# Changing the binding of a variable in a closure

**URL:** <https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297>\
**Category:** Statistics\
**Tags:** question, closure\
**Created:** [July 12, 2019, 6:54pm UTC](https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297 "2019-07-12T18:54:17Z")\
**Posts on this page:** 6\
**Page:** 1

<div class="post-metadata">

**Author:** ![dmbates](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dmbates/32/44_2.png) [@dmbates](https://discourse.julialang.org/u/dmbates)\
**Post date:** [July 12, 2019, 6:54pm UTC](https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297/1 "2019-07-12T18:54:17Z")

</div>

I am labeling this question as “statistics” because the motivation comes from a statistical application. The tl: dr version is that I want to change the binding of a variable in a closure within a loop. The function itself is defined outside the loop. I don’t have a lot of flexibility in defining the function because I am passing it to derivative evaluation functions in the `ForwardDiff` package and those must be unary functions. Right now my approach is to declare the variable’s name as global before I define the function. I’m wondering if there is a more aesthetically pleasing approach.

So the background is that a nonlinear regression model function depends upon parameters and observed data. Generally the parameters are a relatively short vector and the data are columns in a table, like a `DataFrame`. One of the simple (but real) examples in Bates and Watts (Wiley, 1988), _Nonlinear Regression Analysis and Its Applications_ is data from an enzyme kinetics experiment

```julia
julia> data = DataFrame(conc = repeat([0.02,0.06,0.11,0.22,0.56,1.1], inner=2),
           rate = float([76,47,97,107,123,139,159,152,191,201,207,200]))
12×2 DataFrame
│ Row │ conc │ rate │
│ │ Float64 │ Float64 │
├─────┼─────────┼─────────┤
│ 1 │ 0.02 │ 76.0 │
│ 2 │ 0.02 │ 47.0 │
│ 3 │ 0.06 │ 97.0 │
│ 4 │ 0.06 │ 107.0 │
│ 5 │ 0.11 │ 123.0 │
│ 6 │ 0.11 │ 139.0 │
│ 7 │ 0.22 │ 159.0 │
│ 8 │ 0.22 │ 152.0 │
│ 9 │ 0.56 │ 191.0 │
│ 10 │ 0.56 │ 201.0 │
│ 11 │ 1.1 │ 207.0 │
│ 12 │ 1.1 │ 200.0 │

```

for which the Michaelis-Menten model for reaction rate as a function of substrate concentration would be appropriate.

```julia
julia> function MicMen(pars, data)
           Vm = pars[1]
           K = pars[2]
           conc = data.conc
           @. Vm * conc / (K + conc)
       end
MicMen (generic function with 1 method)

```

For these data, values `Vm = 200.` and `K = 0.05` give the predicted response

```julia
julia> show(MicMen([200., 0.05], data))
[57.14285714285714, 57.14285714285714, 109.0909090909091, 109.0909090909091, 137.5, 137.5, 162.96296296296296, 162.96296296296296, 183.60655737704917, 183.60655737704917, 191.30434782608697, 191.30434782608697]

```

I want to use methods from `ForwardDiff` to evaluate the Jacobian matrix but the function argument to `ForwardDiff.jacobian!` must be a function of `pars` only. I can use a closure and generate

```julia
f(x) = MicMen(x, data)

```

in a context where data is defined so that is fine.

However, I also want to do this for several related groups of responses, say saved in a `GroupedDataFrame`, where the value of `pars` will change between groups. This is for fitting nonlinear mixed-effects models, described in Pinheiro and Bates (Springer, 2000), _Mixed-Effects Models in S and S-PLUS_. As I loop over the groups I need the binding of `data` in the closure to change to each SubDataFrame. I am currently doing this by assigning a global variable `_sdf` before defining `f` and changing the global variable’s binding in the loop.

Are there better approaches? I feel that I should be able to do something with `let` but I can’t seem to work out how to do so.

---

<div class="post-metadata">

**Author:** ![kristoffer.carlsson](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/kristoffer.carlsson/32/22_2.png) [@kristoffer.carlsson](https://discourse.julialang.org/u/kristoffer.carlsson)\
**Post date:** [July 12, 2019, 7:03pm UTC](https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297/2 "2019-07-12T19:03:52Z")

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Put the variable in some sort of container (like a `Ref`) and update that instead?

---

<div class="post-metadata">

**Author:** ![foobar\_lv2](https://avatars.discourse-cdn.com/v4/letter/f/ee59a6/32.png) [@foobar\_lv2](https://discourse.julialang.org/u/foobar_lv2)\
**Post date:** [July 12, 2019, 7:05pm UTC](https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297/3 "2019-07-12T19:05:41Z")

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When in doubt, forget that closures exist and do it by hand. E.g.

```julia
julia> struct F{T}
       conc::T
       end

julia> function (f::F)(par)
       VM=pars[1]
       K=pars(2)
       conc = f.conc
       @. Vm * conc / (K + conc)
       end

```

Now, each instance of `F`, like e.g. `F(data.conc)`, is a valid input to forward diff. This is how closures are internally implemented.

Changing bindings in closures is supported, but almost always a bad idea. Instead, construct a new closure.

edit: Or do what @kristoffer.carlsson said, depending on what you actually want (I don’t know whether I understood what your issue is).

---

<div class="post-metadata">

**Author:** ![dmbates](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dmbates/32/44_2.png) [@dmbates](https://discourse.julialang.org/u/dmbates)\
**Post date:** [July 12, 2019, 8:51pm UTC](https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297/4 "2019-07-12T20:51:57Z")

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In case someone reading this later needs more detail, I created the `Ref` from the first `SubDataFrame`, then define the function for `ForwardDiff.jacobian!` dereferencing the `Ref`. If `data` is the `GroupedDataFrame` it looks like

```julia
Rdf = Ref(first(data))
f(pars) = MicMen(pars, Rdf[])

```

Then the loop looks like

```julia
for (res, sdf) in zip(results, data)
    Rdf[] = sdf
    ... manipulate the pars to get the appropriate value
    ForwardDiff.jacobian!(res, f, pars, cfg)
end

```

where `cfg` is a (common) `JacobianConfig` and `results` is a preallocated vector of the form

```julia
results = [DiffResults.JacobianResult(sdf.age, pars) for sdf in data]

```

---

<div class="post-metadata">

**Author:** ![Tamas\_Papp](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tamas_papp/32/25949_2.png) [@Tamas\_Papp](https://discourse.julialang.org/u/Tamas_Papp)\
**Post date:** [July 13, 2019, 6:06am UTC](https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297/5 "2019-07-13T06:06:10Z")

</div>

> [@dmbates](#):
>
> Are there better approaches? I feel that I should be able to do something with `let` but I can’t seem to work out how to do so.

Is it the same `pars` for all subdataframes? It is unclear what the role of `pars` is, can you provide an MWE for the complete calculation?

---

<div class="post-metadata">

**Author:** ![dmbates](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dmbates/32/44_2.png) [@dmbates](https://discourse.julialang.org/u/dmbates)\
**Post date:** [July 13, 2019, 6:22pm UTC](https://discourse.julialang.org/t/changing-the-binding-of-a-variable-in-a-closure/26297/6 "2019-07-13T18:22:32Z")

</div>

The values of `pars` change from group to group. The `groups` of observations correspond to experimental units of some sort in the experiment. In a pharmacokinetics study there will be several participants each of whom receives dosages of a drug following which several measurements of serum concentrations over time are made.

tl; dr As an example, in R, `datasets::Theoph` is data from such a study. A compartment model for concentration as a function of time depends upon three scalar parameters for each subject: `ka`, the absorption rate constant, `k`, the elimination rate constant and `V`, the effective volume of distribution, as

```julia
(dose/V) * (ka /(ka - k)) * (exp(-k*t) - exp(-ka*t))

```

Typically these parameters, all of which must be positive, are estimated on the logarithmic scale so the model function is

```julia
function sdOral1C(φ, d)
    k = exp(φ[1]) # elimination rate constant from lk
    ka = exp(φ[2]) # absorption rate constant from lka
    V = exp(φ[3]) # volume of distribution from lV
    t = d.Time
    @. (d.Dose/V) * (ka/(ka - k)) * (exp(-k*t) - exp(-ka*t))
end

```

If we pool all the data we get

```julia
m = fit!(NLregModel(sdOral1C, Theoph, (lk = -2.5, lka = 0.5, lV = -1.0), :Conc))
Nonlinear regression model (NLregModel)

 Residual sum of squares: 274.4491345682938

──────────────────────────────────
      Estimate Std.Error t value
──────────────────────────────────
lk -2.52424 0.110347 -22.88
lka 0.399228 0.117537 3.40
lV -0.724023 0.0485799 -14.90
──────────────────────────────────

```

If we fit each subject’s data separately we get

```julia
julia> fits = [fit!(NLregModel(sdOral1C, sdf, params(m), :Conc)) for sdf in groupby(Theoph, :Subject)]
12-element Array{NLregModel{3,Float64},1}:
 Nonlinear regression model (NLregModel)

 Residual sum of squares: 2.44424021741903

──────────────────────────────────
      Estimate Std.Error t value
──────────────────────────────────
lk -2.30733 0.196063 -11.77
lka 0.151628 0.213966 0.71
lV -0.665909 0.0980873 -6.79
──────────────────────────────────
   
 Nonlinear regression model (NLregModel)

 Residual sum of squares: 0.9965571862947799
...
 Nonlinear regression model (NLregModel)

 Residual sum of squares: 13.463469674674933

──────────────────────────────────
      Estimate Std.Error t value
──────────────────────────────────
lk -2.42548 0.277133 -8.75
lka 0.386278 0.296296 1.30
lV -0.707117 0.127841 -5.53
──────────────────────────────────

```

A mixed-effects model for these data is intermediate to the pooled model and the individual model. Each subject’s parameters are a sum of population values (called “fixed effects”) and subject-specific deviations (or “random effects”) which are assumed to be samples from a multivariate distribution (typically multivariate Gaussian). The purpose is to estimate the fixed effects and the variance-covariance parameters in the distribution of the random effects. This distribution regularizes the optimization of the individual’s parameter values. Instead of a nonlinear least squares fit, the optimization of each individual’s parameters becomes a penalized nonlinear least squares fit where the penalty depends on those variance-covariance parameters.

`NLregModel` is defined in [GitHub - dmbates/NLreg.jl: Nonlinear regression in Julia](https://github.com/dmbates/NLreg.jl) which is also where I am working on an implementation of a nonlinear mixed-effects model.