# Study of the Dynamical Relationships between PM2.5 and PM10 in the Caribbean Area Using a Multiscale Framework

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## Abstract

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## 1. Introduction

## 2. Experimental Data

## 3. Theoretical Framework

#### 3.1. ICEEMDAN

- Compute by EMD the local means of I realizations ${s}^{\left(i\right)}=s+{\beta}_{0}{E}_{1}\left({w}^{\left(i\right)}\right)$ to obtain the first residue:$${r}_{1}=\langle M\left({s}^{\left(i\right)}\right)\rangle $$
- For $k=1$, compute the first mode:$$\tilde{IM{F}_{1}}=s-{r}_{1}$$
- Estimate the second residue as the average of local means of the realizations ${r}_{1}+{\beta}_{1}{E}_{2}\left({w}^{\left(i\right)}\right)$; the second mode is defined as:$$\tilde{IM{F}_{2}}={r}_{1}-{r}_{2}={r}_{1}-\langle M({r}_{1}+{\beta}_{1}{E}_{2}\left({w}^{\left(i\right)}\right))\rangle $$
- For $k=3,...,K$, compute the kth residue:$${r}_{k}=\langle M({r}_{(k-1)}+{\beta}_{(k-1)}{E}_{k}\left({w}^{\left(i\right)}\right))\rangle $$
- Calculate the kth mode:$$\tilde{IM{F}_{k}}={r}_{(k-1)}-{r}_{k}$$
- Repeat step 4 for the next k.

#### 3.2. Hilbert Transform (HT)

#### 3.3. TDIC

- Decompose the two associated time series using ICEEMDAN;
- Determine the HT of each $IMF\left(t\right)$;
- Find the minimum sliding window size (${t}_{d}$) as the maximum instantaneous period (IP) (reciprocal of IF) between the two IMFs at the current position ${t}_{k}$, i.e.,${t}_{d}=max({T}_{1,i}\left({t}_{k}\right),{T}_{2,i}\left({t}_{k}\right))$, where ${T}_{1,i}$ and ${T}_{2,i}$ are IPs;
- Fix the size of the sliding window (SSW) as ${t}_{w}^{n}=[{t}_{k}-n{t}_{d}/2:{t}_{k}+n{t}_{d}/2]$ where n is a multiplication factor usually fixed as unity;
- Find the TDIC of the pair of IMFs as ${R}_{i}\left({t}_{k}^{n}\right)=Corr(IM{F}_{1,i}\left({t}_{w}^{n}\right),IM{F}_{2,i}\left({t}_{w}^{n}\right))$ at any ${t}_{k}$, where $Corr$ is the correlation coefficient of two time series;
- Examine the statistical significance of correlation by t-test;
- Repeat steps 4 to 7 in an iterative manner until the boundary of the sliding window exceeds the endpoints of the time series.

#### 3.4. TDICC

- Decompose the two associated time series using ICEEMDAN;
- Apply HT on the IMFs to calculate the IF, then compute the instantaneous periods);
- Fix the minimum sliding window size for the local correlation computation, which is ${t}_{d}=max({T}_{1,i}\left({t}_{k}\right),{T}_{2,i}({t}_{k}-\tau ))$, where ${T}_{1,i}$ and ${T}_{2,i}$ are instantaneous periods of the two IMFs;
- Find the size of the sliding window (SSW) as ${t}_{w}^{n}=[{t}_{k}-n{t}_{d}/2:{t}_{k}+n{t}_{d}/2]$ for a specific IMF of the first signal (say $PM2.5$) and ${t}_{w}^{n},\tau =[{t}_{k}-\tau -n{t}_{d}/2:{t}_{k}-\tau +n{t}_{d}/2]$ for the corresponding IMF of the second signal, (say $PM10$), where n is any positive number, and is usually selected as 1 [62];
- Determine the running correlation between the two modes along with their statistical significance using the TDICC t-test. This can be repeated until the boundary of the sliding window exceeds the endpoints of the time series.

## 4. Results and Discussion

#### 4.1. Multiscale Decomposition

#### 4.2. Hilbert Spectral Analysis

#### 4.3. TDIC Analysis

#### 4.4. TDICC Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**TDIC plots of $PM10$ vs. $PM2.5$ at different scales. The white spaces imply that the correlation is not significant at the 5% level.

**Figure 8.**TDICC analysis of high-frequency (IMF1). The white spaces imply that the correlation is not significant at the 5% level.

**Figure 9.**TDICC analysis of IMFs at the weekly time scale (IMF2). The white spaces imply that the correlation is not significant at the 5% level.

**Figure 10.**TDICC analysis of IMFs at the ∼2-week time scale (IMF3). The white spaces imply that the correlation is not significant at the 5% level.

**Figure 11.**TDICC analysis of IMFs at the ∼3-week time scale (IMF4). The white spaces imply that the correlation is not significant at the 5% level.

**Figure 12.**TDICC analysis of IMFs at the intra-annual time scale of ∼1.5 months (IMF5). The white spaces imply that the correlation is not significant at the 5% level.

**Figure 13.**TDICC analysis of IMFs at the quarterly time scale (IMF6). The white spaces imply that the correlation is not significant at the 5% level.

**Figure 14.**TDICC analysis of IMFs at the semester time scale (IMF). The white spaces imply that the correlation is not significant at the 5% level.

**Figure 15.**TDICC analysis of IMFs at the annual time scale (IMF8). The white spaces imply that the correlation is not significant at the 5% level.

**Table 1.**Mean period (days) and variability (%) explained for the different modes of the $PM10$ and $PM2.5$ time series.

$\mathit{P}\mathit{M}10$ | $\mathit{P}\mathit{M}2.5$ | |||
---|---|---|---|---|

Modes | Mean Period (Days) | Variability (%) | Mean Period (Days) | Variability (%) |

IMF1 | 3.102 | 22.698 | 2.866 | 17.265 |

IMF2 | 6.403 | 18.962 | 6.844 | 14.432 |

IMF3 | 12.303 | 14.040 | 13.353 | 15.795 |

IMF4 | 23.297 | 7.331 | 24.333 | 6.477 |

IMF5 | 40.555 | 6.553 | 47.608 | 11.200 |

IMF6 | 73.000 | 3.486 | 91.250 | 10.264 |

IMF | 182.500 | 7.914 | 219.000 | 15.535 |

IMF8 | 365.000 | 13.676 | 365.000 | 5.451 |

Residue | 1095.000 | 5.337 | 1095.000 | 3.577 |

**Table 2.**Mean and trend of instantaneous amplitudes along with the mean frequencies of different IMFs. Z represents the Mann–Kendall value.

Parameter | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 | IMF | IMF8 | |
---|---|---|---|---|---|---|---|---|---|

MA | 8.690 | 7.164 | 6.049 | 4.587 | 4.828 | 3.994 | 5.320 | 8.667 | |

$PM10$ | Z value of IA | −2.59 | −8.06 | −4.82 | −1.75 | −10.78 | −15.50 | 15.61 | 28.61 |

MF | 0.305 | 0.155 | 0.077 | 0.043 | 0.024 | 0.013 | 0.005 | 0.003 | |

MA | 1.609 | 1.459 | 1.511 | 0.999 | 1.302 | 1.372 | 1.645 | 0.983 | |

$PM2.5$ | Z value of IA | −12.43 | −4.99 | −4.76 | 4.36 | −3.13 | −23.03 | 0.38 | 49.05 |

MF | 0.324 | 0.145 | 0.072 | 0.041 | 0.021 | 0.011 | 0.005 | 0.002 |

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**MDPI and ACS Style**

Plocoste, T.; Sankaran, A.; Euphrasie-Clotilde, L.
Study of the Dynamical Relationships between *PM*2.5 and *PM*10 in the Caribbean Area Using a Multiscale Framework. *Atmosphere* **2023**, *14*, 468.
https://doi.org/10.3390/atmos14030468

**AMA Style**

Plocoste T, Sankaran A, Euphrasie-Clotilde L.
Study of the Dynamical Relationships between *PM*2.5 and *PM*10 in the Caribbean Area Using a Multiscale Framework. *Atmosphere*. 2023; 14(3):468.
https://doi.org/10.3390/atmos14030468

**Chicago/Turabian Style**

Plocoste, Thomas, Adarsh Sankaran, and Lovely Euphrasie-Clotilde.
2023. "Study of the Dynamical Relationships between *PM*2.5 and *PM*10 in the Caribbean Area Using a Multiscale Framework" *Atmosphere* 14, no. 3: 468.
https://doi.org/10.3390/atmos14030468