Ternary Prediction Classification Model for Structural Dynamics

Ternary Prediction Classification Model:
Construction, Demonstration, and
Discussion
Adam Cone
January 1, 2008
CIEN E9101 section 003
Civil Engineering Research
Supervisor: Prof. George Deodatis

Adam Cone The Ternary Prediction Classification Model:
January 1, 2008 Construction, Demonstration, and Discussion
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Introduction
Mathematical models are created and used to predict diverse phenomena such
as stock prices, reaction rates, population growth, structural dynamics, etc…. A
mathematical model is useful in application only if it accurately predicts the
phenomena that motivated it. The model is an abstract object that makes exact
predictions with unerring consistency, whereas experimental data are steeped in
procedural errors and measurement uncertainty. Because of this discrepancy,
the testing of a mathematical model is not a rigorously defined process, and a
model’s designation as ‘useful in application’ is primarily a matter of experience
and judgment.
The objective of this paper is to construct and demonstrate a rigorously defined
method of determining whether a mathematical model can predict its motivating
phenomenon. I have named this method the Ternary Prediction Classification
Method (TPCM). The TPCM is based on the integration of experimental
uncertainty and pure mathematics. In a sense, the idea is to get the
mathematical operations and the experimental data ‘speaking the same
language’.
In Section 1, I will construct the TPCM. In Section 2, I will demonstrate the
application of the TPCM by looking at an accepted mathematical model (Hooke’s
Law) and some experimental data. I will then contrast the TCPM with a
contemporary standard analysis. Finally, in Section 3, I will discuss the
conclusions reached in Section 2 and address some possible objections to the
TCPM.

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Section 1
1. Testability
Testability distinguishes science from other disciplines. If no experiment
can be done to prove or disprove a statement, the statement is deemed
untestable and is no longer considered a scientific statement. A statement’s
testability can be classified in exactly 1 of the following 6 ways:
(1) True-Testable: There exists an experimental result that proves the statement.
e.g. If I flip this coin enough times, I will eventually get 100 tails in a row.
(2) False-Testable: There exists an experimental result to disprove the
statement.
e.g. It will take exactly 13 seconds for me to run 100 meters.
(3) True-Untestable: There exists no experiment to prove the statement.
e.g. This pencil is exactly 8cm long.
(4) False-Untestable: There exists no experiment to disprove the statement.
e.g. There’s an undetectable cat behind you.
(5) Fully-Untestable: The statement is both true- and false-untestable.
e.g. When I stop observing something, it stops existing.
(6) Fully-Testable: The statement is both true- and false-testable.
e.g. There is exactly one tennis ball in that box.
A prediction’s truth can be characterized as exactly one of the following:
(1) Experiment has proved it.
(2) Experiment has disproved it.
(3) It is fully-testable, but experiment hasn’t proved or disproved it.
(4) It is true-testable only, but experiment hasn’t proven it.
(5) It is false-testable only, but experiment hasn’t disproved it.
(6) It is fully-untestable.
Essentially, experimental proof trumps everything, and untestability renders a
prediction scientifically useless. It’s (3), (4), and (5) that are the messy ones.
Nearly all contemporary models give predictions of these types.
2. Current Techniques of Obtaining Confidence
There are currently three paradigms for predicting outcomes of phenomena at
points where no data has been taken:
• Data-Oriented Modeling
• Theoretical Modeling
• Theory/Data Hybrid Modeling

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Data-Oriented Modeling
This is a characterization of statistical methods such as regression. The
underlying principle is to assume that the data is determined by some function in
a chosen family of functions, the elements of which are uniquely identified by
parameters. For example, an element in the family of linear functions is
determined by a and b, where f(x) = ax + b. Once a family of functions is decided
upon, various methods can be used to determine the parameters, and hence the
specific function, that accurately predicts the data. I call it data-oriented modeling
because it can be (and often is) done without regard for the “underlying
mechanism” of the physical phenomenon. Given a set of data, I can use data-
oriented modeling to try to predict points outside of the data set. How much
confidence I have in a data-oriented model is situation dependent, and there are
various ways of quantifying agreement of the model with the data.
Theoretical Modeling
Another method of predicting data is deriving a mathematical model from “first
principles”, the generic term for understood phenomena at a lower level of
complexity. For instance, if I want to predict the a ball’s position a given time after
I roll it down a hill, I could use classical mechanical theories based on the ball’s
initial speed, the shape of the hill, the friction of the ball with the surface, the
influence of gravity… etc. If the model is based on ideas that I accept, I am
confident that it will predict the data. Theoretical modeling can be (and often is)
done in the absence of actual, collected data, and only compared with data after
it is constructed. Sometimes it isn’t compared with data at all.
Theory/Data Hybrid Modeling
These techniques are often combined. For instance, Hooke’s Law (the
model under examination in Section 2), which predicts the deflection of a spring
under certain conditions, is expressed by the equation F(x) = kx, where F is the
restoring force in Newtons, x is the linear displacement of the spring in meters,
and k is the spring constant in Newtons per meter. Now, F and x, are easy to
intuit, but where does k come from? The spring constant is a material property of
the spring and could be derived, in principle, from atomic theory. In practice,
however, the spring constant is found in a data-oriented, curve-fitting manner that
lacks any explanation, but just seems to fit. These kinds of data/theory hybrid
model are common in the physical sciences.
3. Models: Definition and Standards
I will use the following notation in my definition of a model:
X = the set of controlled variables

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Y = the set of uncontrolled variables that we want to predict
D = the collected data, consisting of pairs (x,y) in XxY.
Definition: A model M is a map from X->Y.
A map is a rule that associates elements of one set with elements of
another set. A map inputs an element in the domain, and outputs a unique
element in the image. For instance, f(x) = x2
is map. The map from a dog to its
weight is another map.
In order to make predictions outside of the data set, we need a
model—some kind of map from the variables that we control to the variables me
want to predict. The whole point of a model is that, because it is defined
abstractly, it can input values for controlled variables (elements of the domain
set) where no data has been taken, and output values for uncontrolled variables
(elements of the image set). These output values can be interpreted as
predictions about what we will observe in situations where we haven’t collected
any data.
Agreement with data
Of course, given a phenomenon with specified controlled and uncontrolled
variables, there are many possible models. Roughly speaking, a model’s
usefulness is judged by the degree to which its predictions agree with the
experimentally collected data. There are many methods of gauging the strength
of a model in predicting a data set. “Goodness of fit” is a popular one in statistics.
Often, visual inspection of a graph of the collected data and the model’s
prediction is considered sufficient.
Regardless of the metric employed, a model must, by some measure,
agree with the data. A model that doesn’t agree with the data is not a
scientifically useful object, because it does not help us to predict or understand
the system’s behavior.
Agreement with theory
No matter what model we choose, and no matter how well it predicts the
data, no matter how many data points we have tested it against, there is one
central difficulty in obtaining confidence in any model: We do not know what’s
going on between the data points. No matter how confident I am about a model,
until you have actually performed the experiment, you cannot claim to know what
is happening between data points. To me, it is a central problem in any scientific
endeavor, and its effects are extremely insidious. Any time you see experimental
data represented as a line (or any continuous curve), I ask myself “was it
possible to take so many data points that their frequency exceeded the resolution
of the paper/screen?” If the answer is no, the presenter of the data is not showing

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me the data, but rather a fit (a special case of a model) of the data, which may or
may not be justified, and of which he isn’t showing me the derivation. Data is
acquired as discrete observations, between which there are gaps. It’s what’s
going on in these gaps that is of interest to an investigator.
Curve fitting is essentially based on a kind of mathematical continuity
assumption in the data and in the phenomenon itself. The idea is that if two
points are close in the domain, they are also close in the image. In many physical
scenarios, this agrees strongly with our experience and intuition. The more water
I pour into a cup, the harder the cup is to pick up. This continuity assumption is
not true in all cases. Consider walking toward a cliff. My heart beats 70 times per
minute when I’m 10 meters away from the precipice, 70 beats per minute when
I’m 5 meters away, and so on. Even when I’m a millimeter away, my heart still
beats at 70 beats per minute. Then, all of sudden, when I actually walk that extra
millimeter, my heart beats 0 times per minute. Now, if I fit a curve to this data, I
would predict lower heart rates between 1mm and 0mm away from the precipice.
This is clearly ridiculous, and this prediction occurs because the continuous
model does not account for the mechanics of the process. This and other
phenomena have discontinuities or jumps. If we make continuity assumptions
and fit curves to model these phenomena, we run into trouble because our model
simply has no chance of predicting these discontinuities.
Usually, curve-fit type models are justified by physical intuition, which tells
us that nothing special is happening in the phenomenon. I would not use a curve
fit for the cliff experiment for precisely this reason. Physical intuition is useful, but
it’s subjective, and I can’t actually know whether there are discontinuities when
taking data.
Even more insidiously, even if the function is continuous, I still don’t know
how it actually looks. For example, the following phenomenon is observed with
cats: when they fall from apartment buildings, they sustain the most damage from
falls that are either a moderate distance or extremely high. The current theory for
this is that cats instinctively right-themselves in the air to land properly and
minimize damage. From very small heights, they haven’t sustained much
damage for obvious reasons, but from slightly higher heights, they don’t have
time to right themselves, and land awkwardly. From a certain, higher range of
heights, the fall is longer, but the cat can right itself and land properly. Of course,
above a certain distance, how the cat (soon to be pancake) lands is irrelevant. A
curve fit could never predict this phenomenon. Curve fits insidiously assume the
nature of the phenomenon, when that is precisely what I am trying to
ascertain—not a very attractive or convincing way to develop a model.
This seems like an insurmountable problem. If I accept that one domain
point being close to another does not imply similarity of their image points, it
seems like there is no way to interpolate with any confidence at all. Not so. The
way to truly have confidence in a prediction outside of the collected data set is to

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have a theoretically constructed model that agrees with my understanding of
related phenomena. Now the model is not based on any continuity assumptions
about maps I cannot possibly know, but it is a derivation of other principles I am
convinced of. I will refer to such a model as theoretically consistent because it
agrees with related theory of the phenomenon.
In general, you have no idea about the relevance of your certain data set
to data points some distance away. Curve fitting is like driving a car by looking in
the rearview mirror—I assume that conditions remain constant (Herb Brody).
Since, in general, I don’t know the conditions, this is a poor idea.
Upshot
In order for a model to be credible, and to truly inspire confidence in its predictive
capabilities, it must be both consistent with the data and consistent with other,
related theory. If the former fails, the model cannot even be trusted on the
collected data, so we surely cannot trust it outside the data domain. If the latter
fails, it seeems deliberately constructed to agree with the data, and that it is
contrived, and unlikely to indicate much between the data points. Again, both of
these must hold.
Here’s a key point: not only are the data and theory conditions necessary,
they are, in a sense, the best I can do. If I want to predict something, I need to
have a theoretical ground for the prediction, and a track record that the model
has succeeded where it has been tested. I can never know with certainty what
happens outside the collected data without actually taking more data to include
that point by performing the experiment. A model that meets both these criteria
is, in a sense, minimally, appropriately, and optimally justified—I expect nothing
more, and I accept nothing less.
4. Degrees of Confidence in Observation
Let’s examine 2 simple observations:
1. Determining whether there is a pencil on the table
2. Determining the length of the pencil

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Photo 1: a pencil on a table
1. One day, I walk into a room and see a pencil on a table (see Photo 1), and I
say so. If someone asks me to justify my statement, to prove it somehow, I can
only stutter, and maybe repeat myself in different wording. Could I be incorrect
about there being pencils on the table? Sure. Maybe these are actually oranges
that are genetically engineered to look like pencils, but any botanist would
categorize them as oranges. Maybe I’m hallucinating. Maybe I was just reading
about pencils on tables and I can’t distinguish my imagination from my
surrounding reality. Maybe someone set up pencil holograms on the table. These
are not particularly constructive criticisms. By that standard, observations are
useless—this document may not contain any text…. The certainty I have in
observing that a pencil is on a table is about as good as I’m going to get, from a
scientific perspective. I have the ability to observe a pencil and it’s location, and
that is what I observe, so that is what I claim. The bottom line is that there is
nothing fundamentally unknowable about whether there is a pencil on the table.
2.

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Photo 2: pencil and tape measure
Photo 3: graphite end of pencil

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Photo 4: eraser end of pencil
Now, suppose I want to determine the length of the pencil. Well, this requires the
use of some calibrated equipment. So I put the pencil near a tape measure (see
Photo 2), and observe that the graphite end rests between the 3in and 4in ticks
(see Photo 3), and the eraser end rests between the 10in and 11in ticks (see
Photo 4). From this information, I don’t actually know exactly how long the pencil
is. This is somewhat frustrating: If someone asks you how many cars you have,
they don’t want to hear “Between 0 and 18”, they want to hear “2”. Well, suppose
I claimed that the exact length of the pencil was 7.13in. How could anyone verify
that? Well, maybe we could get a more accurate tape measure, with more ticks.
The number of ticks would still be finite, so the same problem would occur, just at
a finer scale. In other words, we could decrease the range of possible values, but
it would still be a range. No matter what instrument we use, we can only bound
the length above and below by some numbers, based on the fidelity of our
measuring device. The actual length of the pencil is unknowable, as long as
length is a continuous quantity. The only way we could know the pencil’s exact
length is if, one day, we discovered that length was, in fact, a discrete quantity.
Then length would no longer be a measurement continuum, and we could,
potentially, find the pencil’s exact length.
More formally stated, the assertion that the length is of any particular
length is true-untestable. Therefore, the exact length of the pencils is not a
question science can answer.
If someone asks me if there is a pencil on the table, I would answer “Yes.”
If someone asks me exactly how long the pencil is, I would answer “I don’t know.”
This doesn’t seem like a satisfactory answer; I measured the damn pencil, right?
Well, the problem is not with my measurement, but with the question: if someone
expects a unique number as an answer, they are in for a disappointment.

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The contemporary method of dealing with this ambiguity is to associate a
probability distribution with the measurement. The idea is that, if I were to take
multiple, independent measurements of the pencil’s length, and if I were to report
a unique number for each of these measurements, I would obtain a variety of
different values. The probability distribution that I have seen used to model the
variation in measurements is the normal distribution. The normal distribution is
essentially a bell-curve that tapers off to zero in either direction. The way to
communicate the pencil’s length, according to this model, is to say something
like: “the length of the pencil has a normal distribution with a mean of 6.921in,
and a standard deviation of 0.229in. There is a 95% probability that the pencil’s
length is between 6.4722in and 7.3698in.”
This statement is fully-untestable, since there is no way to prove or disprove this
statement. No matter how many measurements someone else makes of the
pencil’s length, the statement cannot be proved or contradicted, because no
definitive statement about the pencil was made.
However, if we change our perspective on the length of the pencil and,
instead of seeking an exact length, seek only a range of values, then we can
obtain a definitive and accurate answer to the question ‘how long is the pencil?’
Consider Figure 1:
Figure 1: disc location
In Figure 1, which pair of lines is the black disc between:
• green and orange
• orange and red
• red and blue
• blue and grey
The black disc is between the red line and the blue line. This is not a matter of
opinion. This is not a probabilistic statement. This is a definitive, unambiguous,

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fully-testable statement, and, as far as a human being can be sure of anything,
he can be sure of this. This cannot even really be considered a measurement.
This is basic object recognition.
Now, consider the following two postulates:
(1) The length of the pencil must be communicated as an element of a range
of values.
(2) A human being can be confident of his basic visual perception.
If we accept these two postulates, we can determine, definitively, the length of
the pencil. From Photo 3, we conclude that the graphite end of the pencil is
between the 3in tick, and the 4in tick, with the same certainty that we concluded
that the black disc is between the red line and the blue line in Figure 1. Similarly,
from Photo 4, we conclude that the eraser end of the pencil is between the 10in
tick and the 11in tick with the same certainty that the black disc is between the
red line and the blue line in Figure 1. Therefore, we conclude that the pencil is at
least 10in – 4in = 6in long, and at most 11in – 3in = 8in long. Based on that
measuring tape and this measurement, the pencil’s length is between 6in and
8in.
Again, although we don’t know the exact length of the pencil, we do know
that it’s length is an element, of the set of lengths between 6in and 8in. From now
on, I will denote such sets as (a,b), a<b, and call them segments. Sets in which
the endpoints are included will be denoted [a,b], a<b, and will be called intervals.
Sets that include one endpoint and not the other will be called half-open intervals
and will be denoted either by (a,b], a<b, or by [a,b), a<b, as circumstances
dictate. So the length of the pencils is an element of the segment (6in, 8in),
according to our measuring device.
Here’s an important corollary of the above observation. If someone reports
a measurement continuum quantity, like length or time, as a unique number, the
measurement cannot be taken seriously. For example, if you ask me how far I
ran today, and I say “5 miles”, bells should go off, and you should notice that
something is amiss. How could I ever verify that I ran 5 miles? It’s a true-
untestable claim, and meaningless without being expressed as an element of a
segment or interval. We can, in everyday conversation, attribute meaning to this
number by associating range with it (‘Ted said he ran 5 miles last night, so he
probably ran between 4 and 6 miles.’)
With continuum quantities, we naturally think in terms of segments and
intervals, as we must. In science and mathematics, this fundamental point is
often neglected. However, the nature of continuum quantities, and the
unknowability of the length of the pencils, provides a freedom, and a framework
for better model construction and testing.

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5. Points vs. Cells: Data Representation
Representing continuum data as points is a misrepresentation, because it
is literally impossible to measure a point in a continuum. Remember that anytime
you see a point-representation of continuum data, you must (rigorously,
speaking) disregard it as meaningless. So, we don’t have to worry about how to
represent individual data points, since we don’t observe points in a continuum.
Instead, we must only worry about representing intervals or segments of data.
Suppose you measure the length of a plant and it is 8cm mark the plant
endpoint is on. You might be tempted to say that, in this case, we actually can
claim that the plant height is exactly 8cm. We cannot claim this either—the tick
itself has a finite, nonzero width, and so even measurements that are “on the
tick” cannot be reported as points. So, what can we say about this data? Well,
certainly the height is between 7cm and 9cm exclusive, since it is either between
7cm and 8cm exclusive, between 8cm and 9cm exclusive, or exactly 8cm. We
can, however, do better than that. We can actually say, with as much certainty as
the ruler can provide, that the plant endpoint is closer to 8cm than to 7cm, and
closer to 8cm than 9cm. Therefore, we can report this data as (7.5cm, 8.5cm),
where we exclude the endpoints because the plant is clearly not in the middle of
either (7cm, 8cm) or (8cm, 9cm).
Here’s another potential pitfall. If the endpoint is between 6cm and 7cm,
but it is clearly much closer to 6cm. Can we not say that the height is in the
segment (6cm,6.5cm)? If you were taking the data, and you planned to do the
analysis, and the only reason you’re doing this project is to convince yourself of
something, then you can record whatever you feel is reasonable, because your
standard is the only relevant one. However, if you plan to convince other people
of your conclusions, you better make observations that everyone can agree with.
Although these are soft comments, it seems extremely unreasonable to me that
the point is not between 6cm and 7cm. If someone disagrees with that, I probably
wouldn’t care what he thought of my project anyway. However, we get into more
subjective territory with the whole ‘it’s closer to 6cm than it is to 7cm’ business.
The only reason we could claim it for the “on the tick” case was that we literally
couldn’t differentiate the endpoint from the tick.
For an example of interval data representation, consider the photos in
Figure 2:

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Figure 2: Photos of plant height measurements taken in consecutive weeks.
Using intervals instead of unique numbers, the data taken from these
photos is represented in Table 1.
Measurement Taken (weeks) Plant Height (in)
[0, 1] [1, 2]
[1, 2] [2, 3]
[2, 3] [3, 4]
[3, 4] [3, 4]
[4, 5] [4, 5]
[5, 6] [5, 6]
Table 1: Plant height data
Using interval representation, a graph of the plant’s height over time is
represented in Figure 3:

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Figure 3: plant height vs. time; interval data representation
Each of the rectangles bounded by dashed lines is called a cell. Each of
the shaded green rectangles bounded by solid green lines in called a cell block,
and is a union of adjacent cells.
Multidimensional Data
Often in science, we have data whose domain and range have multiple
dimensions. For instance, we may vary time of day and record temperature and
humidity (1 dimensional domain; 2 dimensional image). Also, we may vary height
and weight, and measure percentage body fat (2 dimensional domain; 1
dimensional image). Some data sets are actually characterized as ‘high-
dimensional’, and can have domains and ranges of 40 dimensions or more. Does
this idea of cell decomposition hold for these circumstances as well? Yes—the
idea extends smoothly to high-dimensional data sets. Unfortunately, as is often a
problem in the analysis of high-dimensional data, visualization is impossible for
data sets of dimension greater than 3.
Definition of Cell Structure

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How should we interpret a cell block? Roughly, it means that, “at some point in
the cell block domain, there is a corresponding point in the cell image.” Let’s
state this more rigorously.
First, let’s rigorously define a cell in a data set with a 1 dimensional domain X
and a 1 dimensional image Y.
Definition: Let XxY denote the data space, the set of all points (x,y) such that x is
in X and y is in Y. Then every possible data point is an element of the data
space.
Definition: If A is a subset of X and B is a subset of Y, then let AxB denote the set
of all points (a,b) such that a ∈ A and b ∈ B.
Definition: A cell is a subset of the data space of the form (a1,a2)x(b1,b2), where
a1 ∈ A, a2 ∈ A, b1 ∈ B, b2∈ B, a1 < a2, and b1 < b2.
Definition: A cell block is a union of adjacent cells.
The principle advantages of cell structure are:
- it doesn’t misrepresent data as points in a continuum
- it extends to discrete and measurement continuous quantities
- it extends to any number of dimensions
Cell structure is a standardized, accurate way of representing data of all kinds.
On the other hand, it is not always practical. For example, in visualization of large
data sets, this is inappropriate, since the cells will be too small. However, I
maintain that, although often this is an ineffective way of actually communicating
data, it always the right way to conceptualize and analyze data. In fact, not using
cell structures can often lure us into conceptualizing the data in an inaccurate
way. Representing data as very small discs gives us the impression that the data
is composed of points, and actually induces us to think of the data as
mathematical points, which they are not. Accurate data representation is crucial,
and, when cell structure can be used, I believe it should.
Much of the above discussion may seem hair-splitting. If someone tells us
their age is 45, we do not think that they actually came out of their mother’s
womb exactly 45 earth-orbits ago. The problem is that, if we represent the age as
a point, which is exactly what we do when we write “45 years old”, and then we
do analysis on the point, and not the actual range, our analysis will be flawed,
and will lead to inaccurate conclusions. It is important to represent the exact data
you have if you’re going to do analysis on it. This becomes increasingly important
as a measurement is subjected to more mathematical operations.

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Ranges of data should be reported explicitly, they should be accurate, and
they should not leave the reader with mistaken impressions.
Interpretation of Cell Structure Data
In some data sets there are two points (x,y) and (x,z), where z is not equal
to y. Visually, this means that the two points are “right on top of each other”. This
is nonsense, if taken literally, since it implies that a different thing happened
under the same conditions. Figures 4 and 5 illustrate this phenomenon in both
contemporary and interval form.
Figure 4: Different results under the same condition in contemporary data point
representation form

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Figure 5: Different results under the same condition in interval representation form
When I see data like that in Figure 4, I assume that one of two things
happened: (1) The investigator didn’t account for all relevant variables, and
something was changing in another data dimension that he happened to not
measure, or (2) The investigator wasn’t actually able to repeat the experiment
because he couldn’t resolve the domain well enough, even though he measured
sufficiently significant variables. (1) has nothing to do with data representation, or
even continuum assumptions, and you can only avoid it either by excellent
intuition and luck, or trial and error. (2), however, is not a possible error when
using a cell decomposition of the product space.
When we see cell blocks directly above one another (as in Figure 5), or
even intersecting, we have no such contradiction. Suppose, as in Figure 5, that
we have 2 cell blocks directly on top of one another. This simply means that, at 2
different points in the domain segment, there were 2 different corresponding
points in the image segment, and that they both corresponded to different points
in the image. Now, because they’re all in the same domain segment, we don’t
know the order in which these points occur. There is nothing inconsistent here,
and the cell structure has accurately represented our data, but it is an indication
of one of two potential problems with our data collection process:

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(1) Our uncertainty segment is too large to resolve important system dynamics.
(2) Our data-taking process is unreliable, and doesn’t provide consistent
measurements.
Whichever one of these is the case, something needs to be reworked. This is a
nice feature of cell structure: it can give us clear indications as to when the
experimental design needs improvement.
6. What “Ought to” Happen: Model Construction
Let’s say we’re pouring water into a cup at some rate R in m3
/s, and that the
water has a density of D in kg/m3
. We want to know the mass of water in the cup
after a time T in s. This seems pretty straightforward, doesn’t it? Our intuition tells
us that
“well, if a volume of R is entering the cup per second, and if a volume of R
corresponds to a mass of RD kg, then a mass of RD kg should enter the cup per
second. So, I guess that, after a time T, we will have RDT kg of water in the cup.”
Well, let’s remember that we’re dealing with measurement continua in the fluid
volume and mass, not to mention in time. So, what will my predictions look like?
Will they be testable? If so, in what sense?
We don’t know the exact density of water—we can only know that it is an
element of some segment (D1, D2). Furthermore, the exact rate R is unknowable;
we can only know that it is an element of some other segment (R1, R2). Now, our
model can only input points, so we have to talk about a point in time, even
though a point cannot be observed. After a time T, our model predicts that the
mass of water in the glass will be an element of the segment (D1R1T, D2R2T). So,
our point model has predicted a segment because we only gave it segments to
work with. Is this output testable? Remember that the data we compare these
predictions with will be, not just segments in the mass dimension, but also in the
time dimension. In other words, we can only compare these predictions with
cells.
7. The Ternary Prediction Classification Method
We have 3 possibilities with the interaction of prediction segment P and a
data cell C (C’s domain must intersect P’s domain for the comparison to be
applicable), which are illustrated in Figure 6:
(1) C ⊂ P; the prediction is validated by the data
(2) C ∩ P ≠ ∅ and C P ≠ ∅; the prediction is consistent with the data
(3) C ∩ P = ∅; the prediction is invalidated by the data

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Figure 6: comparison of theoretical prediction (blue swath) and experimenally collected
data (red). C ⊂ P (validation) is illustrated by the lower-left cell; C ∩ P ≠ ∅ and C P ≠ ∅
(consistency) is illustrated by the upper right cell; and C ∩ P = ∅ (invalidation) is
illustrated by the upper left cell.
This prediction is fully-testable: if (1) holds the prediction has been
experimentally proven, while if (3) holds, the prediction has been experimentally
disproven. (1) experimentally proves the statement because, regardless of which
point in C is actually the “true” mass of water in the cup, that point is in the range
predicted by P, so the prediction was correct. (3) disproves the statement
because, regardless of which point in C is the “true” mass of water in the cup, it is
not in the range predicted by P, so the prediction was incorrect. We get into
messy territory in case (2). In both of these cases, the model is consistent with
the data in that its predictions are not mutually exclusive with the data. In other
words, the “true” mass of water in the cup could lie in both the prediction
segment and the cell slice, but might lie in the cell slice and not the prediction
segment. Of course, different cases may hold for different time points, and
different sections of the overall predictive band, so we may not be able to
characterize the entire model as simply of 1 of the 3 above types.

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In summary, given a phenomenon X, and a prediction about collected data from
phenomenon X, the TPCM uses interval data representation to determine
whether the prediction was correct, incorrect, or whether the comparison was
inconclusive.

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Section 2
I will present the Objective and Method for an experiment to determine whether
Hooke’s Law accurately predicts the deflection of a spring. I will present the two
different analyses for this experiment: one a least-squares minimization; the
other the TPCM.
Objective: Given a spring, and a tensile force acting on the spring, predict the
extension of the spring from its static equilibrium using Hooke’s Law.
Method: A spring was hung from a rigid structure, and a container was hung
from the spring (See Figure 1). A small amount of weight was added to the
container to assure that the spring was in the linear-elastic deflection range.
Discrete volumes of water were successively added to the container, and, after
each addition, the length of the spring was measured.
A measuring cup was used to measure the additions of water. A tape measure
was used to measure the length of the spring.
Least Squares Minimization Analysis
Data:
Water Added past Equilibrium Point (mL) Spring Length (cm)
0.0 11.65
112.5 15.15
225.0 19.20
337.5 23.25
450.0 27.25
Table 2: Water Volume and Spring Length Data
Analysis:
First, the data is transformed from units of volume and length, to the
corresponding units of tensile force and extension beyond the equilibrium point.
For a volume of water V in mL, the tensile force in Newtons is given by T =
(9.81m/s2
)V/(1,000mL/kg). For a change in length of ΔL in cm, the deflection x in
meters is given by ΔL/(100cm/m).
Tensile Force Applied After Equilibrium
Point (N)
Spring Deflection Relative to
Equilibrium Point (m)
0 0
1.1036 0.035
2.2073 0.0755

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3.3109 0.1160
4.4145 0.1560
Table 3: Tensile Force and Deflection Data
Now, the Objective (see above) is to predict the deflection of the spring, given an
applied force. According to Hooke’s Law, T = kx, where T is the tensile force
applied to the spring, x is the resulting deflection, and k is called the spring
constant. To achieve the objective of predicting elongation, we must determine k
for this spring.
Using a least-squares, linear fit, k = 28.5828 N/m. Based on this spring constant
and Hooke’s Law, the theoretically predicted deflection for a tensile force of
5.5181N is 0.1931m. The observed deflection for an applied tensile force of
5.181N is 0.1975m.
Conclusion:
Since 0.1931m ≠ 0.1975m, Hooke’s Law failed to predict the spring’s deflection
for a given load.
Furthermore, of the 6 total data points taken, Hooke’s Law only predicted one of
them accurately (Table 3). This is surprising because the spring constant was
actually based on these data.
Tensile Force (N) Predicted
Deflection (m)
Observed Deflection
(m)
Predicted =
Observed?
0 0 0 Yes
1.1036 0.0386 0.0350 No
2.2073 0.0772 0.0755 No
3.3109 0.1158 0.1160 No
4.4145 0.1545 0.1560 No
5.5181 0.1931 0.1975 No
Table 4: Predicted and Observed Deflection Results.
Since the only result that Hooke’s Law correctly predicted was the one it is made
to by definition, Hooke’s Law was not effective in predicting the data for this
experiment.
Ternary Predication Classification Method
Data:
Water Added past Equilibrium Point (mL) Spring Length (cm)
0 [11.5,12]
[100,125] [15,15.5]

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[200, 250] [19,19.5]
[300, 375] [23,23.5]
[400, 500] [27,27.5]
Table 5: Water Volume and Spring Length Data
Analysis:
First, the data is transformed from units of volume and length, to the
corresponding units of tensile force and deflection relative to the equilibrium
point. For a volume of water V in mL, the tensile force in Newtons is given by T =
(9.81m/s2
)V/(1,000mL/kg). For a change in length of ΔL in cm, the deflection x in
meters is given by ΔL/(100cm/m).
Tensile Force Applied After
Equilibrium Point (N)
Spring Deflection Relative to
Equilibrium Point (m)
0 0
[0.981, 1.22625] [0.03,0.04]
[1.962, 2.4525] [0.07,0.08]
[2.943, 3.67875] [0.11,0.12]
[3.924,4.905] [0.15,0.16]
Table 6: Tensile Force and Deflection Data
The Objective (see above) is to predict the deflection of the spring, given an
applied force. According to Hooke’s Law, T = kx, where T is the tensile force
applied to the spring, x is the resulting deflection, and k is called the spring
constant. To achieve the objective of predicting elongation, we must determine k
for this spring.
Based on the data in Table 2, k ∈ [24.5250N/m, 32.7000N/m].
Now, suppose I add an additional volume of water v ∈ [100mL, 125mL] to the
container. What, based on my original data, and Hooke’s Law, do I predict the
resulting elongation from equilibrium to be? According to Hooke’s Law
T = kx
⇒ x = T/k
Now, after the addition of another v ∈ [100mL, 125mL], T = [4.905N, 6.13125N].
Based on the data above, k ∈ [24.5250N/m, 32.7000N/m]. Therefore,
Xpredict ∈ [0.1500m, 0.2500m]
The experimentally collected data for this applied tensile force is Xpredict ∈ [0.19m,
0.20m].

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Figure 7: Tensile Force vs. Spring Deflection using interval data representation

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Conclusion: As seen in Table 8 and Figure 7, Hooke’s Law successfully
predicted the phenomenon of linear-elastic spring deflection. In TPCM
terminology, the data validated Hooke’s Law.
Tensile Force (N) Predicted
Deflection (m)
Observed
Deflection (m)
Prediction
Accuracy
0 {0} {0} ⊂
[0.981, 1.22625] [0.03, 0.05] [0.03,0.04] ⊂
[1.962, 2.4525] [0.06, 0.10] [0.07,0.08] ⊂
[2.943, 3.67875] [0.09, 0.15] [0.11,0.12] ⊂
[3.924,4.905] [0.12, 0.20] [0.15,0.16] ⊂
[4.905N, 6.13125N] [0.15, 0.25] [0.19, 0.20] ⊂
Table 7: Predicted and Observed Deflection Results.

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Section 3
The analyses of Section 2 indicate the usefulness of the TPCM. In the least-
squares minimization analysis, no information about the measurement
uncertainty was included in the data or in the curve-fit, so the only way to
compare the predictions to the data was to see if they were exactly equal. Since
they were not exactly equal, the only available conclusion to reach was that
Hooke’s Law was not in agreement with the data.
In the TPCM analysis, information about the uncertainty was included both in the
data and the data analysis, so there were a range of agreement values, and
therefore Hooke’s Law was validated by the collected data.
The content of Section 2 is meant to illustrate the following point: using TPCM we
can eliminate probabilistic, untestable methods of accepting a mathematical
model, and instead use a relatively objective, well-defined method that is
conceptually and computationally more in line with the mechanics of the
experimental procedure, to arrive at a definitive conclusion about a mathematical
model’s predictions.
There are several possible objections to the above content, some of which I will
now address.
1. My analysis in Section 2 assumed that the measuring cup and the tape
measure were perfectly accurate. My analysis did not account for calibration
error. Accounting for calibration error would not have fundamentally changed the
TPCM analysis.
2. The least-squares minimization analysis did not use existing, accepted
probability distributions to model the experimental uncertainty. In my experience,
these probability distributions are not actually used all the time in scientific work.
In fact, in my experiences in lab research and applied mathematics, I hardly saw
these tools applied at all. The best attempts I saw to account for experimental
uncertainty were simply error bars with questionable origins. I am using the
analysis I have been exposed to.
‘Usefulness’ of Large Prediction Intervals
One objection to using intervals instead of unique numbers is that, if the intervals
are too large, they are not useful. For instance, consider the following statement:
‘The distance between New York and San Francisco is in the range [1picometer,
134 light-years].’ This statement is true, but for any practical purpose (e.g.
determining how much fuel does a Boeing 747 require to fly from New York to
San Francisco), the statement is not helpful.

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To illustrate this issue, l will consider the problem of determining how many small
balls fit inside a cylinder of known dimensions (see Figure 8). First, I will use a
typical, unique number approach, and then I will use an interval approach. Table
8 lists the quantities used for each analysis.
Quantity Unique Number Interval
Sphere Diameter 0.55cm [0.5cm, 0.6cm]
Cylinder Height 21cm [20.5cm, 21.5cm]
Cylinder Base Diameter 10.5cm [10cm, 11cm]
Glass thickness 1cm [0.95cm, 1.05cm]
Sphere Packing Density ρ 0.64 [0.55, 0.67]
Table 8: Quantities used in both computational approaches to sphere packing problem.
Figure 8: schematic of cylinder and sphere dimensions
Quantity Unique Number Interval
Vsphere = 4π(d/2)3
0.0871cm3
[0.0654cm3
, 0.1131cm3
]
Vcylinder = πH((D-2t)/2)2
1,192cm3
[1004cm3
, 1398 cm3
]
Vall spheres = ρ Vcylinder 763cm3
[553cm3
, 937cm3
]
Nspheres = Vall spheres/ Vsphere 8,754 [4,886, 14,314]
Table 9: results of unique number and interval analysis

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Given these predictions, I am tempted to say that 8,754 spheres is a useful
number, while [4,886 spheres, 14,314 spheres] is too large to be useful. I am
tempted to look at the prediction of 8,754 spheres and guess that there will
actually be between 7,000 and 10,000 spheres that fit in the cylinder, so the
maximum error I would expect to see, based on this range is (10,000-
7,000)/7,000 = 43%. The error from the interval, however, could be as large as
(14,314 – 4,886)/4,886 = 193%. This at least partially explains why the unique
number feels useful, and the range feels less useful.
What I actually did with the unique number was to convert it into an interval. This
conversion process was done with my intuition, which is based largely on my
experience. The unique number seemed more useful, because my intuition
converted the number into a smaller interval than the actual interval analysis
generated. However, does this mean that the unique number actually was more
useful, or only that it feels more useful? If a number is known with low accuracy
and reported with high accuracy, has it somehow become more useful? A
number’s usefulness is practically synonymous with the relative size of the
interval associated with the number, and my intuition-generated interval was the
result of a kind of intentional blindness about how the number was generated. It
was a way of spuriously increasing the apparent usefulness of the number.
The 193% error seems over-conservative because I cannot intuit the error
amplification that occurs when mathematical operations are performed. The
operation of cubing the sphere’s diameter took the original error of (0.6cm –
0.5cm)/0.5cm = 20% in the diameter and returned an error of 73% in the sphere’s
volume. These errors amplified over the course of the calculation, resulting in the
193% error in the final prediction. Because I can’t intuitively predict this
mathematically-generated error amplification, I somehow intuit the 193% error to
be over-conservative.
The criticism that interval-arithmetic generates over-conservative and useless
predictions overlooks the error-amplification that is inherent in the application of
mathematical operations. Human intuition has its place, but it is easy to misapply
intuition gained at the supermarket and in the woodshop to mathematical
formulas that involve numerous operations on complex quantities.
Consider the game Chinese Whispers, in which a short message is whispered
from player to player in sequence. The original message is compared with the
message the last player reports, and the result is often amusing because of how
different the two messages are. The amusement comes from the breaking of our
expectation of fidelity in communication. Generally, if a message is
communicated once or twice, the errors are small and insignificant. However,
large numbers of sequential transmissions produce comically large errors, hence
the fun of the game. The game demonstrates a kind of verbal error-amplification
that is analogous to the quantitative error-amplification described above. The

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basic phenomenon is that we extend our intuition for errors inherent in one or two
operations to errors inherent in large numbers of operations.
To carry the analogy further, suppose someone actually needed the message at
the end of the Chinese Whispers sequence, and was not familiar with the game.
He may take “Show me the monkey grease from the cherished v-neck of
Brooklyn” literally, or with a very small variation, based on his experience.
Suppose an experienced player were to say “There’s a huge uncertainty in that
statement, it may have come from almost anything!”, the experienced player my
well be rebuffed with something like “It may be off by 2 or 3 words, but I think
you’re being over-conservative. What you’re saying isn’t useful to me.” The
experienced player knows that, depending on how many people had been
playing the game, the original sentence may have been “Replies from postal
scrutiny are difficult to launch or flounder on.” The frusteration with the lack of
certainty does not change the fact that the certainty is simply not there.
Furthermore, but precisely computing our uncertainty, instead of using a kind of
global intuition, we have an idea of exactly what to do to most efficiently tighten
the prediction interval. For instance, since we tracked the uncertainty in each of
the dimensions used in the sphere packing problem, we know that improvements
in the measurement accuracy of t will do relatively little to tighten the final
interval, whereas improvements in the sphere diameter would be very helpful.
By accepting the size of rigorously-computed intervals, we can achieve a better
understanding of the process by which they were generated, and increase our
ability to make more accurate estimates.

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