From c10ed326cde70b5196fea16296e620fabd15a4db Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Tue, 30 Nov 2021 16:46:13 +0100
Subject: [PATCH 01/29] Draft of Leanings module

---
 src/CausalityTools.jl     |  1 +
 src/Leanings/Leanings.jl  |  5 +++
 src/Leanings/penchants.jl | 92 +++++++++++++++++++++++++++++++++++++++
 3 files changed, 98 insertions(+)
 create mode 100644 src/Leanings/Leanings.jl
 create mode 100644 src/Leanings/penchants.jl

diff --git a/src/CausalityTools.jl b/src/CausalityTools.jl
index 9a5c11d4d..9dbbb8f36 100644
--- a/src/CausalityTools.jl
+++ b/src/CausalityTools.jl
@@ -15,6 +15,7 @@ module CausalityTools
     include("CrossMappings/CrossMappings.jl")
     include("SMeasure/smeasure.jl")
     include("PredictiveAsymmetry/PredictiveAsymmetry.jl")
+    include("Leanings/Leanings.jl")
     include("example_systems/ExampleSystems.jl")
 
     using Requires 
diff --git a/src/Leanings/Leanings.jl b/src/Leanings/Leanings.jl
new file mode 100644
index 000000000..20858c23c
--- /dev/null
+++ b/src/Leanings/Leanings.jl
@@ -0,0 +1,5 @@
+using Reexport
+
+@reexport module Leanings
+    include("penchants.jl")
+end
\ No newline at end of file
diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
new file mode 100644
index 000000000..684ed095c
--- /dev/null
+++ b/src/Leanings/penchants.jl
@@ -0,0 +1,92 @@
+
+x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
+y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
+export lean
+
+"""
+    lean_ex(C, E, l)
+
+Computes the leaning of the `l`-assignment ``\\{C, E\\} = \\{X, Y\\}, \\{X_{t-l}, Y_t\\}``.
+"""
+function lean(X, Y, l = 1)
+    @assert length(X) == length(Y)
+    L = length(X)
+    
+    # Leave reasonably many for probability computations. `l` can't be too large.
+    @assert l < L ÷ 2
+
+    #P(y_t = 1 | x_{t-1} = 1) = n_ec/n_c
+
+    # Number of time the state y_t = 1 AND x_{t-1} = 1 appears in {X, Y}
+    n_ecs = Vector{Int}(undef, 0)
+    n_cs = Vector{Int}(undef, 0)
+    n_es = Vector{Int}(undef, 0)
+
+    states = unique([X; Y])
+    
+    # Define an iterator over all penchants
+    penchants = Iterators.product(states, states)
+    @show states
+    @show penchants |> collect
+    
+
+      
+
+    for penchant in penchants
+        @show penchant
+        sx, sy = penchant[1], penchant[2]
+            
+        # Number of times the assumed cause has appeared
+        n_c = 0
+
+        # Number of times the assumed effect has appeared
+        n_e = 0
+
+        # Number of time the state y_t = a AND x_{t-1} = b appears in {X, Y},
+        # where (a, b) is one of the length(states)^2 possible states.
+        n_ec = 0
+        for t = l+1:L
+            effect = Y[t]
+            cause = X[t-l]
+
+            if effect == sx && cause == sy
+                n_ec += 1
+            end
+            
+
+            if effect == sx
+                n_e += 1
+            end
+
+            if cause == sy
+                n_c += 1
+            end
+        end
+        push!(n_ecs, n_ec)
+        push!(n_cs, n_c)
+        push!(n_es, n_e)
+    end
+
+    @show Ps_cond = n_ecs ./ n_cs
+    @show Ps_e = n_es ./ (L - l)
+    @show Ps_c = n_cs ./ (L - l)
+    
+    # for t = l+1:L
+    #     yt = Y[t]
+    #     xtl = X[t-l]
+    #     @show Y[t], X[t-l]
+    #     if yt == 1 && xtl == 1
+    #         n_ec += 1
+    #     end
+
+    #     if X[t-l] == 1
+    #         n_c += 1
+    #     end
+    # end
+
+    # @show n_ec
+    # @show n_c
+
+    # p = n_ec/n_c
+end
+

From 1711df73939495b628f5434bf986fd6f8042d9c6 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 05:48:01 +0100
Subject: [PATCH 02/29] Leanings, with tests and docs

---
 docs/make.jl                  |   3 +
 docs/src/leanings.md          |   9 ++
 src/Leanings/penchants.jl     | 166 +++++++++++++++++++++++-----------
 test/methods/test_leanings.jl |   9 ++
 test/runtests.jl              |   1 +
 5 files changed, 135 insertions(+), 53 deletions(-)
 create mode 100644 docs/src/leanings.md
 create mode 100644 test/methods/test_leanings.jl

diff --git a/docs/make.jl b/docs/make.jl
index 1c618fa45..a28af6007 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -48,6 +48,9 @@ PAGES = [
         "generalized_entropy.md",
         "info_estimators.md",
     ],
+    "Exploratory" => [
+        "leanings.md"
+    ],
 
     "example_systems.md",
     "Utilities" => [
diff --git a/docs/src/leanings.md b/docs/src/leanings.md
new file mode 100644
index 000000000..99587e144
--- /dev/null
+++ b/docs/src/leanings.md
@@ -0,0 +1,9 @@
+# Leanings
+
+Leanings are probability-based exploratory causal inference measures that rely on the estimation of probabilities directly from the data. Here, we provide 
+implementations of the *penchants and leanings* algoriths from McCracken and Weigel (2016).
+
+```@docs
+mean_observed_penchant
+mean_leaning
+```
\ No newline at end of file
diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 684ed095c..0cd6db50b 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -1,39 +1,41 @@
 
 x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
 y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
-export lean
+export mean_observed_penchant, mean_leaning
 
-"""
-    lean_ex(C, E, l)
+using Statistics
 
-Computes the leaning of the `l`-assignment ``\\{C, E\\} = \\{X, Y\\}, \\{X_{t-l}, Y_t\\}``.
-"""
-function lean(X, Y, l = 1)
-    @assert length(X) == length(Y)
-    L = length(X)
-    
-    # Leave reasonably many for probability computations. `l` can't be too large.
-    @assert l < L ÷ 2
+# We just assume that the time series have been pre-binned or symbolized
+# However, `get_states`` might turn out much more complicated if we decide 
+# to implement more complicated cause-effect assignments
+get_states(x, y) = unique([x; y])
 
-    #P(y_t = 1 | x_{t-1} = 1) = n_ec/n_c
+# Getting the penchants might also become more complex for other 
+# complicated cause-effect assignments, so separate it out in a function.
+function get_penchants(x, y)
+    states = get_states(x, y)
 
-    # Number of time the state y_t = 1 AND x_{t-1} = 1 appears in {X, Y}
-    n_ecs = Vector{Int}(undef, 0)
-    n_cs = Vector{Int}(undef, 0)
-    n_es = Vector{Int}(undef, 0)
+    # All possible 2-permutations (with repetitions) of the unique
+    # states in `x ∪ y`.
+    return Iterators.product(states, states)
+end
 
-    states = unique([X; Y])
-    
-    # Define an iterator over all penchants
-    penchants = Iterators.product(states, states)
-    @show states
-    @show penchants |> collect
+function penchant_counts(x, y, l = 1)
+    @assert length(x) == length(y)
+    n = length(x)
     
+    # Leave reasonably many for probability computations. `l` can't be too large.
+    @assert l < n ÷ 2
+
+    # Number of time the state y_t = 1 AND x_{t-1} = 1 appears in {x, y}
+    ns_ec = Vector{Int}(undef, 0)
+    ns_c = Vector{Int}(undef, 0)
+    ns_e = Vector{Int}(undef, 0)
 
-      
+    # For each possible penchant, count occurrences.
+    penchants = get_penchants(x, y)
 
-    for penchant in penchants
-        @show penchant
+    @inbounds for penchant in penchants
         sx, sy = penchant[1], penchant[2]
             
         # Number of times the assumed cause has appeared
@@ -42,12 +44,14 @@ function lean(X, Y, l = 1)
         # Number of times the assumed effect has appeared
         n_e = 0
 
-        # Number of time the state y_t = a AND x_{t-1} = b appears in {X, Y},
-        # where (a, b) is one of the length(states)^2 possible states.
+        # Number of times the cause AND effect has appeared simulaneously
         n_ec = 0
-        for t = l+1:L
-            effect = Y[t]
-            cause = X[t-l]
+
+        for t = l+1:n
+            # "we required that the cause must precede the effect". This is the 
+            # standard l-assignment
+            effect = y[t]
+            cause = x[t-l]
 
             if effect == sx && cause == sy
                 n_ec += 1
@@ -62,31 +66,87 @@ function lean(X, Y, l = 1)
                 n_c += 1
             end
         end
-        push!(n_ecs, n_ec)
-        push!(n_cs, n_c)
-        push!(n_es, n_e)
+        push!(ns_ec, n_ec)
+        push!(ns_c, n_c)
+        push!(ns_e, n_e)
+    end
+
+    return ns_c, ns_e, ns_ec
+end
+
+
+function mean_observed_ρ(ρs, κs, ns_ec, L, weighted = false)
+    ρ = 0.0
+    ct = 0
+    @inbounds for i = 1:length(κs)
+        if κs[i] != 0.0
+            if weighted
+                ρ += ρs[i] * ns_ec[i]
+            else
+                ρ += ρs[i]
+            end
+            ct += 1
+        end
     end
 
-    @show Ps_cond = n_ecs ./ n_cs
-    @show Ps_e = n_es ./ (L - l)
-    @show Ps_c = n_cs ./ (L - l)
+    if weighted
+        ρ / L 
+    else 
+        return ρ / ct
+    end
+end
+
+"""
+    mean_observed_penchant(x, y, l, weighted = false) → ρ̄
+
+Computes the *mean observed penchant* of the `l`-assignment ``\\{C, E\\} = \\{x, y\\}, \\{x_{t-l}, y_t\\}``.
+
+If `weighted == true`, then compute the *weighted mean observed penchant*.
+
+[^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
+"""
+function mean_observed_penchant(x, y, l = 1; weighted = false)
+    n = length(x)
+    ns_c, ns_e, ns_ec = penchant_counts(x, y, l)
+
+    L = n - l # "library length", resulting from shifting the time series l steps
+    Ps_e = ns_e ./ L
+    Ps_c = ns_c ./ L
+    κs = ns_ec ./ ns_c
+    ρs = κs.* (1 .+ Ps_c ./ (1 .- Ps_c)) .- (Ps_e) ./ (1 .- Ps_c)
     
-    # for t = l+1:L
-    #     yt = Y[t]
-    #     xtl = X[t-l]
-    #     @show Y[t], X[t-l]
-    #     if yt == 1 && xtl == 1
-    #         n_ec += 1
-    #     end
-
-    #     if X[t-l] == 1
-    #         n_c += 1
-    #     end
-    # end
-
-    # @show n_ec
-    # @show n_c
-
-    # p = n_ec/n_c
+    # Mean observed penchant. We can determine which penchants are observed 
+    # by looking for zeros in `κs`. If `κs[i] == 0`, then the `i`-th penchant
+    # is unobserved. Why? When the penchant is unobserved, there are no 
+    # instances of the cause and effect occurring together, so 
+    # `P(effect | cause) = P(effect, cause) / P(cause) = 0 / P(cause) = 0` 
+    # As explained in section D in the paper, including unobserved penchants
+    # would be meaningless for causal inference.
+    mean_observed_ρ(ρs, κs, ns_ec, L, weighted)
 end
 
+"""
+
+    mean_leaning(x, y, l = 1, weighted = true) → ρ̄
+
+Compute the *mean observed leaning* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`,
+based on the standard `l`-assignment of the `l`-assignment ``\\{C, E\\} = \\{x, y\\}, \\{x_{t-l}, y_t\\}``.
+
+If `ρ̄ > 0`, then the probability that `x` drives `y` is higher than the probability 
+that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` is higher than the probability 
+that `x` drives `y`.
+
+If `weighted == true`, then each penchant is weighted by the number of times it appears in 
+the data, which yields the *weighted mean observed leaning*. This the default behavior, since 
+"...[the weighted mean observed leaning] accounts for the frequency of observed cause-effect pairs 
+within the data, which is assumed to be a predictor of causal influence" (McCracken & Weigel, 2016).
+
+This implementation *does not* take care of the tolerance domains discussed in the paper. 
+Pre-process your time series, using appropriate binning or symbolization schemes, 
+so that `x` and `y` both consists of discrete symbols. 
+
+[^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
+"""
+function mean_leaning(x, y, l = 1, weighted = true)
+    lean(x, y, l, weighted = weighted) - lean(y, x, l, weighted = weighted)
+end
\ No newline at end of file
diff --git a/test/methods/test_leanings.jl b/test/methods/test_leanings.jl
new file mode 100644
index 000000000..a2aa56c9d
--- /dev/null
+++ b/test/methods/test_leanings.jl
@@ -0,0 +1,9 @@
+
+x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
+y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
+
+@test mean_observed_penchant(x, y, 1, weighted = false) ≈ 1.0
+@test mean_observed_penchant(y, x, 1, weighted = false) ≈ 1/7
+
+@test mean_observed_penchant(x, y, 1, weighted = true) ≈ 1.0
+@test mean_observed_penchant(y, x, 1, weighted = true) ≈ 3/63
\ No newline at end of file
diff --git a/test/runtests.jl b/test/runtests.jl
index 58099aebe..1c3c61e7a 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -8,6 +8,7 @@ include("methods/test_smeasure.jl")
 include("methods/test_joint_distance_distribution.jl")
 include("methods/test_predictive_asymmetry.jl")
 include("methods/test_crossmapping.jl")
+include("methods/test_leanings.jl")
 
 include("systems/discrete/test_discrete_systems.jl")
 include("systems/continuous/test_continuous_systems.jl")

From 677a6760b6629a20ff452fd0e348f88de3991228 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 05:49:25 +0100
Subject: [PATCH 03/29] Correct function name

---
 src/Leanings/penchants.jl | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 0cd6db50b..3a5186256 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -148,5 +148,6 @@ so that `x` and `y` both consists of discrete symbols.
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
 function mean_leaning(x, y, l = 1, weighted = true)
-    lean(x, y, l, weighted = weighted) - lean(y, x, l, weighted = weighted)
+    return mean_observed_penchant(x, y, l, weighted = weighted) - 
+        mean_observed_penchant(y, x, l, weighted = weighted)
 end
\ No newline at end of file

From 208e4a5edc74403b1aa98030a1505b1d173d1ab2 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 05:51:08 +0100
Subject: [PATCH 04/29] Use testset

---
 test/methods/test_leanings.jl | 10 ++++++----
 1 file changed, 6 insertions(+), 4 deletions(-)

diff --git a/test/methods/test_leanings.jl b/test/methods/test_leanings.jl
index a2aa56c9d..1f5fea0e1 100644
--- a/test/methods/test_leanings.jl
+++ b/test/methods/test_leanings.jl
@@ -2,8 +2,10 @@
 x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
 y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
 
-@test mean_observed_penchant(x, y, 1, weighted = false) ≈ 1.0
-@test mean_observed_penchant(y, x, 1, weighted = false) ≈ 1/7
+@testset "Leanings" begin
+    @test mean_observed_penchant(x, y, 1, weighted = false) ≈ 1.0
+    @test mean_observed_penchant(y, x, 1, weighted = false) ≈ 1/7
 
-@test mean_observed_penchant(x, y, 1, weighted = true) ≈ 1.0
-@test mean_observed_penchant(y, x, 1, weighted = true) ≈ 3/63
\ No newline at end of file
+    @test mean_observed_penchant(x, y, 1, weighted = true) ≈ 1.0
+    @test mean_observed_penchant(y, x, 1, weighted = true) ≈ 3/63
+end
\ No newline at end of file

From a1c620bdf80e7cd3ed79e5a9028c3e732ce5ff4b Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:14:05 +0100
Subject: [PATCH 05/29] Remove example arrays

---
 src/Leanings/penchants.jl | 2 --
 1 file changed, 2 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 3a5186256..34f457cb4 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -1,6 +1,4 @@
 
-x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
-y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
 export mean_observed_penchant, mean_leaning
 
 using Statistics

From 1b0cd8de27d9e986a294c3f13b12d864b1a3290d Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:14:13 +0100
Subject: [PATCH 06/29] Add return statement

---
 src/Leanings/penchants.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 34f457cb4..89e56940f 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -88,7 +88,7 @@ function mean_observed_ρ(ρs, κs, ns_ec, L, weighted = false)
     end
 
     if weighted
-        ρ / L 
+        return ρ / L 
     else 
         return ρ / ct
     end

From 0310e53ba9dfe861a7787c89aad1ea7f84120ef4 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:14:31 +0100
Subject: [PATCH 07/29] Add data requirement to docstring

---
 src/Leanings/penchants.jl | 22 +++++++++++++++-------
 1 file changed, 15 insertions(+), 7 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 89e56940f..48a5ffae0 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -130,18 +130,26 @@ end
 Compute the *mean observed leaning* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`,
 based on the standard `l`-assignment of the `l`-assignment ``\\{C, E\\} = \\{x, y\\}, \\{x_{t-l}, y_t\\}``.
 
-If `ρ̄ > 0`, then the probability that `x` drives `y` is higher than the probability 
-that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` is higher than the probability 
-that `x` drives `y`.
-
 If `weighted == true`, then each penchant is weighted by the number of times it appears in 
 the data, which yields the *weighted mean observed leaning*. This the default behavior, since 
 "...[the weighted mean observed leaning] accounts for the frequency of observed cause-effect pairs 
 within the data, which is assumed to be a predictor of causal influence" (McCracken & Weigel, 2016).
 
-This implementation *does not* take care of the tolerance domains discussed in the paper. 
-Pre-process your time series, using appropriate binning or symbolization schemes, 
-so that `x` and `y` both consists of discrete symbols. 
+## Intepretation
+
+If `ρ̄ > 0`, then the probability that `x` drives `y` is higher than the probability 
+that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` is higher than the probability 
+that `x` drives `y`.
+
+## Data requirements
+
+Both `x` and `y` *must* be discrete, and the number discrete states can't be too high 
+relative to the number of points in `x` and `y`. Otherwise, probabilities cannot be reliably estimated. 
+If the function returns `NaN`, then you probably either haven't discretized your time series, 
+or the partition is too fine-grained given the numbers of points available.
+
+Hence, this implementation *does not* take care of the tolerance domains discussed in the paper. 
+Pre-process your time series using appropriate binning or symbolization schemes.
 
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """

From 92b561e24aa492c888328195a5177ca92b465840 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:30:21 +0100
Subject: [PATCH 08/29] `weighted` is a keyword

---
 src/Leanings/penchants.jl | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 48a5ffae0..b61406291 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -153,7 +153,8 @@ Pre-process your time series using appropriate binning or symbolization schemes.
 
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
-function mean_leaning(x, y, l = 1, weighted = true)
+function mean_leaning(x, y, l = 1; weighted = true)
     return mean_observed_penchant(x, y, l, weighted = weighted) - 
         mean_observed_penchant(y, x, l, weighted = weighted)
-end
\ No newline at end of file
+end
+

From 944816a2d14e1f4343779eadc286711698533dee Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:40:33 +0100
Subject: [PATCH 09/29] Update test

---
 test/methods/test_leanings.jl | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/test/methods/test_leanings.jl b/test/methods/test_leanings.jl
index 1f5fea0e1..659f754f3 100644
--- a/test/methods/test_leanings.jl
+++ b/test/methods/test_leanings.jl
@@ -1,11 +1,11 @@
 
-x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
-y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
-
 @testset "Leanings" begin
-    @test mean_observed_penchant(x, y, 1, weighted = false) ≈ 1.0
-    @test mean_observed_penchant(y, x, 1, weighted = false) ≈ 1/7
 
-    @test mean_observed_penchant(x, y, 1, weighted = true) ≈ 1.0
-    @test mean_observed_penchant(y, x, 1, weighted = true) ≈ 3/63
+    x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
+    y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
+    @test penchant(x, y, 1, weighted = false) ≈ 1.0
+    @test penchant(y, x, 1, weighted = false) ≈ 1/7
+
+    @test penchant(x, y, 1, weighted = true) ≈ 1.0
+    @test penchant(y, x, 1, weighted = true) ≈ 3/63
 end
\ No newline at end of file

From ab7447ca3d2a541f83762f6492991aca529ea57e Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:40:51 +0100
Subject: [PATCH 10/29] Simplify and shorten function names

---
 src/Leanings/penchants.jl | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index b61406291..e4e16a4b7 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -1,5 +1,5 @@
 
-export mean_observed_penchant, mean_leaning
+export penchant, lean
 
 using Statistics
 
@@ -95,7 +95,7 @@ function mean_observed_ρ(ρs, κs, ns_ec, L, weighted = false)
 end
 
 """
-    mean_observed_penchant(x, y, l, weighted = false) → ρ̄
+    penchant(x, y, l; weighted = false) → ρ̄
 
 Computes the *mean observed penchant* of the `l`-assignment ``\\{C, E\\} = \\{x, y\\}, \\{x_{t-l}, y_t\\}``.
 
@@ -103,7 +103,7 @@ If `weighted == true`, then compute the *weighted mean observed penchant*.
 
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
-function mean_observed_penchant(x, y, l = 1; weighted = false)
+function penchant(x, y, l = 1; weighted = false)
     n = length(x)
     ns_c, ns_e, ns_ec = penchant_counts(x, y, l)
 
@@ -125,7 +125,7 @@ end
 
 """
 
-    mean_leaning(x, y, l = 1, weighted = true) → ρ̄
+    lean(x, y, l = 1; weighted = true) → ρ̄
 
 Compute the *mean observed leaning* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`,
 based on the standard `l`-assignment of the `l`-assignment ``\\{C, E\\} = \\{x, y\\}, \\{x_{t-l}, y_t\\}``.
@@ -153,8 +153,8 @@ Pre-process your time series using appropriate binning or symbolization schemes.
 
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
-function mean_leaning(x, y, l = 1; weighted = true)
-    return mean_observed_penchant(x, y, l, weighted = weighted) - 
-        mean_observed_penchant(y, x, l, weighted = weighted)
+function lean(x, y, l = 1; weighted = true)
+    return penchant(x, y, l, weighted = weighted) - 
+        penchant(y, x, l, weighted = weighted)
 end
 

From 1b8d401155a78debb86e4ffdea5049d1ca7cb92c Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:53:55 +0100
Subject: [PATCH 11/29] Docstring fix

---
 src/Leanings/penchants.jl | 7 +++++--
 1 file changed, 5 insertions(+), 2 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index e4e16a4b7..26d2f6b0f 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -97,10 +97,13 @@ end
 """
     penchant(x, y, l; weighted = false) → ρ̄
 
-Computes the *mean observed penchant* of the `l`-assignment ``\\{C, E\\} = \\{x, y\\}, \\{x_{t-l}, y_t\\}``.
+Computes the *mean observed penchant* (McCracken & Weigel, 2016)[^McCrackenWeigel2016] 
+of the `l`-assignment ``\\{\\text{cause}, \\text{effect}\\} =\\{x_{t-l}, y_t\\}``.
 
 If `weighted == true`, then compute the *weighted mean observed penchant*.
 
+See also data requirements discussed in [`lean`](@ref).
+
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
 function penchant(x, y, l = 1; weighted = false)
@@ -128,7 +131,7 @@ end
     lean(x, y, l = 1; weighted = true) → ρ̄
 
 Compute the *mean observed leaning* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`,
-based on the standard `l`-assignment of the `l`-assignment ``\\{C, E\\} = \\{x, y\\}, \\{x_{t-l}, y_t\\}``.
+based on the standard `l`-assignment of the `l`-assignment ``\\{\\text{cause}, \\text{effect}\\} =\\{x_{t-l}, y_t\\}``.
 
 If `weighted == true`, then each penchant is weighted by the number of times it appears in 
 the data, which yields the *weighted mean observed leaning*. This the default behavior, since 

From 28920fc746f48993f423d0917293a4f36fe381ca Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:55:46 +0100
Subject: [PATCH 12/29] Change function names in docs

---
 docs/src/leanings.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/docs/src/leanings.md b/docs/src/leanings.md
index 99587e144..543b12144 100644
--- a/docs/src/leanings.md
+++ b/docs/src/leanings.md
@@ -4,6 +4,6 @@ Leanings are probability-based exploratory causal inference measures that rely o
 implementations of the *penchants and leanings* algoriths from McCracken and Weigel (2016).
 
 ```@docs
-mean_observed_penchant
-mean_leaning
+penchant
+lean
 ```
\ No newline at end of file

From 7c5d6cc9a32193cde3a00fb0ca6c0aa38f9be7d9 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 06:56:09 +0100
Subject: [PATCH 13/29] Update version

---
 Project.toml | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/Project.toml b/Project.toml
index 0f3f19575..7464bda9c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -41,10 +41,9 @@ Neighborhood = "0.2.3"
 Reexport = "0.2, 1"
 Requires = "^1"
 SimpleDiffEq = "^1"
-StaticArrays = "^1"
 StatsBase = "^0.33"
 TimeseriesSurrogates = "^1.2"
-TransferEntropy = "^1.3.2"
+TransferEntropy = "^1.3.3"
 Wavelets = "0.9"
 julia = "^1.6"
 

From c790973cba7ac5c6b8d80b6b1f91c4255026c222 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:09:44 +0100
Subject: [PATCH 14/29] Exclude failing tests for now (they are not relevant
 for this PR)

T
---
 test/runtests.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/runtests.jl b/test/runtests.jl
index 1c3c61e7a..ad534b9d5 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -6,7 +6,7 @@ import DynamicalSystemsBase: DiscreteDynamicalSystem, ContinuousDynamicalSystem
 
 include("methods/test_smeasure.jl")
 include("methods/test_joint_distance_distribution.jl")
-include("methods/test_predictive_asymmetry.jl")
+#include("methods/test_predictive_asymmetry.jl")
 include("methods/test_crossmapping.jl")
 include("methods/test_leanings.jl")
 

From 78ea28f54825fccf16927cb572d8874619e925df Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:10:03 +0100
Subject: [PATCH 15/29] Rename variables for clarity

---
 src/Leanings/penchants.jl | 9 ++++-----
 1 file changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 26d2f6b0f..62530902a 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -34,7 +34,7 @@ function penchant_counts(x, y, l = 1)
     penchants = get_penchants(x, y)
 
     @inbounds for penchant in penchants
-        sx, sy = penchant[1], penchant[2]
+        state_cause, state_effect = penchant[1], penchant[2]
             
         # Number of times the assumed cause has appeared
         n_c = 0
@@ -51,16 +51,15 @@ function penchant_counts(x, y, l = 1)
             effect = y[t]
             cause = x[t-l]
 
-            if effect == sx && cause == sy
+            if effect == state_effect && cause == state_cause
                 n_ec += 1
             end
-            
 
-            if effect == sx
+            if effect == state_effect
                 n_e += 1
             end
 
-            if cause == sy
+            if cause == state_cause
                 n_c += 1
             end
         end

From 16c06d60b1993f6b7e830d5614037f9cabd08772 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:10:12 +0100
Subject: [PATCH 16/29] Documentation improvements

---
 src/Leanings/penchants.jl | 15 +++++++--------
 1 file changed, 7 insertions(+), 8 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 62530902a..d784dcaff 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -129,20 +129,19 @@ end
 
     lean(x, y, l = 1; weighted = true) → ρ̄
 
-Compute the *mean observed leaning* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`,
-based on the standard `l`-assignment of the `l`-assignment ``\\{\\text{cause}, \\text{effect}\\} =\\{x_{t-l}, y_t\\}``.
+Compute the *mean observed leaning* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`.
+
+If `ρ̄ > 0`, then the probability that `x` drives `y` is higher than the probability 
+that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` is higher than the probability 
+that `x` drives `y`.
+
+## Weighting
 
 If `weighted == true`, then each penchant is weighted by the number of times it appears in 
 the data, which yields the *weighted mean observed leaning*. This the default behavior, since 
 "...[the weighted mean observed leaning] accounts for the frequency of observed cause-effect pairs 
 within the data, which is assumed to be a predictor of causal influence" (McCracken & Weigel, 2016).
 
-## Intepretation
-
-If `ρ̄ > 0`, then the probability that `x` drives `y` is higher than the probability 
-that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` is higher than the probability 
-that `x` drives `y`.
-
 ## Data requirements
 
 Both `x` and `y` *must* be discrete, and the number discrete states can't be too high 

From 7b74daf37abe3724d0f1b311ceb91038d7916c9a Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:15:31 +0100
Subject: [PATCH 17/29] Specify possible ranges for penchants and leanings

---
 src/Leanings/penchants.jl | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index d784dcaff..cc91af65e 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -96,7 +96,7 @@ end
 """
     penchant(x, y, l; weighted = false) → ρ̄
 
-Computes the *mean observed penchant* (McCracken & Weigel, 2016)[^McCrackenWeigel2016] 
+Computes the *mean observed penchant* `ρ̄ ∈ [-1, 1]` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] 
 of the `l`-assignment ``\\{\\text{cause}, \\text{effect}\\} =\\{x_{t-l}, y_t\\}``.
 
 If `weighted == true`, then compute the *weighted mean observed penchant*.
@@ -129,7 +129,7 @@ end
 
     lean(x, y, l = 1; weighted = true) → ρ̄
 
-Compute the *mean observed leaning* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`.
+Compute the *mean observed leaning* `λ ∈ [-2, 2]` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`.
 
 If `ρ̄ > 0`, then the probability that `x` drives `y` is higher than the probability 
 that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` is higher than the probability 

From d916cd70ec5fed55bdb9123b1cde2f6b9f1be84e Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:15:39 +0100
Subject: [PATCH 18/29] Add test for lean

---
 test/methods/test_leanings.jl | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/test/methods/test_leanings.jl b/test/methods/test_leanings.jl
index 659f754f3..a3990f3ec 100644
--- a/test/methods/test_leanings.jl
+++ b/test/methods/test_leanings.jl
@@ -8,4 +8,6 @@
 
     @test penchant(x, y, 1, weighted = true) ≈ 1.0
     @test penchant(y, x, 1, weighted = true) ≈ 3/63
+
+    @test lean(x, y, 1, weighted = true) ≈ 60/63
 end
\ No newline at end of file

From bfec698155aeae316ca45b7a558790de3843da0e Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:29:49 +0100
Subject: [PATCH 19/29] Definition of causal penchant

---
 src/Leanings/penchants.jl | 31 +++++++++++++++++++++++++++----
 1 file changed, 27 insertions(+), 4 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index cc91af65e..9001e2f92 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -96,12 +96,35 @@ end
 """
     penchant(x, y, l; weighted = false) → ρ̄
 
-Computes the *mean observed penchant* `ρ̄ ∈ [-1, 1]` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] 
-of the `l`-assignment ``\\{\\text{cause}, \\text{effect}\\} =\\{x_{t-l}, y_t\\}``.
+Computes the *mean observed penchant* `ρ̄` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] 
+of the `l`-assignment ``\\{\\text{C}, \\text{E}\\} = \\{\\text{cause}, \\text{effect}\\} =\\{x_{t-l}, y_t\\}``.
 
 If `weighted == true`, then compute the *weighted mean observed penchant*.
 
-See also data requirements discussed in [`lean`](@ref).
+## Definition 
+
+The *causal penchant* , or causal tendency, is a causal indicator defined as 
+``\\rho_{CE} = P(E|C) - P(E|\\bar{C})``, where ``\\rho_{CE} \\in [-1, 1]`` (McCracken & Weigel, 2016). 
+If ``\\rho_{CE} > 0``, then ``C`` causes or drives ``E``, and if 
+``\\rho_{CE} \\leq 0`, then the direction of influence is undetermined.
+
+A direct formula for ``\\rho_{CE}`` can be obtained using Bayes' theorem.
+
+```math
+P(E|C) = P(C|E) \\dfrac{P(E)}{P(C)} 
+```
+Using the definitions of probability complements, one arrives at the following 
+expression (see the original paper for a detailed derivation):
+
+```math
+\\rho_{CE} = P(E|C) \\left[1 + \\dfrac{P(C)}{1-P(C)} \\right] - \\dfrac{P(E)}{1 - P(C)}.
+```
+
+Applying appropriate discretization schemes, these probabilities can be estimated 
+directly from time series using simple counting, which makes the method fast and 
+well-suited for exploratory causal inference analysis.
+
+See also notes on data requirements discussed in [`lean`](@ref).
 
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
@@ -157,5 +180,5 @@ Pre-process your time series using appropriate binning or symbolization schemes.
 function lean(x, y, l = 1; weighted = true)
     return penchant(x, y, l, weighted = weighted) - 
         penchant(y, x, l, weighted = weighted)
-end
+end 
 

From 2d199903a2f2e85e018b95a3d7569cf12a81d0a9 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:45:44 +0100
Subject: [PATCH 20/29] Rephrase docs

---
 src/Leanings/penchants.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 9001e2f92..3e8436515 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -124,7 +124,7 @@ Applying appropriate discretization schemes, these probabilities can be estimate
 directly from time series using simple counting, which makes the method fast and 
 well-suited for exploratory causal inference analysis.
 
-See also notes on data requirements discussed in [`lean`](@ref).
+See also [`lean`](@ref) and data requirements discussed therein.
 
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """

From 4c9fbeb935118a88f3fc6924816ee818a8b5375a Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 10:47:24 +0100
Subject: [PATCH 21/29] All tests involving Dataset construction fails,
 reintroduce the file

---
 test/runtests.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/runtests.jl b/test/runtests.jl
index ad534b9d5..1c3c61e7a 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -6,7 +6,7 @@ import DynamicalSystemsBase: DiscreteDynamicalSystem, ContinuousDynamicalSystem
 
 include("methods/test_smeasure.jl")
 include("methods/test_joint_distance_distribution.jl")
-#include("methods/test_predictive_asymmetry.jl")
+include("methods/test_predictive_asymmetry.jl")
 include("methods/test_crossmapping.jl")
 include("methods/test_leanings.jl")
 

From 096fabab88b4e4b24c2d387b216ae8a061af35a2 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 15:36:27 +0100
Subject: [PATCH 22/29] Trigger new CI run

---
 test/methods/test_leanings.jl | 1 -
 1 file changed, 1 deletion(-)

diff --git a/test/methods/test_leanings.jl b/test/methods/test_leanings.jl
index a3990f3ec..8cd1cb682 100644
--- a/test/methods/test_leanings.jl
+++ b/test/methods/test_leanings.jl
@@ -1,6 +1,5 @@
 
 @testset "Leanings" begin
-
     x = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0]
     y = [0, 0, 0, 1, 0, 0, 1, 0, 0, 1]
     @test penchant(x, y, 1, weighted = false) ≈ 1.0

From f7c35911d7d4e999a318c7fe35bbd7731533b792 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 15:39:52 +0100
Subject: [PATCH 23/29] Require DelayEmbeddings 2.0.2 (breaks on Julia 1.7
 otherwise)

---
 Project.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project.toml b/Project.toml
index 7464bda9c..683d8f348 100644
--- a/Project.toml
+++ b/Project.toml
@@ -29,7 +29,7 @@ Wavelets = "29a6e085-ba6d-5f35-a997-948ac2efa89a"
 
 [compat]
 ChaosTools = "^2.0"
-DelayEmbeddings = "^2.0"
+DelayEmbeddings = "^2.0.2"
 Distances = "^0.10"
 Distributions = "^0.24, 0.25"
 DynamicalSystemsBase = "^2.0"

From e21791f51a458da0ac4019d0565c7c4668ebbbb7 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 15:41:55 +0100
Subject: [PATCH 24/29] Change heading in menu

---
 docs/src/leanings.md | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/docs/src/leanings.md b/docs/src/leanings.md
index 543b12144..c95a1409f 100644
--- a/docs/src/leanings.md
+++ b/docs/src/leanings.md
@@ -1,9 +1,10 @@
-# Leanings
+# Penchants and leanings
 
-Leanings are probability-based exploratory causal inference measures that rely on the estimation of probabilities directly from the data. Here, we provide 
-implementations of the *penchants and leanings* algoriths from McCracken and Weigel (2016).
+Penchants and leanings are probability-based exploratory causal inference measures that rely on the estimation of probabilities directly from the data. Here, we provide implementations of the *penchants and leanings* 
+algoriths from McCracken and Weigel (2016).
 
 ```@docs
 penchant
 lean
-```
\ No newline at end of file
+```
+

From 2d29f3cc42b1996a517671060997667b04bb721b Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 15:42:03 +0100
Subject: [PATCH 25/29] Latex typo

---
 src/Leanings/penchants.jl | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 3e8436515..728e18107 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -106,7 +106,7 @@ If `weighted == true`, then compute the *weighted mean observed penchant*.
 The *causal penchant* , or causal tendency, is a causal indicator defined as 
 ``\\rho_{CE} = P(E|C) - P(E|\\bar{C})``, where ``\\rho_{CE} \\in [-1, 1]`` (McCracken & Weigel, 2016). 
 If ``\\rho_{CE} > 0``, then ``C`` causes or drives ``E``, and if 
-``\\rho_{CE} \\leq 0`, then the direction of influence is undetermined.
+``\\rho_{CE} \\leq 0``, then the direction of influence is undetermined.
 
 A direct formula for ``\\rho_{CE}`` can be obtained using Bayes' theorem.
 
@@ -152,11 +152,12 @@ end
 
     lean(x, y, l = 1; weighted = true) → ρ̄
 
-Compute the *mean observed leaning* `λ ∈ [-2, 2]` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] between scalar time series `x` and `y`.
+Compute the *mean observed leaning* `λ ∈ [-2, 2]` (McCracken & Weigel, 2016)[^McCrackenWeigel2016] 
+between scalar time series `x` and `y`.
 
 If `ρ̄ > 0`, then the probability that `x` drives `y` is higher than the probability 
-that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` is higher than the probability 
-that `x` drives `y`.
+that `y` drives `x`. Vice versa, if `ρ̄ > 0`, then the probability that `y` drives `x` 
+is higher than the probability that `x` drives `y`.
 
 ## Weighting
 

From 9bdb54b77fa577280248e3b61932c983560b3566 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 16:21:33 +0100
Subject: [PATCH 26/29] Add penchants and leanings to overview

---
 docs/src/index.md    | 4 ++++
 docs/src/leanings.md | 2 +-
 2 files changed, 5 insertions(+), 1 deletion(-)

diff --git a/docs/src/index.md b/docs/src/index.md
index 2c927bfe3..42281bbcd 100644
--- a/docs/src/index.md
+++ b/docs/src/index.md
@@ -35,6 +35,10 @@ We also provide estimators for generalized entropy and mutual information, thoug
 - Generalized entropy ([`genentropy`](@ref)) and fast histograms ([`binhist`](@ref))
 - Mutual information ([`mutualinfo`](@ref))
 
+### Exploratory methods
+
+- [Penchants and leanings](@ref leanings).
+
 ## Contributions/issues
 
 Do you want additional methods or example systems to be implemented? Make a PR to the 
diff --git a/docs/src/leanings.md b/docs/src/leanings.md
index c95a1409f..b75ca0aa0 100644
--- a/docs/src/leanings.md
+++ b/docs/src/leanings.md
@@ -1,4 +1,4 @@
-# Penchants and leanings
+# [Penchants and leanings](@id leanings)
 
 Penchants and leanings are probability-based exploratory causal inference measures that rely on the estimation of probabilities directly from the data. Here, we provide implementations of the *penchants and leanings* 
 algoriths from McCracken and Weigel (2016).

From 235957acc5dd24a40aaee4c35da36a5feb9a6138 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 17:35:49 +0100
Subject: [PATCH 27/29] `weighted` should be true by default for both `lean`
 and `penchant``

---
 src/Leanings/penchants.jl | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

diff --git a/src/Leanings/penchants.jl b/src/Leanings/penchants.jl
index 728e18107..662792d7b 100644
--- a/src/Leanings/penchants.jl
+++ b/src/Leanings/penchants.jl
@@ -128,7 +128,7 @@ See also [`lean`](@ref) and data requirements discussed therein.
 
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
-function penchant(x, y, l = 1; weighted = false)
+function penchant(x, y, l = 1; weighted = true)
     n = length(x)
     ns_c, ns_e, ns_ec = penchant_counts(x, y, l)
 
@@ -179,7 +179,6 @@ Pre-process your time series using appropriate binning or symbolization schemes.
 [^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
 """
 function lean(x, y, l = 1; weighted = true)
-    return penchant(x, y, l, weighted = weighted) - 
-        penchant(y, x, l, weighted = weighted)
+    return penchant(x, y, l, weighted = weighted) - penchant(y, x, l, weighted = weighted)
 end 
 

From 9042531665cc54275e2b7370b9cb65ad3759f0a1 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Kristian=20Agas=C3=B8ster=20Haaga?= <kahaaga@gmail.com>
Date: Wed, 1 Dec 2021 18:30:10 +0100
Subject: [PATCH 28/29] example

---
 src/Leanings/example.jl | 57 +++++++++++++++++++++++++++++++++++++++++
 1 file changed, 57 insertions(+)
 create mode 100644 src/Leanings/example.jl

diff --git a/src/Leanings/example.jl b/src/Leanings/example.jl
new file mode 100644
index 000000000..6c8201ff9
--- /dev/null
+++ b/src/Leanings/example.jl
@@ -0,0 +1,57 @@
+using CausalityTools, DynamicalSystems, Plots, Statistics, Distributions; gr()
+
+npts, Ttr = 10000, 5000
+x, y = columns(trajectory(henon2(c_xy = 0.0), npts - 1, Ttr = Ttr))
+xr, yr = rand(Uniform(-1, 1), npts), rand(Uniform(-1, 1), npts)
+plot()
+plot!(x, label = "x")
+plot!(y, label = "y")
+
+import Entropies: symbolize, SymbolicPermutation
+
+m = 2; τ = -1
+n = 5
+weighted = false
+nreps = 50
+ls = 0:10
+leans_wn = zeros(nreps, length(ls))
+leans_henon_c0 = zeros(nreps, length(ls))
+leans_henon_c2 = zeros(nreps, length(ls))
+
+function partition(x, n::Int)
+    xmin = minimum(x) # the reference point
+    Δx = (maximum(x) - xmin) / n
+    [floor(Int, (xᵢ - xmin) / Δx) for xᵢ in x]
+end
+
+for i = 1:nreps 
+    xr, yr = rand(Uniform(-1, 1), npts), rand(Uniform(-1, 1), npts)
+    X = partition(xr, n)#symbolize(xr, SymbolicPermutation(m = m, τ = τ))
+    Y = partition(yr, n)#symbolize(yr, SymbolicPermutation(m = m, τ = τ))
+    for (j, l) in enumerate(ls)
+        leans_wn[i, j] = lean(X, Y, l, weighted = weighted)
+    end
+
+    x, y = columns(trajectory(henon2(c_xy = 0.0), npts - 1, Ttr = Ttr))
+    X = partition(x, n)#symbolize(x, SymbolicPermutation(m = m, τ = τ))
+    Y = partition(y, n)#symbolize(y, SymbolicPermutation(m = m, τ = τ))
+    for (j, l) in enumerate(ls)
+        leans_henon_c0[i, j] = lean(X, Y, l, weighted = weighted)
+    end
+
+    x, y = columns(trajectory(henon2(c_xy = 2.0), npts - 1, Ttr = Ttr))
+    X = partition(x, n)#symbolize(x, SymbolicPermutation(m = m, τ = τ))
+    Y = partition(y, n)#symbolize(y, SymbolicPermutation(m = m, τ = τ))
+    for (j, l) in enumerate(ls)
+        leans_henon_c2[i, j] = lean(X, Y, l, weighted = weighted)
+    end
+end
+
+mean_leans_wn = dropdims(mean(leans_wn, dims = 1), dims = 1)
+mean_leans_henon_c0 = dropdims(mean(leans_henon_c0, dims = 1), dims = 1)
+mean_leans_henon_c2 = dropdims(mean(leans_henon_c2, dims = 1), dims = 1)
+
+plot(xlabel = "l", ylabel = "λx→y", ylims = (-1.1, 1.1))
+#plot!(ls, mean_leans_wn, label = "white noise", marker = :star)
+plot!(ls, mean_leans_henon_c0, label = "henon (c_xy = 0.0)", marker = :circle)
+#plot!(ls, mean_leans_henon_c2, label = "henon (c_xy = 2.0)", marker = :star5)
\ No newline at end of file

From ce542deb39d26083c01848844fc6d6d1f63c5de8 Mon Sep 17 00:00:00 2001
From: Kristian Haaga <kahaaga@gmail.com>
Date: Mon, 6 Dec 2021 15:58:28 +0100
Subject: [PATCH 29/29] Add example system

---
 .../discretemaps/nonlinear_sindriver_2d.jl    | 34 +++++++++++++++++++
 1 file changed, 34 insertions(+)
 create mode 100644 src/example_systems/discretemaps/nonlinear_sindriver_2d.jl

diff --git a/src/example_systems/discretemaps/nonlinear_sindriver_2d.jl b/src/example_systems/discretemaps/nonlinear_sindriver_2d.jl
new file mode 100644
index 000000000..ead84e60a
--- /dev/null
+++ b/src/example_systems/discretemaps/nonlinear_sindriver_2d.jl
@@ -0,0 +1,34 @@
+function eom_nonlinear_sindriver_2d(dx, x, p, n)
+    a, b, c, t, Δt = (p...,)
+    x, y = x[1], x[2]
+    𝒩 = Normal(0, 1)
+    
+    dx[1] = sin(t)
+    dx[2] = a*x * (1 - b*x) + c*rand(𝒩)
+    p[end-1] += Δt # update t
+
+    return
+end
+
+"""
+    nonlinear_sindriver_2d(;u₀ = 0, a = 1.0, b = 1.0, c = 2.0, Δt = 1)
+
+A discrete, nonlinear 2D system from McCracken & Weigel (2016)[^McCrackenWeigel2016],
+governed by the following equations of motion:
+
+```math
+\\begin{aligned}
+x_{t+1} = \\sin(t) \\\\
+y(t+1) &= a x_t (1 - B x_t) + C \\eta_t
+\\end{aligned}
+```
+
+where ``\\eta_t \\sim \\mathcal{N}(\\mu = 0, \\sigma = 1)``, ``A, B, C \\in [0, 1]``.
+
+[^McCrackenWeigel2016]: McCracken, J. M., & Weigel, R. S. (2016). Nonparametric causal inference for bivariate time series. Physical Review E, 93(2), 022207.
+"""
+function nonlinear_sindriver_2d(;u₀ = 0, a = 1.0, b = 1.0, c = 2.0, Δt = (1/30*π))
+    @assert a >= 0; @assert b >= 0; @assert c >= 0
+
+    DiscreteDynamicalSystem(eom_nonlinear_sindriver2d, [0, u₀], [a, b, c, 0, Δt])
+end
\ No newline at end of file