# Uncertainty Analysis of Greenhouse Gas (GHG) Emissions Simulated by the Parametric Monte Carlo Simulation and Nonparametric Bootstrap Method

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

#### 2.2. Uncertainty Analysis by the Parametric MCS Method

#### 2.3. Unertainty Analysis by the Nonparametric Bootstrap Method

## 3. Results and Discussion

_{2}-eq/kg (kwh for electricity, liter for diesel), respectively. There are six different batches of the dataset, and all are subsets of the dataset of n = 72 (Table A1). Each subset has a differing length of n, where n is 12, 24, 36, 48, 60, and 72.

#### 3.1. The Effect of the Size of the Dataset (n) on the Uncertainty of the Model Output

_{ab}, EF × (f

_{ab}+ 2 × (1 − f

_{ab}))] was assumed, where EF is an emission factor and “f

_{ab}” is a multiplication factor to generate uniform variates for A. The value of f

_{ab}varies from 0.0 to 1.0. Scenario S3 computes the mean and variance of Z by bootstrapping both X and A. Table 1 shows the summary of the three different scenarios described above.

#### 3.2. The Effect of Treating the Emission Factor (Coefficient Vector) as a Random Variable on the Uncertainty of the Model Output

_{ab}on the U and mean GHG emission for the bootstrap method. The value of U decreases by approximately 90% when f

_{ab}varies from 0.0 to 1. This indicates that the variability of the coefficient affects the U. When f

_{ab}was 0.0, the U was 8.8%. When f

_{ab}reached 0.9, no further changes occurred to the U. Thus, treating the coefficient as a random variable increases the uncertainty of Z or U. This is because the variance of the dataset increases due to increased variance of the coefficient.

_{2}-eq/head-year at f

_{ab}of 0.0, and 438.5 kg CO

_{2}-eq/head-year at f

_{ab}of 1.0. This indicates that treating the coefficient in the mathematical model as a random variable does not alter the mean of Z. It affects adversely, however, the uncertainty of Z. The varying coefficient case yielded higher uncertainty of Z over the constant coefficient case.

#### 3.3. The Effect of Probability Destribution on the Uncertainty of the Model

#### 3.4. Comparison of the Confidence Interval Computation Method

#### 3.5. The Effect of Different Uncertainty Analysis Methods on the Uncertainty of the Model Output

## 4. Conclusions

- There is a certain number of observations of a dataset required before an asymptotic value of the model output can be reached. In this study, the number of observations was 36 for Scenario S3 and 60 for scenarios S1 and S2.
- Bootstrapping reduces the variance and standard error of Z. This is because bootstrapping generates a bootstrapped dataset resembling the population of the original dataset. The variance of Z was the smallest in Scenario S3 (bootstrapping both A and X), followed by in Scenario S2 (bootstrapping X only), and the largest was observed in Scenario S1 (no bootstrapping).
- Uncertainty analysis of GHG emissions should be based on the nonparametric bootstrap method, not the parametric MCS method, when the estimated PDF of the original dataset is incorrect.
- A mathematical model for estimating GHG emissions of a system should consider treating the GHG emission factor as a random variable.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Month | Feed for Dry Cows ^{1} | Feed for Lactating Cows ^{1} | Straw ^{1} | Soybean Meal ^{1} | Electricity ^{2} | Diesel ^{3} |
---|---|---|---|---|---|---|

Mon. | X1 | X2 | X3 | X4 | X5 | X6 |

1 | 114 | 299 | 54 | 13 | 161 | 5 |

2 | 260 | 411 | 13 | 17 | 113 | 5 |

3 | 278 | 320 | 34 | 18 | 121 | 3 |

4 | 236 | 676 | 23 | 46 | 160 | 22 |

5 | 248 | 741 | 38 | 14 | 225 | 5 |

6 | 183 | 384 | 33 | 16 | 216 | 5 |

7 | 358 | 443 | 35 | 13 | 109 | 6 |

8 | 172 | 786 | 27 | 19 | 125 | 10 |

9 | 83 | 294 | 33 | 14 | 134 | 5 |

10 | 73 | 221 | 18 | 16 | 27 | 4 |

11 | 8 | 241 | 16 | 19 | 35 | 5 |

12 | 57 | 239 | 5 | 4 | 138 | 13 |

13 | 114 | 343 | 26 | 17 | 103 | 4 |

14 | 260 | 365 | 25 | 14 | 109 | 3 |

15 | 278 | 367 | 33 | 19 | 112 | 3 |

16 | 236 | 422 | 31 | 12 | 128 | 5 |

17 | 248 | 380 | 32 | 19 | 120 | 4 |

18 | 183 | 405 | 30 | 20 | 133 | 4 |

19 | 358 | 405 | 29 | 18 | 131 | 5 |

20 | 172 | 389 | 31 | 22 | 119 | 4 |

21 | 83 | 387 | 30 | 15 | 111 | 4 |

22 | 73 | 390 | 29 | 12 | 112 | 5 |

23 | 8 | 418 | 34 | 21 | 126 | 5 |

24 | 57 | 407 | 29 | 23 | 121 | 5 |

25 | 114 | 377 | 24 | 20 | 121 | 3 |

26 | 260 | 348 | 22 | 16 | 99 | 4 |

27 | 278 | 371 | 28 | 21 | 116 | 4 |

28 | 236 | 389 | 29 | 18 | 119 | 5 |

29 | 248 | 383 | 29 | 17 | 118 | 4 |

30 | 183 | 396 | 30 | 19 | 123 | 4 |

31 | 358 | 400 | 34 | 21 | 123 | 4 |

32 | 172 | 403 | 33 | 23 | 126 | 6 |

33 | 83 | 406 | 34 | 28 | 126 | 4 |

34 | 73 | 425 | 34 | 16 | 129 | 4 |

35 | 8 | 415 | 36 | 17 | 133 | 4 |

36 | 57 | 401 | 29 | 14 | 116 | 4 |

37 | 114 | 390 | 26 | 19 | 115 | 4 |

38 | 260 | 358 | 22 | 19 | 106 | 5 |

39 | 278 | 417 | 18 | 23 | 145 | 5 |

40 | 236 | 432 | 33 | 21 | 141 | 5 |

41 | 248 | 455 | 32 | 15 | 144 | 4 |

42 | 183 | 436 | 32 | 18 | 136 | 6 |

43 | 358 | 349 | 26 | 17 | 108 | 4 |

44 | 172 | 401 | 27 | 18 | 122 | 5 |

45 | 83 | 376 | 29 | 18 | 115 | 4 |

46 | 73 | 410 | 29 | 22 | 122 | 5 |

47 | 8 | 361 | 29 | 10 | 112 | 5 |

48 | 57 | 378 | 29 | 24 | 121 | 5 |

49 | 114 | 408 | 31 | 17 | 122 | 5 |

50 | 260 | 355 | 31 | 9 | 112 | 4 |

51 | 278 | 366 | 26 | 15 | 107 | 4 |

52 | 236 | 405 | 34 | 13 | 123 | 5 |

53 | 248 | 340 | 32 | 23 | 109 | 3 |

54 | 183 | 384 | 31 | 18 | 116 | 2 |

55 | 358 | 454 | 34 | 21 | 136 | 3 |

56 | 172 | 406 | 32 | 21 | 119 | 5 |

57 | 83 | 416 | 33 | 15 | 123 | 4 |

58 | 73 | 423 | 29 | 20 | 119 | 4 |

59 | 8 | 380 | 27 | 19 | 118 | 4 |

60 | 57 | 383 | 29 | 17 | 123 | 4 |

61 | 114 | 355 | 30 | 14 | 107 | 4 |

62 | 260 | 351 | 25 | 16 | 109 | 1 |

63 | 278 | 381 | 28 | 22 | 117 | 7 |

64 | 236 | 381 | 31 | 15 | 123 | 6 |

65 | 248 | 444 | 34 | 25 | 138 | 5 |

66 | 183 | 403 | 27 | 19 | 122 | 4 |

67 | 358 | 406 | 33 | 16 | 120 | 3 |

68 | 172 | 454 | 38 | 20 | 142 | 5 |

69 | 83 | 448 | 37 | 26 | 126 | 4 |

70 | 73 | 380 | 23 | 22 | 115 | 3 |

71 | 8 | 386 | 35 | 22 | 114 | 4 |

72 | 57 | 375 | 22 | 14 | 113 | 4 |

EF 1 | 0.38 | 0.64 | 0.95 | 0.71 | 0.50 | 3.3 |

^{1}kg CO

_{2}-eq/kg;

^{2}kWh electricity/head-month;

^{3}kg diesel/head-month.

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**Figure 4.**Cumulative distribution function (CDF) plots of the empirical distribution and fitted (theoretical) distributions against the input values of x for the lactating cow feed data.

Scenario | Var(Z) Calculation | Treatment of A Vector |
---|---|---|

S1 | Error propagation method (no bootstrapping) | Keeping A vector constant |

S2 | Bootstrapping X matrix only | Keeping A vector constant |

S3 | Bootstrapping both X matrix and A vector | Treating A vector as a random variable |

Goodness-of-Fit Criterion | Probability Distribution | ||||
---|---|---|---|---|---|

normal | lognormal | Weibull | gamma | loglogistic | |

AIC | 845.9 | 826.5 | 864.9 | 830.9 | 803.5 |

BIC | 850.5 | 831.1 | 869.5 | 835.4 | 808.1 |

Column (Data) Name | Distribution | Mean | Sd ^{1} | Shape | Scale |
---|---|---|---|---|---|

Dry cow feed | Weibull | 1.57 | 189.82 | ||

Lactating cow feed | loglogistic | 12.28 | 390.54 | ||

Straw | normal | 29.36 | 6.36 | ||

Soy bean meal | loglogistic | 6.97 | 17.81 | ||

Electricity | loglogistic | 10.92 | 120.97 | ||

Diesel | loglogistic | 6.20 | 4.36 |

^{1}Standard deviation.

Statistic | Percentile | Normal | Basic |
---|---|---|---|

95% CI | (418.13–458.57) | (418.55–458.00) | (416.88–457.32) |

CI width | 40.44 | 39.45 | 40.44 |

U (%) | 4.64 | 4.51 | 4.64 |

**Table 5.**Uncertainty of GHG emissions of the three uncertainty analysis methods: no simulation, nonparametric bootstrap and parametric Monte Carlo simulation (MCS) (unit: kg CO

_{2}-eq/head-year).

Statistic | Uncertainty Analysis Method | ||
---|---|---|---|

No Simulation | Nonparametric Bootstrap | Parametric MCS | |

Mean | 438.27 | 437.72 | 435.36 |

Bias | 0 | −0.55 | −2.91 |

Standard error | 61.66 | 10.07 | 105.47 |

95% CI | (317.41–559.14) | (418.13–458.57) | (263.64–679.29) |

CI width | 241.73 | 40.44 | 415.65 |

U (%) | 27.60 | 4.64 | 47.74 |

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**MDPI and ACS Style**

Lee, K.M.; Lee, M.H.; Lee, J.S.; Lee, J.Y. Uncertainty Analysis of Greenhouse Gas (GHG) Emissions Simulated by the Parametric Monte Carlo Simulation and Nonparametric Bootstrap Method. *Energies* **2020**, *13*, 4965.
https://doi.org/10.3390/en13184965

**AMA Style**

Lee KM, Lee MH, Lee JS, Lee JY. Uncertainty Analysis of Greenhouse Gas (GHG) Emissions Simulated by the Parametric Monte Carlo Simulation and Nonparametric Bootstrap Method. *Energies*. 2020; 13(18):4965.
https://doi.org/10.3390/en13184965

**Chicago/Turabian Style**

Lee, Kun Mo, Min Hyeok Lee, Jong Seok Lee, and Joo Young Lee. 2020. "Uncertainty Analysis of Greenhouse Gas (GHG) Emissions Simulated by the Parametric Monte Carlo Simulation and Nonparametric Bootstrap Method" *Energies* 13, no. 18: 4965.
https://doi.org/10.3390/en13184965