Spatial Durbin Model with Expansion Using Casetti’s Approach: A Case Study for Rainfall Prediction in Java Island, Indonesia
Abstract
:1. Introduction
2. Materials and Methods
2.1. Inverse Distance Weight Matrix
- is the latitude of the location, ,
- is the longitude location,
- is the latitude of the jth location, j = 1, 2, 3, …, N, and
- is the longitude of the location.
2.2. Moran Index
- There is no spatial autocorrelation between locations, vs.
- There is spatial autocorrelation between locations,
- With .
- is the Moran Index value,
- is the number of observation locations,
- is the value of the observation variable at the ith location,
- is the value of the observation variable at jth location,
- is the average of the number of variables, and
- is an element of the standardized weight matrix between locations and .
- Decision Rule:
2.3. Spatial Durbin Model
- is the vector of dependent variables of size (n × 1),
- is the matrix of independent variables of size (n × k),
- = ,
- = ,
- is the spatial lag coefficient of the dependent variable,
- is the constant parameter,
- is the vector of regression parameters of size (k × 1),
- is the spatial lag parameter vector of covariate variable of size (k × 1),
- is the spatial weight matrix of size (n × n), and
- is the error vector of size (n × 1).
2.4. Spatial Expansion with Casetti’s Approach
- is the vector of dependent variables of size .
- is the matrix of independent variables of size .
- is the location information that contains elements representing the latitude and longitude of each observation, of size .
- is the expansion of the identity matrix of size .
- is the matrix of size contains parameter estimators for all explanatory variables at each observation.
- is the parameter expressed by of size .
- is the Kronecker product.
- is the error vector of size .
- is the location matrix with .
2.5. Spatial Durbin Model with Expansion Using Casetti’s Approach
2.6. Parameter Estimation
2.7. Mean Absolute Percentage Error (MAPE)
2.8. Knowledge Discovery in Databases Methodology
3. Real Data Application
3.1. Research Location
3.2. Data Description
3.3. RShiny for SDM with Expansion Using Casetti’s Approach
- Description of model: This section explains the formulation of the SDM with expansion using Casetti’s approach.
- Import data: In this section, users can upload data files in .csv or .txt format. The data should contain the coordinate of location (latitude and longitude), dependent variable and exogenous variables.
- Vector and matrixConstruct vectors and matrices based on the data, as follows:
- a.
- Vector defines the dependent variable at each location.
- b.
- Matrix and represent the exogenous variables.
- c.
- Matrix consists of location coordinate entries in latitude and longitude.
- d.
- Matrix is an identity matrix with a size of as many as four exogenous variables according to the matrix or matrix .
- e.
- Matrix is the result of calculating the inverse distance weight matrix using the equation with input location coordinates (latitude and longitude).
- f.
- Kronecker is the expression obtained from the multiplication of the Kronecker with the identity matrix of exogenous variables .
- g.
- Matrix is the product of matrix , , and .
- h.
- Matrix is the combination of vector 1, matrix , and the product of matrix and .
- Result of prediction: This includes the calculation of results of parameter estimation , , , and obtaining the prediction results , absolute error, and MAPE.
- Download data: This menu allows users to download the prediction calculation data.
- Created by: This section lists the names of the RShiny development team.
3.4. KDD for SDM with Expansion Using Casetti’s Approach
3.5. Result of Moran’s Index and Scatterplot
3.6. Prediction Result of SDM with Expansion Using Casetti’s Approach
4. Discussion
5. Conclusions
6. Patents
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
No | Locations | SDM with Expansion Using Casetti’s Approach for Predicting Rainfall |
---|---|---|
1 | Serang City | |
2 | Pandeglang | |
3 | Tangerang City | |
4 | South Tangerang City | |
5 | Kepulauan Seribu | |
6 | Central Jakarta | |
7 | Bekasi City | |
8 | Bogor City | |
9 | Indramayu | |
10 | Karawang | |
… | … | … |
64 | Ponorogo |
Appendix B
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No. | Locations | Latitude | Longitude | Climate Variables (Averages) | ||||
---|---|---|---|---|---|---|---|---|
Rainfall (mm) | Air Temperature (°C) | Humidity (%) | Solar Irradiation (W/m2) | Surface Pressure (kPa) | ||||
1 | Serang City | −6.15 | 106 | 166.53 | 27.08 | 81.47 | 17.63 | 100.51 |
2 | Pandeglang | −6.3092 | 106.1047 | 173.91 | 25.49 | 84.50 | 17.63 | 98.27 |
3 | Tangerang City | −6.171389 | 106.640556 | 175.80 | 27.20 | 81.44 | 17.63 | 100.81 |
4 | South Tangerang City | −6.28577727 | 106.7122607 | 178.34 | 24.66 | 85.15 | 17.63 | 96.82 |
5 | Kepulauan Seribu | −5.662900 | 106.568300 | 202.67 | 27.80 | 80.03 | 18.25 | 100.96 |
6 | Central Jakarta | −6.170000 | 106.820000 | 175.80 | 27.20 | 81.44 | 17.63 | 100.81 |
7 | Bekasi City | −6.241586 | 106.992416 | 175.80 | 27.20 | 81.44 | 17.63 | 100.81 |
8 | Bogor City | −6.899541 | 107.533867 | 178.34 | 24.66 | 85.15 | 17.63 | 96.82 |
9 | Indramayu | −6.327583 | 108.324936 | 184.87 | 26.40 | 80.98 | 18.75 | 99.66 |
10 | Karawang | −6.32273 | 107.337579 | 190.09 | 25.25 | 83.46 | 18.28 | 97.86 |
… | … | … | … | … | … | … | … | … |
64 | Ponorogo | −7.8686 | 111.4619 | 163.89 | 24.98 | 82.55 | 19.48 | 97.43 |
No | Climate Variables | p-Value | |||
---|---|---|---|---|---|
1 | (rainfall) | 0.703 | −0.016 | 0.004 | * |
2 | (air temperature) | 0.376 | −0.016 | 0.004 | * |
3 | (humidity) | 0.527 | −0.016 | 0.004 | * |
4 | (solar irradiation) | 0.806 | −0.015 | 0.005 | * |
5 | (surface pressure) | 0.337 | −0.016 | 0.004 | * |
Coefficient | Parameter Estimated Value | ||
---|---|---|---|
(air temperature) | −7.994 | 11.969 | |
17.524 | |||
(humidity) | 2.312 | −0.227 | |
7.295 | |||
(solar irradiation) | −0.570 | −6.869 | |
−0.570 | |||
(surface pressure) | 0.191 | −8.393 | |
0.456 |
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Andriyana, Y.; Falah, A.N.; Ruchjana, B.N.; Sulaiman, A.; Hermawan, E.; Harjana, T.; Lim-Polestico, D.L. Spatial Durbin Model with Expansion Using Casetti’s Approach: A Case Study for Rainfall Prediction in Java Island, Indonesia. Mathematics 2024, 12, 2304. https://doi.org/10.3390/math12152304
Andriyana Y, Falah AN, Ruchjana BN, Sulaiman A, Hermawan E, Harjana T, Lim-Polestico DL. Spatial Durbin Model with Expansion Using Casetti’s Approach: A Case Study for Rainfall Prediction in Java Island, Indonesia. Mathematics. 2024; 12(15):2304. https://doi.org/10.3390/math12152304
Chicago/Turabian StyleAndriyana, Yudhie, Annisa Nur Falah, Budi Nurani Ruchjana, Albertus Sulaiman, Eddy Hermawan, Teguh Harjana, and Daisy Lou Lim-Polestico. 2024. "Spatial Durbin Model with Expansion Using Casetti’s Approach: A Case Study for Rainfall Prediction in Java Island, Indonesia" Mathematics 12, no. 15: 2304. https://doi.org/10.3390/math12152304
APA StyleAndriyana, Y., Falah, A. N., Ruchjana, B. N., Sulaiman, A., Hermawan, E., Harjana, T., & Lim-Polestico, D. L. (2024). Spatial Durbin Model with Expansion Using Casetti’s Approach: A Case Study for Rainfall Prediction in Java Island, Indonesia. Mathematics, 12(15), 2304. https://doi.org/10.3390/math12152304