Standard Nonlinear Models
-------------------------

The `a F' command also accepts other kinds of models besides lines and
polynomials.  Some common models have quick single-key abbreviations;
others must be entered by hand as algebraic formulas.

Here is a complete list of the standard models recognized by `a F':

`1'
     Linear or multilinear.  a + b x + c y + d z.
`2-9'
     Polynomials.  a + b x + c x^2 + d x^3.
`e'
     Exponential.  a exp(b x) exp(c y).
`E'
     Base-10 exponential.  a 10^(b x) 10^(c y).
`x'
     Exponential (alternate notation).  exp(a + b x + c y).
`X'
     Base-10 exponential (alternate).  10^(a + b x + c y).
`l'
     Logarithmic.  a + b ln(x) + c ln(y).
`L'
     Base-10 logarithmic.  a + b log10(x) + c log10(y).
`^'
     General exponential.  a b^x c^y.
`p'
     Power law.  a x^b y^c.
`q'
     Quadratic.  a + b (x-c)^2 + d (x-e)^2.
`g'
     Gaussian.  (a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2).

All of these models are used in the usual way; just press the
appropriate letter at the model prompt, and choose variable names if
you wish.  The result will be a formula as shown in the above table,
with the best-fit values of the parameters substituted.  (You may find
it easier to read the parameter values from the vector that is placed
in the trail.)

All models except Gaussian and polynomials can generalize as shown to
any number of independent variables.  Also, all the built-in models
have an additive or multiplicative parameter shown as `a' in the above
table which can be replaced by zero or one, as appropriate, by typing
`h' before the model key.

Note that many of these models are essentially equivalent, but express
the parameters slightly differently.  For example, `a b^x' and the
other two exponential models are all algebraic rearrangements of each
other.  Also, the "quadratic" model is just a degree-2 polynomial with
the parameters expressed differently.  Use whichever form best matches
the problem.

The HP-28/48 calculators support four different models for curve
fitting, called `LIN', `LOG', `EXP', and `PWR'.  These correspond to
Calc models `a + b x', `a + b ln(x)', `a exp(b x)', and `a x^b',
respectively.  In each case, `a' is what the HP-48 identifies as the
"intercept," and `b' is what it calls the "slope."


If the model you want doesn't appear on this list, press `'' (the
apostrophe key) at the model prompt to enter any algebraic formula,
such as `m x - b', as the model.  (Not all models will work,
though--see the next section for details.)

The model can also be an equation like `y = m x + b'.  In this case,
Calc thinks of all the rows of the data matrix on equal terms; this
model effectively has two parameters (`m' and `b') and two independent
variables (`x' and `y'), with no "dependent" variables.  Model
equations do not need to take this `y =' form.  For example, the
implicit line equation `a x + b y = 1' works fine as a model.

When you enter a model, Calc makes an alphabetical list of all the
variables that appear in the model.  These are used for the default
parameters, independent variables, and dependent variable (in that
order).  If you enter a plain formula (not an equation), Calc assumes
the dependent variable does not appear in the formula and thus does
not need a name.

For example, if the model formula has the variables `a,mu,sigma,t,x',
and the data matrix has three rows (meaning two independent
variables), Calc will use `a,mu,sigma' as the default parameters, and
the data rows will be named `t' and `x', respectively.  If you enter
an equation instead of a plain formula, Calc will use `a,mu' as the
parameters, and `sigma,t,x' as the three independent variables.

You can, of course, override these choices by entering something
different at the prompt.  If you leave some variables out of the list,
those variables must have stored values and those stored values will
be used as constants in the model.  (Stored values for the parameters
and independent variables are ignored by the `a F' command.)  If you
list only independent variables, all the remaining variables in the
model formula will become parameters.

If there are `$' signs in the model you type, they will stand for
parameters and all other variables (in alphabetical order) will be
independent.  Use `$' for one parameter, `$$' for another, and so on.
Thus `$ x + $$' is another way to describe a linear model.

If you type a `$' instead of `'' at the model prompt itself, Calc will
take the model formula from the stack.  (The data must then appear at
the second stack level.)  The same conventions are used to choose
which variables in the formula are independent by default and which
are parameters.

Models taken from the stack can also be expressed as vectors of two or
three elements, `[MODEL, VARS]' or `[MODEL, VARS, PARAMS]'.  Each of
VARS and PARAMS may be either a variable or a vector of variables.
(If PARAMS is omitted, all variables in MODEL except those listed as
VARS are parameters.)

When you enter a model manually with `'', Calc puts a 3-vector
describing the model in the trail so you can get it back if you wish.


Finally, you can store a model in one of the Calc variables `Model1'
or `Model2', then use this model by typing `a F u' or `a F U'
(respectively).  The value stored in the variable can be any of the
formats that `a F $' would accept for a model on the stack.


Calc uses the principal values of inverse functions like `ln' and
`arcsin' when doing fits.  For example, when you enter the model `y =
sin(a t + b)' Calc actually uses the easier form `arcsin(y) = a t +
b'.  The `arcsin' function always returns results in the range from
-90 to 90 degrees (or the equivalent range in radians).  Suppose you
had data that you believed to represent roughly three oscillations of
a sine wave, so that the argument of the sine might go from zero to
3*360 degrees.  The above model would appear to be a good way to
determine the true frequency and phase of the sine wave, but in
practice it would fail utterly.  The righthand side of the actual
model `arcsin(y) = a t + b' will grow smoothly with `t', but the
lefthand side will bounce back and forth between -90 and 90.  No
values of `a' and `b' can make the two sides match, even
approximately.

There is no good solution to this problem at present.  You could
restrict your data to small enough ranges so that the above problem
doesn't occur (i.e., not straddling any peaks in the sine wave).  Or,
in this case, you could use a totally different method such as Fourier
analysis, which is beyond the scope of the `a F' command.
(Unfortunately, Calc does not currently have any facilities for taking
Fourier and related transforms.)