fit a time series with a form containing time as a TimeObject

Question

I created a TimeSeries from TimeObjects (i.e.the list timemeting, times of the measurements) and the measured values (anglesarray). I want to fit the data to a function of time. The function contains another function (SunPosition) requiring the time in the form of a TimeObject, the same as used in making the TimeSeries. Just calling it "t" as done in the examples for fitting a timeseries does not work. Do I have to deconstruct the timeobjects to get a simpler variable and construct again when calling the function? That seems putting the horse behind the wagon.

timemeting= {DateObject[{2022, 7, 17, 7, 54, 33.}, "Instant"], 
 DateObject[{2022, 7, 17, 8, 16, 8.}, "Instant"], 
 DateObject[{2022, 7, 17, 8, 56, 4.}, "Instant"], 
 DateObject[{2022, 7, 17, 9, 36, 20.}, "Instant"], 
 DateObject[{2022, 7, 17, 10, 16, 1.}, "Instant"], 
 DateObject[{2022, 7, 17, 10, 16, 55.}, "Instant"], 
 DateObject[{2022, 7, 17, 10, 55, 51.}, "Instant"], 
 DateObject[{2022, 7, 17, 10, 56, 13.}, "Instant"], 
 DateObject[{2022, 7, 17, 11, 36, 8.}, "Instant"], 
 DateObject[{2022, 7, 17, 12, 16, 10.}, "Instant"], 
 DateObject[{2022, 7, 17, 14, 16, 21.}, "Instant"], 
 DateObject[{2022, 7, 17, 14, 57, 3.}, "Instant"], 
 DateObject[{2022, 7, 17, 15, 37, 4.}, "Instant"], 
 DateObject[{2022, 7, 17, 16, 16, 6.}, "Instant"], 
 DateObject[{2022, 7, 17, 16, 56, 3.}, "Instant"], 
 DateObject[{2022, 7, 17, 17, 36, 26.}, "Instant"], 
 DateObject[{2022, 7, 17, 18, 16, 22.}, "Instant"], 
 DateObject[{2022, 7, 17, 19, 34, 8.}, "Instant"], 
 DateObject[{2022, 7, 17, 19, 35, 44.}, "Instant"]}
anglesarray= {-83.95, -79.106, -68.979, -58.889, -49.1, -48.725, \
-38.725, -39.15, -29.31, -19.25, 10.595, 20.688, 30.924, 40.635, \
50.53, 60.993, 71.066, 90.977, 91.1841}

NonlinearModelFit[TimeSeries[Transpose[{timemeting, anglesarray}]], 
 ArcTan[Cos[#1] (Cos[#3] Sin[\[Beta]] Sin[#2] + 
         Cos[\[Beta]] Sin[#3]) + 
      Sin[#1] (-Cos[\[Gamma]] Cos[#2] Cos[#3] - 
         Sin[\[Gamma]] (Cos[\[Beta]] Cos[#3] Sin[#2] - 
            Sin[\[Beta]] Sin[#3])), -Cos[#2] Cos[#3] Sin[\[Gamma]] + 
      Cos[\[Gamma]] (Cos[\[Beta]] Cos[#3] Sin[#2] - 
         Sin[\[Beta]] Sin[#3])] & @ 
   Flatten[{gardenloc["Latitude"] Degree + \[Alpha], 
     QuantityMagnitude[SunPosition[gardenloc, t]] Degree}]/
  Degree, {\[Alpha], \[Beta], \[Gamma]}, t]

Answer 1

There is a syntax issue that needs to addressed first. You have

ArcTan[Cos[#1] (Cos[#3] Sin[β] Sin[#2] + 
         Cos[β] Sin[#3]) + 
      Sin[#1] (-Cos[γ] Cos[#2] Cos[#3] - 
         Sin[γ] (Cos[β] Cos[#3] Sin[#2] - 
            Sin[β] Sin[#3])), -Cos[#2] Cos[#3] Sin[γ] + 
      Cos[γ] (Cos[β] Cos[#3] Sin[#2] - 
         Sin[β] Sin[#3])] & @ 
   Flatten[{gardenloc["Latitude"] Degree + α, 
     QuantityMagnitude[SunPosition[gardenloc, t]] Degree}]/
  Degree

but that work work as you expect because the result of Flatten[{gardenloc["Latitude"] Degree + α, QuantityMagnitude[SunPosition[gardenloc, t]] Degree}] is going to be a List and you are sending it to a Function which is expecting 3 arguments. So you need to modify @ to @@ (Apply).

With that in mind, the main issue is that NonlinearModelFit isn't built to use TimeObjects as a point to fit data to. So you need to convert the TimeObject to a number. For example, using AbsoluteTime:

atimes = AbsoluteTime /@ timemeting - AbsoluteTime[timemeting[[1]]]

(I set the first time to be 0 so I don't have to worry too much about working with numbers with large exponents.) Now up a function to use as our model. But now we need to convert t back to the correct TimeObject for SunPosition to use). Also notice that I used the pattern t_?NumericQ to ensure that my function doesn't evaluate until I have a numeric value for t.

func[t_?NumericQ, {α_, β_, γ_}] := 
 Module[{time = FromAbsoluteTime[t + AbsoluteTime[timemeting[[1]]]]}, 
  ArcTan[Cos[#1] (Cos[#3] Sin[β] Sin[#2] + 
          Cos[β] Sin[#3]) + 
       Sin[#1] (-Cos[γ] Cos[#2] Cos[#3] - 
          Sin[γ] (Cos[β] Cos[#3] Sin[#2] - 
             Sin[β] Sin[#3])), -Cos[#2] Cos[#3] Sin[γ] + 
       Cos[γ] (Cos[β] Cos[#3] Sin[#2] - 
          Sin[β] Sin[#3])] & @@ 
    Flatten[{gardenloc["Latitude"] Degree + α, 
      QuantityMagnitude[SunPosition[gardenloc, time]] Degree}]/
   Degree 
  ]

and from here now we can use NonlinearModelFit as before (I used gardenloc = Here of course you would set it to the appropriate value for you).

nlm = NonlinearModelFit[
  TimeSeries[Transpose[{atimes,anglesarray}]], func[t, {α, β, γ}], {α, β, γ}, t];

nlm["BestFitParameters"]
(* {α -> 1.7518, β -> 3.01411, γ -> -0.062247} *)