## vim: nowrap tabstop=2 shiftwidth=2 expandtab textwidth=10000 #acl All:read <> {{{#!wiki important 1. It's been death by a thousand cuts, but the [[#simgrading|grader tarball]] ready to go (updated 11 Nov 17:00 with scientific notation tests on output) 1. Here is [[#|the change log|&action=info]] for this assignment write-up. I will try to be descriptive in my log messages. 1. You can also [[#|subscribe|&action=subscribe]] to this page and receive Emails when changes are made. 1. There have been two minor updates to `grader.sh`. One to correct a title typo, and one small fix to the $n$ curve, I had written `n<=40` in my solution whereas the text has `n<40`. }}} In the Fall of 2019, I heard the following radio story: [[https://www.cpr.org/show-segment/plastic-week-continues-with-eliminating-single-use-plastic-like-cups-bags-and-takeout-containers/|Eliminating Single Use Plastic...]]. In it we learn that a single use aluminum container is recycled $70\%$ of the time, to which the program host says (surprisingly):<> ||'''Host:''' || "There's probably a diet-rite that I was drinking in the early 90s whose aluminum is still in the system, is what you're saying."|| ||'''Guest:'''|| "That's correct."|| Really? :\ = Conceptual Model = We want to know the fraction remaining of a single use recyclable aluminum container after a certain number of "generations". The probability that a single use item in generation $g$ will be recycled and "live on" to generation $g+1$ is $p$. When an item is recycled, we envision it being divided into $r$ equal parts; each contributing to the production of $r$ new (distinct) items in generation $g+1$. This cycle continues (each item in generation $g+1$ has probability $p$ of continuing on to generation $g+2$ as $r$ new distinct items. The initial generation will be $g=0$, so any item in the use-recycle-use chain at generation $g = 1,2,3,4\ldots$ will have $\frac{1}{r^g}$ of the original ($g=0$) item within it. = Specification Model = The parameters $p$ and $r$ will be provided to your `SIM` via command line parameters. The confidence intervals will be calculated to the half width $\pm0.1$. = Project Requirements = Your `SIM` will calculate a confidence interval for the average remaining fraction $f$ of an item after a certain number of generations. Your `SIM` will use '''Algorithm 8.1.2''' for the calculation. = Input = The command line parameters provided to your `SIM` will be: || Argument || Value || || `GENS` || The number of generations to simulate || || $t$ || `idfNormal()` value to use for confidence interval construction, called $t^{*}_\infty$ the text || || $p$ || The probability that an item in generation $g$ is recycled and reused in generation $g+1$ || || $r$ || The (equally sized) number of parts an item is broken into during the recycling process, each goes into a new distinct item.|| || `RANDOM.DAT` || As usual, a source of random values $Uniform(0,1)$.|| = Output = You will `OUTPUT` the following values from `SIM` (in this order, and according to the course [[Assignments/Requirements|submission requirements]]): 1. the number of simulations run in order to achieve the confidence interval required ($n$) 1. the lower bound of the confidence interval, '''use eight digit precision and [[Assignments/ScientificNotation|scientific notation]] output'''<> 1. the upper bound of the confidence interval, '''use eight digit precision and [[Assignments/ScientificNotation|scientific notation]] output''' = Documentation = Of course, your documentation should be handed off according to [[Assignments/Requirements|the usual submission requirements]]. 1. Document the location of your Algorithm 8.1.2 implementation (source and lines) as well as your "Welford object" (if you are using one). 2. Provide a paragraph or two of well written and thoughtful analysis, address the following topics: a. Does the simulation, as written, address the fundamental question? If so, what assumptions are being made? If not, what do you believe is ''not'' being modeled correctly and how would you implement remedies? a. Estimate what fraction of the pop can used by the radio show host in the 90s is still with us today in the use-recycle-use chain. This can be done in one of two ways: i. Using some math and the results of small `GENS` and $r$ `SIM` runs, i. Making some much longer `SIM` runs with appropriate parameters (beware of overflow and you are going to need '''a lot''' of random numbers and an efficiency bump!) '''Either approach yields an equal amount of credit''', but you must '''explain your methods clearly'''. You are not required to use the ''graded'' `SIM` implementation to support your findings. You may augment your `SIM` with optional invocation magic (just so long as it still works with `grader.sh`) or hack up another analysis. Choosing a language or library with an `idfBinomial()` variate would be very wise! = Hints and SIM Testing = 1. Beware of tracking each individual item at each generation with an object - with $p=0.70$, $r=10$ you will have about $75\times10^6$ individual items in a data structure by the ninth generation. Yikes :o It is better to maintain the following '''two pieces of information''' as a "generation" loop plods forward, breaking when the $f$ falls to zero or the required `GENS` have been simulated: the ''number of items'' in the current generation, and of course the ''current generation'' $g$. In this case, the remaining fraction of the original item is \[ f=\frac{\mbox{items}}{r^g} \] {{{#!wiki warning Even with this more efficient approach, we must beware of numerical issues! Depending on the random numbers (of course), you can still overflow a 32-bit integer tracking the number of items currently in the recycling chain. '''Use a 64-bit integer, or a language with arbitrary precision integers (Python, for instance) for your `items` variable.''' }}} Ideally, we would use a $Binomial(\mbox{items},p)$ random variate to determine how many progress to the next generation. But this would require a numerical inversion of the incomplete beta function (Appendix D), which is beyond the scope of this course. Your `SIM` can simply "flip" a $Bernoulli(p)$ coin for each item. == grader.sh == <> I am providing to students the same tarball the grader will use for testing your `SIM`. Here is how to use it: First, download <> to your Mines Linux account ("alamode" machines!) and unroll it in a temporary directory. {{{ $ ls XXX-student.* alrecycling-student.tar.bz2 $ mkdir tmp $ cd tmp $ tar xjf ../alrecycling-student.tar.bz2 }}} Second, set the SIMGRADING environmental variable with: {{{ $ source ~khellman/SIMGRADING/setup.sh ~khellman/SIMGRADING }}} Now go to the directory holding your !AlRecycling `SIM` and execute the `grader.sh` script from the `alrecycling-grader.tar.bz2` resource. {{{ $ cd ~/sim/alrsim $ ls SIM SIM $ ~/tmp/AlRecycling/grader.sh : : : }}} You will need to read any messages from the script carefully, and hit `ENTER` several times throughout its course. This script checks for: a. missing tracefiles a. truncated tracefiles a. and the difference between `SIM` results and expected results The latter test generates 3 PDF files for your inspection (one $n$ curve comparison, and two confidence interval comparisons). For plots generating CDF comparison curves: the <> is the result of '''your `SIM`''' with $N=200$ data points; the <> are the expected results from an $N=500$ trial, and the <> are the results of three separate trials with $N=100$ for comparison. Your `SIM`'s red line should be a better approximation of the blue dots than the blue lines. For plots generating overlapped confidence intervals: the <> are the $200$ intervals generated by '''your `SIM`'''; the <> are $500$ intervals from a good implementation, and the <> are the results of three separate trials with $N=100$ intervals. Your `SIM`'s <> should not be any more varied from the <> than the <> are. Here is an example of one of the plots generated by `grader.sh`: {{attachment:_s_n-2,70,8.svg|ns for 2 gens, 0.70%, 8 parts|width="60%"}} {{attachment:_s_cis-2,70,8.svg|CIs for 2 gens, 0.70%, 8 parts|width="60%"}} <> = Submit Your Work = <> <> {{{#!wiki important To ease the logistics of grading, there are two Rubrics and two Submission Slots in the course interface. '''Submit the same tarball or zip file for both!''' The course grader will handle the computational assessment, and your instructor will grade the analysis portion of your submission. }}} = Computational Rubric (part 1) = This work is worth 50 points. || Requirements || Points || Notes || || Meets [[Assignments/Requirements|simulation course project requirements]] || 10 || || || Algorithm 8.1.2 and Welford documentation || 5 || || || Output meets requirements (eight significant digits) || 5 || Some libraries will print in "fixed" notation when the sci notation exponent is 0. || || Calculation of $n$ (`GENS=2`, $r=70\%$, $parts=8$) || 10 || || || Confidence Intervals (`GENS=2`, $r=70\%$, $parts=8$) || 10 || || || Confidence Intervals (`GENS=6`, $r=70\%$, $parts=8$) || 10 || || = Analysis Rubric (part 2) = This work is worth 25 points. || Requirements || Points || Notes || || Meets [[Assignments/Requirements|simulation course project requirements]] || 5 || || || Clarity and prose || 10 || || || Analysis || 10 || ||