Fit two intersecting straight lines in Mathematica

Question

I am trying to do a Kurie analysis of the double beta decay of Strontium-90.

Here we have two straight sections, and I need to obtain a value for the intersection along with an error for it. I cannot fit the linear sections separately and use gaussian error propagation, since the fit parameters are correlated, which leads to a wrong result.

My idea was to fit the entire data with one function which is defined piecewise, since the intersection parameter would then not be correlated to the others.

However, after several attempts and a fine looking fit, Mathematica reports an error of 0.

NonlinearModelFit[data, (a*x + b)*UnitStep[x + k] + (c*x + d)*UnitStep[k - x], 
  {{a, -6.5}, {b, 3}, {c, -1}, {d, 1.8}, {k, 0.55}}, x, Method -> "NMinimize"]

Are there any other ways of doing this?

---

As requested in the comments:

data = {
   {2.160471686147928`, 0.14147102932224526`},{2.0826390189609363`,0.19988842616357125`},
   {2.018773794495269`, 0.2541246545339056`}, {1.9476326649797793`, 0.3042648279519891`},
   {1.8745330031773189`, 0.3792877696740006`}, {1.8048469776435923`, 0.4333264807298175`},
   {1.731446650327849`, 0.4855243148263505`}, {1.6651014082972075`, 0.5645773920773384`},
   {1.5932847071615477`, 0.5972837751080117`}, {1.52139595538298`, 0.6653922271670848`},
   {1.4422671545128036`, 0.7171604890149067`}, {1.3757137824715229`, 0.79262676502781`},
   {1.3073786844775914`, 0.8404843652309524`}, {1.2337051064574602`, 0.8954499357694174`},
   {1.1906240113139641`, 0.9399196660012136`}, {1.1601352767466366`, 0.9890482524748053`},
   {1.0867224338785175`, 1.0227388598388758`}, {1.0206577309807057`, 1.0542107134579233`},
   {0.9477230432778146`, 1.1301493061594161`}, {0.8733917792374216`, 1.1877273944557323`},
   {0.8065623408414323`, 1.2154971541338453`}, {0.736758267792002`, 1.3121152330650923`},
   {0.6676435767247085`, 1.3823323706871722`}, {0.635101745155093`, 1.3868057675840053`},
   {0.599382518347645`, 1.4179592728730186`}, {0.5622741608125342`, 1.4536070008301036`},
   {0.5321771084417978`, 1.4636936970264636`}, {0.490818693322041`, 1.6204857554395113`},
   {0.4613963222976981`, 1.6851063997905946`}, {0.42109009911055906`,1.8551603288963705`},
   {0.39567303637157025`, 1.945847339995944`}, {0.36747227092332413`, 2.145514603636669`},
   {0.34125816875918824`, 2.333372509561687`}, {0.3124933023702082`, 2.347039270807089`}};

Answer 1

Something strange happens when you allow your lines in your model to have a gap at point k. You also specified UnitStep functions wrongly - see my version below. You do realize your model function allows for gap?

It is better to have less parameters in the model. Obviously your data assume that lines meet without gap. Why not to explicitly specify this excluding one parameter?

Solve[a*x + b == c x + d /. x -> k, d]

{{d -> b + a k - c k}}

Now your model is

g[x_, a_, b_, c_, k_] := 
(a x + b) UnitStep[k - x] + (c x + (b + a k - c k)) UnitStep[x - k]

But I would better define it as:

g[x_, a_, b_, c_, k_] := 
Piecewise[{{a x + b, x < k}, {b + (a - c) k + c x, x >= k}}]

Using your data:

nlm = NonlinearModelFit[data, 
  g[x, a, b, c, k], {{a, -3.5}, {b, 3.5}, {c, -1}, {k, 0.5}}, x];

Plot[nlm[x], {x, 0, 3}, Epilog -> {Red, PointSize[.01], Point@data}, 
 Frame -> True, Axes -> False]

nlm["ParameterTable"]