Becoming a member of us right this moment is Keshav Patel, who’s a NSF Graduate Analysis Fellow at College of Utah. He was part of the workforce that completed as runners-up within the 2015 MathWorks Math Modeling Problem (M3C). Keshav can be following up on a earlier weblog publish relating to Half 1 of the 2019 MathWorks Math Modeling Problem (M3C). When you have not learn half 1 but, we encourage you to have a look right here. Over to you Keshav..
Within the first publish, Dr. Wesley Hamilton created a framework for the right way to deal with this downside. Now, let’s check out what actual members submitted for this competitors. We’ve got combed by way of a number of options throughout your complete vary of scores to ask, “what makes a great submission?”
On this publish, we can be inspecting groups for his or her method to the issue, their assumptions, their outcomes, and their submission construction/format. As you learn this weblog, it’s possible you’ll ask your self: which of those fashions is the “greatest”? We hope to indicate by way of these examples the next truth about modeling within the M3C: Mannequin justifications are way more necessary than using high-level arithmetic. Good submissions are those which have good arguments for his or her mannequin which offer justification for the alternatives they made. Stated a manner that may higher attraction to the mathematicians on the market:
Cool math + poor communication = unhealthy mannequin
Curve Becoming Options
A majority of groups approached this downside in a really related method to Wesley, though the precise strategies assorted relying on what knowledge they selected to include and what operate they selected to suit their knowledge to. First, we’ll study a submission that didn’t win, however got here fairly shut.
A Shut Have a look at a “good” Regression Mannequin
The primary workforce we can be inspecting arrange their submission in a way outlined by different math modeling assets: they restated the issue, wrote down assumptions, outlined their variables, after which described their mannequin. First, let’s check out one of the crucial necessary parts of mannequin setup: the assumptions.
Assumptions
Listed below are a couple of of essentially the most noteworthy assumptions on this submission:
- Assumption: The % of the inhabitants that makes use of vaping merchandise is an correct measure of the unfold of nicotine attributable to vaping merchandise. Justification: It will be unreasonable to find out the precise quantity of nicotine used over the previous couple years and predict it for the approaching years. Every vaping product has a special quantity of nicotine in it and, as seen throughout our investigation, current knowledge doesn’t report the quantity of nicotine every person consumed towards a time metric. Nevertheless, the unfold of nicotine might be measured by its reputation within the US market, because the extra folks use nicotine-based vaping merchandise, the extra nicotine is used.
- Assumption: There is no such thing as a new pertinent info relating to the hazards of nicotine-based vaping merchandise or legal guidelines that can have an effect on its reputation. Justification: Many research and studies have already been launched advocating the negatives of utilizing nicotine-based vaping merchandise, however regardless of this, as our analysis confirmed, the recognition of nicotine-based vaping merchandise has continued to extend. Moreover, though the introduction of complete FDA laws within the August of 2016 did trigger a pointy lower within the reputation of nicotine-based vaping merchandise [citation], since most related laws relating to using nicotine-based vaping merchandise has already been handed, and since these merchandise did regain reputation within the aftermath of the laws with the surge of vaping use in 2018, it’s affordable to imagine that future laws won’t have a big affect on the recognition of nicotine-based vaping merchandise.
- Assumption: The carrying capability of the market dimension for nicotine-based vaping merchandise might be estimated utilizing the historic carrying capability of the market dimension for cigarettes. Justification: When analyzing the traits within the reputation of cigarettes, the group seen that the preliminary development in reputation of cigarettes carefully mirrored that of nicotine-based vaping merchandise like e-cigarettes.
The primary assumption explains to the reader how this workforce selected to interpret nicotine use knowledge, the second assumption is a simplifying assumption that justifies the extrapolation of their statistical mannequin, and the third assumption provides a justification for utilizing cigarette knowledge to assist fill in a lacking piece of knowledge for vaping knowledge.
This workforce not solely gave an inventory of assumptions, but additionally gave a justification (with citations!) for many assumptions. Some questions your assumptions ought to assist to reply embrace: how are you deciphering your knowledge? what are you taking into consideration or not taking into consideration once you extrapolate your knowledge? why is your selection of operate for the regression a sound operate to make use of? What would occur when you didn’t make the idea?
Another word about justifications: their first assumption’s justification is kind of a difficulty of availability of knowledge. Because the competitors is just fourteen hours, it is a completely affordable assumption. It will be a good suggestion to revisit this assumption within the “Strengths and Weaknesses” part of your submission. Take into account how your assumptions would change and the way you’ll change your mannequin setup when you had higher entry to related knowledge.
Mannequin
Subsequent, we’ll have a look at the mannequin that was constructed. This workforce particularly compiled the e-cigarette use dataset in the identical manner that Wesley did in final week’s weblog:
Nevertheless, they determined to use a logistic regression to this dataset:
From this curve, they claimed that the general use of e-cigarettes will improve from about 15% in 2018 to about 45% across the 12 months 2025, after which degree off (however by no means lower). From a mathematical perspective, this work and their outcomes are completely legitimate. Now could be the time we must always ask “is that this an inexpensive consequence?” Properly, one of many workforce’s assumptions is that “There is no such thing as a new pertinent info relating to the hazards of nicotine-based vaping merchandise or legal guidelines that can that can have an effect on its reputation“. So, in contrast to the historic knowledge on cigarette utilization, there is no such thing as a change to the legal guidelines or how nicotine is consumed to suppose that the quantity of cigarette utilization will ever decline. As a result of I can join the workforce’s assumptions to their outcomes, I consider it is a affordable consequence. My private biases would lead me to consider that their result’s pessimistic; fortunately, inspecting the submission based mostly on argumentation helps to take private biases out the equation.
Earlier than we transfer on to the Strengths and Weaknesses part, there are two extra feedback value making. First, it will have been preferable to plot the logistic curve and the information on the identical determine. Second, it’s value taking a second both on this part or within the Strengths and Weaknesses part to say the impacts of the outcomes on the actual world. In any case, math modeling is all about answering actual world questions!
Strengths and Weaknesses
Given the time restriction within the M3C, you could have needed to make a variety of simplifying assumptions that you simply in any other case wouldn’t have, or it’s possible you’ll not have had time to analysis the subject or knowledge extra completely. The Strengths and Weaknesses part is an efficient time to acknowledge limitations to your mannequin and recommend potential enhancements.
Within the submission we’ve been analyzing, the workforce mentioned how their mannequin is an enchancment over a polynomial regression due to the bodily interpretation. Whereas the workforce doesn’t go into additional particulars on what “bodily interpretation” means, I take it to imply that the polynomial regression concludes that the cigarette utilization both turns into damaging or will increase indefinitely. Neither of those outcomes could be affordable when fascinated with numbers of individuals, so it is a good factor to make a remark of. Wesley additionally talked about this truth in Half 1 of our weblog sequence.
Their level on sensitivity evaluation is one {that a} choose could push again on. Any parameters which are estimated (by utilizing earlier literature, regression, instinct, and so on.) are inherently not precise. So, it’s often good apply to research how your outcomes would change in case your parameters are elevated or decreased in some vary. For the context of the M3C, a spread of 5% or 10% out of your estimated worth is customary, however in apply, your vary could also be based mostly on different components, resembling the usual deviation of your estimated worth.
Additional Follow
- What similarities and variations do you word within the assumptions? For those who had opposing assumptions to a different workforce, take into account a) how effectively you justified your assumption, b) how your mannequin must change when you used a special assumption, and c) which assumption you’ll fairly use.
- What similarities and variations do you word within the figures? How does the formatting look (i.e. is the textual content sufficiently big, are the completely different curves clearly labeled, is there spacing between desk entries, and so on.)? For those who regarded solely at a workforce’s figures and tables (and their captions), may you perceive the workforce’s outcomes?
- Take into account every workforce’s mannequin. Is it clear what they’re attempting to do? Are the variables clearly marked or labeled indirectly? Is there a facet of their mannequin or their outcomes that go towards the workforce’s assumptions?
Analyzing Different Regression Fashions
As talked about earlier, a number of groups used a regression method to deal with this downside. Nevertheless, there are a couple of similarities that many non-winning submissions share:
- Poor communication of their assumptions and variables
- Poor abstract of the mathematical mannequin
- Poor formatting or placement of necessary parts
Discover that none of those have something to do with the precise arithmetic! Recall our handy-dandy method as you write your submission and analyze others:
Cool math + poor communication = unhealthy mannequin
Let’s have a look at some examples of regression fashions that had some communication shortcomings.
Within the first instance, this workforce carried out a linear regression on the offered datasets. It is a good method as a result of it’s fairly easy, so not a variety of particulars are wanted. Additionally, if a workforce argues that ten years just isn’t a very long time, then it’s affordable to imagine that almost all different regression fashions may look fairly much like a line. The next is a screenshot of their outcomes:
The determine is pretty easy, is effectively labeled, and permits us to instantly examine cigarette use and e-cigarette use over time. One place the place their submission was not as sturdy was in one in all their key assumptions, which is given under:
- Assumption: Nicotine/Tobacco product utilization traits could have a linear sample within the coming decade. Justification: Each Regular Common and Exponential lines-of-best-fit proved to be extremely problematic of their potential to undertaking nicotine product utilization. Subsequently, a linear pattern should be assumed.
Whereas it’s glorious to incorporate this type of assumption, their justification is sort of obscure. As I learn this justification, I’m left questioning why the strategies they talked about are “problematic” and why linear pattern just isn’t “problematic”? After Half 1 of our weblog sequence and the submission above, we all know that there are points associated to “bodily interpretation” of the regression strategies, however the linear regression runs into the identical problem. To strengthen the justification, this workforce ought to focus on extra explicitly what was “extremely problematic” about these different strategies. Additionally, there are a plethora of different suits to attempt (logarithmic, for instance), so the workforce might also wish to touch upon why the linear mannequin is the very best.
Additional Follow
- Do you suppose the above assumption is an efficient assumption to make for this downside? If sure, rewrite the justification to enhance the argument. If no, write down a special assumption and justification and take into account how this workforce’s mannequin would possibly change consequently.
The following submission we’ll study was given a mid-level rating. The workforce begins out with an inventory of superb assumptions together with some very temporary justifications, like
- Assumption: Teenagers had the identical entry to cigarettes as they do to vaping. Justification: This enables for equal comparisons of the 2 types of nicotine transmission.
- Assumption: As soon as a well being problem is found, the vaping development price will lower equally to the lower of cigarette utilization after 1964. Justification: This may be assumed due to the recognized detrimental results of nicotine.
The primary justification is sort of obscure. As a reader, I’m not sure as to what “equal comparability” means. Are they attempting to outline what a “cigarette person” and a “vape person” are so the dependent variables might be in contrast? Or, are they attempting to equate a particular kind of cigarette buy to a vape buy? Additionally, as I learn the remainder of the report, I’m not sure as to precisely how “entry” components into their mannequin. These are parts of assumptions that seem in loads of different submissions, so having them just isn’t a foul thought in any respect. Nevertheless, if we take into account our submission as a sequence of logical arguments, it is very important take into account how your assumptions move into the later components.
The workforce then describes their mannequin; they created a compound curiosity method for the expansion in proportion of each cigarette and vape utilization. They then undergo the calculations crucial to succeed in their consequence, as proven under:
It is a good place to spotlight that your submission just isn’t the identical as your homework. Whereas your instructors could care concerning the nuts and bolts of your computations, judges wish to see simply sufficient work that your outcomes are reproducible. On this case, a compound curiosity mannequin is one thing I really feel doesn’t warrant house for computations. This house would possibly higher be utilized in including a desk or determine, or in including extra particulars to the mannequin rationalization or justifications.
For those who do really feel that your mannequin is sort of advanced, you would possibly take into account giving a brief pattern situation. It’s good to maintain these pattern calculations in your notes after which tentatively embrace them within the report, but when your report is simply too lengthy then pattern calculations are good issues to think about eradicating first.
Different Mathematical Strategies
As talked about above, a majority of groups tackled this downside utilizing a regression method. This has many benefits, one in all which is that it’s simple to implement and write about. Nevertheless, if you want your mannequin to extra deeply clarify how particular person components construct to population-level dynamics, then extra superior mathematical strategies might be useful. For this part of the weblog, we’ll briefly study extra superior modeling frameworks. When used appropriately, these fashions can permit a workforce to make significant connections between the inputs and outputs of the mannequin.
Nevertheless, this isn’t a suggestion to construct a extremely concerned mannequin that your workforce just isn’t snug with. Loads of excessive scoring and successful submissions use easy statistical/mathematical approaches and stable arguments. So, in case your workforce is uncomfortable utilizing a particular kind of math, then don’t use it!
Bizarre Differential Equations Mannequin
Subsequent, the workforce spends fairly a little bit of the submission (maybe an excessive amount of) explaining how they compute an necessary parameter in SIR fashions, referred to as
R0, from the out there knowledge. This parameter is a measurement for the way many individuals on common a single infectious particular person finally ends up infecting. Lastly, they offer the next plot as their predominant consequence:
Just like the figures within the submissions that make the most of regression, we will visually see the rise within the “Contaminated” group (comparable to energetic customers of e-cigarettes) as much as 10 years, then a slight decline. The group doesn’t go right into a sensitivity evaluation, they usually have brief part on strengths and weaknesses for his or her total submission on the finish, which doesn’t focus on any future instructions.
As we talked about earlier, a sensitivity evaluation, significantly on this all-important
R0parameter, is perhaps a good suggestion to incorporate to indicate how a lot variability exists in your mannequin. Additionally, it’s a good suggestion to refer again to your assumptions and focus on how they match up along with your outcomes, and the place you can alter your assumptions or conduct additional testing sooner or later. Desirous about the logical arguments you’re making in addition to the issue outdoors of the context of the competitors are issues readers would like to see!
Additional Follow
- What similarities and variations do you word within the assumptions? What assumptions are made within the SIR fashions that aren’t made within the regression fashions, and vice versa?
- What similarities and variations do you word within the figures? How does the formatting look (i.e. is the textual content sufficiently big, are the completely different curves clearly labeled, is there spacing between desk entries, and so on.)? For those who regarded solely at a workforce’s figures and tables (and their captions), may you perceive the workforce’s outcomes?
- Take into account every workforce’s mannequin. Is it clear what they’re attempting to do? Are the variables clearly marked or labeled indirectly? Is there a facet of their mannequin or their outcomes that go towards the workforce’s assumptions?
Closing Ideas
We hope that these two “You’ve Acquired to be Modeling Me” weblog posts provide you with a roadmap for getting began with this 12 months’s M3C. In fact, every competitors consists of three components. Elements 2 and three are rather more open ended, will typically require further analysis, and actually profit from groups dividing up duties. Keep tuned for future assets from me and Wesley for these components as effectively!
What’s definitely relevant to all components of the M3C (and math modeling usually) is that the arguments which are made in your submission are simply as necessary (if no more necessary) than the arithmetic. High quality submissions assist your reader perceive how your modeling decisions issue into your outcomes and spotlight strengths and weaknesses of your work. Who knew writing could be so helpful in doing math?
