[Docs] Proof-reading (#154)

* Move from exemples to examples

* Anticonomotony -> anticomonotony

* Typo

* one more typo

* Fix first example

README.md

@@ -56,8 +56,7 @@ D = SklarDist(C,(X₁,X₂,X₃)) # The final distribution
 simu = rand(D,1000) # Generate a dataset
 
 # You may estimate a copula using the `fit` function:
-D̂ = fit(SklarDist{FrankCopula,Tuple{Gamma,Normal,LogNormal}}, simu)
-# Increase the number of observations to get a beter fit (or not?)  
+D̂ = fit(SklarDist{ClaytonCopula,Tuple{Gamma,Normal,LogNormal}}, simu)
 ```
 
 The list of availiable copula models is *very* large, check it out on our [documentation](https://lrnv.github.io/Copulas.jl/stable) ! 

docs/make.jl

@@ -42,9 +42,9 @@ makedocs(;
         ],
         "Dependence measures" => "dependence_measures.md",
         "Examples" => [
-            "exemples/fitting_sklar.md",
-            "exemples/turing.md",
-            "exemples/other_usecases.md"
+            "examples/fitting_sklar.md",
+            "examples/turing.md",
+            "examples/other_usecases.md"
         ],
 
         "Dev Roadmap" => "dev_roadmap.md",

docs/src/dependence_measures.md

@@ -33,7 +33,7 @@ Spearman's Rho can be obtained through `ρ(C::Copula)`. Its value only depends o
     There exists several multivariate extensions of Spearman's rho. The one implemented here is the one we just defined what ever the dimension $d$, be careful as the normalization might differ from other places in the literature.
 
 !!! note "Specific values of tau and rho"
-    Kendall's $\tau$ and Spearman's $\rho$ have values between -1 and 1, and are -1 in case of complete anticonomotony and 1 in case of comonotony. Moreover, they are 0 in case of independence. This is 
+    Kendall's $\tau$ and Spearman's $\rho$ have values between -1 and 1, and are -1 in case of complete anticomonotony and 1 in case of comonotony. Moreover, they are 0 in case of independence. This is 
     why we say that they measure the 'strength' of the dependency.
 
 !!! tip "More-that-bivariate cases"

docs/src/getting_started.md

@@ -110,12 +110,10 @@ X₃ = LogNormal(0,1)
 C = ClaytonCopula(3,0.7) # A 3-variate Clayton Copula with θ = 0.7
 D = SklarDist(C,(X₁,X₂,X₃)) # The final distribution
 
-# This generates a (3,1000)-sized dataset from the multivariate distribution D
-simu = rand(D,1000)
+simu = rand(D,1000) # Generate a dataset
 
-# While the following estimates the parameters of the model from a dataset: 
-D̂ = fit(SklarDist{FrankCopula,Tuple{Gamma,Normal,LogNormal}}, simu)
-# Increase the number of observations to get a beter fit (or not?)  
+# You may estimate a copula using the `fit` function:
+D̂ = fit(SklarDist{ClaytonCopula,Tuple{Gamma,Normal,LogNormal}}, simu)
 ```
 
 !!! info "About fitting methods"

src/ArchimedeanCopula.jl

@@ -8,13 +8,13 @@ Constructor:
 
     ArchimedeanCopula(d::Int,G::Generator)
 
-For some Archimedean [`Generator`](@ref) `G::Generator` and some dimenson `d`, this class models the archimedean copula wich has this generator. The constructor checks for validity by ensuring that `max_monotony(G) ≥ d`. The ``d``-variate archimedean copula with generator ``\\phi`` writes: 
+For some Archimedean [`Generator`](@ref) `G::Generator` and some dimenson `d`, this class models the archimedean copula which has this generator. The constructor checks for validity by ensuring that `max_monotony(G) ≥ d`. The ``d``-variate archimedean copula with generator ``\\phi`` writes: 
 
 ```math
 C(\\mathbf u) = \\phi^{-1}\\left(\\sum_{i=1}^d \\phi(u_i)\\right)
 ```
 
-The default sampling method is the Radial-simplex decomposition using the williamson transformation of ``\\phi``. 
+The default sampling method is the Radial-simplex decomposition using the Williamson transformation of ``\\phi``. 
 
 There exists several known parametric generators that are implement in the package. For every `NamedGenerator <: Generator` implemented in the package, we provide a type alias ``NamedCopula{d,...} = ArchimedeanCopula{d,NamedGenerator{...}}` to be able to manipulate the classic archimedean copulas without too much hassle for known and usefull special cases. 
 
@@ -35,7 +35,7 @@ pdf(C,spl)           # pdf
 loglikelihood(C,spl) # llh
 ```
 
-Bonus: If you know the williamson d-transform of your generator and not your generator itself, you may take a look at [`WilliamsonGenerator`](@ref) that implements them. If you rather know the frailty distribution, take a look at `WilliamsonFromFrailty`.
+Bonus: If you know the Williamson d-transform of your generator and not your generator itself, you may take a look at [`WilliamsonGenerator`](@ref) that implements them. If you rather know the frailty distribution, take a look at `WilliamsonFromFrailty`.
 
 References:
 * [williamson1955multiply](@cite) Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581
@@ -94,7 +94,7 @@ end
 # end
 
 function Distributions._rand!(rng::Distributions.AbstractRNG, C::CT, x::AbstractVector{T}) where {T<:Real, CT<:ArchimedeanCopula}
-    # By default, we use the williamson sampling. 
+    # By default, we use the Williamson sampling. 
     Random.randexp!(rng,x)
     r = rand(rng,williamson_dist(C))
     sx = sum(x)
@@ -104,7 +104,7 @@ function Distributions._rand!(rng::Distributions.AbstractRNG, C::CT, x::Abstract
     return x
 end
 function Distributions._rand!(rng::Distributions.AbstractRNG, C::CT, A::DenseMatrix{T}) where {T<:Real, CT<:ArchimedeanCopula}
-    # More efficient version that precomputes the williamson transform on each call to sample in batches: 
+    # More efficient version that precomputes the Williamson transform on each call to sample in batches: 
     Random.randexp!(rng,A)
     n = size(A,2)
     r = rand(rng,williamson_dist(C),n)

src/Generator/UnivariateGenerator/InvGaussianGenerator.jl

@@ -12,7 +12,7 @@ Constructor
 The Inverse Gaussian copula in dimension ``d`` is parameterized by ``\\theta \\in [0,\\infty)``. It is an Archimedean copula with generator :
 
 ```math
-\\phi(t) = \\exp{\\frac{(1-\\sqrt{1+2θ^{2}t}}{θ}}.
+\\phi(t) = \\exp{\\frac{1-\\sqrt{1+2θ^{2}t}}{θ}}.
 ```
 
 More details about Inverse Gaussian Archimedean copula are found in :

src/Generator/WilliamsonGenerator.jl

@@ -3,15 +3,15 @@
     i𝒲{TX}
 
 Fields:
-* `X::TX` -- a random variable that represents its williamson d-transform
+* `X::TX` -- a random variable that represents its Williamson d-transform
 * `d::Int` -- the dimension of the transformation. 
 
 Constructor
 
     WilliamsonGenerator(X::Distributions.UnivariateDistribution, d)
     i𝒲(X::Distributions.UnivariateDistribution,d)
 
-The `WilliamsonGenerator` (alias `i𝒲`) allows to construct a d-monotonous archimedean generator from a positive random variable `X::Distributions.UnivariateDistribution`. The transformation, wich is called the inverse williamson transformation, is implemented in [WilliamsonTransforms.jl](https://www.github.com/lrnv/WilliamsonTransforms.jl). 
+The `WilliamsonGenerator` (alias `i𝒲`) allows to construct a d-monotonous archimedean generator from a positive random variable `X::Distributions.UnivariateDistribution`. The transformation, which is called the inverse Williamson transformation, is implemented in [WilliamsonTransforms.jl](https://www.github.com/lrnv/WilliamsonTransforms.jl). 
 
 For a univariate non-negative random variable ``X``, with cumulative distribution function ``F`` and an integer ``d\\ge 2``, the Williamson-d-transform of ``X`` is the real function supported on ``[0,\\infty[`` given by:
 

src/MiscellaneousCopulas/EmpiricalCopula.jl

@@ -8,7 +8,7 @@ Constructor
 
     EmpiricalCopula(u;pseudos=true)
 
-The EmpiricalCopula in dimension ``d`` is parameterized by a pseudo-data matrix wich should have shape (d,N). Its expression is given as :  
+The EmpiricalCopula in dimension ``d`` is parameterized by a pseudo-data matrix which should have shape (d,N). Its expression is given as :  
 
 ```math
 C(\\mathbf x) = \\frac{1}{N}\\sum_{i=1}^n \\mathbf 1_{\\mathbf u_i \\le \\mathbf x}

src/SklarDist.jl

@@ -27,12 +27,10 @@ X₃ = LogNormal(0,1)
 C = ClaytonCopula(3,0.7) # A 3-variate Clayton Copula with θ = 0.7
 D = SklarDist(C,(X₁,X₂,X₃)) # The final distribution
 
-# This generates a (3,1000)-sized dataset from the multivariate distribution D
-simu = rand(D,1000)
+simu = rand(D,1000) # Generate a dataset
 
-# While the following estimates the parameters of the model from a dataset: 
-D̂ = fit(SklarDist{FrankCopula,Tuple{Gamma,Normal,LogNormal}}, simu)
-# Increase the number of observations to get a beter fit (or not?)  
+# You may estimate a copula using the `fit` function:
+D̂ = fit(SklarDist{ClaytonCopula,Tuple{Gamma,Normal,LogNormal}}, simu)
 ```
 
 References: