Forecasting cauliflower prices in Nepal: a comparative analysis using seasonal time series and nonlinear models

Figures & data

Table 1. Review of related work.

Author	Commodity	Forecasting model
Geweke and Porter-Hudak (1983)	Postwar monthly economic time series	Logarithmic Periodogram
Wang et al. (2011)	Household food	ARIMA
Yercan and Adanacioglu (2012)	Tomato	SARIMA (1,0,0) (1,1,1)12
Luo et al. (2013)	Cucumber	SARIMA (1,0,1) (1,1,1)12
da Veiga et al. (2014)	Dairy products	ARIMA, Holt-Winters
Choudhury and Jones (2014)	Crop yield	ARMA
Otu et al. (2014)	Inflation Rate	SARIMA
Arunraj et al. (2016)	Food consumption	SARIMA
Sellam and Poovammal (2016)	Rice	Regression Analysis
Souza et al. (2016)	Soybean	SARIMA
Esther and Magdaline (2017)	Pulses	ARIMA
Zhang et al. (2018)	Soybean	QR-RBF Neural Network
Bisht and Kumar (2019)	Pulses	GARCH
Vibas and Raqueno (2019)	Fruit, vegetable	ARIMA, SARIMA, ARIMAX
Mutwiri et al. (2019)	Tomato	SARIMA
Selvanayagam et al. (2019)	Vegetable	LSTM RNN, ARIMA
Rajeswari and Suthendran (2019)	Crop price	Hybrid Association rule-based Decision Tree (HADT)
Zahara et al. (2019)	Consumer price index	Nesterov Adam Optimization Algorithm
Wamalwa and Muchemi (2020)	Maize	SARIMA, ETS
Divisekara et al. (2021)	Red lentils	SARIMA (2,1,2) (0,1,1)52
Christensen et al. (2021)	Food crises	Logistic Regression, Neural Network
Purohit et al. (2021)	Agricultural crop prices	ARIMA, ETS, SVM, MLP, LSTM
Sarangi et al. (2021)	Consumer food price index	Backpropagation learning-based ANN
Oswari et al. (2022)	Food crop farmer index price	ARIMA, LSTM
Deléglise et al. (2022)	Financial services indicators	Machine Learning Techniques
Guo et al. (2022)	Price	AttLSTM-ARIMA-BP Model
Majhi et al. (2023)	Food price index	SARIMA, ETS, Fb Prophet
Giri and Giri (2023)	Cucumber	SARIMA, SSES

Figure 1. Schematic representation of an LSTM RNN Cell.

Figure 2. Research methodology of the study.

Figure 3. a) Time-series plot of prices of cauliflower (April 2013 to March 2023); b) Seasonal, trend and residual decomposition of time-series data.

Table 2. Unit root test for prices of cauliflower.

ADF Test	Test statistic	Critical value (5%)	p Value	Significance level	Remarks
Before seasonal differencing	−1.309568	−2.888954	0.624784	0.05	Accept H0 (series non-stationary)
After seasonal differencing	−4.217568	−2.892480	0.000615	0.05	Reject H0 (series stationary)

Figure 4. ACF and PACF plots of seasonally differenced data.

Table 3. Model estimation with AIC, BIC, RMSE and MAPE criterion for SARIMA (p, 0, q) (P, 1, Q)₁₂.

SARIMA model	Akaike information criterion (AIC)	Bayesian information criterion (SBC)	R-Squared	RMSE	MAPE	Remarks
(1,0,1) (2,1,1)12	852.0677	867.4538	0.252694	18.205972	121.187292	SMA (12): 0.978 > 0.05, AR (1): 0.543 > 0.05, MA (1): 0.106 > 0.05
(1,0,1) (2,1,0)12	877.9378	890.7596	0.020949	20.838961	127.936046	AR (1): 0.857 > 0.05
(0,0,1) (2,1,0)12	875.973	886.2305	0.020242	20.846500	127.661862	All regressors significant
(0,0,1) (3,1,0)12	855.4423	868.2641	0.228847	18.494553	133.348233	All regressors significant
(0,0,1) (4,1,0)12	853.476454	868.8623	0.256740	18.156537	130.667024	SAR (48): 0.091 > 0.05

Table 4. Parameter estimators of SARIMA (0, 0, 1) (3, 1, 0)₁₂.

Variable	Coefficient	Std. error	t-Statistic	p Value^a
C	Exp (0.51241) = 1.6693	0.0995	5.1497	***
MA (1)	0.5124	0.100	5.150	***
SAR (12)	−1.3043	0.087	−14.494	***
SAR (24)	−1.0712	0.115	−9.349	***
SAR (36)	−0.5510	0.085	−6.452	***
SIGMASQ	292.4564	46.473	6.293	***

^aLess than 0.05 (*), less than 0.005 (**) and less than 0.0005 (***).

Figure 5. a) Residual ACF and PACF plots of SARIMA (0 0, 1) (3, 1, 0)12; b) residual time-series plot.

Table 5. Number of neurons and Epochs considered in five LSTM RNN models.

Model	No. of neurons	No. of Epochs	RMSE	MAPE
LSTM1	100	50	31.803930	42.945598
LSTM2	100	60	45.500532	90.419521
LSTM3	100	40	33.462227	61.681581
LSTM4	150	50	40.360049	78.319300
LSTM5	150	40	48.156613	104.973604

Figure 6. Loss per Epoch during training of LSTM₁ model.

Table 6. Comparison of in-sample and out-sample performances of SARIMA, LSTM RNN and Fb Prophet models.

	Performance metrics	SARIMA forecast	LSTM RNN forecast	Fb Prophet forecast
In-sample (training)	R-squared	0.3339	−0.1261	0.6592
	MAE	15.3870	24.0741	10.5941
	RMSE	20.4719	31.8039	13.4144
	MAPE	36.6410	42.9456	27.5700
Out-sample (testing)	R-squared	0.6434	0.0525	0.5744
	MAE	10.5913	15.6799	11.4158
	RMSE	13.0242	21.2287	14.2280
	MAPE	21.4064	39.1171	23.6645

Figure 7. Comparison of actual vs. predicted test data of different models.

Figure 8. a) Forecasts using SARIMA model, b) Forecasts using Fb Prophet model and c) Forecasts using LSTM RNN model.

Table 7. Future forecasts of cauliflower prices using different models.

Date	SARIMA forecasts	Fb Prophet forecasts	LSTM RNN forecasts	Date	SARIMA forecasts	Fb Prophet forecasts	LSTM RNN forecasts
April, 2023	54.852	59.402	61.189	April, 2024	46.126	63.460	51.036
May, 2023	47.740	70.323	70.778	May, 2024	43.712	74.598	56.999
June, 2023	64.186	79.737	77.485	June, 2024	62.650	83.952	58.345
July, 2023	89.759	97.678	78.577	July, 2024	90.987	102.802	55.319
August, 2023	93.591	110.415	70.772	August, 2024	96.270	116.240	49.086
September, 2023	90.741	106.965	57.378	September, 2024	89.908	112.755	42.383
October, 2023	81.796	93.475	44.773	October, 2024	77.431	98.791	37.687
November, 2023	54.032	68.624	36.038	November, 2024	45.016	71.944	34.656
December, 2023	43.692	58.899	31.124	December, 2024	38.171	61.252	33.381
January, 2024	35.951	46.890	30.886	January, 2025	36.915	48.922	33.868
February, 2024	29.232	42.992	35.080	February, 2025	30.302	44.409	35.965
March, 2024	39.481	54.280	42.587	March, 2025	41.624	56.721	39.192

Geweke, J., & Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4(4), 221–238. https://doi.org/10.1111/j.1467-9892.1983.tb00371.x

 Google Scholar

Wang, Y., Ye, X., & Huo, Y. (2011). Prediction of household food retail prices based on ARIMA Model [Paper presentation]. Proceedings of 2011 International Conference on Multimedia Technology. Hangzhou. https://doi.org/10.1109/ICMT.2011.6002376

 Google Scholar

Yercan, M., & Adanacioglu, H. (2012). An analysis of tomato pricesat wholesale level in Turkey: an application of SARIMA model. Custos e Agronegocio, 8, 52–75. Retrieved from: http://www.custoseagronegocioonline.com.br/numero4v8/Tomato%20english.pdf

 Web of Science ®Google Scholar

Luo, C. S., Zhou, L. Y., & Wei, Q. F. (2013). Application of SARIMA Model in Cucumber Price Forecast. Applied Mechanics and Materials, 373–375, 1686–1690. https://doi.org/10.4028/www.scientific.net/amm.373-375.1686

 Google Scholar

da Veiga, C. P., da Veiga, C. P., Catapan, A., Tortato, U., & Silva, W. V. (2014). Demand forecasting in food retail: A comparison between the Holt-Winters and ARIMA models. WSEAS Transactions on Business and Economics, 11(1), 608–614. https://www.researchgate.net/publication/286314562_Demand_forecasting_in_food_retail_A_comparison_between_the_Holt-Winters_and_ARIMA_models

 Google Scholar

Choudhury, A., & Jones, J. (2014). Crop yield prediction using time series models. Journal of Economics and Economic Education Research, 15(3), 53–67. Retrieved from https://www.researchgate.net/publication/288387590_Crop_yield_prediction_using_time_series_models

 Google Scholar

Otu, A. A., Osuji, G. A., Opara, J., Mbachu, H. I., & Iheagwara, A. I. (2014). Application of Sarima models in modelling and forecasting Nigeria’s inflation rates. American Journal of Applied Mathematics and Statistics, 2(1), 16-28. https://doi.org/10.12691/ajams-2-1-4

 Google Scholar

Arunraj, N. S., Ahrens, D., & Fernandes, M. (2016). Application of SARIMAX model to forecast daily sales in the food retail industry. International Journal of Operations Research and Information Systems, 7(2), 1–21. https://doi.org/10.4018/IJORIS.2016040101

 Google Scholar

Sellam, V., & Poovammal, E. (2016). Prediction of crop yield using regression analysis. Indian Journal of Science and Technology, 9(38), 1–5. https://doi.org/10.17485/ijst/2016/v9i38/91714

 Google Scholar

Souza, R. C. T. de, Guedes Filho, O., Santos, M. A. R. dos, & Coelho, L. dos S. (2016). Monthly closing price forecasting of soybean grain in Paraná using SARIMA modeling with intervention. Journal of Geospatial Modeling, 2(1), 27–44. https://doi.org/10.22615/2526-1746-jgm-2.1-5887

 Google Scholar

Esther, W., & Magdaline, K. (2017). Forecasting pulses production in Kenya using ARIMA model. Asian Journal of Economics, Business and Accounting, 2(3), 1–8. https://doi.org/10.9734/AJEBA/2017/32414

 Google Scholar

Zhang, D., Zang, G., Li, J., Ma, K., & Liu, H. (2018). Prediction of soybean price in China using QR-RBF neural network model. Computers and Electronics in Agriculture, 154, 10–17. https://doi.org/10.1016/j.compag.2018.08.016

 Web of Science ®Google Scholar

Bisht, M., & Kumar, R. (2019). Estimating volatility in prices of pulses in India: An application of GARCH model. Economic Affairs, 64(3), 513–516. https://doi.org/10.30954/0424-2513.3.2019.6

 Google Scholar

Vibas, V. M., & Raqueño, A. R. (2019). A Mathematical Model for Estimating Retail Price Movements of Basic Fruit and Vegetable Commodities Using Time Series Analysis. International Journal of Advance Study and Research Work, 2(7), 01–05. https://doi.org/10.5281/zenodo.3333529

 Google Scholar

Mutwiri, M., Gichuhi, R., & Wabwoba, F. (2019). Forecasting of tomatoes wholesale prices of Nairobi in Kenya: Time series analysis using SARIMA model. International Journal of Statistical Distributions and Applications, 5(3), 46–53. https://doi.org/10.11648/j.ijsd.20190503.11

 Google Scholar

Selvanayagam, T., Suganya, S., Palendrarajah, P., Manogarathash, M. P., Gamage, A., & Kasthurirathna, D. (2019). Agro-genius: Crop prediction using machine learning. International Journal of Innovative Science and Research Technology, 4(10), 243–249. https://ijisrt.com/assets/upload/files/IJISRT19OCT1880.pdf

 Google Scholar

Rajeswari, S., & Suthendran, K. (2019). Developing an agricultural product price prediction model using HADT algorithm. International Journal of Engineering and Advanced Technology, 9(1S4), 569–575. https://doi.org/10.35940/ijeat.A1126.1291S419

 Google Scholar

Zahara, S., Sugianto, S. & Ilmiddaviq, M. B. (2019 Consumer price index prediction using Long Short Term Memory (LSTM) based cloud computing [Paper presentation]. The 5th International Conference on Technology and Vocational Teachers (ICTVT 2019), 14–15 September 2019. Yogyakarta, Indonesia. https://doi.org/10.1088/1742-6596/1456/1/012022

 Google Scholar

Wamalwa, T. M., & Muchemi, L. (2020). An artificial neural network model for predicting maize prices in Kenya. Africa Journal of Physical Sciences, 4, 54–59. http://jsep.uonbi.ac.ke/ojs/index.php/ajps/article/view/1016/922

 Google Scholar

Divisekara, R., Jayasinghe, G., & Kumari, K. (2021). Forecasting the red lentils commodity market price using SARIMA models. SN Business & Economics, 1(1). https://doi.org/10.1007/s43546-020-00020-x

 Google Scholar

Christensen, C., Wagner, T., & Langhals, B. (2021). Year-independent prediction of food insecurity using classical and neural network machine learning methods. AI, 2(2), 244–260. https://doi.org/10.3390/ai2020015

 Google Scholar

Purohit, S. K., Panigrahi, S., Sethy, P. K., & Behera, S. K. (2021). Time series forecasting of price of agricultural products using hybrid methods. Applied Artificial Intelligence, 35(15), 1388–1406. https://doi.org/10.1080/08839514.2021.1981659

 Web of Science ®Google Scholar

Sarangi, P. K., Sinha, D., Sinha, S., & Mittal, N. (2021). Machine Learning Approach for the Prediction of Consumer Food Price Index. Proceedings of [Paper presentation]. 2021 9th International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions) (ICRITO). Noida, India. https://doi.org/10.1109/ICRITO51393.2021.9596527

 Google Scholar

Oswari, T., Yusnitasari, T., Kusumawati, R., & Setiawan, I. (2022). Prediction analysis of food crop farmer index price during Covid-19 pandemic using ARIMA and LSTM. Journal of Management Information and Decision Sciences, 25(2S), 1–10. https://www.abacademies.org/articles/prediction-analysis-of-food-crop-farmer-index-price-during-covid19-pandemic-using-arima-and-lstm-13360.html

 Google Scholar

Deléglise, H., Interdonato, R., Bégué, A., d’Hôtel, E. M., Teisseire, M., & Roche, M. (2022). Food security prediction from heterogenous data combining machine and deep learning methods. Expert Systems with Applications, 190, 116189. https://doi.org/10.1016/j.eswa.2021.116189

 Web of Science ®Google Scholar

Guo, Y., Tang, D., Tang, W., Yang, S., Tang, Q., Feng, Y., & Zhang, F. (2022). Agricultural price prediction based on combined forecasting model under spatial-temporal influencing factors. Sustainability, 14(17), 10483. https://doi.org/10.3390/su141710483

 Web of Science ®Google Scholar

Majhi, S. K., Bano, R., Srichandan, S. K., Acharya, B., Al-Rasheed, A., Alqahtani, M. S., & Soufiene, B. O. (2023). Food price index prediction using time series models: A study of Cereals, Millets and Pulses. Research Square, https://doi.org/10.21203/rs.3.rs-2999898/v1

 PubMedGoogle Scholar

Giri, A., & Giri, V. R. (2023). Comparative assessment of SARIMA and SSES models for forecasting cucumber prices in Nepal. Advances in Applied Sciences, 8(3), 106–121. https://doi.org/10.11648/j.aas.20230803.17

 Google Scholar

Availability of supporting data

All data generated or analysed during this study are included in this article and its Supplementary Information files.