Monday, March 14, 2011

Mathematical Models & Future Predictions 2

Here is part two of our series on the strengths and weaknesses of mathematical models.
Have we put too much trust in mathematical models?  Have some scientists and mathematicians tried to downplay the powerlessness of mathematical models in predicting outcomes of non-linear phenomena?
Mathematical Modeling streamlines some processes:

Conceptualizes a thought
Thoughts are placed through a simulated rigor.  The basic definition of a mathematical model is explained by Kundzewicz, Afouda & Szolgay in their 1987 paper entitled, Mathematical Modeling,
Mathematical models are a means of representing essential aspects of reality (process, phenomenon, object, element, system, etc.) with the help of mathematical constructs. Mathematical models typically offer convenience and cost advantages over other means of obtaining the required information on reality.
The thought is subjected to various external realities.  These realities, can be tested with relative ease and speed.  Modifications of the model can test different scenarios on the same hypothesis.  There is no question of the usefulness of mathematical models when utilized in this fashion.
"God made the natural numbers. Everything else is the work of man."
Leopold Kronecker

Types Of Mathematical Models
Of course not all mathematical models are the same.  There are several types:
1.  Static Models - this is a "snapshot" of a process which is frozen in time in order to understand it.
2.  Dynamic Models - describe a process in time.  They attempt to describe future events based on what has happened in the past.
3.  Linear Models - These are models where there is a clear cause and effect relationship between events.
4.  Nonlinear Models - These are models were there is no way of predicting an effect from any cause or groups of causes.  The outcome or effect can vary even when the same factors are used, thus creating havoc in any accurate predicting.
5.  Stationary Models - this term is synonymous with static models
6.  Non-stationary Models - this term is synonymous with dynamic models
7.  Deterministic Models - this term is synonymous with linear models, emphasizing the predictable aspect of the model - a clear cause an effect system, where if you plug in the correct variables, one can predict the outcome
8.  Stochastic Models - these models are the opposite of deterministic models in that there is only a probability not certainty in an outcome or prediction.  These are the models that use Bayesian statistics.  All attempts at modeling non-linear systems use stochastic models.
9.  Discrete Models - these models are synonymous with stationary models.  They take a snapshot in time.
10. Continuous Models - these models are synonymous to dynamic models and with non-stationery models

Problems With Mathematical Models
We were amused with this quote in wikipedia under mathematical models,
Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
This is a sophisticated way of saying that any phenomena that is non-linear, "chaotic," emergent, etc., whatever term is desired have VERY limited susceptibility  to mathematical modeling.  
"Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone."
Bertrand Russell
The model may be insensitive to the variables.  This means that the model might not have taken into account important information which would greatly affect the outcome of a phenomena or event.  This would also imply that the model is too simple to be an effective explanation of the underlying structure of what is being studied.  

On the other hand, if the model is built to be very complex in order to take into account ALL the important information, it might become unwieldily and too computer and time intensive to run, thus making it impractical and expensive to use.

All these factors only apply to linear systems, phenomena that can does not have chaotic outcomes.  For these there are no mathematical models that can predictably work.  

We will not get right now into Godel's Incompleteness Theorems.  These touch on this issue of the limitations of mathematical models, but would lead us to far astray.

Mathematical Model foundationally is a philosophical and logical construct. It is created in the mind of man.  Although philosophy attempts understanding and logic is based on the variables that can be quantified, both constructs are highly subject to perceptions, understanding, resource limits and bias. Thus the Logic of Failure can be a failure of logic.

Logic of Failure:
We must here discuss what a strange attractor is.  In complexity or chaotic theory, a strange attractor is set toward which a complex system gravitates or is attracted to, but which has an outcome that cannot be predicted.  When the model is run again, each time there is what is called non-periodic behavior, thus producing a non-repeating pattern.  Some of the known strange attractors are:
1.  Henon Attractor
2.  Rosseler Attractor
3.  Lorenz Attractor
4.  Tamari Attractor

Models that did not consider the strange attractors:

Visual representation
of a strange attractor
The Challenger Disaster: On January 28th in 1986 after a cold spell in Florida the Challenger lifted off to a highly publicized event. Unbeknownst to the occupants in the Space Shuttle Cockpit was that the O-Rings or the “Toric joints” had become brittle due to exposure below the glass transition temperatures of 40 degrees Fahrenheit overnight. They failed 73 seconds into flight. The failure of the O-Ring resulted in high temperature gasses expulsion onto the external fuel tank leading to a catastrophic “System Anomaly!”

The Risk analysis done prior to the Challenger flight and based on previous such space flight was calculated at 1 in 100,000. But later analysis showed it was realistically at 1 in 100. Given that the Challenger mission was the tenth mission and no previous failure data was available to asses the real risk of failure which actually would have been 1 in 10 (Since this was the tenth such flight). The O-Ring issue had not been previously addressed but had been in consideration for the Launch parameters. These parameters were set at 50 degrees Fahrenheit.  The overnight lower temperatures however were not considered into the equation. The noted Richard Feynman made the dramatic announcement of the etiology of the failure before a nation in mourning.

If you cannot see the embedded video here is the link:

A visualization of the Rossler Attractor
H1N1 Epidemic: In the Spring of 2009 the CDC (Center for Disease Control made a bold and provocative declaration that the estimated numbers of the H1N1 cases in the country were “upwards of 100,000. At that point and time only 7415 had been confirmed. These lofty projections were based on a mathematical modeling done by two supercomputers. Both computers fed similar data drew the same conclusion. The “Rubbish in rubbish out” motto is self-explanatory. The model was projected on the basis of an exponential rise in cases based on previously drawn comparators of the Influenza Epidemic of 1918. The World Health Organization predicted dire outcomes of a worldwide pandemic and issued orders for quarantine of suspected individuals. Fortunately that proved to be untrue but in the process the world panicked. Conjecture and axioms have a way of loading up and ganging on the unsuspecting scientists and lay people alike. Airline traffic declined hotels lost booking, conferences were cancelled with unanticipated huge economic losses. Those forced to travel wore unwieldy masks to protect themselves furthering fear. This was not a case of “red-face” or “black-eye.” It was a debacle promulgated by a scientific folly.

Dr. David Ho
HIV Progression: Two high profile science papers in the prestigious journal Nature were published in 1995 related to HIV and AIDS. During the media buzz of the coming Armageddon (HIV) stated large population decimation as a result of this infectious virus.  Human existence was at stake. The premise was to treat all patients infected with HIV even if they were asymptomatic. The two studies by Drs David Ho and George Shaw called the Ho/Shaw predictive model used the “viral load and the quantitative value of the CD4 T cells (Immune Cells) as their parameters for their therapeutic assertions. The rise of the viral load was one parameter and the fall of the CD4 cells was the other. There was a reciprocal relationship between the two and thus the model predicted that in order to prevent an asymptomatic carrier from full blown AIDS they should be “hit hard and hit early” with a protease inhibitor.

There was no Control Group used for either of the studies done separately by the doctors. Large numbers of patients were treated using the model. Eventually data gleaned from these treatments showed that the initial drop in the viral load was associated with a rise in CD4 cells but over a prolonged period of close observation the reverse happened and the viral load increased. So there was no validation of the conceptual design that had not been rendered through a properly designed scientific study with a control group to determine real efficacy. The unintended consequence of this undertaking of treating asymptomatic patients was that it created in short order multiple mutations in the virus that became resistant to the protease inhibitor. Now those with the disease would not get help. After many years and unnecessary treatment this treatment method was abandoned.  The discussion as to whether the HIV virus is the cause of AIDS is outside the scope of this article.  We will assume that it is.

Since then, the debate has risen again with a study done in 2005 by Vivek Jain and Steven G. Deeks, entitled, When To Start Antiretroviral Therapy, which recommends early treatment even asymptomatic patients who have been exposed to the HIV virus.  This debate is still far from over with a currnet article in Current Infectious Disease Reports by Frank S. Rhame entitled, When To Start Antiretroviral Therapy.  Now the trend is to return toward early treatment of asymptomatic patients, even from birth, who have been exposed to the HIV virus.

Looking at the weather phenomena, where the changeability and lack of predictability makes it the best of all possible jobs and the worst of all models that lend themselves to prediction. You can predict and be wrong almost daily and still keep your job as a meteorologist. The blame of that inconsistency then falls squarely on the unmanageable and unpredictable mother-nature. The butterfly effect signifies a small change in the initial condition that can lead to a massive change in the matured outcome at play in weather as in other natural phenomena. A cold front draping north to south moving west to east suddenly picks up moisture from the gulf and causes a deluge in the northeastern United States is blamed on computer modeling probabilities.

So the real world scenario differs from a computer modeled world. In the real world the systems are not structured, linear and deterministic. In fact they are the polar opposites.  Real world simulates itself. It is non-deterministic. Things are unpredictable and volatile.

In our third part of this series we will conclude discussing the limitations of mathematical models.  Stay tuned.

Co-writtren by JEDIMedicine and PlusUltraTech


Anonymous said...

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Что вы думаете по этому поводу.
Пардоньте если не в тот раздел )

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Anonymous said...

Всем привет. Наткнулся недавно на статью, в которой говорится что ученые американской Академии наук (NAS) утверждают, что конец света наступит 22 сентября 2012 года. Причем это будет "конец света" в прямом смысле.
Что вы думаете по этому поводу.
Пардоньте если не в тот раздел )

Guillermo Santamaria said...

The last statement was in Russian I translate it: Hello. Recently came across an article stating that scientists of the American Academy of Sciences (NAS) argue that the world would end on Sept. 22, 2012. And it would be "doomsday " in the literal sense.
What do you think about it.
Pardon if not in that section) My response will be in Google Russian: Я хотел бы видеть ссылку на эту научную статью. Есть ли у вас это как ссылку на Интернет?

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