Dimensionality reduction and mathematical modeling after reading blog!

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author ： Miss Xiaoyu , University of Connecticut ,Datawhale member

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        The article comes from Zhihu's question ： How to evaluate  
       
       2021 
       
        American Undergraduate Mathematical Modeling Contest ？ How's the game this year ？
      
      
      
      

       
       
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Let's join the fun and reduce the dimension to attack the US race , Write down the ideas of the first two questions .2018 When I was a sophomore in, I won an American Competition H, I've read it now PhD 了 , Look at what I did at that time... Like shit . After reading several answers, I found that many people even have a wrong understanding of the topic , Private opinion The most important thing in the game is to solve the problem , Instead of piling up a pile of fancy things Fancy Methods .

A topic

First question ： describe breakdown of ground litter and woody fibers stay  fungal activity in multiple species of fungi Under the circumstances

To solve this problem, we must first understand what to use to quantify the decomposition ？ What affects the decomposition ？fungal activity What is fungal activity ？ What kinds of fungi are they ？

Since we are not mushroom experts , The information we can get basically comes from A trait-based understanding of wood decomposition by fungi This article , Download it to see what information there is . Although we can't copy the methods in the paper , But it can help us understand some basic concepts .

stay Introduction We found a sentence in it ：

Models traditionally used microbial biomass as a proxy of decomposer activity (3, 6), treating the microbial community as a single homogeneous group or a small number of functionally distinct pools (7).

Traditional methods generally use microbial biomass Biomass to represent decomposer activity , This question is obviously fungi Namely decomposer, therefore fungi The amount of should represent the fungal activity in this problem . After reading the article , We found that in the experiment hyphal density( Mycelial density ) It should correspond to this biomass .

To characterize the growth of each isolate, we quantified its hyphal extension rate and hyphal density (presented in ref. 34)

So what is used to quantify the decomposition ？ That's what the paper says ：

Here, we explore which fungal characteristics predict wood decomposition rate across a range of common wood decomposer fungi.

Namely wood decomposition rate. Concrete , This is calculated based on the number of pieces of wood eaten by fungi .

we measured the mass loss of wood blocks when colonized by each fungus to estimate a standardized wood decomposition rate

The next question is what kinds of fungi are there ? From the second question, add... To the model Fungi Interactions between different species , We can generally speculate that the first question does not need to consider a variety of fungi The internal role of .

OK First, we have a clear idea , The topic wants us to build a mathematical model （ The regression model ） To characterize the independent variable （ Amount of fungi fungi biomass） And dependent variables ( Decomposition rate wood decomposition rate) The relationship between , This is what the traditional method does . I think it is possible to find some basic assumptions about the relationship between the two variables in the following two articles .

K. L. McGuire, K. K. Treseder, Microbial communities and their relevance for ecosystem models: Decomposition as a case study. Soil Biol. Biochem. 42, 529–535 (2010)

S. Manzoni, A. Porporato, Soil carbon and nitrogen mineralization: Theory and models across scales. Soil Biol. Biochem. 41, 1355–1379 (2009).

Second questions ： Consider the interaction between different fungi , Add... To the model Fig1 and Fig2 Information about

First, understand the remaining questions in the previous question , What are the types of fungi ? In the article, we find that ：

First, we use a database of 22 fungal traits previously measured in each of 34 wood rot fungi collected from a wide geographic range across North America (25, 34–36) to identify potential drivers of wood decomposition.

Here the author uses 22 Study on the characteristics of fungi , What are the characteristics of fungi ？ Why is there no type of fungus in it ？ We continue to explore from the paper , Find such a sentence ：

However, in recent years, the development of traitbased approaches has begun to transform our understanding of broad-scale functional patterns. By linking traits to ecosystem functioning, these approaches have been used in plant and animal ecology to infer the functioning of novel communities without prior knowledge of the taxa that are present.

It's probably a new kind of traitbased Research methods , Knowing only the characteristics of a group of fungi （ There is no need to consider their kind ）, Instead of knowing the species of fungi . So perhaps the core of our question is not to consider specific species, Instead, consider a bunch of people with the same traits Different classes of fungi （ For example, fast-growing bacteria , Antifreeze bacteria …… wait ）, With their common trait As a basis for species .

So these traits What is it ？ From the paper, we found that 22 Kind of (2+11+9)traits It can be divided into three categories :

Hyphal extension rate and hyphal density reflect hyphal morphology and growth strategy,  ecological performance traits (11 in total) relate to combative ability.tolerance of a range of temperature and moisture conditions (25),  and finally, the production of oxidative and hydrolytic enzymes (9 traits) promotes nutrient acquisition from organic resources (26).

But pay attention to the problem. It says , We only focus on growth rate and moisture tolerance Just fine ：

For this MCM Problem you should focus on just two traits of a fungus: the growth rate of the fungus and the fungus’ tolerance to moisture.

OK So the question is clear , Here we need to join growth rate and moisture tolerance To the univariate regression model we established in the first question , Turn it into a multiple regression model , So in the original traditional method , Introduce the methods in this paper , Therefore, we need to use the data of this article ：

​ ​https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1909166117/-/DCSupplemental​​

Pay attention to the results of your final model , To compare the original paper Fig.3, See if it is consistent with this figure .

Third questions , The analysis model , And describe the long-term and short-term relationship between different kinds of fungi

Second, after building the model , Analyze the relationship between the next few independent variables （ Relevance and independence ）, Let's see what the relationship is , Do some statistical analysis according to these relationships . For example, whether it is found that there is a trade-off between independent variables , It can be explained by the competition between different bacteria, etc ……

It is important to pay attention to these points ：

• The dynamic relationship between long-term and short-term , Consider their changing trends （ You can refer to the method of extracting trend from time series ）. Consider their competitive relationship , The author carried out pair wise The competition comparison is in this article Combative ability In the data .
• test rapid fluctuations Influence on each variable sensitivity （ Introduce noise to test hypothesis ）
• Analyze the impact of climate change . Then add the independent variable temperature 、 Humidity and other indicators . Use the... In this paper Temperature and moisture niche data .

……

The latter two questions also need to use the regression model established by the first three questions to make some predictions . After clarifying the basic concepts , There's a way of thinking .

B topic

B For a long time , I can't understand it directly , Let's talk about my understanding of the topic first .

There are several objects in the title ：

• EOC： Understood as the rescue command center , It should be some fixed location sites , however Google No location found , I don't know whether it's one or many .
• SSA drones： Drones with cameras and sensors , Used to monitor the fire brigade , Understood as a drone used near the fire brigade （ Understood as a 0 distance , The camera can't see very far ）
• Radio Repeater drones： Drone with repeater , Used to help EOC Drones communicating with the fire brigade , You can stay away from the fire brigade （20KM）
• Firefighters ： Fire brigade

I searched SSA What is it , Here's an explanation . Situational awareness is happening around you, at or near your physical location, It should be the one with close observation . I think it's like the American special forces , A small drone with its own .

Situational Awareness is a concept closely involved with physical security information management (PSIM, see the white papers on this subject). It is usually defined as being aware of what is happening around you, at or near your physical location, or at some other location where you are supposed to be or where there are assets or people that you must protect. 

ref: https://www.titan-vision.com/pages/hot-topics/total-situational-awareness.html

A few confused questions ：

• Is there any communication between the two UAVs ？ I didn't see in the question .
• Where do drones start flying ? It's the fire brigade. Take it and fly , Or from the EOC Fly over ？ I think the fire brigade should fly with it , And it can also be roughly inferred from the hovering of the aircraft below .

Presumably, the situation is like this ,K A drone with repeater hovers at a distance EOC Fixed distance (20km) Inside the circle , Make sure to add the communication radius of the fire brigade （5km flat ground ,2km The city ） To be able to talk to EOC communication . Because every fire brigade has one SSA Unmanned aerial vehicle (uav) , So directly abstract the fire brigade team as SSA,SSA You need to follow the fire brigade .

You have to buy a lot SSA Unmanned aerial vehicle (uav) , And a few drones with repeaters . The problem is that drones with repeaters and remote sensing are expensive for $10000 , It should be the kind of large one used to transport things ,SSA A small drone with a camera for surveillance , Similar to Dajiang ,Amazon Search should be very cheap ？

Akme Corporation's prototype WileE–15.2X hybrid drone is projected to cost approximately $10,000 (AUD) when equipped with either a radio repeater or video & telemetry capability.

The loss function can define a 【 Out of union SSA（ Fire brigade ） Number 】【 The area of the fire that was not rescued in time 】 And 【 The cost of buying two drones 】 A weighted , Just reflect the requirements in the title ：

Your model should 【balance capability and safety】 with 【economics】,

You can change these three weights to do it trade-off, See what happens .

Then I think of the following constraints ：

• Fire brigade （SSA） Distance of （5km flat ground ,2km The city ）+   Repeater distance (20km) Less than and EOC Distance of
• At least... Is required within the distance of each fire point 1 A fire brigade
• (SSA) Fire brigade （SSA） The time to get to the fire point needs to be less than K minute

These constraints do not need to be fully satisfied , Use Lagrange relaxation to find a lower bound .

About whether SSA And repeaters use a drone , I prefer not to use a drone , Otherwise, first ask balance Economy and security cannot be done , Just focus on safety .B The idea of the question is based on my correct understanding , If I don't understand correctly , Then the idea is wrong , Let's refer to it .

My understanding of the second question is , To randomly generate some extreme fire situations , Then look at the first question, which constraints have been broken . According to those broken constraints , See how it should be adjusted .

……

I know that meisai is very difficult for undergraduates without systematic scientific research training , But that's the meaning of this kind of competition , Be able to concentrate on solving a practical problem alone in three days , It's a painful thing, but it'll be enjoyable after it's over . Post a sentence from my favorite pop drummer ："If you want the fucking core part, earn it! "

Sorting is not easy to , spot Fabulous Three even ↓