∑MathAdvanced

Copulas & Dependency Structures

Key Points

•
A copula is a function that glues together marginal distributions to form a multivariate joint distribution while isolating dependence from the margins.
•
Sklar's theorem states that any joint CDF F can be written as F( $x_{1}$ , ..., $x_{d}$ ) = C( $F_{1}$ ( $x_{1}$ ), ..., $F_{d}$ ( $x_{d}$ )) for some copula C.
•
By working on the rank scale (uniform [0,1] variables), copulas model dependence independently of the shapes of the marginals.
•
Gaussian and Archimedean copulas (like Clayton and Gumbel) are popular families that capture different dependence and tail behaviors.
•
Empirical copulas estimate the copula directly from data using ranks, avoiding assumptions on marginals.
•
Kendall’s tau and Spearman’s rho depend only on the copula and can be used to fit copula parameters.
•
Sampling from a copula is easy: sample uniforms with the copula’s dependence, then push them through inverse CDFs of desired marginals.
•
C++ implementations typically involve rank transforms, random sampling, and likelihood evaluation, each with clear time and space complexities.

Prerequisites

→Probability and Random Variables — Understanding CDFs, PDFs, and transformations is essential for copulas and Sklar’s theorem.
→Order Statistics and Ranks — Empirical copulas and rank-based measures rely on ranks and their properties.
→Linear Algebra (Cholesky Decomposition) — Sampling from Gaussian copulas requires factoring correlation matrices.
→Numerical Optimization — Fitting copula parameters via pseudo-likelihood involves optimizing over parameters.
→Dependence Measures — Kendall’s tau and Spearman’s rho are central to parameter estimation and model diagnostics.

Detailed Explanation

Tap terms for definitions

01Overview

Copulas are mathematical tools that separate the shape of individual variables (their marginal distributions) from the way those variables move together (their dependence). Imagine you want to describe heights and weights in a population: each has its own marginal distribution, but the interesting part is how they co-vary. Copulas let you express the joint distribution as a composition of the marginals and a dependence function called a copula. Sklar’s theorem guarantees that, for any multivariate joint cumulative distribution function (CDF), there exists a copula that ties the marginals together. This is powerful in practice: you can choose the best-fitting marginals for each variable and then choose a copula to model the dependence structure—like picking the shape of individual puzzle pieces and the way they interlock independently. Different copula families capture different dependence features: Gaussian copulas represent symmetric dependence without tail emphasis, while Archimedean copulas (Clayton, Gumbel, Frank) can capture asymmetric or tail-heavy relationships. In statistics, finance, hydrology, and machine learning, copulas are used for simulation, risk aggregation, imputation, and building flexible multivariate models. They are especially useful when tail dependence or non-linear association matters more than simple linear correlation.

02Intuition & Analogies

Think of marginals as the flavors of two scoops of ice cream—say vanilla and chocolate—and the copula as the way you swirl them together in a cup. The flavors themselves don’t tell you how they’re mixed; the swirl pattern does. If the scoops are placed side by side with a clean boundary, that’s like weak dependence; if they’re intensely marbled, that’s strong dependence. Copulas describe the swirl pattern in a standardized space where both flavors are converted to the same intensity scale: ranks mapped to [0,1]. Why ranks? Because absolute values (like centimeters or kilograms) should not affect dependence; only the ordering (who’s bigger) should. By mapping each variable through its CDF, every observation becomes a uniform number between 0 and 1. Now the joint behavior of these uniforms—how often they are simultaneously small or large—is exactly the copula. Some copulas concentrate mass near (0,0) and (1,1), meaning the variables tend to be simultaneously extreme (tail dependence). Others spread mass symmetrically with no tail emphasis, like the Gaussian copula. Once you’ve modeled the swirl in [0,1]^d, you can pour it back through any set of inverse CDFs to get real-world units again. This modularity is the key benefit: you can change marginals without touching dependence, and vice versa.

03Formal Definition

Let X = (

X_{1}

\dots

X_{d}

) be a random vector with joint CDF F(

x_{1}

\dots

x_{d}

) and continuous marginal CDFs

F_{j}

(

x_{j}

). A copula is a function C: [0,1]^d

\to

[0,1] that is grounded, d-increasing, and has uniform[0,1] marginals: C(1,

\dots

, 1,

u_{j}

, 1,

\dots

, 1) =

u_{j}

. Sklar’s theorem states there exists a copula C such that F(

x_{1}

\dots

x_{d}

) = C(

F_{1}

(

x_{1}

\dots

F_{d}

(

x_{d}

)). If all

F_{j}

are continuous, then C is unique and the random vector U = (

U_{1}

\dots

U_{d}

) with

U_{j} = F_{j} (X_{j}

) has joint distribution C and uniform marginals. When densities exist, the joint density can be written as f(

x_{1}

\dots

x_{d}

) = c(

F_{1}

(

x_{1}

\dots

F_{d}

(

x_{d}

))

\prod_{j = 1}^{d}

f_{j}

(

x_{j}

), where c is the copula density c(u) =

\frac{\partial ^{d} C ( u )}{\partial u _{1} \dots \partial u _{d}}

. Dependence measures such as Kendall’s tau and Spearman’s rho depend only on C, not on the marginals. Popular copula families include elliptical copulas (Gaussian, t) and Archimedean copulas defined via a generator function

φ

with C(u) =

φ^{- 1}

(

φ

(

u_{1}

) +

\dots

φ

(

u_{d}

)). Tail dependence coefficients

λ_{L}

λ_{U}

quantify joint extreme behavior and are determined by the copula.

04When to Use

Use copulas when you need to model or simulate multivariate data with flexible and possibly different marginals, but you also care deeply about the dependence pattern (nonlinear association, asymmetry, or tail co-movement). Typical scenarios include: (1) Risk management and finance where joint extreme losses matter—select a copula with upper tail dependence (e.g., Gumbel) or lower tail dependence (e.g., Clayton). (2) Environmental modeling (rainfall, river flows) where simultaneous extremes cause floods—copulas help capture tail dependence beyond correlation. (3) Imputation and simulation: generate synthetic datasets with desired marginals and dependence for stress testing or bootstrapping. (4) Mixed or non-Gaussian data: when linear correlation is misleading or marginals are heavy-tailed or skewed, copulas provide a principled approach. (5) Feature engineering: transform to copula space (ranks) for robust dependence analysis invariant to monotone transformations. Copulas are less useful when data dimensionality is extremely high and simple parsimonious dependence structures suffice, or when continuous marginals cannot be reasonably estimated (heavy ties).

⚠️Common Mistakes

Common pitfalls include: (1) Ignoring marginal modeling. Skipping good marginal fits can distort the dependence when back-transforming; use appropriate parametric or nonparametric marginal estimates before applying a copula. (2) Confusing correlation with dependence. Linear correlation does not capture tail dependence; a Gaussian copula may fit correlation but miss extremes. (3) Mishandling ties. Empirical copulas assume continuous marginals; with ties, use average ranks or jitter, and be explicit about the tie strategy. (4) Using the wrong copula family. Choose based on tail behavior: Clayton (lower tails), Gumbel (upper tails), t-copula (symmetric with tail dependence), Gaussian (no tail dependence). (5) Numerical issues in likelihoods. Pseudo-observations u very close to 0 or 1 can cause log(0) or division by zero in densities; clamp u to (\epsilon, 1-\epsilon). (6) Overfitting with small samples. Parameter estimates from weak signals or small n can be unstable; use rank-based methods (Kendall’s tau) or penalization. (7) Forgetting identifiability with non-continuous marginals. Sklar’s theorem gives non-unique copulas for discrete marginals; interpret with care and prefer pseudo-likelihood methods tailored for discreteness.

Key Formulas

Sklar's Theorem (CDF Decomposition)

F (x_{1}, \dots, x_{d}) = C (F_{1} (x_{1}), \dots, F_{d} (x_{d}))

Explanation: Any joint CDF can be written as a copula applied to its marginal CDFs. When marginals are continuous, the copula is unique.

Density Factorization

f (x_{1}, \dots, x_{d}) = c (F_{1} (x_{1}), \dots, F_{d} (x_{d})) j = 1 \prod d f_{j} (x_{j})

Explanation: When densities exist, the joint density equals the copula density evaluated at marginal CDF values times the product of marginal densities. This cleanly separates dependence and marginals.

Empirical Copula

C_{n} (u_{1}, \dots, u_{d}) = \frac{1}{n} i = 1 \sum n 1 {U_{i 1} \leq u_{1}, \dots, U_{i d} \leq u_{d}}

Explanation: This nonparametric estimator counts the fraction of rank-transformed points that fall below a target point u in [0,1]^d. It requires only ranks, not parametric assumptions.

Gaussian Copula

C_{Gauss} (u, v; ρ) = Φ_{ρ} (Φ^{- 1} (u), Φ^{- 1} (v))

Explanation: Apply the bivariate normal CDF with correlation $ρ$ to the normal quantiles of u and v. It captures symmetric dependence without tail dependence.

Clayton Copula

C_{Clayton} (u, v; θ) = (u^{- θ} + v^{- θ} - 1)^{- 1/ θ}, θ > 0

Explanation: An Archimedean copula focusing on lower-tail dependence. Larger $θ$ strengthens association near zero.

Clayton Density

c_{Clayton} (u, v; θ) = (1 + θ) (uv)^{- (1 + θ)} (u^{- θ} + v^{- θ} - 1)^{- (2 + 1/ θ)}

Explanation: The density obtained by differentiating the Clayton copula. It appears in likelihoods for parameter estimation from pseudo-observations.

Kendall's Tau via Copula

τ = 4 \int_{[0, 1]^{2}} C (u, v) d C (u, v) - 1

Explanation: Kendall’s tau can be computed from the copula alone, showing its marginal invariance. It often leads to simple parameter relationships.

Spearman's Rho via Copula

ρ_{S} = 12 \int_{0}^{1} \int_{0}^{1} (C (u, v) - uv) d u d v

Explanation: Spearman’s rho equals 12 times the average deviation of the copula from independence (uv). It depends only on the copula.

Clayton Parameter–Tau Link

τ_{Clayton} = \frac{θ}{θ + 2}, θ = \frac{2 τ}{1 - τ}

Explanation: Kendall’s tau has a simple one-to-one relationship with the Clayton parameter, enabling quick estimation from ranks.

Tail Dependence Coefficients

λ_{U} = u \to 1^{-} lim Pr (U_{2} > u ∣ U_{1} > u), λ_{L} = u \to 0^{+} lim Pr (U_{2} \leq u ∣ U_{1} \leq u)

Explanation: These limits quantify the probability of joint extremes. Gaussian copulas have $λ_{U}$ = $λ_{L}$ = 0 for $∣ ρ ∣$ <1, while t-copulas have nonzero tail dependence.

Copula Pseudo-Likelihood

L (θ) = i = 1 \sum n lo g c (U_{i 1}, \dots, U_{i d}; θ)

Explanation: Log-likelihood for copula parameters based on pseudo-observations. Maximizing it estimates the dependence parameters independent of marginals.

Probability Integral Transform

U_{j} = F_{j} (X_{j}), X_{j} = F_{j}^{- 1} (U_{j})

Explanation: Mapping through the CDF yields uniforms, and inverse CDF maps uniforms back to the original scale. This is the basis for copula simulation.

Complexity Analysis

Core operations with copulas typically involve rank computation, evaluation of empirical copulas, parameter estimation, and sampling. Computing ranks for n observations per dimension via sorting costs O(n log n) time and O(n) space; with d dimensions, this becomes O(d n log n). Constructing an empirical copula query

C_{n}

(u) requires scanning all n pseudo-observations to count how many lie coordinatewise below u, leading to O(n) time per query and O(n) space for storage. Multiple queries can be accelerated by preprocessing (e.g., 2D Fenwick trees) at the cost of O(n log n) construction and O(lo

g^{2}

n) queries. For parametric families, evaluating the pseudo log-likelihood L(

θ)

= ∑ log c(

U_{i}

;

θ)

is O(n) per evaluation since each density call is O(1) in fixed dimension. Gradient-based optimization then scales with the number of iterations k as O(k n). Estimating Kendall’s tau naively is O(

n^{2}

) because it checks all pairs; it can be improved to O(n log n) using inversion counting with a Fenwick tree after sorting, but the constants are larger. Sampling from a Gaussian copula requires drawing n standard normals and applying a fixed Cholesky factor, which is O(n

d^{2}

) for dense correlation in d dimensions (O(n) in 2D), plus O(n d) to apply univariate CDFs. Memory usage is O(n d) to store samples or pseudo-observations, and O(

d^{2}

) for covariance/correlation matrices. Numerical stability considerations (clamping uniforms away from 0 and 1) add negligible cost but prevent infinities in logs or powers.

Code Examples

Empirical Copula in 2D from Data (Ranks and Queries)

1 #include <iostream>
2 #include <vector>
3 #include <algorithm>
4 #include <cmath>
5 #include <numeric>
6 #include <limits>
7 
8 // Compute average ranks (1..n) with tie handling by averaging tied positions
9 static std::vector<double> average_ranks(const std::vector<double>& x) {
10     int n = (int)x.size();
11     std::vector<int> idx(n);
12     std::iota(idx.begin(), idx.end(), 0);
13     std::stable_sort(idx.begin(), idx.end(), [&](int a, int b){ return x[a] < x[b]; });
14     std::vector<double> rank(n);
15     const double eps = 1e-12;
16     int i = 0;
17     while (i < n) {
18         int j = i + 1;
19         // find tie block [i, j)
20         while (j < n && std::fabs(x[idx[j]] - x[idx[i]]) <= eps) j++;
21         // average rank over positions i..j-1
22         double r1 = i + 1;            // ranks start at 1
23         double r2 = j;                // inclusive end
24         double avg = (r1 + r2) / 2.0; // average of consecutive integers
25         for (int k = i; k < j; ++k) rank[idx[k]] = avg;
26         i = j;
27     }
28     return rank;
29 }
30 
31 struct EmpiricalCopula2D {
32     std::vector<double> U, V; // pseudo-observations in (0,1)
33 
34     // Construct from raw data vectors x and y
35     EmpiricalCopula2D(const std::vector<double>& x, const std::vector<double>& y) {
36         int n = (int)x.size();
37         if ((int)y.size() != n || n == 0) throw std::runtime_error("Input vectors must have same positive length");
38         auto rx = average_ranks(x);
39         auto ry = average_ranks(y);
40         U.resize(n); V.resize(n);
41         // Use n+1 in denominator to keep pseudo-observations strictly in (0,1)
42         for (int i = 0; i < n; ++i) {
43             U[i] = rx[i] / (n + 1.0);
44             V[i] = ry[i] / (n + 1.0);
45         }
46     }
47 
48     // Empirical copula C_n(u,v) = (1/n) sum 1{U_i <= u, V_i <= v}
49     double C(double u, double v) const {
50         int n = (int)U.size();
51         int cnt = 0;
52         for (int i = 0; i < n; ++i) if (U[i] <= u && V[i] <= v) cnt++;
53         return (double)cnt / n;
54     }
55 
56     // Spearman's rho computed from ranks: correlation of U and V scaled to [-1,1]
57     double spearman_rho() const {
58         int n = (int)U.size();
59         double mean = 0.5; // mean of uniform(0,1)
60         double num = 0.0, du = 0.0, dv = 0.0;
61         for (int i = 0; i < n; ++i) {
62             double au = U[i] - mean;
63             double av = V[i] - mean;
64             num += au * av;
65             du += au * au;
66             dv += av * av;
67         }
68         double corr = num / std::sqrt(du * dv);
69         return corr; // equals Spearman's rho for pseudo-observations
70     }
71 };
72 
73 int main() {
74     // Example data (height-like and weight-like fictitious numbers)
75     std::vector<double> x = {1.2, 0.9, 1.5, 2.0, 1.7, 1.1, 1.8, 1.3};
76     std::vector<double> y = {2.1, 1.8, 2.4, 2.9, 2.5, 2.0, 2.7, 2.2};
77 
78     EmpiricalCopula2D ec(x, y);
79 
80     // Query the empirical copula at a few grid points
81     for (double u : {0.25, 0.5, 0.75}) {
82         for (double v : {0.25, 0.5, 0.75}) {
83             std::cout << "C_n(" << u << "," << v << ") = " << ec.C(u, v) << "\n";
84         }
85     }
86     std::cout << "Spearman rho (ranks): " << ec.spearman_rho() << "\n";
87     return 0;
88 }
89

This program constructs an empirical copula from bivariate data by converting each margin to pseudo-observations using average ranks. It then evaluates the empirical copula C_n(u, v) at grid points and reports Spearman’s rho computed on the rank scale. Average ranks mitigate the effect of ties, and dividing by n+1 keeps values strictly in (0,1), which is numerically safer for later copula density computations.

Time: O(n log n) to compute ranks and O(n) per C_n(u,v) querySpace: O(n) to store pseudo-observations

Sampling from a Gaussian Copula and Transforming to Chosen Marginals

1 #include <iostream>
2 #include <vector>
3 #include <random>
4 #include <cmath>
5 #include <utility>
6 
7 // Standard normal CDF using erf
8 static inline double Phi(double x) {
9     return 0.5 * (1.0 + std::erf(x / std::sqrt(2.0)));
10 }
11 
12 // Sample n pairs from a bivariate Gaussian copula with correlation rho
13 // Returns U pairs (uniforms in (0,1))
14 std::vector<std::pair<double,double>> sample_gaussian_copula(int n, double rho, std::mt19937_64& rng) {
15     if (rho <= -1.0 || rho >= 1.0) throw std::runtime_error("rho must be in (-1,1)");
16     std::normal_distribution<double> nd(0.0, 1.0);
17     double a11 = 1.0;
18     double a21 = rho;
19     double a22 = std::sqrt(1.0 - rho * rho); // Cholesky of [[1,rho],[rho,1]]
20 
21     std::vector<std::pair<double,double>> uv;
22     uv.reserve(n);
23     for (int i = 0; i < n; ++i) {
24         double z1 = nd(rng);
25         double z2 = nd(rng);
26         double y1 = a11 * z1;                // = z1
27         double y2 = a21 * z1 + a22 * z2;     // correlated with y1
28         double u = Phi(y1);
29         double v = Phi(y2);
30         // Clamp defensively to avoid exact 0 or 1
31         const double eps = 1e-12;
32         u = std::min(1.0 - eps, std::max(eps, u));
33         v = std::min(1.0 - eps, std::max(eps, v));
34         uv.emplace_back(u, v);
35     }
36     return uv;
37 }
38 
39 // Example inverse CDFs (marginals) with closed forms
40 // Exponential(lambda): F^{-1}(u) = -ln(1-u)/lambda
41 static inline double inv_exp(double u, double lambda) {
42     return -std::log(1.0 - u) / lambda;
43 }
44 // Log-logistic with scale s>0 and shape k>0: F^{-1}(u) = s * (u/(1-u))^{1/k}
45 static inline double inv_loglogistic(double u, double s, double k) {
46     return s * std::pow(u / (1.0 - u), 1.0 / k);
47 }
48 
49 int main() {
50     std::mt19937_64 rng(12345);
51     int n = 10;
52     double rho = 0.7; // positive dependence
53     auto uv = sample_gaussian_copula(n, rho, rng);
54 
55     // Transform uniforms to chosen marginals
56     double lambda = 1.5;     // exponential rate
57     double s = 2.0, k = 1.2; // log-logistic params
58 
59     std::cout << "u\tv\tx=Exp^{-1}(u)\ty=LogLogistic^{-1}(v)\n";
60     for (auto [u, v] : uv) {
61         double x = inv_exp(u, lambda);
62         double y = inv_loglogistic(v, s, k);
63         std::cout << u << "\t" << v << "\t" << x << "\t" << y << "\n";
64     }
65     return 0;
66 }
67

This program samples dependent uniforms from a bivariate Gaussian copula using a Cholesky factor of the correlation matrix and the standard normal CDF. It then maps the uniforms to custom marginals via closed-form inverse CDFs (exponential and log-logistic). This demonstrates the core simulation pipeline: copula for dependence, inverse CDFs for marginals.

Time: O(n) for 2D sampling (each sample uses O(1) operations)Space: O(n) to store the samples

Estimate Clayton Copula Parameter from Kendall’s Tau and Compute Log-Likelihood

1 #include <iostream>
2 #include <vector>
3 #include <algorithm>
4 #include <cmath>
5 #include <numeric>
6 #include <stdexcept>
7 
8 // Compute normalized ranks to pseudo-observations (n+1 in denominator)
9 static std::vector<double> pseudo_obs(const std::vector<double>& x) {
10     int n = (int)x.size();
11     std::vector<int> idx(n);
12     std::iota(idx.begin(), idx.end(), 0);
13     std::stable_sort(idx.begin(), idx.end(), [&](int a, int b){ return x[a] < x[b]; });
14     std::vector<double> r(n);
15     int i = 0;
16     const double eps = 1e-12;
17     while (i < n) {
18         int j = i + 1;
19         while (j < n && std::fabs(x[idx[j]] - x[idx[i]]) <= eps) j++;
20         double avg = ( (i + 1) + j ) / 2.0;
21         for (int k = i; k < j; ++k) r[idx[k]] = avg;
22         i = j;
23     }
24     std::vector<double> U(n);
25     for (int k = 0; k < n; ++k) U[k] = r[k] / (n + 1.0);
26     return U;
27 }
28 
29 // Naive O(n^2) Kendall's tau for teaching/demo
30 static double kendall_tau_naive(const std::vector<double>& x, const std::vector<double>& y) {
31     int n = (int)x.size();
32     if ((int)y.size() != n) throw std::runtime_error("Mismatched sizes");
33     long long concordant = 0, discordant = 0;
34     for (int i = 0; i < n; ++i) {
35         for (int j = i + 1; j < n; ++j) {
36             double a = (x[i] - x[j]) * (y[i] - y[j]);
37             if (a > 0) concordant++;
38             else if (a < 0) discordant++;
39             // ties ignored for simplicity
40         }
41     }
42     long long total = concordant + discordant;
43     if (total == 0) return 0.0;
44     return (double)(concordant - discordant) / (double)total;
45 }
46 
47 // Clayton copula density c(u,v; theta)
48 static double clayton_density(double u, double v, double theta) {
49     const double eps = 1e-12;
50     u = std::min(1.0 - eps, std::max(eps, u));
51     v = std::min(1.0 - eps, std::max(eps, v));
52     double up = std::pow(u, -theta);
53     double vp = std::pow(v, -theta);
54     double t = up + vp - 1.0;
55     double logc = std::log1p(theta) - (1.0 + theta) * (std::log(u) + std::log(v)) - (2.0 + 1.0/theta) * std::log(t);
56     return std::exp(logc);
57 }
58 
59 // Log-likelihood for pseudo-observations under Clayton copula
60 static double clayton_loglik(const std::vector<double>& U, const std::vector<double>& V, double theta) {
61     int n = (int)U.size();
62     double ll = 0.0;
63     for (int i = 0; i < n; ++i) {
64         const double eps = 1e-12;
65         double u = std::min(1.0 - eps, std::max(eps, U[i]));
66         double v = std::min(1.0 - eps, std::max(eps, V[i]));
67         double up = std::pow(u, -theta);
68         double vp = std::pow(v, -theta);
69         double t = up + vp - 1.0;
70         ll += std::log1p(theta) - (1.0 + theta) * (std::log(u) + std::log(v)) - (2.0 + 1.0/theta) * std::log(t);
71     }
72     return ll;
73 }
74 
75 int main() {
76     // Fictitious dependent data
77     std::vector<double> x = {1.1, 0.8, 2.2, 1.7, 1.3, 2.0, 0.9, 1.5, 1.8, 1.2};
78     std::vector<double> y = {3.0, 2.6, 4.1, 3.5, 3.1, 3.9, 2.7, 3.3, 3.7, 3.0};
79 
80     // Step 1: Estimate Kendall's tau
81     double tau = kendall_tau_naive(x, y);
82     std::cout << "Kendall tau (naive): " << tau << "\n";
83 
84     // Step 2: Estimate Clayton parameter via tau: theta = 2 * tau / (1 - tau)
85     double theta_hat = (tau < 0.999) ? (2.0 * tau / (1.0 - tau)) : 1e6; // guard near tau=1
86     if (theta_hat <= 0.0) theta_hat = 1e-6; // constrain to positive domain
87     std::cout << "Theta_hat (Clayton): " << theta_hat << "\n";
88 
89     // Step 3: Build pseudo-observations and compute log-likelihood
90     auto U = pseudo_obs(x);
91     auto V = pseudo_obs(y);
92     double ll = clayton_loglik(U, V, theta_hat);
93     std::cout << "Log-likelihood at theta_hat: " << ll << "\n";
94 
95     // Optional: evaluate density of the first pair
96     std::cout << "c(U1,V1;theta_hat) = " << clayton_density(U[0], V[0], theta_hat) << "\n";
97     return 0;
98 }
99

This example estimates Kendall’s tau from raw data using a simple O(n^2) algorithm, converts it to the Clayton parameter via the closed-form relationship, and evaluates the pseudo log-likelihood on rank-based pseudo-observations. Clamping uniforms to (ε, 1−ε) avoids infinities in logs and powers. For larger datasets, replace the naive tau with an O(n log n) inversion-counting method for scalability.

Time: O(n^2) with naive Kendall’s tau; O(n) for log-likelihood evaluationSpace: O(n) for storing ranks and pseudo-observations

1	#include <iostream>
2	#include <vector>
3	#include <algorithm>
4	#include <cmath>
5	#include <numeric>
6	#include <limits>
7
8	// Compute average ranks (1..n) with tie handling by averaging tied positions
9	static std::vector<double> average_ranks(const std::vector<double>& x) {
10	int n = (int)x.size();
11	std::vector<int> idx(n);
12	std::iota(idx.begin(), idx.end(), 0);
13	std::stable_sort(idx.begin(), idx.end(), [&](int a, int b){ return x[a] < x[b]; });
14	std::vector<double> rank(n);
15	const double eps = 1e-12;
16	int i = 0;
17	while (i < n) {
18	int j = i + 1;
19	// find tie block [i, j)
20	while (j < n && std::fabs(x[idx[j]] - x[idx[i]]) <= eps) j++;
21	// average rank over positions i..j-1
22	double r1 = i + 1; // ranks start at 1
23	double r2 = j; // inclusive end
24	double avg = (r1 + r2) / 2.0; // average of consecutive integers
25	for (int k = i; k < j; ++k) rank[idx[k]] = avg;
26	i = j;
27	}
28	return rank;
29	}
30
31	struct EmpiricalCopula2D {
32	std::vector<double> U, V; // pseudo-observations in (0,1)
33
34	// Construct from raw data vectors x and y
35	EmpiricalCopula2D(const std::vector<double>& x, const std::vector<double>& y) {
36	int n = (int)x.size();
37	if ((int)y.size() != n \|\| n == 0) throw std::runtime_error("Input vectors must have same positive length");
38	auto rx = average_ranks(x);
39	auto ry = average_ranks(y);
40	U.resize(n); V.resize(n);
41	// Use n+1 in denominator to keep pseudo-observations strictly in (0,1)
42	for (int i = 0; i < n; ++i) {
43	U[i] = rx[i] / (n + 1.0);
44	V[i] = ry[i] / (n + 1.0);
45	}
46	}
47
48	// Empirical copula C_n(u,v) = (1/n) sum 1{U_i <= u, V_i <= v}
49	double C(double u, double v) const {
50	int n = (int)U.size();
51	int cnt = 0;
52	for (int i = 0; i < n; ++i) if (U[i] <= u && V[i] <= v) cnt++;
53	return (double)cnt / n;
54	}
55
56	// Spearman's rho computed from ranks: correlation of U and V scaled to [-1,1]
57	double spearman_rho() const {
58	int n = (int)U.size();
59	double mean = 0.5; // mean of uniform(0,1)
60	double num = 0.0, du = 0.0, dv = 0.0;
61	for (int i = 0; i < n; ++i) {
62	double au = U[i] - mean;
63	double av = V[i] - mean;
64	num += au * av;
65	du += au * au;
66	dv += av * av;
67	}
68	double corr = num / std::sqrt(du * dv);
69	return corr; // equals Spearman's rho for pseudo-observations
70	}
71	};
72
73	int main() {
74	// Example data (height-like and weight-like fictitious numbers)
75	std::vector<double> x = {1.2, 0.9, 1.5, 2.0, 1.7, 1.1, 1.8, 1.3};
76	std::vector<double> y = {2.1, 1.8, 2.4, 2.9, 2.5, 2.0, 2.7, 2.2};
77
78	EmpiricalCopula2D ec(x, y);
79
80	// Query the empirical copula at a few grid points
81	for (double u : {0.25, 0.5, 0.75}) {
82	for (double v : {0.25, 0.5, 0.75}) {
83	std::cout << "C_n(" << u << "," << v << ") = " << ec.C(u, v) << "\n";
84	}
85	}
86	std::cout << "Spearman rho (ranks): " << ec.spearman_rho() << "\n";
87	return 0;
88	}
89

1	#include <iostream>
2	#include <vector>
3	#include <random>
4	#include <cmath>
5	#include <utility>
6
7	// Standard normal CDF using erf
8	static inline double Phi(double x) {
9	return 0.5 * (1.0 + std::erf(x / std::sqrt(2.0)));
10	}
11
12	// Sample n pairs from a bivariate Gaussian copula with correlation rho
13	// Returns U pairs (uniforms in (0,1))
14	std::vector<std::pair<double,double>> sample_gaussian_copula(int n, double rho, std::mt19937_64& rng) {
15	if (rho <= -1.0 \|\| rho >= 1.0) throw std::runtime_error("rho must be in (-1,1)");
16	std::normal_distribution<double> nd(0.0, 1.0);
17	double a11 = 1.0;
18	double a21 = rho;
19	double a22 = std::sqrt(1.0 - rho * rho); // Cholesky of [[1,rho],[rho,1]]
20
21	std::vector<std::pair<double,double>> uv;
22	uv.reserve(n);
23	for (int i = 0; i < n; ++i) {
24	double z1 = nd(rng);
25	double z2 = nd(rng);
26	double y1 = a11 * z1; // = z1
27	double y2 = a21 * z1 + a22 * z2; // correlated with y1
28	double u = Phi(y1);
29	double v = Phi(y2);
30	// Clamp defensively to avoid exact 0 or 1
31	const double eps = 1e-12;
32	u = std::min(1.0 - eps, std::max(eps, u));
33	v = std::min(1.0 - eps, std::max(eps, v));
34	uv.emplace_back(u, v);
35	}
36	return uv;
37	}
38
39	// Example inverse CDFs (marginals) with closed forms
40	// Exponential(lambda): F^{-1}(u) = -ln(1-u)/lambda
41	static inline double inv_exp(double u, double lambda) {
42	return -std::log(1.0 - u) / lambda;
43	}
44	// Log-logistic with scale s>0 and shape k>0: F^{-1}(u) = s * (u/(1-u))^{1/k}
45	static inline double inv_loglogistic(double u, double s, double k) {
46	return s * std::pow(u / (1.0 - u), 1.0 / k);
47	}
48
49	int main() {
50	std::mt19937_64 rng(12345);
51	int n = 10;
52	double rho = 0.7; // positive dependence
53	auto uv = sample_gaussian_copula(n, rho, rng);
54
55	// Transform uniforms to chosen marginals
56	double lambda = 1.5; // exponential rate
57	double s = 2.0, k = 1.2; // log-logistic params
58
59	std::cout << "u\tv\tx=Exp^{-1}(u)\ty=LogLogistic^{-1}(v)\n";
60	for (auto [u, v] : uv) {
61	double x = inv_exp(u, lambda);
62	double y = inv_loglogistic(v, s, k);
63	std::cout << u << "\t" << v << "\t" << x << "\t" << y << "\n";
64	}
65	return 0;
66	}
67