This research was conducted using data from earthquake disasters site (the United States Geological Survey, USGS) recorded on the period January 1, 1910 to December 31, 2022 on the south coast of Java. This data consists of location, depth of earthquake (≤ 60 km), time of occurrence of the earthquake, and magnitude (≥ 5 Mw). The criteria for the depth and strength of the earthquake data are based on the characteristics of earthquakes that have the potential to cause tsunamis, which were obtained from earthquake events that caused tsunamis in the past. The south coast of Java is an area of investigation, bounded by longitudes
where Mw is a unit of magnitude of earthquakes; it is called moment magnitude.
Map of South Coast of Java
To find forecasting for earthquakes, the Semi-Markov process is used, which is a development of the Markov stochastic process. Further explanation is given as follows.
A stochastic process is a collection of random variables
The semi-Markov process is an extension of the stochastic Markov process. In a semi-Markov process, the Markov property is no longer fulfilled, meaning that predicting the future state is not only based on the current state but also on the length of time in the current state before moving to the future state. Consider a stochastic process with a state space
The transition matrix is denoted by
where
The time during which the semi-Markov process remains in state
where
The core matrix is denoted by
where
After calculating the core matrix, continue by adding up each row of the core matrix for each time diference (holding time) to get the mass waiting time function. That is
where
where
where
The interval transition probability matrix
where
is a diagonal matrix where the
Based on Bruin
where
Deterministic Forecasting Algorithm
| Step 0 | Begin |
|---|---|
| Step 1 | Use the past n data to determine F̂RM (n is the number of total data) |
| Step 2 | Determine F̂RM |
| Step 3 | F̂RM1(i) = F̂RM(i), ∀i = 1, …, k (k is the number of future time intervals that can be forecasted) |
| Step 4 | Mⱼ = {t-th order maximum in F̂RM1(j)}, ∀j = 1, …, k |
| Step 5 | F̂RM2(j) = [ F̂RM1(j) / Mⱼ ], ∀j = 1, …, k (The highest integer number that is less than the real number x is obtained by using [x].) |
| Step 6 | For all element x ∈ F̂RM2(j) with x ≥ 1 will be replaced by 1 and name the resulted matrices FRMD(j), ∀j = 1, …, k |
| Step 7 | An earthquake with magnitude m in region r will occur in the j-th time period if FRMD_rm(j) = 1; else, no earthquake with magnitude m in region r will occur in the j-th time period |
| Step 8 | End |
To determine the level of accuracy of earthquake forecasting results, a validation process must be carried out. Many indicators can be used to measure the level of accuracy of forecasting, including Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Deviation (MAD), and Mean Absolute Percentage Error (MAPE). The forecasting results are always attempted to be close to actual events. In other words, the error value produced by the forecasting results should be small. Mean Absolute Percentage Error (MAPE) can be calculated by using the absolute error for each period divided by the actual observed value for that period. The advantage of the MAPE approach is that it can express the percentage error in forecasting results regarding actual demand during a certain period, providing information on whether the percentage error is too high or too low, thereby enhancing accurate. However, a significant deficiency of MAPE is that the calculation will yield an infinite or undefined value if the actual value is zero or near zero (see
Validation Algorithm
| Step 0 | Begin |
|---|---|
| Step 1 | d = 0 |
| Step 2 | n₂(0) = 0 |
| Step 3 | d = 1 |
| Step 4 | Use the (n₁ + n₂(d)) first data to determine F̂RM and FRM, where n₁ is the number of the first data that used to forecast the next n₂ data and n₂(d) is the number of the actual data that occurred during d periods time after the first n₁ data. |
| Step 5 | Determine F̂RM and FRM |
| Step 6 | If FRM,ᵢⱼ ≠ 0, then: MAPE(d) = [ Σᵢ₌₁ʳ Σⱼ₌₁ᵐ ( |FRM,ᵢⱼ − F̂RM,ᵢⱼ| / FRM,ᵢⱼ ) ] / (r · m) × 100% ... (9) |
| Step 7 | d = d + 1 |
| Step 8 | MAPE = ( Σ_{d=1}^{p} MAPE(d) ) / p ... (10) |
| Step 9 | End |
The results of the MAPE calculation are usually in percentage form. The smaller the percentage, the better the accuracy level. The MAPE calculation criteria to measure the accuracy of the forecasting results are given in
MAPE Criteria for Model Evaluation
| MAPE | Forecasting power |
|---|---|
| < 10% | Highly accurate forecasting |
| 10% – 20% | Good forecasting |
| 20% – 50% | Reasonable forecasting |
| > 50% | Weak and inaccurate forecasting |