#1 changes to README for 1.0, and added elastic net example

Author: mauriciogtec

README.md

@@ -36,9 +36,10 @@ Let's verify the result
 
 
 ```julia
-x = -linspace(-10.0, 10.0, 100)
-envelop = eval_envelop.(sampler.envelop, x)
-target = f.(x)
+# Plot the results and compare to target distribution
+x = range(-10.0, 10.0, length=100)
+envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
+target = [f(xi) for xi in x]
 
 histogram(sim, normalize = true, label = "Histogram")
 plot!(x, [target envelop], width = 2, label = ["Normal(μ, σ)" "Envelop"])
@@ -56,12 +57,14 @@ plot!(x, [target envelop], width = 2, label = ["Normal(μ, σ)" "Envelop"])
 f(x) = β^α * x^(α-1) * exp(-β*x) / gamma(α)
 support = (0.0, Inf)
 
+# Build the sampler and simulate 10,000 samples
 sampler = RejectionSampler(f, support)
 @time sim = run_sampler!(sampler, 10000) 
 
-x = linspace(0.0, 5.0, 100)
-envelop = eval_envelop.(sampler.envelop, x)
-target = f.(x)
+# Plot the results and compare to target distribution
+x = range(0.0, 5.0, length=100)
+envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
+target = [f(xi) for xi in x]
 
 histogram(sim, normalize = true, label = "Histogram")
 plot!(x, [target envelop], width = 2, label = ["Gamma(α, β)" "Envelop"])
@@ -71,8 +74,6 @@ plot!(x, [target envelop], width = 2, label = ["Gamma(α, β)" "Envelop"])
 
 
 
-
-
 ![](img/example2.png)
 
 ## Truncated distributions and unknown normalisation constant
@@ -85,12 +86,14 @@ We don't to provide an exact density--it will sample up to proportionality--and
 f(x) = β^α * x^(α-1) * exp(-β*x) / gamma(α)
 support = (1.0, 3.5)
 
+# Build the sampler and simulate 10,000 samples
 sampler = RejectionSampler(f, support)
 @time sim = run_sampler!(sampler, 10000) 
 
-x = linspace(0.01, 8.0, 100)
-envelop = eval_envelop.(sampler.envelop, x)
-target = f.(x)
+# Plot the results and compare to target distribution
+x = range(0.01, 8.0, length=100)
+envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
+target = [f(xi) for xi in x]
 
 histogram(sim, normalize = true, label = "histogram")
 plot!(x, [target envelop], width = 2, label = ["target density" "envelop"])
@@ -99,7 +102,33 @@ plot!(x, [target envelop], width = 2, label = ["target density" "envelop"])
       0.007766 seconds (181.82 k allocations: 3.024 MiB)
 
 
+![](img/example3.png)
+
+## Elastic Net distribution
+
+The following example arises from elastic net regression and smoothing problems. In these cases, the integration constants are not available analytically.
+
+```julia
+# Define function to be sampled
+function f(x, μ, λ1, λ2)
+      δ = x - μ
+      nl = λ1 * abs(δ) + λ2 * δ^2
+      return exp(-nl)
+end
+support = (-Inf, Inf)
+
+# Build the sampler and simulate 10,000 samples
+μ, λ1, λ2 = 0.0, 2.0, 1.0
+sampler = RejectionSampler(x -> f(x, μ, λ1, λ2), support, max_segments = 5)
+@time sim = run_sampler!(sampler, 10000);
 
+# Plot the results and compare to target distribution
+x = range(-2.3, 2.3, length=100)
+envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
+target = [f(xi, μ, λ1, λ2) for xi in x]
 
+histogram(sim, normalize = true, label = "histogram")
+plot!(x, [target envelop], width = 2, label = ["target density" "envelop"])
+```
 
-![](img/example3.png)
+![](img/example4.png)