# Comparing Prophet and Deep Learning to ARIMA in Forecasting Wholesale Food Prices

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset Description and Preparation

- Training dataset: years 2013–2017;
- Validation dataset: year 2018;
- Test dataset: years 2019–March 2020.

- Since the time series for all products have a long window with no data in the second half of year 2013, we do not consider this empty period and start right after it;
- When occasional weeks with no data occur, we take the average of the preceding and following week prices and interpolate.

#### 2.2. ARIMA Models

- for an exact MA(q), ACF is zero for lags larger than q;
- for an exact AR(p), PACF is zero for lags larger than p.

#### 2.3. Prophet

#### 2.4. Neural Networks

- reshape data so that at each time t, ${\mathbf{x}}_{\mathbf{t}}$ is a $n\times 9$ tensor containing the n last values of each time series in Table 4;
- set ${y}_{t}={\Delta}_{t+1}={p}_{t+1}-{p}_{t}$ as the variable to be used in the cost function.

## 3. Results

#### 3.1. ARIMA Results

#### 3.2. Prophet Results

#### 3.2.1. Prophet Grid Search

#### 3.2.2. Prophet Forecasting

#### 3.3. Neural Networks Results

#### 3.3.1. NN Grid Search

- a number $l\in \{1,2,3\}$ of LSTM layers;
- a number ${n}_{u}\in \{32,64,96\}$ of LSTM neurons in every layer, with normal Glorot weight initialization [42]. Each layer has the same number of units;
- following each LSTM layer, a dropout layer with dropout rate $r\in \{0.1,0.2,0.3\}$. The dropout rate is taken to be equal for all layers;
- an output layer consisting of a single neuron with linear activation function and normal Glorot initialization;
- an MSE cost function and Adam optimization algorithm [43], with learning rate (or stepsize) $\alpha \in \{0.0005,0.001\}$;
- an early stopping procedure, monitoring the cost function on the validation set with a patience of 5 epochs, while also having an upper bound of 150 training epochs.

- two one-dimensional convolutional layers (conv1D) with $f\in \{10,20,30\}$output filters each, kernel size ${k}_{s}\in \{2,4\}$ and relu (rectified linear unit) activation function. The hyperparameters $f,{k}_{s}$ were taken to be the same for both layers;
- in each of the conv1D layers, padding was set to same, causal, or no padding at all. The corresponding hyperparameters are dubbed pad${}_{1}$, pad${}_{2}$;
- an average pooling layer in between the convolutional layers, with pool size equal to 2 and no padding;
- following the above layers, LSTM layers are added following the same structure as for the first class of models.

#### 3.4. NN Forecasting

#### 3.5. Result Comparison

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NN | Neural Network |

LSTM | Long Short–Term Memory |

NN | Neural Network |

CNN | Convolutional Neural Network |

GAM | Generalized Additive Model |

ARIMA | Autoregressive Integrated Moving Average |

ADF | Augmented Dikey–Fuller |

BIC | Bayesian Information Criterion |

ACF | Autocorrelation Function |

PACF | Partial Autocorrelation Function |

RMSE | Root Mean Squared Error |

MAE | Mean Absolute Error |

MAPE | Mean Absolute Percent Error |

ME | Mean Error |

## Appendix A. Univariate Deep Learning Models

Product | l | ${\mathit{n}}_{\mathit{u}}$ | r | $\mathit{\alpha}$ | Train RMSE | Valid RMSE |
---|---|---|---|---|---|---|

1 | 2 | 96 | 0.2 | 0.001 | 0.116 | 0.0674 |

2 | 2 | 96 | 0.3 | 0.0005 | 0.195 | 0.183 |

3 | 2 | 64 | 0.3 | 0.001 | 0.224 | 0.162 |

Product | l | ${\mathit{n}}_{\mathit{u}}$ | r | $\mathit{\alpha}$ | f | ${\mathit{k}}_{\mathit{s}}$ | pad${}_{1}$ | pad${}_{2}$ | n | Train RMSE | Valid RMSE |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 96 | 0.3 | 0.0005 | 30 | 2 | valid | same | 8 | 0.112 | 0.0600 |

2 | 2 | 96 | 0.1 | 0.001 | 10 | 2 | causal | causal | 12 | 0.188 | 0.179 |

3 | 1 | 64 | 0.2 | 0.001 | 30 | 2 | same | causal | 8 | 0.229 | 0.154 |

Product | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|

class | A | B | A | B | A | B |

RMSE | 0.0630 | 0.0591 | 0.186 | 0.166 | 0.226 | 0.207 |

MAE | 0.0468 | 0.0449 | 0.131 | 0.122 | 0.155 | 0.158 |

MAPE | 0.0281 | 0.0257 | 0.0175 | 0.0163 | 0.0221 | 0.0220 |

ME | 0.00605 | 0.00747 | −0.0981 | −0.0654 | 0.0530 | 0.0262 |

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**Figure 1.**Price distribution (after z-score filtering) for product 2: to the left, boxplot, and to the right, violin plot.

**Figure 2.**(

**a**) Unit price and (

**b**) sold quantity time series for product 1 after resampling with weekly frequency.

**Figure 3.**(

**a**) Unit price and (

**b**) sold quantity time series for product 2 after resampling with weekly frequency.

**Figure 4.**(

**a**) Unit price and (

**b**) sold quantity time series for product 3 after resampling with weekly frequency.

**Figure 5.**Autocorrelation function and partial autocorrelation function for $\Delta log\left({p}_{t}\right)$ of product 2.

**Figure 6.**Prophet fit over the entire product 1 dataset. The actual fit is shown in light blue, the red line shows the trend function, while trend changepoints are indicated by red dashed vertical lines.

**Figure 7.**Comparison of Prophet forecasts and observed data for price variations in the test set, for product 2. The corresponding RMSE is 0.220.

**Figure 8.**Trajectory of training and validation losses, as a function of the training epoch, for product 1. Please note that the cost functions are calculated on rescaled data, therefore they cannot be directly compared to the values appearing on the first line of Table 8.

**Figure 9.**Comparison of NN forecasts and observed data for price variations in the test set, for product 3. The corresponding RMSE is 0.200.

Product | 1 | 2 | 3 |
---|---|---|---|

Mean (€) | 1.99 | 7.64 | 7.27 |

Std (€) | 0.36 | 0.40 | 0.26 |

Product | Model | Train | Valid | Test |
---|---|---|---|---|

1 | Prophet | 240 | 52 | 62 |

ARIMA & NN | 211 | |||

2 | Prophet | 241 | 52 | 62 |

ARIMA & NN | 211 | |||

3 | Prophet | 242 | 52 | 62 |

ARIMA & NN | 212 |

Product | Series | ADF Test Statistics | p-Value | Lags Used |
---|---|---|---|---|

1 | $log\left({p}_{t}\right)$ | −1.10 | 0.713 | 4 |

$\Delta log\left({p}_{t}\right)$ | −12.3 | $6.63\times {10}^{-23}$ | 3 | |

2 | $log\left({p}_{t}\right)$ | −2.84 | 0.0530 | 7 |

$\Delta log\left({p}_{t}\right)$ | −11.5 | $4.03\times {10}^{-21}$ | 6 | |

3 | $log\left({p}_{t}\right)$ | −2.29 | 0.175 | 7 |

$\Delta log\left({p}_{t}\right)$ | −11.5 | $4.12\times {10}^{-21}$ | 6 |

**Table 4.**A slice of the dataset used to generate input data for the NN models, for a given product. The columns contain the following information: quant is the number of units sold, customers is the number of different customers served, orders is the number of orders, on sale indicates how many orders had a price discount, avg_cost was the average cost of the product for the wholesaler, w _cos and w_sin are as defined in Equation (4), p_std is the standard deviation of the product prices, and price_avg is the weighted average of the product sale price.

Quant | Customers | Orders | On Sale | cost_avg | w_cos | w_sin | p_std | price_avg |
---|---|---|---|---|---|---|---|---|

10 | 1 | 1 | 0 | 1.40 | 0.990 | −0.141 | 0 | 1.41 |

0 | 0 | 0 | 0 | 1.40 | 1.000 | −0.0214 | 0 | 1.55 |

70 | 6 | 6 | 0 | 1.40 | 0.993 | 0.120 | 0.0690 | 1.69 |

220 | 17 | 18 | 0 | 1.40 | 0.971 | 0.239 | 0.0580 | 1.75 |

230 | 14 | 15 | 0 | 1.39 | 0.935 | 0.353 | 0.0685 | 1.86 |

**Table 5.**Selected ARIMA models for the three products, associated RMSE on the entire training + validation set, and performances on the test set (RMSE, MAE and MAPE). Please note that MAPE is computed for price time series ${p}_{t}$, not for the $\Delta \left(t\right)$ time series.

Product | Model | Train + Val RMSE | Test RMSE | Test MAE | Test MAPE | Test ME |
---|---|---|---|---|---|---|

1 | ARIMA (2, 1, 0) | 0.097 | 0.0758 | 0.173 | 0.215 | 0.00521 |

2 | ARIMA (0, 1, 2) | 0.232 | 0.0581 | 0.132 | 0.159 | 0.0140 |

3 | ARIMA (3, 1, 1) | 0.211 | 0.0348 | 0.0178 | 0.0222 | 0.0507 |

**Table 6.**Prophet results: RMSE on the validation set, RMSE, MAE, MAPE and ME calculated on the test set. The metrics were calculated as described in the main text. Please note that the MAPE is computed for the price times series ${p}_{t}$, not for the $\Delta \left(t\right)$ time series.

Product | $\mathit{\tau}$ | $\mathit{\sigma}$ | Valid RMSE | Test RMSE | Test MAE | Test MAPE | Test ME |
---|---|---|---|---|---|---|---|

1 | 0.5 | 0.01 | 0.0831 | 0.00812 | 0.0694 | 0.0414 | −0.0152 |

2 | 0.1 | 0.01 | 0.293 | 0.220 | 0.165 | 0.0224 | 0.0456 |

3 | 0.5 | 1.0 | 0.215 | 0.350 | 0.301 | 0.0424 | −0.217 |

Product | l | ${\mathit{n}}_{\mathit{u}}$ | r | $\mathit{\alpha}$ | Train RMSE | Valid RMSE |
---|---|---|---|---|---|---|

1 | 3 | 32 | 0.1 | 0.001 | 0.0964 | 0.0612 |

2 | 3 | 32 | 0.1 | 0.001 | 0.157 | 0.176 |

3 | 1 | 64 | 0.1 | 0.001 | 0.143 | 0.148 |

Product | l | ${\mathit{n}}_{\mathit{u}}$ | r | $\mathit{\alpha}$ | f | ${\mathit{k}}_{\mathit{s}}$ | pad${}_{1}$ | pad${}_{2}$ | n | Train RMSE | Valid RMSE |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 64 | 0.3 | 0.0005 | 20 | 2 | causal | causal | 8 | 0.0770 | 0.0553 |

2 | 3 | 64 | 0.1 | 0.0005 | 20 | 2 | no | same | 12 | 0.175 | 0.165 |

3 | 1 | 32 | 0.2 | 0.001 | 20 | 2 | same | same | 8 | 0.148 | 0.132 |

Product | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|

class | A | B | A | B | A | B |

RMSE | 0.0617 | 0.0613 | 0.179 | 0.162 | 0.219 | 0.200 |

MAE | 0.0498 | 0.0511 | 0.130 | 0.126 | 0.157 | 0.150 |

MAPE | 0.0299 | 0.0305 | 0.0174 | 0.0168 | 0.0221 | 0.0212 |

ME | 0.0138 | 0.00138 | −0.0767 | −0.0321 | −0.0260 | 0.0262 |

**Table 10.**Comparison of model performances on the test dataset. NN-A and NN-B indicate class A and class B neural network models, respectively, while no-change denotes the model where the price is forecasted to be equal to the latest value of the series.

Product | Metric | ARIMA | Prophet | NN-A | NN-B | No-Change |
---|---|---|---|---|---|---|

1 | RMSE | 0.0758 | 0.0812 | 0.0617 | 0.0613 | 0.0972 |

MAE | 0.0581 | 0.0694 | 0.0498 | 0.0511 | 0.0715 | |

MAPE | 0.0348 | 0.0414 | 0.0299 | 0.0305 | 0.0429 | |

ME | 0.00521 | −0.0152 | 0.0138 | 0.00138 | −0.0066 | |

2 | RMSE | 0.173 | 0.220 | 0.179 | 0.162 | 0.268 |

MAE | 0.132 | 0.165 | 0.135 | 0.126 | 0.204 | |

MAPE | 0.0178 | 0.0224 | 0.0181 | 0.0168 | 0.0276 | |

ME | 0.0140 | 0.0456 | −0.0767 | −0.0321 | 0.00811 | |

3 | RMSE | 0.215 | 0.350 | 0.219 | 0.200 | 0.354 |

MAE | 0.159 | 0.391 | 0.157 | 0.150 | 0.376 | |

MAPE | 0.0222 | 0.0424 | 0.0221 | 0.0212 | 0.0425 | |

ME | 0.0507 | −0.217 | −0.0260 | 0.0262 | −0.0211 | |

avg MAPE | 0.0249 | 0.0354 | 0.0234 | 0.0228 | 0.0377 |

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## Share and Cite

**MDPI and ACS Style**

Menculini, L.; Marini, A.; Proietti, M.; Garinei, A.; Bozza, A.; Moretti, C.; Marconi, M.
Comparing Prophet and Deep Learning to ARIMA in Forecasting Wholesale Food Prices. *Forecasting* **2021**, *3*, 644-662.
https://doi.org/10.3390/forecast3030040

**AMA Style**

Menculini L, Marini A, Proietti M, Garinei A, Bozza A, Moretti C, Marconi M.
Comparing Prophet and Deep Learning to ARIMA in Forecasting Wholesale Food Prices. *Forecasting*. 2021; 3(3):644-662.
https://doi.org/10.3390/forecast3030040

**Chicago/Turabian Style**

Menculini, Lorenzo, Andrea Marini, Massimiliano Proietti, Alberto Garinei, Alessio Bozza, Cecilia Moretti, and Marcello Marconi.
2021. "Comparing Prophet and Deep Learning to ARIMA in Forecasting Wholesale Food Prices" *Forecasting* 3, no. 3: 644-662.
https://doi.org/10.3390/forecast3030040