Microservice-Oriented Platform for Internet of Big Data Analytics: A Proof of Concept

. 2019 Mar 6;19(5):1134. doi: 10.3390/s19051134

Algorithm 1 Monte Carlo Approximation of Double Integral

Input: N: the total number of trials in Monte Carlo estimation.

Output: The approximation result of the predefined double integral $\int_{a}^{b} \int_{g (x)}^{h (x)} f (x, y) d y d x$ .

1:
$valid_points \leftarrow 0$ ▹ for counting the number of points inside the valid x–y region.
2:
$function_evaluations \leftarrow 0$ ▹ for summing up individual evaluations of $f (x, y)$ .
3:
$X \leftarrow$ a random number between a and b
4:
$Y \leftarrow$ a random number between $MIN {g (x)}$ and $MAX {h (x)}$ , $x \in [a, b]$
5:
for $i \leftarrow 1, N$ do
6:
if $g (X) < = Y$ and $Y < = h (X)$ then ▹ If the random point $(X, Y)$ is located inside the valid x–y region.
7:
$valid_points \leftarrow valid_points + 1$
8:
$function_evaluations \leftarrow function_evaluations + f (X, Y)$
9:
end if
10:
end for
11:
area $\leftarrow (b - a) \cdot (MAX {h (x)} - MIN {g (x)}) \cdot valid_points / N$ , $x \in [a, b]$ ▹ Estimating the area of the valid x–y region.
12:
$volume \leftarrow area \cdot function_evaluations / valid_points$ ▹ Note that the function was evaluated valid_points times.
13:
returnvolume ▹ The estimated volume as the double integral result.