Finding a common parameter to two models for two small data sets

Question

I am aware that similar questions have been asked previously. I am new to Mathematica. I tried all the suggested solutions but unfortunately did not manage to solve my problem. Any help would be highly appreciated.

I have two extremely small datasets:

dat1={{2, 1.4725}, {3, 1.4612}, {4, 1.4587}, {5, 1.4580}}

dat2={{2, 1.4733}, {3, 1.4608}, {4, 1.4591}}

I have two models:

mod1 = a + b Exp[-c x]

mod2 = a + d Exp[-e x]

The parameter a has to be the same for both the models, but needs to be obtained from fitting dat1 and dat2 to mod1 and mod2, respectively. How do I find an optimal fit for all five parameters {a,b,c,d,e} from these two small data sets?

Answer 1

An opinion: You have no business attempting to fit 6 parameters (a, b, c, d, e, and error variance) with just 8 points.

A solution ignoring the above opinion and with hope that you get more data:

Create a single data set where the points are identified as to which dataset they belong:

dat1 = {{2, 1.4725}, {3, 1.4612}, {4, 1.4587}, {5, 1.4580}};
dat2 = {{2, 1.4733}, {3, 1.4608}, {4, 1.4591}};
data = Join[Join[List /@ {1, 1, 1, 1}, dat1, 2], Join[List /@ {2, 2, 2}, dat2, 2]]
(* {{1, 2, 1.4725}, {1, 3, 1.4612}, {1, 4, 1.4587}, {1, 5, 1.458}, {2, 2,
   1.4733}, {2, 3, 1.4608}, {2, 4, 1.4591}} *)

Fit a model:

nlm = NonlinearModelFit[data, 
   a + Boole[y == 1] b Exp[-c x] + Boole[y == 2] d Exp[-e x], {a, b, c, d, e}, {y, x}];
nlm["BestFitParameters"]
(*{a -> 1.45814, b -> 0.331115, c -> 1.5689, d -> 0.437684, e -> 1.68206} *)
Show[ListPlot[{dat1, dat2}], 
 Plot[{nlm[1, x], nlm[2, x]}, {x, Min[data[[All, 2]]], Max[data[[All, 2]]]}]]

Answer 2

Hi not elegant solution(but works)

dat1 = {{2, 1.4725}, {3, 1.4612}, {4, 1.4587}, {5, 1.4580}};
dat2 = {{2, 1.4733}, {3, 1.4608}, {4, 1.4591}};
fun[a_, b_, c_, d_, e_] :=
Module[{rez1, rez2}, 
 rez1 = Total[(dat1[[All, 2]] - (a + b*Exp[-c*dat1[[All, 1]]]))^2];
rez2 = Total[(dat2[[All, 2]] - (a + d*Exp[-e*dat2[[All, 1]]]))^2];
rez1 + rez2]
NMinimize[fun[a, b, c, d, e], {a, b, c, d, e}]