NonlinearModelFit and plot with multiple parameters function

Question

i ve got quite simple function:

$f(t) = x1 * a(t) + x2 * b(t)$

and table of results like this one:

t | f(t) | a(t) | b(t)
----------------------
1 |  2   |   5  |  3
----------------------
2 |  4   |   9  | -4 
----------------------
3 |  10  |  12  | 5
----------------------

And I need this to solve with NonlinearModelFit, Ive looked across the internet, but I was not able to find solve and plot function like this..

My (wrong, not functionall solution):

data = {{2,5,3}, {4,9,-4}, {10,12,5}
// This line will ends with error: Number of coordinates (3) is not equal to the number of variables (2).
nls  = NonlinearModelFit[data, x1*y + x2*x, {{y, 0.3}, {x, -0.24}}, {y,z}} 

// Next thing is problem with plot.. offcourse i can do sth like this:
f = { {1, 2}, {2,4}, {3, 10}}

// This plot is incomplete, because I was no able to find
// how to write x in data array :(
Show[ListPlot[f], Plot[nls[t,x,y], ...

I bet someone will find solution immediatelly, but Iam new in Mathematica.. Sorry for the dataset (i cant send full dataset, its bigger).

Thanks

EDIT:

Thank you very much to @jens_bo, I solve that with his help and his response. The main problem was caused by sharing variables across notebooks in mathematica... restart mathematica and everything works great :)

Answer 1

Note: You should check if the code I am using matches your problem, since I had to guess some unclear parts (parenthesis and x,y,z naming).

Since I don't know anything about $a(t)$ and $b(t)$ I threat $f(t)$ as a function of two unknowns $a$ and $b$, or $f(a,b)$.

If we want to fit the data we can use NonlinearFit, but NonlinearFit takes data of the form {{x11,x12,...,y1},{x21,x22,...,y2},...}} if you have multiple variables. So in your case that would be {{a(t1),b(t1),f(t1)},{a(t2),b(t2),f(t2)},...}, so:

data = {{5, 3, 2}, {9, -4, 4}, {12, 5, 10}};

nls = NonlinearModelFit[data, x1*a + x2*b, {{x1, 0.3}, {x2, -0.24}}, {a, b}]

Now, in order to check the fit we can use a 3D plot (since your function only depends on two variables).

Show[
     ListPointPlot3D[data, PlotStyle -> Directive[Red, PointSize -> .025]], 
     Plot3D[nls[a, b], {a, -15, 15}, {b, -15, 15},PlotStyle ->Opacity[.75]]
    ]

gives you:

If you have more than two dependencies or you just want to see the deviations of the model from the data in a more presentable (2D) way, you can use the residuals of the fit and maybe plot them vs. the times $t$.

times = {1, 2, 3};
ListPlot[Transpose@{times, nls["FitResiduals"]}, Joined -> True, PlotMarkers -> Automatic]

Answer 2

In the format you propose, the dependence of $f$ on $t$ is left implicit in your model. $a(t)$ and $b(t)$ seem to be some quadratic function of $t$, but as @jens_bo mentioned, you need to tell us or find out more about $a(t)$ and $b(t)$ to obtain a fit of $f$ as a function of $t$.

Data format: Data needs to be presented to all fitting functions as a list of data points, each data point being itself a list of numbers, with the values of the independent variables coming first, and the dependent variable last. For example, see this format specified in the documentation page for NonLinearModelFit. In other words, data to be fit needs to conform to the pattern {{x1 , y1, f1}, {x2 , y2, f2}, ...}. Your data has the value of $f$ first instead.

In this case, it would be easy to fix this by hand, but a programmatic solution could be the following:

data = {{2, 5, 3}, {4, 9, -4}, {10, 12, 5}}
rearrangeddata = Transpose@RotateLeft[Transpose@data, 1]

(* Out: {{5, 3, 2}, {9, -4, 4}, {12, 5, 10}}, i.e.

a    b    f
-----------
5    3    2
9   -4    4
12   5   10

*)

Fitting: In all cases, the functions depend linearly on the parameters (see also my answer to question 80998 on the distinction between non-linear and linear fitting), so you don't need to carry out an iterative non-linear fit: a linear model will suffice.

You could then take two approaches to fitting this function. You could obtain a fit of $f$ as a function of variables $a$ and $b$, disregarding their dependence on $t$. $f(a,b)=x1\ a+x2\ b$ from your data.

linearmodelfit = Fit[rearrangeddata, {y, x}, {y, x}]

(* Out: 0.321159 x + 0.613899 y *)

Alternatively, you could obtain a fit of $f$ as a function of the single variable $t$. Even though $f\propto t^2$, the model itself is still linear, so again we don't need to use NonlinearModelFit; LinearModelFit will do just fine.

LinearModelFit[{{1, 2}, {2, 4}, {3, 10}}, {t, t^2}, x][t]

(* Out: 4 - 4 t + 2 t^2 *)