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Diffstat (limited to 'man/calc.texi')
| -rw-r--r-- | man/calc.texi | 38 |
1 files changed, 32 insertions, 6 deletions
diff --git a/man/calc.texi b/man/calc.texi index 9e50629a3b2..e13dd9097d6 100644 --- a/man/calc.texi +++ b/man/calc.texi @@ -23962,7 +23962,13 @@ such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters to be determined. For a typical set of measured data there will be no single @expr{m} and @expr{b} that exactly fit the data; in this case, Calc chooses values of the parameters that provide the closest -possible fit. +possible fit. The model formula can be entered in various ways after +the key sequence @kbd{a F} is pressed. + +If the letter @kbd{P} is pressed after @kbd{a F} but before the model +description is entered, the data as well as the model formula will be +plotted after the formula is determined. This will be indicated by a +``P'' in the minibuffer after the help message. @menu * Linear Fits:: @@ -24055,7 +24061,7 @@ $$ @noindent is on the stack and we wish to do a simple linear fit. Type @kbd{a F}, then @kbd{1} for the model, then @key{RET} to use -the default names. The result will be the formula @expr{3 + 2 x} +the default names. The result will be the formula @expr{3. + 2. x} on the stack. Calc has created the model expression @kbd{a + b x}, then found the optimal values of @expr{a} and @expr{b} to fit the data. (In this case, it was able to find an exact fit.) Calc then @@ -24245,6 +24251,8 @@ case, the model might be @expr{a x + b y + c z}; and in the polynomial case, the model could be @expr{a x + b x^2 + c x^3}. You can get a homogeneous linear or multilinear model by pressing the letter @kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}. +This will be indicated by an ``h'' in the minibuffer after the help +message. It is certainly possible to have other constrained linear models, like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single @@ -24454,6 +24462,18 @@ Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}. Gaussian. @texline @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}. @infoline @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}. +@item s +Logistic @emph{s} curve. +@texline @math{a/(1+e^{b(x-c)})}. +@infoline @mathit{a/(1 + exp(b (x - c)))}. +@item b +Logistic bell curve. +@texline @math{ae^{b(x-c)}/(1+e^{b(x-c)})^2}. +@infoline @mathit{a exp(b (x - c))/(1 + exp(b (x - c)))^2}. +@item o +Hubbert linearization. +@texline @math{{y \over x} = a(1-x/b)}. +@infoline @mathit{(y/x) = a (1 - x/b)}. @end table All of these models are used in the usual way; just press the appropriate @@ -24462,8 +24482,9 @@ 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 +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 @expr{a} in the above table which can be replaced by zero or one, as appropriate, by typing @kbd{h} before the model key. @@ -24603,7 +24624,7 @@ to convert the model into this form. For example, if the model is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x}, and @expr{h(x) = x^2} are suitable functions. -For other models, Calc uses a variety of algebraic manipulations +For most other models, Calc uses a variety of algebraic manipulations to try to put the problem into the form @smallexample @@ -24662,7 +24683,12 @@ The Gaussian model looks quite complicated, but a closer examination shows that it's actually similar to the quadratic model but with an exponential that can be brought to the top and moved into @expr{Y}. -An example of a model that cannot be put into general linear +The logistic models cannot be put into general linear form. For these +models, and the Hubbert linearization, Calc computes a rough +approximation for the parameters, then uses the Levenberg-Marquardt +iterative method to refine the approximations. + +Another model that cannot be put into general linear form is a Gaussian with a constant background added on, i.e., @expr{d} + the regular Gaussian formula. If you have a model like this, your best bet is to replace enough of your parameters with |