diff --git a/docs/make.jl b/docs/make.jl index ad09a2ff..5e0fe5e2 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -41,6 +41,7 @@ makedocs(; "Transformed Copulas" => "transformations.md", ], "Examples" => [ + "examples/lossalae.md", "examples/fitting_sklar.md", "examples/turing.md", "examples/other_usecases.md" diff --git a/docs/src/examples/lossalae.md b/docs/src/examples/lossalae.md new file mode 100644 index 00000000..80cd5c0e --- /dev/null +++ b/docs/src/examples/lossalae.md @@ -0,0 +1,87 @@ +# Loss-Alae fitting example. + +Loss-Alae is a dataset that is provided in the R `copula` package, which documentation can be found [there](https://search.r-project.org/CRAN/refmans/copula/html/loss.html). +This dataset corresponds to claims received by an insurer, where the two variables `loss` and `alae` respectively correspond to the amount of the loss and to associated expenses. +There is a certain dependence structure between the two, and the actual distribution generating this data is of course unknown. +Our task here is to provide a parametric model that approximates this distribution. + +Let us first import the data : +```@example 6 +using Copulas, Distributions, Plots +data = [ + 10.0 24.0 45.0 51.0 60.0 74.0 75.0 78.0 87.0 100.0 115.0 123.0 133.0 140.0 147.0 147.0 165.0 192.0 200.0 300.0 308.0 311.0 326.0 350.0 350.0 400.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 500.0 520.0 578.0 600.0 600.0 634.0 667.0 700.0 750.0 750.0 750.0 750.0 750.0 750.0 750.0 798.0 800.0 833.0 833.0 875.0 900.0 909.0 916.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1075.0 1091.0 1100.0 1122.0 1125.0 1200.0 1200.0 1200.0 1250.0 1250.0 1250.0 1250.0 1250.0 1250.0 1250.0 1250.0 1298.0 1300.0 1300.0 1310.0 1333.0 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Let's move to log scales: +```@example 6 +scatter(log.(loss),log.(alae)) +``` + +This is a much better looking scatter plot. +To fit a full distribution on this dataset, we have to understand what families of marginals and of dependence structure to use. +Let us first look at the marginal behaviors: + +```@example 6 +plot(histogram(log.(loss)),histogram(log.(alae))) +``` + +This histogram look fairly Gaussian, and thus a good first guess for the distributions of the marginals would be `LogNormal` distributions. +Now the dependence structure: + +```@example 6 +ranks = Copulas.pseudos(data) +scatter(ranks[1,:],ranks[2,:]) +``` + +The vertical strides are there because of rounding in the input data. +But still, we observe that there is dependence structure in both tails, and that the dependence structure seems fairly symmetric. +Let us try to fit a few different copulas: + +```@example 6 +fit_gaussian = fit(SklarDist{GaussianCopula,Tuple{LogNormal,LogNormal}}, data) +fit_clayton = fit(SklarDist{ClaytonCopula,Tuple{LogNormal,LogNormal}}, data) +fit_gumbel = fit(SklarDist{GumbelCopula,Tuple{LogNormal,LogNormal}}, data) +fit_frank = fit(SklarDist{FrankCopula,Tuple{LogNormal,LogNormal}}, data) +nothing # hide +``` + +Let's check the negative loglikelihood on each of those models (note that they all have the same number of parameters): +```@example 6 +nllhs = [-loglikelihood(fit,data) for fit in (fit_gaussian,fit_clayton,fit_gumbel,fit_frank)] +``` + +So the Clayton looks a bit better. Let's look at the parametrization: +```@example 6 +fit_clayton +``` + +For the marginals, we can for example check quantile quantile plots (again, on log-scale) + +```@example 6 +n = size(data,2) +plot( + scatter(sort(log.(loss)), log.(quantile.(Ref(fit_clayton.m[1]),(1:n)./(n+1))), label="Loss"), + scatter(sort(log.(alae)), log.(quantile.(Ref(fit_clayton.m[2]),(1:n)./(n+1))), label="Alae") +) +``` + +These quantile-quantile plots are not perfect, we see that both tails are a bit wiggly. +For the dependence structure, we can sample a new dataset from the fitted copula to check if the ranks behaviors looks like what we had before: +```@example 6 +u = rand(fit_clayton.C, 1500) +scatter(u[1,:],u[2,:]) +``` + +There are potential improvements that can be made to this fit: + +- The tail dependency does not look like it is on the right side. To solve that, we could use `SurvivalCopula` to fit a flipped version of the Clayton. +- We could use other marginal proposals than `LogNormal`s and validate (e.g., through likelihood ratio tests) that the fits are OK. +- We could keep marginals and/or the dependence structure empirical, through e.g., `EmpiricalCopula`. \ No newline at end of file