Probabilistic Model of Random Encounter in Obstacle Space
Round 1
Reviewer 1 Report
This paper proposes a method to simulate probabilities of two moving objects seeing/encountering each other that can account for terrain variation and the viewshed. The paper also uses a study area in Wuhan China as an example to demonstrate the method and discuss the results.
Overall, the paper contains original works and shows good efforts to consider geographic contexts. However, the reviewer does have major concern and minor comments.
1. The paper discussed movement probabilities of A and B within time t in Figure 6 and 7, which shows that the objects tends to stay at current location instead of moving and the movement of two objects are independent from each other. Although the second assumption is reasonable, a mobile object usually has some direction preference toward destination. Otherwise, what is the reason for moving if there is no place to go?
2. If A can see B, B can see A since there is no obstacle between them, Therefore, “A encounters B” and “B encounters A” should have the same probabilities by intuition, right? If so, why Figure 8 and Figure 9 shows different probability distribution?
Also, does “A encounters B” means, given A is at location lA at time t, what is the probability that A can see B? If so, the distributions in Figure 9 should be the joint probability and hence 9a and 9b should be the same, right? Otherwise, the movements of A and B are not independent.
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3. The third question is about time dependency. Based on equation 6 (and the following up ones), probability at time t1 will not affect probability at time t2. However, in the real-world, if A and B see each other for the first time, their movement in the future will be affected by it. If not, what will be the reason to study encountering?
Therefore, it seems more reasonable to model the probabilities of A and B encountering for the first time; and then based on the locations of A and B when they encounter, model their reaction considering whether they prefer to meet each other or trying to avoid each other.
4. The final question is about movements probabilities considering obstacles. Results in Figure 7 do not seem to consider terrain while simulating the visit probabilities. For instance, moving up should have slower speed than moving down and there may be some locations that cannot be accessed due to the sudden change in elevation (like one showed in Figure 3, cell (r1, c2) seems not be accessible (or need quite a bit time to “climb up”). Since the paper claims to consider the terrain and obstacles, the simulation of movement probabilities for A and B should consider it.
5. The equations need further edits. For instance, in equation 2, f(x, y) should be f(x, y, z) and it is better to have some parenthesis to separate inner operations. In equation 4, since the accessible locations for A are changing over time, omega of A and B are time dependent. Therefore, Omega A, B should be 5-dimension instead of 6, given that t for A and B is always be the same. Equation 9 and 10 use a combination of discreate and continuous notations for aggregation, which does not seem correct and consistent.
Author Response
Response to Reviewer 1 Comments
This paper proposes a method to simulate probabilities of two moving objects seeing/encountering each other that can account for terrain variation and the viewshed. The paper also uses a study area in Wuhan China as an example to demonstrate the method and discuss the results.
Overall, the paper contains original works and shows good efforts to consider geographic contexts. However, the reviewer does have major concern and minor comments.
Point 1: The paper discussed movement probabilities of A and B within time t in Figure 6 and 7, which shows that the objects tends to stay at current location instead of moving and the movement of two objects are independent from each other. Although the second assumption is reasonable, a mobile object usually has some direction preference toward destination. Otherwise, what is the reason for moving if there is no place to go?
Response 1: As suggested by the reviewer, we have outlined these moving scenarios and reason in the revised manuscript for the sake of better understanding and clarity. (See Line 394 from top, 4.2 Calculation of positional probability)
“The motion of two objects is assumed to be independent of each other, and moving objects usually have a certain directional preference for the destination. Some scenarios that are consistent with the hypothesis of this experiment, such as human movement, including an orienteering event, the potential movements of missing person and recreationalists (e.g., hunters and hikers) [36], and also apply to mating behaviors between animals, predation relationships, and so on.”
Point 2: If A can see B, B can see A since there is no obstacle between them, Therefore, “A encounters B” and “B encounters A” should have the same probabilities by intuition, right? If so, why Figure 8 and Figure 9 shows different probability distribution?
Also, does “A encounters B” means, given A is at location lA at time t, what is the probability that A can see B? If so, the distributions in Figure 9 should be the joint probability and hence 9a and 9b should be the same, right? Otherwise, the movements of A and B are not independent.
Response 2: After examining the reviewer’s comments carefully, we must admit that we have not expressed our meaning clearly in the previous manuscript. Our explanation has been added to the article and the revisions were shown in the following.
“The reachable domain range of mobile objects A and B are different, and the distribution probability of mobile objects A and B on their reachable domain are also different, so the probability map based on the reachable domain of the mobile object is different, that is, the probability map of A meeting B and B meeting A are different [17]. And individual differences are also part of the reason that leads to different probability maps, such as height and vision (the farthest distance can be seen).In order to facilitate the calculation, this paper only sets the difference in individual height.” (See Line 503 from top, 5 Results )
Response 2: Many thanks to reviewer for your valuable suggestions. This is a very significant question. It's correct that A encounters B means the probability that A can see B at given A is at location lA at time t.
Response 2: Thanks for the reviewer’ question, we found these remarks most useful to clarify our points and We discuss this aspect now clearly in the revision. The probability calculation flow chart has been added to the article, Equation 8 belongs to the joint probability, and Formula 10 is the edge probability. So we can't analyze the independence with the joint probability in this article. (See Line 350 from top, 3.3.3 Calculating the probability that an individual at a specific location met another )
Point 3: The third question is about time dependency. Based on equation 6 (and the following up ones), probability at time t1 will not affect probability at time t2. However, in the real-world, if A and B see each other for the first time, their movement in the future will be affected by it. If not, what will be the reason to study encountering?
Therefore, it seems more reasonable to model the probabilities of A and B encountering for the first time; and then based on the locations of A and B when they encounter, model their reaction considering whether they prefer to meet each other or trying to avoid each other.
Response 3: This is a very significant question. Thanks for the reviewer’ question, the answer to the question makes the proposed method more explicit.
“The method proposed in this paper is based on the assumption that individual motion is independent of each other. In fact, the occurrence of encounter events has an impact on the independence of individual actual movements, but does not affect the assumption that the null hypothesis (i.e., individual motion is independent of each other) is correct, because the null hypothesis itself is the hypothesis to be tested. This means that the null hypothesis needs to be tested. ”
As an extension of the proposed method in practical applications, the test of the null hypothesis (i.e., individual motion is independent of each other) can be used to test whether two individuals tend to move independently of each other, or show mutual attraction or avoidance.
“Our research is based on a moment and our subsequent work will calculate the probability of encounter in a continuous period of time, and then use the sequence time to analyze the independence. This method that how to infer independence in the sequence moment has been discussed [17].”(See Line 584 from top, 7 Conclusions )
Point 4: The final question is about movements probabilities considering obstacles. Results in Figure 7 do not seem to consider terrain while simulating the visit probabilities. For instance, moving up should have slower speed than moving down and there may be some locations that cannot be accessed due to the sudden change in elevation (like one showed in Figure 3, cell (r1, c2) seems not be accessible (or need quite a bit time to “climb up”). Since the paper claims to consider the terrain and obstacles, the simulation of movement probabilities for A and B should consider it.
Response 4: This is a very interesting and significant question. Many thanks to reviewer for your valuable suggestions. Our explanation has been added to the article. The revisions were shown in the followings.
“In the classic time geographic model, the velocity vmax is only related to the individual's movement ability. However, practically vmax is, importantly, influenced by the characteristics of the environment through which the object moves, such as topography, land cover, and the presence of barriers [36]. For example, Doherty proposed a method to calculate the pedestrian speed defined by the mathematical function of the terrain slope in the field-based time geography [37].For the convenience of calculation, we use the inverse proportional weight to directly assign the position probability, without considering the influence of terrain on moving speed, but there are corresponding methods in the literature [37].”(See Line 569 from top, 6 Discussion )
Point 5: The equations need further edits. For instance, in equation 2, f(x, y) should be f(x, y, z) and it is better to have some parenthesis to separate inner operations. In equation 4, since the accessible locations for A are changing over time, omega of A and B are time dependent. Therefore, Omega A, B should be 5-dimension instead of 6, given that t for A and B is always be the same. Equation 9 and 10 use a combination of discreate and continuous notations for aggregation, which does not seem correct and consistent.
Response 5: Many thanks to reviewer for your careful review. The parenthesis have been added to the formula and there are some explanations for our formulas.
“f(x, y) is the elevation function of the surface, expressing the elevation value at the two-dimensional position (x, y), such as zA = f (xA, yA), zB = f (xB, yB). ”(See Line 233 from top, 3.2.2 Building the event that individuals were at specific locations at a specific time )
It’s correct that this function should be three-dimensional and the three-dimensional coordinates can be represented by (x, y, f(x, y)), so we use f(x, y) and there is no effect on our calculations.
Response 5: “The Equation 4 is a 6-D space. That is, the Cartesian product of the three-dimensional variable of lA and the three-dimensional variable of lB: lA(x, y, z) × lB(x, y, z). The time t is a constant rather than a variable, so it is not seen as one dimension. Although lA and lB are variables in three-dimensional space, the lA and lB are units based on the surface of the discrete expression, and the number of the unit is based on the unary, such as lA[1] and lA[2], instead of the two tuple (row, column). And lA and lB can be expressed in 1 dimension. Thus, a one-dimensional array of lA and lB can be traversed by two parallel operation symbols.”
Response 5: “Yin et al. [15] introduced a perception distance threshold and proposed a model of encounter probability based on this distance threshold. Irregular encounter areas are infinitely subdivided into regular grids and then calculated by integral. ”(See Line 123 from top, 2.1 Encounter in probabilistic time geography )
In the Equation 9, “both lA and lB are three-dimensional spatial points, so the corresponding integral is triple and the Equation is not contradictory. Discrete type and explanation of Equation 9 and Equation 10 are derived in section 4.3. ”(See Line 340 from top)
In the Equation 10, “This probability is an edge probability of Equation (8), where A is in a known location and B is in all possible locations nearby at the same time.”(See Line 346 from top)
“It is possible to convert the encounter probability formula from continuous-type to discrete-type, thus realizing the conversion of the probability formula of the encounter from the mathematical model to the model that can be calculated in the computer. This conversion mainly includes two aspects: converting the point of continuous space and the probability on it into a discrete unit and the probability on it; converting the integral operation into a sum operation. Equation (8) is the probability of encounter of two individuals at a given pair of points, which can be the center point of the unit in discrete space, so that this formula can be directly applied to discrete spaces.”(See Line 417 from top)
“Since lA and lB are represented by a one-dimensional tuple, the two summation operators can traverse the one-dimensional array of lA and lB, respectively. Obviously, Equation 14 is also the summation form of joint probability.”(See Line 429 from top)
The flow chart of probability calculations has been added to the article. (See Line 350 from top)
Author Response File: 
Author Response.docx
Reviewer 2 Report
This paper was an interesting read. It seems to build its basis on the existing literature in the field of time geography.
The paper should be proofread again and some of the tables have to be rearranged as they are hard to read.
I think that the schema for building the events would be a valuable addition to the paper (3.2.2 etc.) as well as a flowchart for a step by step probability calculations.
I like how the authors discuss the results and how they plan to develop the model further.
Overall I enjoyed the paper.
Some additional comments:
49:- spatio-temporal no need for capitalising.
Be consistent with spatio-temporal.spatiotemporal
63: t - make it italic c
65: what is this: ---- ? Quotation marks?
107: used is misspelled
123: YIN is capitalised.
128: rewrite this sentence
172: table 1 is hard to read. It should be edited.
176: what do you mean by: GIS provides them with different methods…?
278: missing words at the end of the sentence
339: additional instead of addition
339-341 - Possibly rewrite
344 = the, instead of The
367: sentences such as this one are not necessary.
374: can a classical time geography calculate anything? Please rewrite this sentence
419: why does the DEM is just in a grey scale?
450: we need a legend to the DEM values as well.
Author Response
Response to Reviewer 2 Comments
This paper was an interesting read. It seems to build its basis on the existing literature in the field of time geography.
The paper should be proofread again and some of the tables have to be rearranged as they are hard to read.
I think that the schema for building the events would be a valuable addition to the paper (3.2.2 etc.) as well as a flowchart for a step by step probability calculations.
I like how the authors discuss the results and how they plan to develop the model further.
Overall I enjoyed the paper.
Point 1:The paper should be proofread again and some of the tables have to be rearranged as they are hard to read.
I think that the schema for building the events would be a valuable addition to the paper (3.2.2 etc.) as well as a flowchart for a step by step probability calculations.
Response 1: Many thanks to reviewer for your valuable suggestions. The revised version has added the schema for building the events and a flowchart for a step by step probability calculations.(See Line 243 and 350 from top)
Point 2:49:- spatio-temporal no need for capitalising.
Be consistent with spatio-temporal.spatiotemporal
Response 2: This is a very significant suggestion. It is necessary to make spatio-temporal and spatiotemporal to be consistent. We have used the spatio-temporal all in the revised version.
Point 3: 63: t - make it italic c
Response 3: We are sorry for our carelessness. It is now corrected in the revised version.(see Line 63 from top )
Point 4: 65: what is this: ---- ? Quotation marks?
Response 4: We are sorry for our carelessness. It is a wrong symbol, we deleted and rewritten the sentence in the revised version.(see Line 65 from top)
Point 5: 107: used is misspelled
Response 5: Used is now corrected in the revised version.(see Line 107 from top)
Point 6：123: YIN is capitalised.
Response 6: Yin is now corrected in the revised version.(see Line 123 from top)
Point 7：128: rewrite this sentence
Response 7: This sentence has been rewritten. (see Line 128 from top)
“ Can animals and natural enemies see each other in the wild? Is there a significant interaction between the two species? ”
Point 8：172: table 1 is hard to read. It should be edited.
Response 8: Many thanks to reviewer for your careful review. The table 1 is now have been edited in the revised version. (see Line 172 from top )
Point 9：176: what do you mean by: GIS provides them with different methods…?
Response 9: After examining the reviewer’s comments carefully, we must admit that we have not expressed our meaning correctly in the previous manuscript. So we deleted this sentence in the revised version.(see Line 176 from top )
Point 10：278: missing words at the end of the sentence
Response 10: Many thanks to reviewer for your careful review. We have rewritten the sentence in the revised version.(see Line 291 from top )
“It is also possible to build the event that an individual A at a specific location lA met another individual B at time t, which is denoted as the event”
Point 11：339: additional instead of addition
Response 11: As suggested by the reviewer, the addition is now rewritten as additional.(see Line 360 from top )
Point 12：339-341 - Possibly rewrite
Response 12: As suggested by the reviewer, we have rewritten the sentence in the revised version. (see Line 360 from top )
“Furthermore, mapping the probability of the location of all lA in the reachable domain to create a probability map of individual A encountering B [26].”
Point 13：344 = the, instead of The
Response 13: Done.(see Line 365 from top )
Point 14：367: sentences such as this one are not necessary.
Response 14: Good idea. We deleted this sentence in the revised version.(see Line 389 from top )
Point 15：374: can a classical time geography calculate anything? Please rewrite this sentence
Response 15: Thanks for the reviewer’ question. We must admit that we have not expressed our meaning correctly in the previous manuscript. The any time has been corrected to given moment in time and the maximum speed of the individual is now added to the sentence.(see Line 398 from top )
“Classical time geography [21] can calculate the reachable domain of an individual at given moment in time based on the spatio-temporal information of the starting and ending points and the maximum speed of the individual.”
Point 16：419: why does the DEM is just in a grey scale?
Response 16: Thanks for the reviewer’ question, the answer to the question makes the proposed method more explicit.
“in order to show the probability of encounter on the plane in the results, the grayscale map better reflects the range of the reachable domain instead of elevation changes.”(see Line 440 from top )
Point 17：450: we need a legend to the DEM values as well.
Response 17: This is a very significant suggestion which makes the pictures more complete and makes the results more meaningful. The DEM values is now added to the map.(see Line 471 and 481 from top )
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The authors have responded to the reviewer's comment/question with respect. The reviewer value the efforts. The methods in the paper has been clarified significantly.
One comment on the transition from continuous space and probabilities to discrete units is to further clarify that: "The paper consider continuous space and time and provide the analytical framework. For simulation, considering the discrete nature of computer environment, the paper converts the original formulas into discrete case (see Equation ** and **)."
A final note, future works could use MC simulation to account for two cases after meeting for the first time (1) getting away from each other; and (2) moving toward each other. These would be more useful to modeling the two corresponding cases (1) animal movements and (2) human movements (e.g rescuing people as mentioned in the article).
Author Response
Response to Reviewer 1 Comments
The authors have responded to the reviewer's comment/question with respect. The reviewer value the efforts. The methods in the paper has been clarified significantly.
Point 1: One comment on the transition from continuous space and probabilities to discrete units is to further clarify that: "The paper consider continuous space and time and provide the analytical framework. For simulation, considering the discrete nature of computer environment, the paper converts the original formulas into discrete case (see Equation ** and **)."
Response 1: Many thanks to reviewer for your valuable suggestions. We found these remarks most useful to clarify our points. As suggested by the reviewer, we have outlined some explanations in the revised manuscript for the sake of better understanding and clarity.
“Because of the additive nature of integrals, i.e., an integral over a surface equals the sum of integrals over disjoint units that cover the surface, we can treat each of the unit regions in space-time separately and add them together once the computation is done [37].” (See Line 408 from top, 4.3 Calculation method of encounter probability )
“ And this paper considers continuous space and time and provides the analytical framework for quantitatively measuring the likelihood of two individuals encountering in an obstacle space. For simulation, considering the discrete nature of computer environment, the paper converts the original formulas into discrete case (see Equation (14) and Equation (15) ). So far, time geography has relatively perfect methods to measure random encounters. ” (See Line 542 from top, 7 Conclusions )
Point 2: A final note, future works could use MC simulation to account for two cases after meeting for the first time (1) getting away from each other; and (2) moving toward each other. These would be more useful to modeling the two corresponding cases (1) animal movements and (2) human movements (e.g rescuing people as mentioned in the article).
Response 2: This is a very significant suggestion. Many thanks to reviewer for your careful review. Based on the valuable comments made by the reviewer, we
add this sentence in the revised manuscript.
“A final note, future works could use MC simulation to account for two cases after meeting for the first time (1) getting away from each other; and (2) moving toward each other. These would be more useful to modeling the two corresponding cases (1) animal movements and (2) human movements (e.g rescuing people as mentioned in the article).” (See Line 564 from top, 7 Conclusions )
Author Response File: Author Response.docx
