the location of the asymptote; the sign of a determines
increase or decrease; the sign of b determines whether increase/decrease
is for ascending or descending x.
How to fit
first estimate x0; next set x-x0  t and plot y against exp(t): y=a(log b+t).
Linear least squares gives a and slope=a log b. Find b as exp(slope/a)..
Example
Perception (e.g., perceived loudness is proportional to the log of the air
pressure); computing science (execution time of algorithms sometimes
grows proportional with log of data size)
Remarks
Alternative parameterisation: y = y0 +log(x-x0), x>x0 or y0+log(x0-x), x<x0
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
7
Behavior
Unbounded increase left, decrease right – or vice versa
Suggested
parameterisation
y = a x2 + b x + c
Parameters
a determines curvature; b left-right symmetry; c absolute height.
How to fit
first estimate apex (xa,ya); find point (xa+p,yshift) for arbitrary p. Then a =
(yshift-ya)/p2; b = -2axa; c = ya+b2/4a
Example
Free falling and thrown objects have parabolic trajectories; stopping distance for
braking cars is quadratic in speed; air resistance on a moving object is (roughtly)
parabolically dependent; potential energy for an oscillating system; area of a surface
given a characteristic dimension.
Remarks
Any even degree polynomial has the behavior of unbounded increase or decrease both
left and right; they can have inflection points and therefore multiple local extrema. For
large |x|, only the highest power dominates, so left branch and right branch tend to be
mirror symmetric for large |x|.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
8
Behavior
Asymptotic increase left, decrease right – or vice versa
Suggested
parameterisation
y=a+b tan 2 (c(x-x0))
Parameters
The location of the asymptotes determines c and x0 The height of the apex
is a; b determines the steepness.
How to fit
Let xa1 and xa2 the locations of the asymptotes. Then x0=(xa1+xa2)/2;
c=/(xa2-xa1). The height of the apex is a; b tunes the steepness.
Example
No known examples.
Remarks
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Unbounded increase (decrease) left and right – monotonically or not
Suggested
parameterisation
y = a x3 + b x2 + c x + d
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or
Parameters
How to fit
Brute-force method: substitute at least 4 points (xi,yi) into y=ax3+bx2+cx+d
and solve linear set of equations for a,b,c,d in least-squares sense.
Example
For all coefficients ≠0: no common applications known. For only a ≠ 0: the
volume or the mass of an object, given its characteristic dimension.
Remarks
Although the cubic function (or higher, odd-degree polynomial functions) have little practical
application, cubic parameter curves x=Fx(t), y=Fy(t) , 0t 1 (so called splines) form the working horse
of most of computer aided geometric design: since they have 4 parameters, they can satisfy two
continuity constraints on both ends (values and tangents), and form a smooth curve consisting of
piecewise cubic curves.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Asymptotic increase (decrease) left and right
Suggested
parameterisation
y = c tan (a(x-b))+d
Parameters
How to fit
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a and b shift the curve to the left or the right; a and c together determine
the steepness in the inflection point; d is a vertical offset
First find a and b such that asymptotes are on the right locations, use that
tan(x) has asymptotes for /2 and -/2. Next adjust c to get the steepness
right; finally adjust d to get the intersection with x-axis right.
Example
tan functions often occur in relation to geometric problems involving angles
or ratios of lengths. E.g., find the height of a building from the length of its
shadow and the angle of the sun above the horizon.
Remarks
An alternative parameterization is, for instance, y = x/((x-x0)(x-x1)). This function has two vertical
asymptotes, for x=x0 and x1 respectively. However, it also has horizontal asymptotes; in our
taxonomy it would therefore classify as ‘saturation both left and right, increasing or decreasing
everywhere’
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
11
Behavior
Asymptotic decrease left, saturating increase right – or vice versa
Suggested
parameterisation
several; simplest in use: y = c((a/x)12-2(a/x)6)
Parameters
c is a scale factor, determining the depth of the ‘dip’; a is the x-value for
which the minimum is reached.
How to fit
Find the location of the dip; its x-coordinate is a. Next substitute the y value
of the minimum to adjust c.
Example
The force between particles (atoms, molecules) if often a combination of
attraction at long distance andd repulsion at short distance. This form for
the interaction was proposed by, and named after E. Lennard-Jones.
Remarks
http://en.wikipedia.org/wiki/Lennard-Jones_potential
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Linear increase left, saturating decrease right – or vice versa
Suggested
parameterisation
y = C x * logistic curve (x)
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Parameters
C is overall scale factor; should be 1 if x and y are in same units.Logistic
curve can have various parameterizations.
How to fit
Find overall scale factor C from slope left hand part; next divide by Cx
and find parameters for logistic curve as described with logistic curve.
Example
Income as a function of selling price: if the price is to low, income is low
despite large volume; if the price is too high, market (share) will be too
small.
Remarks
There are no immediate interpretations of the curve with x  -x of with y  -y.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Suggested
parameterisation
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saturation both left and right, increasing or decreasing everywhere. When
monotonous, a logistic curve is a likely candidate.
in standard form: y=1/(1+exp(-x))
Parameters
Standard version has no parameters: asymptotes for y=0 and y=1.
Standard inflection point for x=0 and slope 1.
How to fit
For arbitrary asymptotes, inflection point and slope, use Richards
generalised logistic curve; see http://en.wikipedia.org/wiki/Generalised_logistic_curve
Example
Applications in ecology (population growth), chemistry (autocatalyse),
neural networks, medicine (tumor growth), physics (Fermi distribution),
economy (price elasticity)
Remarks
Depending on the application, other parametrizations can be y=a arctan (bx +c)+d, or piecewise
linear (ramp function) . If the function can have a vertical asymptote (that is, is not monotonous), a
two-branch hyperbola like y = 1/x could be tried.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
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saturation left or right, and a vertical asymptote; monotonically increasing or
decreasing.
Suggested
parameterisation
one branch of an (orthogonal) hyperbola y=a+b/(x-c)
Parameters
a and c define asymptotes; b defines slope.
How to fit
Find vertical asymptote; this defines c. Find horizontal asymptote; this
defines a. Slope is controlled by b; sign of b defines which the quadrants.
Example
In physics, the product of P and V (pressure and volume) for an amount of
gas with constant temperature is constant. Both are non-negative, so only
one branch of the hyperbola.
Remarks
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
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Behavior
monotonically increasing or decreasing with inflection point, no saturation
Suggested
parameterisation
y=x 1/3
Parameters
How to fit
Example
Characteristic dimension of an object with volume x
Remarks
Compare with y=x1/2=xon;y defined for x>0. Applications of square root:
mechanics (fall time of a point mass for given height); applications
involving Pythagoras’ theorem; characteristic dimension of a surface with
given area.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
Suggested
parameterisation
Parameters
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saturation both left and right, approached from the same side; (1) no
vertical asymptotes
y=a exp (-(x-b)2/2c2) (Gaussian), or y = a / (1+(x-b)2/2c2) (Lorentzian)
b is location of maximum; c relates to width of half maximum; a is height of
maximum.
How to fit
Example
Gaussian: statistics, normal distribution; Lorentzian: distribution of energy in
spectra, forced resonance; geometric distribution of light from a point
source over a surface.
Remarks
The Lorentzian has a ‘thicker’ tail than the Gaussian. It counts as a pathological distribution in
statistics, because it has no mean and its variance is infinity.
A Core Course on Modeling
Week 4-Dealing with mathematical relations
     From graph shape to functional relation    
Behavior
17
saturation both left and right, approached from the same side; (2) vertical
asymptote
Suggested
parameterisation
y = 1 / ((x-b)2/2c2)
Parameters
b is location of vertical asymptote; c relates to width of peak
How to fit
Example
Remarks
Distribution of light from a point source over a surface containing the light
source