# High-Dimensional Radial Symmetry of Copula Functions: Multiplier Bootstrap vs. Randomization

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- draw $\tau $ as an independently distributed vector of Bernoulli random variables and set$$\left(\right)open="("\; close=")">{\mathbf{U}}_{1}^{\tau},\dots ,{\mathbf{U}}_{n}^{\tau}.$$
- draw n independent standard uniform random variables ${\eta}_{1},\dots ,{\eta}_{n}$ and set, for $i=1,\dots ,n$$${\mathbf{U}}_{i}^{\tau ,\eta}={\mathbf{U}}_{i}^{\tau}-{n}^{-1}{\mathbf{1}}_{d}{\eta}_{i}.$$
- For $i=1,\dots ,n$ and $j=1,\dots ,d$ compute$${V}_{ij}^{\tau ,\eta}={\displaystyle \frac{1}{n}}\sum _{k=1}^{n}\mathbb{I}\left(\right)open="("\; close=")">{U}_{kj}^{\tau ,\eta}\le {U}_{ij}^{\tau ,\eta}$$
- Compute$$\begin{array}{ccc}\hfill {C}_{n}^{\tau ,\eta}\left(\mathbf{u}\right)& =& {\displaystyle {\displaystyle \frac{1}{n}}\sum _{i=1}^{n}\mathbb{I}\left(\right)open="("\; close=")">{\mathbf{V}}_{n}^{\tau ,\eta}\le \mathbf{u}}\hfill \end{array}\hfill {S}_{n}^{\tau}\left(\right)open="("\; close=")">{\mathbf{V}}_{1}^{\tau ,\eta},\dots ,{\mathbf{V}}_{n}^{\tau ,\eta}\\ =& {\displaystyle \frac{1}{n}}\sum _{i=1}^{n}{\left(\right)}^{{C}_{n}^{\tau}}-{\overline{C}}_{n}^{\tau ,\eta}\left(\right)open="("\; close=")">{\mathbf{V}}_{i}^{\tau ,\eta}\hfill & 2$$
- repeat steps 1–4 for a large number of times M and compute the approximate p-value$$\begin{array}{ccc}\hfill {\widehat{P}}_{rand}& =& {\displaystyle {\displaystyle \frac{1}{M}}\sum _{m=1}^{M}\mathbb{I}\left(\right)open="("\; close=")">{S}_{n}^{\left[m\right],rand}{S}_{n}.}\hfill \end{array}$$

**Lemma**

**1.**

**Lemma**

**2.**

**Hypothesis**

**1.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proposition**

**1.**

## 3. Simulation Study

#### 3.1. Elliptical Family

#### 3.2. Archimedean Family

## 4. Empirical Application

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs and Auxiliary Results

**Lemma**

**A1.**

**Proof.**

#### Appendix A.1. Proof of Lemma 1

**Proof.**

- (A)
- Maneageability of the ${f}_{ni}$’s with envelopes ${F}_{ni}={n}^{-1/2}$ is a consequence of the monotonicity of the ${f}_{ni}$, as discussed in [23].
- (B)
- If $i\ne j$, using the independence of ${\tau}_{i}$ and ${\tau}_{j}$, it follows that ${\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{f}_{ni}\left(\mathbf{u}\right){f}_{nj}\left(\mathbf{v}\right){\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{f}_{nj}\left(\mathbf{v}\right)$. In the case $i=j$ we obtain:$$\begin{array}{ccc}\hfill & & {\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{f}_{ni}\left(\mathbf{u}\right){f}_{ni}\left(\mathbf{v}\right)-{\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{f}_{ni}\left(\mathbf{u}\right)\hfill & {\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{f}_{ni}\left(\mathbf{v}\right)\end{array}$$Then, we the following expectation can be computed:$$\begin{array}{c}\hfill {\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{\mathbb{G}}_{n}^{\tau}\left(\mathbf{u}\right){\mathbb{G}}_{n}^{\tau}\left(\mathbf{v}\right)\\ =& {\displaystyle \frac{1}{4}}\left(\right)open="["\; close="]">{C}_{n}\left(\right)open="("\; close=")">\mathbf{u}\wedge \mathbf{v}+{\overline{C}}_{n}\left(\right)open="("\; close=")">\mathbf{u}\wedge \mathbf{v}\hfill \end{array}$$Using Lemma A1 it follows that$$\begin{array}{c}\hfill \underset{n\to \infty}{lim}{\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{\mathbb{G}}_{n}^{\tau}\left(\mathbf{u}\right){\mathbb{G}}_{n}^{\tau}\left(\mathbf{v}\right)\\ =& {\displaystyle \frac{1}{4}}\left(\right)open="["\; close>C\left(\right)open="("\; close=")">\mathbf{u}\wedge \mathbf{v}+\overline{C}\left(\right)open="("\; close=")">\mathbf{u}\wedge \mathbf{v}\hfill \end{array}$$
- (C)
- ${lim\; sup}_{n\to \infty}{\sum}_{i=1}^{n}{\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{F}_{ni}^{2}$.
- (D)
- ${\sum}_{i=1}^{n}{\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{F}_{ni}^{2}\mathbb{I}\left(\right)open="("\; close=")">{F}_{ni}\u03f5=\mathbb{I}\left(\right)open="("\; close=")">{n}^{-1/2}\u03f5$ for each $\u03f5$.
- (E)
- ${\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{\left(\right)}^{{f}_{ni}}2={\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{f}_{ni}{\left(\mathbf{u}\right)}^{2}$$$\begin{array}{c}\hfill {\rho}_{n}{\left(\right)}^{\mathbf{u}}2=\sum _{i=1}^{n}{\mathbb{E}}_{\tau}\left(\right)open="["\; close="]">{\left(\right)}^{{f}_{ni}}2\\ .\end{array}$$Using the same line of reasoning of [22] in their proof of Lemma 4.1 point (E), we obtain$$\begin{array}{c}\hfill {\rho}_{n}{\left(\right)}^{\mathbf{u}}2\\ =& {\displaystyle \frac{1}{2}}{C}_{n}\left(\mathbf{u}\right)+{\displaystyle \frac{1}{2}}{\overline{C}}_{n}\left(\mathbf{u}\right)+{\displaystyle \frac{1}{2}}{C}_{n}\left(\mathbf{v}\right)+{\displaystyle \frac{1}{2}}{\overline{C}}_{n}\left(\mathbf{v}\right)\hfill \end{array}$$$$\begin{array}{c}\hfill \rho {\left(\right)}^{\mathbf{u}}2\\ =& {\displaystyle \frac{1}{2}}C\left(\mathbf{u}\right)+{\displaystyle \frac{1}{2}}\overline{C}\left(\mathbf{u}\right)+{\displaystyle \frac{1}{2}}C\left(\mathbf{v}\right)+{\displaystyle \frac{1}{2}}\overline{C}\left(\mathbf{v}\right)\hfill \end{array}$$

#### Appendix A.2. Proof of Lemma 2

**Proof.**

#### Appendix A.3. Proof of Lemma 3

**Proof.**

#### Appendix A.4. Proof of Lemma 4

**Proof.**

#### Appendix A.5. Proof of Proposition 1

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**Figure 1.**Average Linear correlation, Spearman’s $\rho $, and Kendall’s $\tau $, computed using a rolling window of one year. Recessions and slowdown periods are from Eurostat’s Business Cycle Clock (BCC).

**Table 1.**Running Times of ${S}_{n}^{mult}$, ${S}_{n}^{rand}$, as estimated from 1000 replicates, in the $n=250$, $d=100$ case, under the Frank copula model.

Mean (s) | Max (s) | Min (s) | |
---|---|---|---|

${S}_{n}^{plain}$ | 2.23 | 2.06 | 4.42 |

${S}_{n}^{rand}$ | 10.54 | 9.96 | 24.61 |

**Table 2.**Rejection percentages at 5% significance level, as estimated from 1000 replicates, for the tests based on ${S}_{n}^{mult}$ and ${S}_{n}^{rand}$ under the normal copula.

d = 2 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.055 | 0.045 | 0.025 | 0.050 | 0.034 | 0.026 | 0.038 | 0.036 | 0.020 |

${S}_{n}^{rand}$ | 0.062 | 0.059 | 0.044 | 0.059 | 0.048 | 0.056 | 0.041 | 0.043 | 0.033 |

d = 5 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.039 | 0.012 | 0.001 | 0.028 | 0.028 | 0.000 | 0.039 | 0.038 | 0.022 |

${S}_{n}^{rand}$ | 0.092 | 0.066 | 0.048 | 0.071 | 0.073 | 0.052 | 0.056 | 0.051 | 0.046 |

d = 10 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.006 | 0.001 | 0.000 | 0.007 | 0.010 | 0.000 | 0.030 | 0.024 | 0.004 |

${S}_{n}^{rand}$ | 0.215 | 0.073 | 0.058 | 0.127 | 0.064 | 0.051 | 0.075 | 0.053 | 0.056 |

d = 50 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

${S}_{n}^{rand}$ | 0.877 | 0.149 | 0.097 | 0.684 | 0.087 | 0.070 | 0.328 | 0.066 | 0.058 |

d = 100 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

${S}_{n}^{rand}$ | 0.982 | 0.171 | 0.100 | 0.942 | 0.101 | 0.073 | 0.725 | 0.070 | 0.059 |

**Table 3.**Rejection percentages at 5% significance level, as estimated from 1000 replicates, for the tests based on ${S}_{n}^{mult}$ and ${S}_{n}^{rand}$ under the Student-t copula with $\nu =4$ degrees of freedom.

d = 2 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.040 | 0.049 | 0.022 | 0.043 | 0.046 | 0.024 | 0.054 | 0.054 | 0.033 |

${S}_{n}^{rand}$ | 0.050 | 0.058 | 0.049 | 0.061 | 0.055 | 0.046 | 0.059 | 0.068 | 0.046 |

d = 5 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.036 | 0.011 | 0.000 | 0.040 | 0.018 | 0.000 | 0.046 | 0.039 | 0.029 |

${S}_{n}^{rand}$ | 0.104 | 0.064 | 0.050 | 0.088 | 0.059 | 0.048 | 0.065 | 0.055 | 0.058 |

d = 10 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.006 | 0.002 | 0.000 | 0.016 | 0.006 | 0.000 | 0.032 | 0.036 | 0.005 |

${S}_{n}^{rand}$ | 0.193 | 0.085 | 0.075 | 0.116 | 0.081 | 0.067 | 0.081 | 0.060 | 0.071 |

d = 50 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

${S}_{n}^{rand}$ | 0.818 | 0.152 | 0.087 | 0.631 | 0.097 | 0.075 | 0.305 | 0.069 | 0.059 |

d = 100 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

${S}_{n}^{rand}$ | 0.959 | 0.196 | 0.098 | 0.900 | 0.114 | 0.088 | 0.674 | 0.085 | 0.058 |

**Table 4.**Rejection percentages at 5% significance level, as estimated from 1000 replicates, for the tests based on ${S}_{n}^{mult}$ and ${S}_{n}^{rand}$ under the Frank copula.

d = 2 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.053 | 0.052 | 0.016 | 0.042 | 0.039 | 0.017 | 0.044 | 0.048 | 0.030 |

${S}_{n}^{rand}$ | 0.061 | 0.052 | 0.040 | 0.050 | 0.052 | 0.040 | 0.050 | 0.054 | 0.048 |

d = 5 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.016 | 0.012 | 0.000 | 0.134 | 0.028 | 0.001 | 0.781 | 0.452 | 0.031 |

${S}_{n}^{rand}$ | 0.081 | 0.074 | 0.042 | 0.302 | 0.123 | 0.057 | 0.882 | 0.614 | 0.103 |

d = 10 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.049 | 0.003 | 0.000 | 0.722 | 0.157 | 0.000 | 1.000 | 1.000 | 0.087 |

${S}_{n}^{rand}$ | 0.533 | 0.249 | 0.057 | 0.976 | 0.789 | 0.109 | 1.000 | 1.000 | 0.779 |

d = 50 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.993 | 1.000 | 0.008 |

${S}_{n}^{rand}$ | 0.998 | 0.996 | 0.313 | 1.000 | 1.000 | 0.917 | 1.000 | 1.000 | 1.000 |

d = 100 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.968 | 0.000 |

${S}_{n}^{rand}$ | 1.000 | 0.999 | 0.606 | 1.000 | 1.000 | 0.989 | 1.000 | 1.000 | 1.000 |

**Table 5.**Rejection percentages at 5% significance level, as estimated from 1000 replicates, for the tests based on ${S}_{n}^{mult}$ and ${S}_{n}^{rand}$ under the Clayton copula.

d = 2 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.071 | 0.087 | 0.052 | 0.122 | 0.178 | 0.120 | 0.235 | 0.460 | 0.446 |

${S}_{n}^{rand}$ | 0.082 | 0.112 | 0.106 | 0.130 | 0.209 | 0.209 | 0.265 | 0.495 | 0.544 |

d = 5 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.085 | 0.075 | 0.000 | 0.378 | 0.392 | 0.019 | 0.983 | 0.987 | 0.738 |

${S}_{n}^{rand}$ | 0.167 | 0.232 | 0.142 | 0.534 | 0.603 | 0.396 | 0.990 | 0.993 | 0.897 |

d = 10 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.052 | 0.008 | 0.000 | 0.810 | 0.635 | 0.002 | 1.000 | 1.000 | 0.929 |

${S}_{n}^{rand}$ | 0.402 | 0.493 | 0.210 | 0.942 | 0.975 | 0.614 | 1.000 | 1.000 | 1.000 |

d = 50 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.998 | 1.000 | 0.084 |

${S}_{n}^{rand}$ | 0.856 | 0.993 | 0.712 | 0.994 | 1.000 | 0.998 | 1.000 | 1.000 | 1.000 |

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.704 | 0.000 |

${S}_{n}^{rand}$ | 0.898 | 0.994 | 0.927 | 0.996 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

**Table 6.**Rejection percentages at 5% significance level, as estimated from 1000 replicates, for the tests based on ${S}_{n}^{mult}$ and ${S}_{n}^{rand}$ under the Gumbel copula.

d = 2 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.066 | 0.090 | 0.047 | 0.092 | 0.163 | 0.106 | 0.237 | 0.457 | 0.416 |

${S}_{n}^{rand}$ | 0.099 | 0.128 | 0.085 | 0.146 | 0.236 | 0.213 | 0.305 | 0.537 | 0.537 |

d = 5 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.082 | 0.075 | 0.001 | 0.388 | 0.393 | 0.026 | 0.977 | 0.983 | 0.751 |

${S}_{n}^{rand}$ | 0.353 | 0.360 | 0.187 | 0.700 | 0.701 | 0.427 | 0.994 | 0.996 | 0.929 |

d = 10 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.058 | 0.013 | 0.000 | 0.816 | 0.621 | 0.001 | 1.000 | 1.000 | 0.923 |

${S}_{n}^{rand}$ | 0.696 | 0.666 | 0.278 | 0.988 | 0.986 | 0.694 | 1.000 | 1.000 | 1.000 |

d = 50 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.996 | 1.000 | 0.086 |

${S}_{n}^{rand}$ | 0.896 | 0.996 | 0.757 | 0.999 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

d = 100 | n = 50 | n = 100 | n = 250 | ||||||

$\tau $ | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5 | 0.75 |

${S}_{n}^{mult}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.708 | 0.000 |

${S}_{n}^{rand}$ | 0.890 | 0.997 | 0.928 | 0.995 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

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## Share and Cite

**MDPI and ACS Style**

Billio, M.; Frattarolo, L.; Guégan, D.
High-Dimensional Radial Symmetry of Copula Functions: Multiplier Bootstrap vs. Randomization. *Symmetry* **2022**, *14*, 97.
https://doi.org/10.3390/sym14010097

**AMA Style**

Billio M, Frattarolo L, Guégan D.
High-Dimensional Radial Symmetry of Copula Functions: Multiplier Bootstrap vs. Randomization. *Symmetry*. 2022; 14(1):97.
https://doi.org/10.3390/sym14010097

**Chicago/Turabian Style**

Billio, Monica, Lorenzo Frattarolo, and Dominique Guégan.
2022. "High-Dimensional Radial Symmetry of Copula Functions: Multiplier Bootstrap vs. Randomization" *Symmetry* 14, no. 1: 97.
https://doi.org/10.3390/sym14010097