# Non-Linear Regression Modelling to Estimate the Global Warming Potential of a Newspaper

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. System Definition and Baseline LCA Model

- Prepress: platemaking processes, processing, folded, and perforated plates.
- Printing: the entire supply process (paper, chemicals, ink…), maintenance, and cleaning of the printing machine.
- Finishing: transportation of the printed product, stacking, packing, and delivery.
- Distribution.

^{2}is considered. As functional units, both FU-1: per kg of paper and FU-2: per unit of newspaper, are used. Both functional units are related by: FU-2 = FU-1 × (weight × page surface × number of pages/2). Other scenarios consider different product specifications: run (copies), weight (gr/m

^{2}), size, location and technology of the paper factory, location of the printing plant, distribution locations, and electricity source and mix. The methodology ReCiPe [15,16] is used in order to calculate the midpoint impacts [17] for different categories of damage: human health, ecosystems, and cost of resources [12,18]. As a result, the article shows the environmental impacts for each of the scenarios.

#### 2.2. Data Used in the Model

- Number of pages: On the one hand, in the newspaper used for this study, the number of pages that can be printed simultaneously varies by multiples of 16. On the other hand, the minimum number of pages that a newspaper usually has, which corresponds to the sports type, is 32 pages and the maximum is 64. In our model, we have therefore considered 32-, 48- and 64-page newspapers.
- Grammage: The grammage represents the mass of paper per printed surface. Newspapers and magazines on high-quality newsprint weigh more than 60 g/m
^{2}. In the case of sports newspapers and daily newspapers, the usual weight is 45 g/m^{2}. In the model, we have considered weights of 42, 45 and 48.8 g/m^{2}. - Height: The surface of the paper that forms a newspaper page is delimited by the width, which depends on the development of the printing rollers and which is usually a fixed parameter; in our case it has a value of 289 mm. The height varies according to the length of the paper rolls used. Three different paper sizes have been considered, resulting in heights of 360, 390 and 420 mm.
- Paper type: The "type of paper used" parameter includes several variables:
- -
- The location of the paper mill, which is related to the local energy mix. In areas of northern Europe and Canada, there is a large amount of hydroelectric power; in central Spain, the energy mix contains mainly fossil fuels.
- -
- The transport to the printing plant, which is usually by train, truck and ship. It is necessary to consider the kilometres covered by each means.
- -
- The printing technology and the raw materials are recycled in different percentages.

- Madrid: 100% recycled deinked pulp.
- Belgium 100% recycled deinked pulp.
- Sweden: 50% recycled deinked pulp.
- Canada: 0% recycled deinked pulp.

- Print run (number of copies): The number of copies to be printed is a parameter that varies depending on the print run requested by each publisher. In the current market, print runs are decreasing because the reading of printed paper is being replaced by reading through digital devices. Printing technology and new machines have been adapted to produce saleable copies with very low print runs. The machine under consideration may have saleable copies from 250 invalid copies. A newspaper has fixed costs for capital, labour, and printing plates that are distributed among the printed copies, so that the unit cost and impacts of a newspaper decrease as the circulation increases. We have considered variations in print runs between 500 and 50,000 copies.

_{2}eq., was used.

#### 2.3. The Multiple Regression Functions

_{i}is the random noise of each observation of the response variable; and $\mathsf{\beta}={({\beta}_{1},{\beta}_{2},\dots ,{\beta}_{p})}^{T}$ are the $p$ coefficients, or parameters, which define the relationship between the input variables and response variable in function $f$.

#### 2.3.1. Description of Non-Linear Regression Models Used

#### 2.3.2. Algorithm for the Estimation of β Unknown Parameters

**β**unknown parameters is carried out with the Levenberg–Marquard (LM) Algorithm. This is a hybrid optimization technique that uses both Gauss–Newton and steepest descent approaches to converge to an optimal solution [21]. It takes advantage of the high speed of the Gauss–Newton algorithm and the high stability of the steepest descent method [22]. In this work, the algorithm finds the best set of unknown parameters in order to minimize error between the response variable (environmental impacts obtained from the non-linear functions) and the actual values (actual observations of the response). Basically, the LM algorithm provides a numerical solution to the problem of minimizing a nonlinear function, over a space of parameters for the function (

**β**).

#### 2.4. Metrics Used to Evaluate Numerical Estimations of the Models

#### 2.5. Method for the Training and Testing of the Four Models Analysed

- Desired variables are selected, and missing values are checked to delete all the sample rows (if missing data exist in some variable of the observation).
- Samples are randomized by rows, to guarantee the data is representative.
- The whole dataset is divided in 10 parts, or folds, of the same dimension (each one with the same number of samples).
- Nine data folds (90% of the total dataset samples) are considered a temporal training dataset and they are used to train the model. The remaining fold (10% of samples) is considered a temporal data test set and it is used to carry out the testing of the model.
- With the training dataset, the models defined by Equations (2)–(5) are adjusted and unknown
**β**parameters are estimated using the Levenberg–Marquardt nonlinear least squares algorithm [19]. - This procedure is repeated 10 times for each model, to rotate the test dataset and obtain 10 evaluations in each case. As results, 10 values of error, calculated following a statistic metric formulation, are obtained. The statistical metrics considered measure the error between the true value of the response and the estimated value obtained by the model. In this study, the error metrics considered were MAE, MAPE and R-squared.
- Finally, the average mean values of the 10 values obtained for each metric and standard deviation are calculated.

## 3. Results

_{2}/kg paper) performed by Model 4 (the model which achieved the best performance). The abscissa axis represents, meanwhile, the true values observed for each estimation carried out by the model. As results, interceptions between true values and estimated values are obtained for each sample of the data and are represented as red crosses. The blue line (with a slope of 45 degrees) represents the best possible estimation. A red cross over the blue line means that observed value and estimated value are equal and a perfect match between model and reality has been achieved in that individual estimation. In this figure, it can be seen that estimations of low impacts are better than estimations of impacts with medium and high magnitudes. However, it is possible to check that all cases are well distributed on both sides of the blue line. This is reflected by the low value results of standard deviation as well. The final

**β**parameters obtained for each model are presented in Table 5.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Life cycle of a printed newspaper. Figure inspired in [14].

**Figure 3.**Representation of each individual variable against the impact. Individual behaviour of impacts vs. (

**a**) number of pages; (

**b**) grammage; (

**c**) heights (

**d**) paper types and (

**e**) print run.

**Figure 5.**10-fold cross-validation applied to evaluate each model. (1) Data selection and cleaning; (2) data randomization by rows; (3) data division; (4) Cross-Validation; (5) Performance determination by metrics.

Row | Pages | Grammage | Height | Paper Type | Print Run | Impact |
---|---|---|---|---|---|---|

Number | (#) | (g/m^{2}) | (mm) | (-) | (#) | (CO_{2} eq/kg) |

1 | 32 | 42 | 360 | 1 | 500 | 9.392 |

2 | 32 | 42 | 360 | 1 | 1000 | 5.506 |

3 | 32 | 42 | 360 | 1 | 2000 | 3.563 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

1102 | 32 | 45 | 420 | 2 | 30000 | 1.314 |

1103 | 32 | 45 | 420 | 2 | 31000 | 1.310 |

1104 | 32 | 45 | 420 | 2 | 32000 | 1.307 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

2203 | 48 | 42 | 390 | 4 | 9000 | 1.756 |

2204 | 48 | 42 | 390 | 4 | 10000 | 1.717 |

2205 | 48 | 42 | 390 | 4 | 11000 | 1.685 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

3304 | 48 | 48.8 | 390 | 1 | 39000 | 1.700 |

3305 | 48 | 48.8 | 390 | 1 | 40000 | 1.698 |

3306 | 48 | 48.8 | 390 | 1 | 41000 | 1.696 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

4405 | 64 | 45 | 360 | 3 | 18000 | 1.560 |

4406 | 64 | 45 | 360 | 3 | 19000 | 1.549 |

4407 | 64 | 45 | 360 | 3 | 20000 | 1.540 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

5506 | 64 | 48.80 | 420 | 4 | 4800 | 1.569 |

5507 | 64 | 48.80 | 420 | 4 | 4900 | 1.568 |

5508 | 64 | 48.80 | 420 | 4 | 5000 | 1.566 |

Iteration | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

1 | 0.094 | 0.038 | 0.036 | 0.039 |

2 | 0.087 | 0.037 | 0.038 | 0.035 |

3 | 0.084 | 0.031 | 0.037 | 0.034 |

4 | 0.084 | 0.034 | 0.036 | 0.038 |

5 | 0.088 | 0.039 | 0.035 | 0.035 |

6 | 0.091 | 0.039 | 0.043 | 0.037 |

7 | 0.092 | 0.040 | 0.038 | 0.038 |

8 | 0.091 | 0.039 | 0.037 | 0.033 |

9 | 0.087 | 0.033 | 0.036 | 0.039 |

10 | 0.087 | 0.041 | 0.034 | 0.040 |

Average | 0.089 | 0.037 | 0.037 | 0.037 |

Standard deviation | 0.003 | 0.003 | 0.002 | 0.002 |

Iteration | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

1 | 5.106 | 1.836 | 1.601 | 1.640 |

2 | 5.193 | 1.765 | 1.690 | 1.590 |

3 | 4.836 | 1.672 | 1.703 | 1.570 |

4 | 4.990 | 1.700 | 1.564 | 1.689 |

5 | 5.033 | 1.747 | 1.593 | 1.615 |

6 | 5.157 | 1.833 | 1.765 | 1.613 |

7 | 5.015 | 1.847 | 1.663 | 1.757 |

8 | 5.087 | 1.803 | 1.665 | 1.478 |

9 | 5.012 | 1.769 | 1.578 | 1.793 |

10 | 5.118 | 1.776 | 1.640 | 1.703 |

Average | 5.055 | 1.775 | 1.646 | 1.645 |

Standard deviation | 0.102 | 0.058 | 0.063 | 0.093 |

Iteration | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

1 | 0.9848 | 0.9942 | 0.9934 | 0.9933 |

2 | 0.9840 | 0.9939 | 0.9932 | 0.9934 |

3 | 0.9853 | 0.9943 | 0.9930 | 0.9935 |

4 | 0.9851 | 0.9942 | 0.9934 | 0.9936 |

5 | 0.9853 | 0.9941 | 0.9933 | 0.9936 |

6 | 0.9845 | 0.9943 | 0.9935 | 0.9931 |

7 | 0.9843 | 0.9944 | 0.9933 | 0.9933 |

8 | 0.9845 | 0.9942 | 0.9934 | 0.9932 |

9 | 0.9848 | 0.9941 | 0.9938 | 0.9934 |

10 | 0.9848 | 0.9941 | 0.9931 | 0.9932 |

Average | 0.9847 | 0.9942 | 0.9933 | 0.9934 |

Standard deviation | 0.0004 | 0.0001 | 0.0002 | 0.0002 |

β Parameter | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

β_{1} | 0.387 | −0.377 | 0.911 | 0.917 |

β_{2} | 0.386 | −0.401 | 0.143 | 0.142 |

β_{3} | 0.018 | 0.000 | 0.574 | 0.573 |

β_{4} | 0.405 | 28.607 | −32.838 | −35.833 |

β_{5} | −0.088 | −1.296 | 0.000 | 3280.310 |

β_{6} | 0.417 | 7.487 | 3287.628 | −0.996 |

β_{7} | −0.082 | −0.271 | −0.996 | - |

β_{8} | 0.283 | 0.142 | - | - |

β_{9} | −45,785,423.699 | 0.573 | - | - |

β_{10} | 3345.275 | −7210.433 | - | - |

β_{11} | −0.999 | 0.000 | - | - |

β_{12} | - | 3368.258 | - | - |

β_{13} | - | −1.000 | - | - |

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**MDPI and ACS Style**

Lozano, A.; Cabrera, P.; Blanco-Marigorta, A.M.
Non-Linear Regression Modelling to Estimate the Global Warming Potential of a Newspaper. *Entropy* **2020**, *22*, 590.
https://doi.org/10.3390/e22050590

**AMA Style**

Lozano A, Cabrera P, Blanco-Marigorta AM.
Non-Linear Regression Modelling to Estimate the Global Warming Potential of a Newspaper. *Entropy*. 2020; 22(5):590.
https://doi.org/10.3390/e22050590

**Chicago/Turabian Style**

Lozano, Alexis, Pedro Cabrera, and Ana M. Blanco-Marigorta.
2020. "Non-Linear Regression Modelling to Estimate the Global Warming Potential of a Newspaper" *Entropy* 22, no. 5: 590.
https://doi.org/10.3390/e22050590