Note

The following implementations and documentation closely follow the work of Gautier Marti: Some contributions to the clustering of financial time series and applications to credit default swaps.

Copula-Based Metrics

The work mentioned above introduces a new approach of representing the random variables that splits apart dependency and distribution without losing any information. It also contains a distance metric between two financial time series based on a novel approach.

According to the author’s classification:

“Many statistical distances exist to measure the dissimilarity of two random variables, and therefore two i.i.d. random processes. Such distances can be roughly classified in two families:

1. distributional distances, […] which focus on dissimilarity between probability distributions and quantify divergences in marginal behaviours,

2. dependence distances, such as the distance correlation or copula-based kernel dependency measures […], which focus on the joint behaviours of random variables, generally ignoring their distribution properties.

However, we may want to be able to discriminate random variables both on distribution and dependence. This can be motivated, for instance, from the study of financial assets returns: are two perfectly correlated random variables (assets returns), but one being normally distributed and the other one following a heavy-tailed distribution, similar? From risk perspective, the answer is no […], hence the propounded distance of this article”.

Tip

Read the original work to understand the motivation behind creating the novel technique deeper and check the reference papers that prove the above statements.

Spearman’s Rho

Following the work of Marti:

“[The Pearson correlation coefficient] suffers from several drawbacks: - it only measures linear relationship between two variables; - it is not robust to noise - it may be undefined if the distribution of one of these variables have infinite second moment.

More robust correlation coefficients are copula-based dependence measures such as Spearman’s rho:

\[\begin{split}\rho_{S}(X, Y) &= 12 E[F_{X}(X), F_{Y}(Y)] - 3 \\ &= \rho(F_{X}(X), F_{Y}(Y))\end{split}\]

and its statistical estimate:

\[\hat{\rho}_{S}(X, Y) = 1 - \frac{6}{T(T^2-1)}\sum_{t=1}^{T}(X^{(t)}- Y^{(t)})^2\]

where \(X\) and \(Y\) are univariate random variables, \(F_{X}(X)\) is the cumulative distribution function of \(X\) , \(X^{(t)}\) is the \(t\) -th sorted observation of \(X\) , and \(T\) is the total number of observations”.

Our method is a wrapper for the scipy spearmanr function. For more details about the function and its parameters, please visit scipy documentation.

Implementation

spearmans_rho(x: array, y: array) → float

Calculates a statistical estimate of Spearman’s rho - a copula-based dependence measure.

Formula for calculation: rho = 1 - (6)/(T*(T^2-1)) * Sum((X_t-Y_t)^2)

It is more robust to noise and can be defined if the variables have an infinite second moment. This statistic is described in more detail in the work by Gautier Marti https://www.researchgate.net/publication/322714557 (p.54)

This method is a wrapper for the scipy spearmanr function. For more details about the function and its parameters, please visit scipy documentation https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.spearmanr.html

Parameters:

• x – (np.array/pd.Series) X vector

• y – (np.array/pd.Series) Y vector (same number of observations as X)

Returns:

(float) Spearman’s rho statistical estimate

Generic Parametric Representation (GPR) distance

Theoretically, Marty defines the distance \(d_{\Theta}\) between two random variables as:

” Let \(\theta \in [0, 1]\) . Let \((X, Y) \in \nu^{2}\) , where \(\nu\) is the space of all continuous real-valued random variables. Let \(G = (G_{X}, G_{Y})\) , where \(G_{X}\) and \(G_{Y}\) are respectively \(X\) and \(Y\) marginal cdfs. We define the following distance

\[d_{\Theta}^{2}(X, Y) = \Theta d_{1}^{2}(G_{X}(X), G_{Y}(Y)) + (1 - \Theta) d_{0}^{2}(G_{X}, G_{Y})\]

where

\[d_{1}^{2}(G_{X}(X), G_{Y}(Y)) = 3 \mathbb{E}[|G_{X}(X) - G_{Y}(Y)|^{2}]\]

and

\[d_{0}^{2}(G_{X}, G_{Y}) = \frac{1}{2} \int_{R} (\sqrt{\frac{d G_{X}}{d \lambda}} - \sqrt{\frac{d G_{Y}}{d \lambda}})^{2} d \lambda "\]

For two Gaussian random variables, the distance \(d_{\Theta}\) is therefore defined by Marti as:

” Let \((X, Y)\) be a bivariate Gaussian vector, with \(X \sim \mathcal{N}(\mu_{X}, \sigma_{X}^{2})\) , \(Y \sim \mathcal{N}(\mu_{Y}, \sigma_{Y}^{2})\) and \(\rho (X,Y)\) . We obtain,

\[d_{\Theta}^{2}(X, Y) = \Theta \frac{1 - \rho_{S}}{2} + (1 - \Theta) (1 - \sqrt{\frac{2 \sigma_{X} \sigma_{Y}}{\sigma_{X}^{2} + \sigma_{Y}^{2}}} e^{ - \frac{1}{4} \frac{(\mu_{X} - \mu_{Y})^{2}}{\sigma_{X}^{2} + \sigma_{Y}^{2}}}) "\]

The use of this distance is referenced as the generic parametric representation (GPR) approach.

From the paper:

“GPR distance is a fast and good proxy for distance \(d_{\Theta}\) when the first two moments \(\mu\) and \({\sigma}\) predominate. Nonetheless, for datasets which contain heavy-tailed distributions, GPR fails to capture this information”.

Tip

The process of deriving this definition as well as a proof that \(d_{\Theta}\) is a metric is present in the work: Some contributions to the clustering of financial time series and applications to credit default swaps.

Implementation

gpr_distance(x: array, y: array, theta: float) → float

Calculates the distance between two Gaussians under the Generic Parametric Representation (GPR) approach.

According to the original work https://www.researchgate.net/publication/322714557 (p.70): “This is a fast and good proxy for distance d_theta when the first two moments … predominate”. But it’s not a good metric for heavy-tailed distributions.

Parameter theta defines what type of information dependency is being tested: - for theta = 0 the distribution information is tested - for theta = 1 the dependence information is tested - for theta = 0.5 a mix of both information types is tested

With theta in [0, 1] the distance lies in range [0, 1] and is a metric. (See original work for proof, p.71)

Parameters:

• x – (np.array/pd.Series) X vector.

• y – (np.array/pd.Series) Y vector (same number of observations as X).

• theta – (float) Type of information being tested. Falls in range [0, 1].

Returns:

(float) Distance under GPR approach.

Generic Non-Parametric Representation (GNPR) distance

The statistical estimate of the distance \(\tilde{d}_{\Theta}\) working on realizations of the i.i.d. random variables is defined by the author as:

” Let \((X^{t})_{t=1}^{T}\) and \((Y^{t})_{t=1}^{T}\) be \(T\) realizations of real-valued random variables \(X, Y \in \nu\) respectively. An empirical distance between realizations of random variables can be defined by

\[\tilde{d}_{\Theta}^{2}((X^{t})_{t=1}^{T}, (Y^{t})_{t=1}^{T}) \stackrel{\text{a.s.}}{=} \Theta \tilde{d}_{1}^{2} + (1 - \Theta) \tilde{d}_{0}^{2}\]

where

\[\tilde{d}_{1}^{2} = \frac{3}{T(T^{2} - 1)} \sum_{t = 1}^{T} (X^{(t)} - Y^{(t)}) ^ {2}\]

and

\[\tilde{d}_{0}^{2} = \frac{1}{2} \sum_{k = - \infty}^{+ \infty} (\sqrt{g_{X}^{h}(hk)} - \sqrt{g_{Y}^{h}(hk)})^{2}\]

\(h\) being here a suitable bandwidth, and \(g_{X}^{h}(x) = \frac{1}{T} \sum_{t = 1}^{T} \mathbf{1}(\lfloor \frac{x}{h} \rfloor h \le X^{t} < (\lfloor \frac{x}{h} \rfloor + 1)h)\) being a density histogram estimating dpf \(g_{X}\) from \((X^{t})_{t=1}^{T}\) , \(T\) realization of a random variable \(X \in \nu\) “.

The use of this distance is referenced as the generic non-parametric representation (GNPR) approach.

As written in the paper:

” To use effectively \(d_{\Theta}\) and its statistical estimate, it boils down to select a particular value for \(\Theta\) . We suggest here an exploratory approach where one can test

1. distribution information (θ = 0),

2. dependence information (θ = 1), and

3. a mix of both information (θ = 0,5).

Ideally, \(\Theta\) should reflect the balance of dependence and distribution information in the data. In a supervised setting, one could select an estimate \(\hat{\Theta}\) of the right balance \(\Theta^{*}\) optimizing some loss function by techniques such as cross-validation. Yet, the lack of a clear loss function makes the estimation of \(\Theta^{*}\) difficult in an unsupervised setting”.

Note

The implementation of GNPR in the ArbitrageLab package was adjusted so that \(\tilde{d}_{0}^{2}\) (dependence information distance) is being calculated using the 1D Optimal Transport Distance following the example in the POT package documentation. This solution was proposed by Marti.

Distributions of random variables are approximated using histograms with a given number of bins as input.

Optimal Transport Distance is then obtained from the Optimal Transportation Matrix (OTM) using the Loss Matrix (M) as shown in Optimal Transport blog post by Marti:

\[\tilde{d}_{0}^{2} = tr (OT^{T} * M)\]

where \(tr( \cdot )\) is trace of a matrix and \(\cdot^{T}\) is a transposed matrix.

This approach solves the issue of defining support for underlying distributions and choosing a number of bins.

Implementation

gnpr_distance(x: array, y: array, theta: float, n_bins: int = 50) → float

Calculates the empirical distance between two random variables under the Generic Non-Parametric Representation (GNPR) approach.

Formula for the distance is taken from https://www.researchgate.net/publication/322714557 (p.72).

Parameter theta defines what type of information dependency is being tested: - for theta = 0 the distribution information is tested - for theta = 1 the dependence information is tested - for theta = 0.5 a mix of both information types is tested

With theta in [0, 1] the distance lies in the range [0, 1] and is a metric. (See original work for proof, p.71)

This method is modified as it uses 1D Optimal Transport Distance to measure distribution distance. This solves the issue of defining support and choosing a number of bins. The number of bins can be given as an input to speed up calculations. Big numbers of bins can take a long time to calculate.

Parameters:

• x – (np.array/pd.Series) X vector.

• y – (np.array/pd.Series) Y vector (same number of observations as X).

• theta – (float) Type of information being tested. Falls in range [0, 1].

• n_bins – (int) Number of bins to use to split the X and Y vector observations. (100 by default)

Returns:

(float) Distance under GNPR approach.

Examples

The following example shows how the above functions can be used:

import pandas as pd
from arbitragelab.codependence import (spearmans_rho, gpr_distance, gnpr_distance,
                                       get_dependence_matrix)

# Getting the dataframe with time series of returns
data = pd.read_csv('X_FILE_PATH.csv', index_col=0, parse_dates = [0])

element_x = 'SPY'
element_y = 'TLT'

# Calculating the Spearman's rho coefficient between two time series
rho = spearmans_rho(data[element_x], data[element_y])

# Calculating the GPR distance between two time series with both
# distribution and dependence information
gpr_dist = gpr_distance(data[element_x], data[element_y], theta=0.5)

# Calculating the GNPR distance between two time series with dependence information only
gnpr_dist = gnpr_distance(data[element_x], data[element_y], theta=1)

# Calculating the GNPR distance between all time series with both
# distribution and dependence information
gnpr_matrix = get_dependence_matrix(data, dependence_method='gnpr_distance',
                                    theta=0.5)

Research Notebooks

The following research notebook can be used to better understand the codependence metrics described above.

• Codependence by Marti

Download Notebook Download Sample Data

Presentation Slides

References

• Marti, G., 2017. Some contributions to the clustering of financial time series and applications to credit default swaps (Doctoral dissertation, Université Paris-Saclay (ComUE)).

• Marti Gautier, 2020. Measuring non-linear dependence with Optimal Transport. Available at personal blog.