Dia 1

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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 taxonomy on the basis of graph shapes
• assume functions fR ? 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!

2
A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
2
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
3
Linear
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y ax b
a slope with the x axis b intercept with the
y-axis
linear least squares (http//en.wikipedia.org/wiki
/Linear_least_squares_(mathematics) )
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)
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
4
Piecewise linear (tent- or V-shape, sharp bend)
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y a abs(x - x0) b
a slope with the x axis b height of the apex
x0 location of the ( or -) apex
first estimate x0 next linear least squares to
find a and b (http//en.wikipedia.org/wiki/Linear_
least_squares_(mathematics) )
Accurate measurements using compensation method
(e.g., Wheatstone bridge for measuring
resistance, capacity, inductance)
Possibly the slopes of left- and right segments
are too different then treat as two separate
lines
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
5
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
Horizontal asymptote left (right) unbounded
increase/decrease right (left) or vice versa
y exp (b (x-x0)) a
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
first estimate a next take log(y-a) b(x-x0) and
estimate b and x0 using linear least squares.
Proportional growth, e.g. financial assets
(compound interest), populations absorption in a
medium (xthickness), radioactive decay (xtime),

Alternative parameterisation y y0 exp bx a
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
6
Vertical asymptote left (right) unbounded
increase/decrease right (left) or vice versa
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y a log (b (x-x0)), b(x-x0) gt0
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.
first estimate x0 next set x-x0 ? t and plot y
against exp(t) ya(log bt). Linear least
squares gives a and slopea log b. Find b as
exp(slope/a)..
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)
Alternative parameterisation y y0 log(x-x0),
xgtx0 or y0log(x0-x), xltx0
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
7
Unbounded increase left, decrease right or vice
versa
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y a x2 b x c
a determines curvature b left-right symmetry c
absolute height.
first estimate apex (xa,ya) find point
(xap,yshift) for arbitrary p. Then a
(yshift-ya)/p2 b -2axa c yab2/4a
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.
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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
8
Asymptotic increase left, decrease right or
vice versa
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
yab tan 2 (c(x-x0))
The location of the asymptotes determines c and
x0 The height of the apex is a b determines the
steepness.
Let xa1 and xa2 the locations of the asymptotes.
Then x0(xa1xa2)/2 c?/(xa2-xa1). The height of
the apex is a b tunes the steepness.
No known examples.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
9
Unbounded increase (decrease) left and right
monotonically or not
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y a x3 b x2 c x d
or

Brute-force method substitute at least 4 points
(xi,yi) into yax3bx2cxd and solve linear set
of equations for a,b,c,d in least-squares sense.
For all coefficients ?0 no common applications
known. For only a ? 0 the volume or the mass of
an object, given its characteristic dimension.
Although the cubic function (or higher,
odd-degree polynomial functions) have little
practical application, cubic parameter curves
xFx(t), yFy(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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
10
Asymptotic increase (decrease) left and right
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y c tan (a(x-b))d
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.
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.
An alternative parameterization is, for instance,
y x/((x-x0)(x-x1)). This function has two
vertical asymptotes, for xx0 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
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
11
Asymptotic decrease left, saturating increase
right or vice versa
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
several simplest in use y c((a/x)12-2(a/x)6)
c is a scale factor, determining the depth of the
dip a is the x-value for which the minimum is
reached.
Find the location of the dip its x-coordinate is
a. Next substitute the y value of the minimum to
adjust c.
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.
http//en.wikipedia.org/wiki/Lennard-Jones_potenti
al
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
12
Linear increase left, saturating decrease right
or vice versa
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y C x logistic curve (x)
C is overall scale factor should be 1 if x and y
are in same units.Logistic curve can have various
parameterizations.
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.
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.
There are no immediate interpretations of the
curve with x ? -x of with y ? -y.
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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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saturation both left and right, increasing or
decreasing everywhere. When monotonous, a
logistic curve is a likely candidate.
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
in standard form y1/(1exp(-x))
Standard version has no parameters asymptotes
for y0 and y1. Standard inflection point for
x0 and slope 1.
For arbitrary asymptotes, inflection point and
slope, use Richards generalised logistic curve
see http//en.wikipedia.org/wiki/Generalised_logis
tic_curve
Applications in ecology (population growth),
chemistry (autocatalyse), neural networks,
medicine (tumor growth), physics (Fermi
distribution), economy (price elasticity)
Depending on the application, other
parametrizations can be ya 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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
14
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
saturation left or right, and a vertical
asymptote monotonically increasing or decreasing.
one branch of an (orthogonal) hyperbola
yab/(x-c)
a and c define asymptotes b defines slope.
Find vertical asymptote this defines c. Find
horizontal asymptote this defines a. Slope is
controlled by b sign of b defines which the
quadrants.
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.
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A Core Course on Modeling
Week 4-Dealing with mathematical relations
? ? ? ? ? From graph shape to functional relation
? ? ? ??
15
monotonically increasing or decreasing with
inflection point, no saturation
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
yx 1/3
Characteristic dimension of an object with volume
x
Compare with yx1/2?xony defined for xgt0.
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.
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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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saturation both left and right, approached from
the same side (1) no vertical asymptotes
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
ya 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.
Gaussian statistics, normal distribution
Lorentzian distribution of energy in spectra,
forced resonance geometric distribution of light
from a point source over a surface.
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.
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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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saturation both left and right, approached from
the same side (2) vertical asymptote
Behavior Suggested parameterisation
Parameters How to fit Example
Remarks
y 1 / ((x-b)2/2c2)
b is location of vertical asymptote c relates to
width of peak
Distribution of light from a point source over a
surface containing the light source