Davi Pereira dos Santos2 and Jo˜ao Pedro Pedroso1,2
1 Faculdade de Ciˆencias, Universidade do Porto, Portugal
2 INESC TEC, Porto, Portugal davi.p.santos@inesctec.pt, jpp@fc.up.pt
Abstract. This paper describes an algorithm for a problem arising in sea exploration, where the aim is to schedule the expedition of a ship for collecting information about the resources on the seafloor. The aim of the algorithms is to decide locations where to collect data by prob-ing. This way, after the expedition the information available is optimally enriched. The evaluation of a solution is done by comparing the esti-mation of the level of the resource on the given surface, which is done by regression using Gaussian processes, with the “true” level. The al-gorithms comprises elements from combinatorial optimization and from continuous optimization, and is influenced by evolutionary computation.
Keywords: Recognition problems·Tour planning·Orienteering· Sur-face exploration·Gaussian processes
1 Introduction
This work describes a problem with origins in sea exploration, though similar problems arise in other contexts. The identification of the contents of the seafloor is important in view of a possible exploitation of some of these resources. The aim of this problem is to schedule the journey of a ship for collecting information about the composition of the seafloor. We consider a bounded surface, through which some resource can be found with a given level. This “true value” is initially unknown, except for a limited number of points for which there is previous empirical information.
Optimal expedition planning involves three subproblems, each corresponding to a different phase in the process: assessment, planning and estimation.
Assessment consists of estimating the amount of information that would be conveyed by probing the surface at each point. This is done by means of an indicator function. Previous work assumed that actual information obtained by probing is not usable at the time of planning; here, we assume that after com-mitting to probing at a certain place, the information obtained can immediately be used to change the course of the following decisions (in particular, the set of points used for building the indicator function is dynamically expanded).
Planning, the next phase in the solution process, consists of deciding on the
is the only one to which we commit. The objective is to maximize the overall informational reward obtained, taking into account that the total duration of the trip is limited to a known bound. Hence, online planning involves using the previously available points together with the points newly probed in this trip, in order to decide the location of the next point to probe — though an estimation of the whole remaining trip is necessary for correctly taking this decision.
The third subproblem is estimation, which is related to the final aim of the problem: an estimation of the resource level available at any point on the surface, based on all the information available at the end of the trip. This is done through regression using both the initially available points and those collected during the expedition.
In this work we detail a hybrid algorithm for tackling this problem, including components of combinatorial optimization, machine learning and evolutionary computation. The objective is to carefully plan a data collection expedition that maximizes the information available at the end of the trip. This implies choosing the most profitable probing points, which are part of a ship trip that must respect a time limit. The trip length is determined by means of an integer programming model for orienteering [1]. We propose the estimation of the resource levels using a Gaussian processes model [4]. Points with high variance are initial candidates for probing, but their position will evolve through random distortion towards a solution which minimizes a measure of the variance allover the relevant surface.
Experiments with simulated data show that the proposed method improves the quality of the ship’s schedule. We use an error measure involving the dis-crepancy between the “true value” and the predicted value, estimated in a fine grid over the relevant surface.
2 Method
The main difficulties for designing an algorithm for this problem concern the nonviability of direct evaluation of the objective function at the time of assessing the quality of a solution. This is due to the fact that the objective—i.e., the error between the “true” value of contents and its evaluation through the function that is used to estimated them, along the relevant surface—is not known at the time some point is considered for probing. We deal with this difficulty in several layers.
We assume that the variance at each point is a surrogate for the interest of probing it, in order to obtain a better overall picture. However, assessing the variance just at the probing points was found to be insufficient; we verified that an estimation of the variance allover the surface was indeed necessary for having a meaningful measure of the interested of probing a point. Hence, when in the algorithm we attempt probing at a given point, the variance summed for a set of points representative of the whole surface is used; in practice, we evaluate it at the same points that are used for evaluating the error, by means of the variance estimated by Gaussian processes. This is depicted in Figure 1.
The main method is provided in Algorithm 1. The best kernel setting φ is selected according to a 5-fold cross-validation procedure that minimizes the
dynamic version
staticversion
Assessment
New data Initial data
Simulated data
Planning:
- combinatorial optimization (route feasibility) - continuous optimization
(probe location)
Estimation
Fig. 1.Information flow and interactions between different parts of the algorithm.
error on the set of previously known dataD. 25 kernel settings were used. They were based on the scikit-learn library’s RationalQuadratic, RBF and Matern [2]. The parameters length scale bounds and alpha bounds were set to respect one of the intervals in the set{(0.00001,0.001), (0.001,0.1), (0.1,10), (10,1000), (1000,100000)}. The values for parameternu boundswere{0.5,1.5,2.5}. A noisy component was added to all kernels with bounds (1e−5,1e−2). The optimizer was allowed to restart 10 times.
The algorithm can be described as follows. A Gaussian Processes model w (with standard deviation functions) based on kernel settingφis induced overD by functionfit. While there is computational resource available, i.e. the elapsed time is within the time limit, the algorithm repeats a 4-step sequence:
– orienteering - insertion of points that maximizesover a grid Gin accor-dance to the allowed trip costs;
– disturbance - one of the trip (a) points (p∈a) is randomly chosen to be distorted according to the bivariate normal distribution3N2(p,
0.005 0 0 0.005
); – simulation - the regression functionw is applied to the distorted trip (a0) to simulate probing values and to induce a new model able to calculate the sum of standard deviation valuesσ;
– TSP - discard of the previous steps results if the distorted trip (a0) is not feasible and the unfeasibility is confirmed for the best TSP solution.
The function cost and the TSP solver take into account the trip duration and probing time.
functionMain(D, T, S, E, v, L, K)
φ←best kernel setting according to 5-fold cross-validation onD G← {(x0+δk, y0+δ`), k= 1, . . . K, `= 1, . . . L} /grid onS σmin← ∞
hw, si ←fit(D, φ) /regression functions:wfor prediction;sfor standard dev.
whileelapsed time<time limit do a←Orienteering(D, T, S, φ, G, a) a0←Disturb(a)
h, si ←fit(D∪apply(w, a0), φ) /applywtoa0 to simulate probing
σ← P
hx,yi∈G
s(x, y)
r←TSP solution visitinga0 if cost(r)< T andσ < σminthen
σmin←σ a←a0 end end
D0←D∪apply(v, a) /start of the testing part hw, i ←fit(D0, φ)
∆←P
(x,y)∈E|v(x, y)−w(x, y)|
return∆
Algorithm 1:Main procedure. Input: previously known dataD, allowed trip costs including duration and probing time T, available area S, testing points E, “true function” evaluatorv, grid dimensionsL×K. Output: estimated error value over the grid∆.
functionOrienteering(D, T, S, φ, G, a) while True do
hw, si ←fit(D, φ) hx, yi ←arg max
hx,yi∈G
s(x, y) /return element with maximums
a0← {hx, yi} ∪a
r←TSP solution visitinga0 if cost(r)>T then
break end a←a0 z←w(x, y)
D.append(hz, x, yi) end
returna
Algorithm 2: Orienteering. Input: previously known data D, allowed trip costs including duration and probing timeT, available areaS, kernel settingφ- grid where to take measurements G, list of points to visit for probinga. Output: expanded list of points to visit for probinga.