calc: Standard Nonlinear Models
11.8.4 Standard Nonlinear Models
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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).
‘s’
Logistic _s_ curve. a/(1 + exp(b (x - c))).
‘b’
Logistic bell curve. a exp(b (x - c))/(1 + exp(b (x - c)))^2.
‘o’
Hubbert linearization. (y/x) = a (1 - x/b).
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, logistics, Hubbert and polynomials can
generalize as shown to any number of independent variables. Also, all
the built-in models except for the logistic and Hubbert curves 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.)
Info Catalog
calc: Error Estimates for Fits
calc: Curve Fitting
calc: Curve Fitting Details