MATHEMATICAL MODELS, FUNCTIONS, TABLES, Algebraic, Spread Sheet, J, and APL Notation

This is a general discussion of mathematical modelling. ( Model Types ) Some knowledge of algebra and the concepts of a mathematical function (math fn) are assumed. The concept of a Mathematical Function (math fn) is credited to EULER (ref: The History of Civilization,Vol. IX, p ).

Approaches to Modelling:

Reverse Polish and infix are some of the other notations used. APL was developed as an algorithmic notation. It was made into a computer executable notation that can be written with an electric typewriter. Traditional mathematical or Algebraic notation evolved when paper, pens, and pencils were the writing tools. Notations that do not require super or subscripts and which employ conventional standard character symbols are much easier to use in text. APL used special symbols, J uses only ASCII symbols as found on most English language computer keyboards.

The late Ken Iverson who invented APL joined with Roger Hui and developed J.

Recall that most algebraic symbols were invented when writing was done with pens, pencils, etc. A typerwriter keyboard has limited space and only the most heavily used symbols such as: ' " ; + - = % / \ | [ ] ( ) { } ! @ # \$ % ^ & * _ . ~ ` < > ? are represented. Many users of mathematics like other symbols than the conventional alphabets. Greek alphabetical symbols are favored by many.

The Roman alphabet used by the English language has no supplimentary accents, etc and forms the basis for systems in the English speaking and other parts of the world. Digital representation of symbols developed standard length digital words. Current systems use an eight bit word that has evolved from shorter versions. Eight bits allows 256 unique combinations. The current standard is called ASCII (American Standard Code for Information Interchange)

Each notation has advantages, supporters and detractors. You should choose one for your personal style. I have long used APL because it fitted my work. APL is relatively simple to understand if you are familiar with conventional Algebra. My inclination now is to convert to J because of increased power and use of only ASCII symbols. However it may not be easy to learn yet another language.

Estimates: ( to TOP of page )

Quantitative estimates of what has or may happen are used to refine or transform information for decision making. Such estimates include the whole range of inputs, outputs; resources, products; costs and revenues of activities, systems, machines, people, etc. Machine processing of information in text form is also possible but is less well developed. Machine processing of pictures is new technology. How it can be used for analysis is just emerging.

The common estimating techniques use simple mathematical models, that are often evaluated mentally, but may become sufficiently complicated and repetitive to warrant more formal processes. The current hierarchy is:

1. Mental (of major importance for all judgement activities)
2. Pencil & Paper (disappearing in favour of the following)
3. Hand Calculator (fast and portable, e.g. indows builtin calculator )
4. Computer (capability of hardware way ahead of applications) Following in MYNET:
[ Easy Calc ] * [ Lotus 123
] * [ J Empty WS ] * [ APL clear ws ] [ for more see TOOLS page.
Remember moving down in the list increases costs that should be exceeded by returns.

Models require assumptions about and measurements of behaviour. The better the models describe the behaviour the better the models will estimate or model the measurements of behaviour. For convenience a model is considered to receive input values that are independent variables and to produce outputs that are the dependent variables.

Models as LAWS

When a model is sufficiently robust and general it may be termed a LAW where the independent and dependent variables may be interchangeable. e.g.. Ohm's LAW can be written with potential, resistance, or current as the dependent variable, etc. Such models are non directional. LAWS usually have a strong theoretical basis.

Reversibility, i.e. inter changeability of the dependent and independent variables may be possible in the model but impossible in nature. E.g.. Time is apparently irreversible, and the second LAW of Thermodynamics decrees increasing ENTROPY.

Fitting a model requires: a. Selection of a mathematical function which closely approximates behaviour; b. the estimation of the calibration constants.

Selection of the mathematical form can be avoided if sufficient calibration data are available. These data can be sorted on one or more of the independent variables and to form a table of the data values of the dependent variable which can then be plotted to reveal function shape. A model form is then selected, preferably one that will allow transformation of the data to producd an approximately linear relationship. The data now be used as parts of calibration arrays, and the coefficients determined.

Tablular Models:

A tabular model allows a great deal of freedom in the shape of a relationship and is particularly suited to situations where the correct mathematical function may be difficult or uncertain. Tables can contain ranges and may be one or 'n' dimensional arrays. The usual table is one or two dimensional because of the limitations of the visual process, paper etc. Human and machine memory is not so limited. Tabular models can also incorporate fuzziness, i.e. give a range of dependent variable values for a set of independent variable values that would produce a single result from a rigid model.

The table technique can be used when there are no mathematical relationships such as in telephone directories, tables of contents, lists of facts, etc. Large tables are often referred to as 'DATA BASES' and computers have led to the development of data and data base processing technology.

A mathematical model constructed as a table of dependent variable values is a tabular function. Intermediate values within the table can be estimated by some form of interpolation.

Lookup tables are also constructed by mathematical functions. These are the familiar function tables of values used to avoid complicated and repetitive calculations. Table lookup may be faster than complicated function evaluation for a computer as it usually is for a person. The values in the table can be efficiently retrieved as required by addressing. Fast and efficient searching of large tables is an essential part of Computer Science and Software Enginering.

MODEL 1. The SUM, Σ, or Conservation model ( to TOP of page )

Y is equal to the SUM of the parts X1, X2 ... .; or
0 = Σ X; where X is a vector of all the pertenant variables within the boundary of the system.

The conservation laws that form the basis for accounting , engineering, science and other aspecsts of life or nature are SUM models. They assume that 'the whole' is equal to the SUM of the parts. For some processes this may not be true and 'the whole' may be greater than the sum of the parts. It can never be less if all the parts have been accounted for.

Y = X1 + X2 + X3 + ... + Xn (Xi may be positive or negative). i is a monotonic index beginning with 1, or 0.

In Lotus 123 this is written as: + X1 cell + X2 cell + X3 + .... etc.

In J this is written as: Y =. +/X
Where: X is a list of the various X's or parts.S

In APL this is written as Y ← +/ X
Where: X is a list of the various X's or parts.

These are verbalized as: Y is assigned the SUM (plus reduction) of the X's.

MODEL 2. The RATE, or LINEAR model ( to TOP of page )

Y is Rate a multiplied by Quantity Q (simple linear model).

Y = aQ ¦ in conventional algebraic notation. The line passes through the origin.

Note: either or both a and Q may single quantities or lists (vectors). If both are lists they must be the same length or there must be some rule for matching one list with another. If a and Q are singletons they define a line in two dimensions. If they are vectors of length two then they define a line in three dimensions, and so on.

In J, Y =. a * Q; ifa and Q> are the same length, then:
(\$a) = \$Q where \$ is the shape primitive and is used to count the number of items in the vectors.

In APL if a and Q are vectors of the same length, then: (ρ a) = ρ Q. where: ρ is the shape primitive etc.
The APL multiplication primitive is x and the model is written as:
Y ← a x Q ¦ assumes a[n] is multipiled by Q[n]

MODEL 3. FIXED plus RATE ( to TOP of page )

Y equals fixed amount = C plus a rate = a multiplied by quantity = Q. This is a more general version of MODEL 2. because it allows the line to be offset from the origin (first degree polynomial). The constant term C defines the amount of the offset. ( See also the Polynomial Model )

Y = C + aQ or Y = aQ + C in Algebraic notation
(precedence rules multiply a by Q and add the result to C.)

When: 'a, Q, C' are singletons the model is a constant plus directly varying amount, i.e. a straight line in two dimensions that is not necessarily through the origin.
In J, Y =. C + a * Q; if a and Q are the same length, then:
(\$a) = \$Q\$ is the shape primitive and is used to count the number of items in the vectors.

In both APL and J this expression can define a family of models if C is also a vector of proper dimensions.

In APL this model can be written as:
Y ← C + a x Q
where: C, a, Q may each be singletons or equal length vectors

If C, a, Q are equal length lists they define a single or family of straight lines, or surfaces. Note that the surfaces are planar even though they may be in more than two dimensions, i.e. a multidimensional linear model.

MODEL 4. The PRODUCT or Π model ( To Top of page. )

Y is the product of several constants and/or variables.

Y = abc . ... . n or Y = a x b x c x. ... . x n.
Note the model can include recriprical or ratio terms:
e.g. Y = a x (b/c) x d x (e/f)

In J this model can be written as: Y =. a * b * c, or Y =. */ X

In APL this model can be written as: Y ← a x b x c, or Y ← x/X
where: X is a vector of numeric quantities.

MODEL 5. The RATIO model ( To Top of page. )

The ratio model is a variant of the product model.
It is shown because it exists as a unique variant of the use of the reduction operator.

Y = abc/def where: a, b, c, d, e, f are seperate variables (algebraic)

i.e. Y equals (a times b times c) divided by (d times e times f), or
Y=a/d x b/e x c/f

In J this model can be written as:
Y =. %/ a,d,b,e,c,f
or simply as Y =. %/X (where: X is a properly ordered list of variables)

The APL model is the same as for J and can be written simply as:
Y ← ÷/a,d,b,e,c,f
or Y ← ÷/X

MODEL 6. The EXPONENTIAL, or e model ( To Top of page. )

Y equals a quantity raised to a power.
Y = Xn
In J this expression is written as Y =. X^n
In APL this expression is written as Y ← X*n

e.g. a J version of Pythagoras' theorm on the properties of a right triangle:
a =. 3
b =. 4
( assignments for a 3, 4, 5 Triangle )
c =. (+/(a^2),b^2))^0.5
(note the parentheses are necessary because J & APL have only order precedence rules)

Length of line (vector) by J shown below:
xyz =. 13 23 16 ( assignments of x, y, z projections)
%: +/ xyz ^2 answer is 30.8869 , ( note: %: is J primitive for square root)

The LOGARITHMIC functions are written similarly to the exponental functions above.
The natural Log of 'a' in J is: ^. a
The log of a to base n is: 10 ^, a

MODEL 7. The CIRCLUAR Ο (transcendental functions) ( To Top of page. )

In J there are 23 variations of the circular function o. to give the triginometric & hyperbolic functions, and their inverses. The monadic version is equivalent to o. π (pi) times.
e.g. o. 0.5 is π times 0.5 or a right angle in radians.

o. % 180 is the number of radians per Degree.

o. % 200 ¦ is the number of radians per Grad.

The common trig functions of the angle a in radians are:
Sine =. 1 o. ; Inverse, Sine-1 =. -1 o.
Cosine =. 2 o. ; Inverse, Cosine-1 =. -2 o.
Tamgent =. 3 o. ; Inverse Tangent-1 =. -3 o.

In APL there are 16 variations of the circular function Ο to give the triginometric & hyperbolic functions, and their inverses. The monadic version is equivalent to π (pi) times.
e.g. Ο 0.5 is pi times 0.5 or a right angle in radians.

Ο ÷ 180 is the number of radians per Degree.

Ο ÷ 200 ¦ is the number of radians per Grad.

The common trig functions of the angle a in radians are:
Sine ← 1 Ο ; Inverse, Sine-1 ← -1 Ο
Cosine ← 2 Ο ; Inverse, Cosine-1 ← -2 Ο
Tamgent ← 3 Ο ; Inverse Tangent-1 ← -3 Ο

The meanings of the other circular fns are given in the APL idiom list page 8. Use Q workspace for acess.

‘cc' 8' ¦ Trig Idioms#

MODEL 8. Polynomial Models 8( To Top of page. )

Y = (an x Xn) + .. ... .. (a2 x X2 ) + (a1 x X) + a0
This is a polynomial of degree n and is often used as an approximation when relationships are non linear. A straight line is a first degree polynomial. Polynomials can be written with the high order term first, or the reverse.

In APL a polynomial can be written as: Y ← X ⊥ A
where: A is a list of an, .... a2,a1,a0 ( The a's are the polynomial coefficients, high order to the left.)
In J the expression c p. x is a polynomial with "coefficients" c.

"Polynomials are important for a number of reasons: (from J Studio Lessons, J home page )

1. A sum of polynomials is a polynomial
2. A product of polynomials is a polynomial
3. The derivative of a polynomial is a polynomial
4. The integral of a polynomial is a polynomial
5. Polynomials serve to approximate many functions, such as sine, cosine, and exponential.
In APL X ⊥ A is the general polynomial including a straight line. A is list of coefficients and can be used to describe a line.

To evaluate A at several points X must be a column (vector) The idiom for making column from a row is (X ø.+,0) Y(X ø.+,0) æ A ¦ is one of the expressions that can be used.

Example: Two degree polynommial: Y = 0.1X2 + 2X + 7
A = coeficients; X = values of x; and Y is list of y's
A ← .1 2 7
X ← 1 1.5 2 2.5 3
Y ← (X ø.+,0) ⊥ A

MODEL 9. Defined Functions, or Programs. ( To Top of page. )

In general a model can consist of a number of primitives, (called verbs in J) that can be executed directly (immediate execution) or stated as as a defined function and used as a program or subroutine.

End to date: 041130, ams