fitting - NonlinearModelFit: fit a vector-valued function to a data vector

I have a vector of data points:

data = 10 Sin[#/10] + RandomVariate@NormalDistribution[] & /@ 
   Range[200];

And I have model that provides the complete data vector as value

matrix = Table[i (Sin[j/10]), {i, 20}, {j, 200}];
weights[a_] := Table[PDF[NormalDistribution[a, 1], i], {i, Range@20}];
model[a_] := weights[a].matrix;

model[10] is a good approximation of data.

They way I currently determine the fit parameters is by using FindMinimum and explicitly minimize the fit penalty:

FindMinimum[
 Total[(data - model[a])^2],
 {{a, 10}}
 ]

This works nice. But the evaluation of fit statistics like error volumes is very tedious because the underlying matrix and the model function is rather complex. When fitting with NonlinearModelFit instead, a lot of those statistics are readily available. But I have struggled to adapt it accordingly:

Something like

NonlinearModelFit[
 data,

 model[a],
 {{a, 1}},
 var
 ]

Unfortunately, there are problems. Can it be adapted? I dont actually need any variables to fit the expressions.

Kind regards