Adonis Diaries

Posts Tagged ‘Legendre

Article #30 of “What is Human Factors in Engineering?”;  December 27, 2005

 “How objective and scientific are experiments?”

If we narrow this article to the statistical analysis of experiments and without going into details suffice us to mention a few controversies.  First, let us do a chronology of the various paradigms in statistics and statistical algorithms.  From a European perspective Pascal is believed to have started probability theory in1654.

LaPlace and Legendre contributed to the Least-Squares algorithm for how to fit a model to data (1750-1810)

Gauss developed the geometry and algebra of the multivariate normal distribution (1800’s)

Galton studied regression between two variables (1885) and Pearson the correlation coefficient in 1895.

Fisher, Snedecor and Sheffe concurrently worked on experimental design and analysis of variance algorithm (ANOVA) to statistically test the population distribution under the assumptions of normality in the 1920’s.

The data analyses of non distribution base samples to fit models to data showing structural features were developed by Thurstone in Factor Analysis, by Young and Householder (1935) in Multidimensional scaling and Cluster analysis algorithms.

Joreskog, K. G developed in 1973 the algorithm of a general method for estimating a linear structural relational equation labeled LISREL that analyses the relationships among latent variables linked to operationalized indicators. This general method considers as special cases path analysis recursive or non recursive as well as Factors analysis.

John Tukey and Mosteller concentrated on studying exploratory data analysis to fit mathematical and geometric models to data showing both structural and residual, and thus complementing confirmatory or inferential analyses.

There are divergent paradigms in the following concepts:  first, the suitability of data measurements according to measurement theory versus the distribution properties of the variable of interest (S. S. Stevens versus I. R. Savage in the 60’s); second, the need to investigate real world data prior to applying any statistical package (data snooping) so that if you perform serious detective work on the data and torture it long enough it will confess and open many ways to understand its underlying behavior (John Tukey); thus increased emphasis on graphs of individual data points and plotting to investigate the preliminary screening so as to ensure that the summary statistics selected are truly relevant to the data at hand. 

Third, the application of the Bayesian approach from the consumer or decision maker viewpoint which provide the final probability against evidence instead of the investigator standard acceptance of a p-value to rejecting a hypothesis (read the “Illusion of Objectivity” by James Berger and Donald Berry, 1988).

Fourth, the selection of an investigator for a statistical package that he is familiar with instead of the appropriate statistics for the research in question;  The acceptance of untenable assumptions on population distributions and computing unrealistic parameters simply because the investigator is not trained to understanding or interpreting alternative statistical methods of nonparametric or distribution freer population methods.

Fifth, there are examples of investigators adopting explanatory statistical packages to torture data into divulging confusing causative variables while, in fact, the science is already well established in the domain to specifically determine exhaustively the causative factors simply because the investigator is not versed in mathematics or physics (“Tom Swift and his electric factor analysis machine by J. Scott Armstrong, 1967).

Sixth, there is a need to the “mathematization of behavioral sciences” (Skelum, 1969) which involves the development of mathematically stated theories leading to quantitative predictions of behavior and to derivation from the axioms of the theory of a multitude of empirically testable predictions. Thus, instead of testing verbal model as to the null hypothesis, an adequate mathematical model account for both variability and regularity in behavior and the appropriate statistical model is implied by the axioms of the model itself.  Another advantage is that attention is turned to measuring goodness of fit, range of phenomena handled by the model and ability to generating counterintuitive predictions.

This discussion is an attempt to emphasize the concept of experimentation as a structured theory and that the current easy and cheap computational potentials should be subservient to the theory so that data are transformed to answer definite and clear questions.  The Human Factors practitioner, whom should be multidisciplinary in order to master the human and physical sciences, is hard hit by the need of performing complex scientific experiments involving human subjects and yet required to yield practical recommendations for the applied engineering fields.

No wonder Human Factors professionals are confused in their purposes and ill appreciated by the other discipline unless a hybrid kind of scientists are generated from a structural combination of engineering discipline and modern experimental methods and statistical algorithms. 

However, Human Factors engineers who have an undergraduate engineering discipline and a higher degree in experimental research and statistical analyses training can be better positioned to handle research involving mathematical modeling of theories in sciences.

The fixed mindedness in adolescents reminds us of the mind fix of old people with the assumption that the mind has the potential flexibility to grow while young.

You may look young masking and old mind or look older and exhibiting a younger mind; it is your choice how much time and energy you are willing to invest for acquiring knowledge.


Article #52, (September 12, 2006)

“Mathematics: a unifying abstraction for Engineering and Physics Phenomena”

A few examples of mechanical and electrical problems will demonstrate that mathematical equations play a unifying abstraction to various physical phenomena of entirely different physical nature.

Many linear homogeneous differential equations with constant coefficients can be solved by algebraic methods and their solutions are elementary functions known from calculus such as the examples in article 51.  For the differential equations with variable coefficients, the functions are non elementary and they fall within two classes and play an important role in engineering mathematics.

The first class consists of linear differential equations such as Bessel, Legendre, and the hyper geometric equations; these equations can be solved by the power series method.

The second class consists of functions defined by integrals which cannot be evaluated in terms of finite many elementary functions such as the Gamma, Beta, and error functions (used in statistics for the normal distribution) and the sine, cosine, and Fresnel integrals (used in optics and antenna theory); these functions have asymptotic expansions in the sense that their series may not converge but numerical values could be computed for large values of the independent variable.

Entirely different physical systems may correspond to the same differential equations, not only qualitatively, but even quantitatively in the sense that, to a given mechanical system, we can construct an electric circuit whose current will give the exact values of the displacement in the mechanical system when suitable scale factors are introduced.

The practical importance of such an analogy between mechanical and electrical systems may be used for constructing an electrical model of a given mechanical system. In many cases the electrical model provides essential simplification because it is much easier to assemble and the values easily measured with accuracy while the construction of a mechanical model may be complicated, expensive, and time-consuming.

An RLC-circuit offers the following correspondence with a mechanical system such as: Inductance (L) to mass (m), resistance (R) to damping constant (c), reciprocal of capacitance (1/C) to spring modulus (k), derivative of electromotive force to the driving force or input force, and the current I(t) to the displacement y(t) or output.

Here are a few elementary examples:

5)      Ohm’s law: Experiments show that the voltage drop (E) in a close circuit when an electric current flows across a resistor (R) is proportional to the instantaneous current (I), or E = R* I.

Also, that the voltage drop across an inductor (L) is proportional to the instantaneous time rate of change of the current, or E = L*dI/dt.

Also, the voltage drop across a capacitor (C) is proportional to the instantaneous electric charge (Q) on the capacitor, or E = Q*1/C.  Note that I(t) = dQ/dt.

6)    Kirchhoff’s second law:  The algebraic sum of all the instantaneous voltage drops around any closed loop is zero, or the voltage impressed on a closed loop is equal to the sum of the voltage drops in the rest of the loop. Thus,

E(t) = R*I + L*dI/dt + Q*1/C.

For example, a capacitor (C = 0.1 farad) in series with a resistor (R = 200 ohms) is charged from a source (E = 12 volts).  Find the voltage V(t) on the capacitor, assuming that at t = 0 the capacitor is completely uncharged.

7)    Hooke’s Law: Experiments show that when a string is stretched then the force generated from the string is proportional to the displacement of the stretch,

or F = k*s.  If a mass (M) is attached to a string, then when the string is stretched further more (y) after the system is in a static equilibrium, then: F = -k*s(0) – k*y.

Newton’s second law for the resultant of all forces acting on a body says that:

Mass * Acceleration = Force, or My” = -k*y.

Furthermore, if we connect the mass to a dashpot, then an additional force come into play, which is proportional to the rate of change of the displacement due to the viscous substance with constant (c).  The equation is then a homogeneous second order differential equation: M*y” + c*y’ + k*y = 0.  Depending on the magnitude of (c) we have 3 different solutions: either 2 distinct real rots, 2 complex conjugate roots, or a real double root [c(2) = 4*M*k)} corresponding respectively to the conditions of  over damping, under dumping, or critical damping.

For example, determine the motions of the mechanical system described in the last equation, starting from y = 1, initial velocity equal zero, M = 1 kg, k =1 for the various damping constant: c = 0, c = 0.5, c = 1, c =1.5, and c = 2.

8)    Laplace’s equation is one of the most important partial differential equations because it occurs in connection with gravitational fields, electrostatic fields, steady-state heat conduction, and incompressible fluid flow.  The solutions of the Laplace equation fall within the potential theory.

For example, find the potential of the field between two parallel conducting plates extending to infinity which are kept at constant potentials; or the potential between two coaxial conducting cylinders; or the complex potential of a pair of opposite charged sources lines of the same strength at two points.





March 2023

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