lunes, 8 de agosto de 2016

R/simecol Simulation Model for the Battle of Iwo Jima

It was about 71 years ago today that the Second World War on the Asian front ended. After the tragic Little Boy and Fat Man detonations over Hiroshima and Nagasaki on the 6th and 9th of August 1945, respectively, the Japanese surrendered in August 14th.

When I taught the Simulation and Modeling course for the Systems Engineering department at the UNEFA, we used to do an exercise in class on simulating the battle of Iwo Jima, one of the most decisive and sanguinary battles in the Pacific. This simulation was based on a System Dynamics model derived from the Lanchester Laws of combat and was implemented in R using the simecol/deSolve/FME packages. Prior to the class, we had covered the topic of Forrester diagrams and System Dynamic models in general1, the Lanchester Laws of combat and of course a bit of background on the use of the simecol and related packages for dynamic model fitting and simulation.

In this post, I will present a brief summary on the Lanchester Laws of Combat, some historical background on the battle of Iwo Jima and then go into the simulation model itself, detailing its implementation and data fitting in R through the said packages. Subsequently, I will contrast the model's output with the real historical data and derive some interpretations. To conclude the post, I will pose some "what if ..." questions on alternative hypothetical scenarios and offer some thoughts about possible applications of simulation models like this one for teaching history in classrooms.

The Lanchester Models of Combat

 Frederick Lanchester was a British engineer well-known for his inventions and contributions to the budding automotive industry at the end of the nineteenth century (he's the guy that invented the accelerator pedal, for example). Lanchester also made some contributions to the theory of aerodynamics and being concerned over the ways in which the aeronautical industry could change warfare (he was more of a pacifist, really), he set out to study the dynamics of warfare and develop mathematical models of battle through systems of differential equations, now known as the Lanchester Laws of Combat. These differential equation models are fairly abstract - they leave out many details that we commonly associate with battles and focus on the way in which number of combat units of two armies decrease over time.
Without loss of generality, we will represent in each time instant $t$ the number of troops or combat units of the red army as $R(t)$ or $R$ and of the green army as $G(t)$ or simply $G$. At the start of the battle, at $t=0$, $R(0)$ and $G(0)$ represent the number of combat units of each army. When $R(t)$ or $G(t)$ reach 0 at any given time, the corresponding army is left without troops and so the battle ends with the opposing army as victorious. Lanchester proposed two different scenarios for battle, according to the warfare technology used:

First Lanchester Law

In the first Lanchester Law, two armies confront each other in hand-to-hand combat. At each time instant, the number of casualties of each army is proportional to the number of confrontations between the combatant units and the effectiveness or lethality coefficient of the enemy army. This is represented mathematically by the following differential equations:

\begin{align*}\dfrac{dR}{dt} &= -g \cdot R \cdot G \\ \dfrac{dG}{dt} &= -r \cdot R \cdot G\end{align*}

Derivatives $\tfrac{dR}{dt}$ and $\tfrac{dG}{dt}$ indicate the way in which the battle unfolds and represent the rate of attrition of the red and green armies, respectively. The $r$ y $g$ parameters are the effectiveness coefficients of each army and represent the number of casualties on the enemy side inflicted during one time instant at each confrontation (represented by $R \cdot G$). This model of combat is also referred to as the the model of "unaimed fire", the word "fire" in this case not necessarily referring to artillery fire but rather to the directionality of discharge of the lethal armament used in combat. In fact, this model, being applied mainly to battle scenarios with no artillery can also be used for battles with low-precision artillery.

If we divide one equation by the other and isolate the red and green terms on each side, we obtain $r \cdot dR = g \cdot dG$. In turn, if we integrate (with respect to $t$) this last equation we can see that at each time instant, the expression $r\cdot R(t) - g \cdot G(t)$ remains constant. The terms $r\cdot R$ and $g \cdot G$ are called the fighting strengths of the reds and greens, respectively. The fighting strengths, which could also be considered as the number of troops eliminated the the red and green armies, determine result of the battle. Since $r\cdot R(t) - g \cdot G(t)$ is constant throughout the whole time, the reds will win if $r\cdot R(t) - g \cdot G(t) > 0$, the greens win if $r\cdot R(t) - g \cdot G(t) < 0$ and if $r\cdot R(t) - v \cdot G(t) = 0$, then both sides will be wiped out simultaneously. Since the fighting strengths are linear in $R$ and $G$, the first Lanchester Law is also known as the linear law of combat.

We can infer some characteristics of combat scenarios based on this linear law of combat:
1. An army's efficiency parameter and its number of troops are equally important factors to its fighting strength (ie. if you double the number of troops but decrease the efficiency to one half, the fighting strength will be the same).
2. If you manage to divide the enemy in two contingents and confront each contingent successively, this will not alter the result of the battle.

Second Lanchester Law

This law supposes a scenario where both sides use aimed artillery fire, hence the casualty infliction mechanism no longer depends on the number of hand-to-hand combat confrontations being produced, but rather on the number of troops firing upon the enemy:

\begin{align*} \dfrac{dR}{dt} &= -g \cdot G \\ \dfrac{dG}{dt} &= -r \cdot R\end{align*}

Again, dividing one equation by the other, isolating the terms corresponding to each army and integrating, we obtain the following constant expression: $r\cdot R^2 -g \cdot G^2=0$. This time, the fighting strengths $r\cdot R^2$ and $g \cdot G^2$ are quadratic on $R$ and $G$, hence the alternative name of "square law of combat" being applied to Lanchester's Second Law. In two armies with equal number of troops, the more efficient one will prevail. If however, one army manages to divide the enemy in two or more contingents and confront each contingent successively, then it will certainly win over the other. This is what Von Clausewitz and Sun Tzu meant by advising generals not to divide their troops because each division would significantly reduce an army's fighting strength. In other words, if an army is numerically superior, then it must combat undivided against a more efficient enemy. As an example of this principle, Hannibal confronted a qualitatively superior enemy at Cannas by dividing him and prevailed. Conversely, the American strategy of flooding the island of troops at Iwo Jima proved effective in managing to establish a beach head.

Applications of the Lanchester combat models and further elaborations

Lanchester originally applied his model to the study of Nelson's naval tactics in the Battle of Traffalgar, that famous naval battle of the Napoleonic Wars. In more recent literature, Lanchester models were applied to the Kursk battle in 1943 involving panzers in the German and Soviet sides, the battle of Ardennes (again in World War II) and the naval conflict in the Atlantic in which the German U-Boats inflicted damage on the Allied convoys. In the Engel 1954 paper we find an application to the Battle of Iwo Jima, based on the data compiled by Cliiford and Moorehouse. On this last point, I must remark that the model fits to these battles has been partially successful due to the fact that under normal battle conditions, it is quite difficult to accurately compile day-to-day data of casualties on both sides.

The Lanchester models themselves can be further generalized to include reinforcement like so:

\begin{align*}\dfrac{dR}{dt} &= f_r(t) - g \cdot G \\ \dfrac{dG}{dt} &= f_g(t) -r \cdot R \end{align*}

Here, $f_r(t)$ and $f_g(t)$ represent the reinforcement rates of the red and green armies respectively. This is the model we will use for our Battle of Iwo Jima simulation, as the American side brought in reinforcements on several occasions, while, as we shall see, the Japanese had none.

If we are willing to relax our assumptions that the battle is based wholly on either an aimed or an unaimed fire combat model, then we get Bracken's generalized Lanchester Model:

\begin{align*} \dfrac{dR}{dt} &= -g \cdot R^q \cdot G^p \\ \dfrac{dG}{dt} &= -r \cdot R^p \cdot G^q\end{align*}

If we define $\alpha=1+p-q$, we get $gG^\alpha-rR^\alpha$ as the constant quantity and we can easily see how the first and second Lanchester models are particular cases of this model. However (see MacKay, 2005), when fitting this model to any particular battle there is an ambiguity in the estimates for $p$ and $q$, as several best fitting estimates could be obtained.

Some historical background on the Battle of Iwo Jima2

 The island of Iwo Jima3 is a tiny island of some 8 square miles situated about 750 miles south from Tokyo in the Japanese mainland and halfway between Saipan and Tokyo. Were it not for this last fact, the place would not have been the stage of a major battle in the Second World War and would probably remain largely unknown today. By the summer of 1944, the Americans had gained control of Saipan and the Mariana islands, from where they staged bombing raids to mainland Japan with long range B-29 Superfortress bombers. However, the losses inflicted upon these bomber planes by Japanese fighter planes deploying from the two airfields in Iwo Jima were unsustainable to the Americans and furthermore, the island's radio station relayed early warning reports to the mainland to ready themselves for the bombing raids. For these reasons, it was of strategic importance to the Americans on their northward multi-pronged move towards Japan to seize control of Iwo Jima. And, naturally, the Japanese knew this.
 For the Japanese, Iwo Jima was more than just strategically significant. Iwo Jima was sacred Japanese soil. On account of this, an eventual American invasion of Iwo Jima would be a heavy blow to the Japanese morale as for more than four thousand years, no foreign country had invaded Japan. Therefore, the Japanese were not ready to surrender the island without first inflicting heavy losses upon the American forces. Emperor Hirohito intrusted Lietenant General Tadamichi Kuribayashi4 with this mission.
Kuribayashi drew up his plan for the defense of the island upon the following premises:
• The Japanese would engage in combat from underground, not on the surface of the island.
• The battle would be defensive, the idea being to seriously erode the American forces.
• No Japanese soldier would survive.
• Every Japanese was to kill 10 Americans before being killed.
Kuribayashi's strategy stood apart from the usual Japanese tactics to which the Americans had become accustomed. He specifically forbade the famous "Banzai" suicide attacks to repel the American disembarkation on the island, which besides, were completely futile to prevent it, since the Japanese would not be receiving air or naval support, and hence no reinforcements. Instead, he began construction of a gallery of underground tunnels connecting a network of bunkers and pillboxes. By the time the Americans arrived, the Japanese had completed over 11 miles of tunnels and placed large artillery pieces protected by thick steel walls from within Mount Suribachi.

For their part, Americans were also preparing for what they designated as Operation Detachment (the invasion of Iwo Jima). According to their intelligence reports, there were some 14,000 Japanese troops on the island (there were really over 21,000). Before D-Day, and based on their experience at Saipan, Tarawa and Peleliu, the Marines requested for 10 days of previous bombings. The Navy finally acquiesced and on the 16th of February, 1945 began a 3-day bombing. Previously, though, B-29s had completed a 72 day bombing campaign.
The bombing campaign on Iwo Jima had been devastating ... but only on the surface. Inexplicably, not only had the Japanese defensive fortifications not diminished, according to an American intelligence report, but they had grown in number from 450 to 750 3 months after the bombings began. For the unsuspecting American forces, the real surprise lay waiting for them underground...

On D-day (February 19th), the Americans disembarked 54,000 troops. During the first moments, many thought that no Japanese had survived the bombings and that the estimates on the number of Japanese troops were exaggerated. Then, as the troops made their way 300 yards into the beach and the spot became congested with troops and equipment (what Kuribayashi was waiting for), all hell broke loose. There was crossfire from Mount Suribachi and some hills to the north of the island. The beach offered no protection, as the fine and ashy volcanic sands made it impossible to dig trenches. As Japanese fire impacted upon the disembarkation boats, the amphibious vehicles got stuck in the fine sands. The beaches of Iwo Jima became an infernal chaos in the carnage that ensued.

 During the first days of battle, it seemed like the Americans had made no progress at all. They were fighting an enemy they couldn't see - an enemy who fought from under the ground and usually at night. Upon burning underground burrows where Japanese were thought to be with flamethrowers, Americans were surprised to see Japanese troops re-emerge from burrows that were thought to be "cleansed" days ago. What the unsuspecting Americans didn't know is that these holes and burrows were part of a vast network of underground tunnels. In the meantime, from the 20th to the 23rd day of February, the Marines attempted to take Suribachi. A small group of Marines eventually managed to raise the American flag on the top of Mount Suribachi, a moment immortalized for posterity in the famous historical photograph by Joe Rosenthal5.
Fighting continued for 36 days after the start of the battle. Once Suribachi was taken, the Americans continued to slowly and painfully advance to the north of the island, encountering fierce pockets of Japanese resistance along the way. One can only surmise about the fierceness of the combat that took place on Iwo Jima by reading the names of certain places on a map of the island, such as "the meat grinder". During the whole of Operation Detachment, the Americans made two more disembarkation of troops: one of 6,000 men and another of 13,000 men on the third and sixth day of the battle, respectively. In total, the Americans sustained 28,000 casualties including about 8,000 dead. The cost of taking this small island was much more than they expected. As for the Japanese, over 20,000 died and about 1000 were taken prisoners. The last two holdouts6 did not surrender until 1951. It is said that Kuribayashi himself died in a suicide raid on the southern part of the island disguised as a common soldier without his rank of General insignia.

Data for the simulation study

The data for this simulation study is based on a report titled "The Iwo Jima Operation" written by Captain Clifford Moorehouse, an American officer. This report contains day-to-day records on the number of American casualties. Together with the data on the number of American troops put ashore on the three disembarkations, this data serves to reconstruct the number of active American troops on the given days of the 36-day battle. Of course, we must assume an estimate number of 21,500 Japanese troops at the start of the battle and 0 active Japanese troops at the end, in between which no estimates exist. The data for the number of active troops on each side is summarized in the following table, in which the "time" column indicates the day of the battle and the "ame" and "jap" columns indicate the active troops for the Americans and the Japanese, respectively:

time ame jap time ame jap time ame jap
0 0 21500 13 59549 NA 25 53347 NA
1 52839 NA 14 59345 NA 26 53072 NA
2 50945 NA 15 59081 NA 27 52804 NA
4 56031 NA 16 58779 NA 28 52735 NA
5 53749 NA 17 58196 NA 29 52608 NA
6 66155 NA 18 57259 NA 30 52507 NA
7 65250 NA 19 56641 NA 31 52462 NA
8 64378 NA 20 54792 NA 32 52304 NA
9 62874 NA 21 55308 NA 33 52155 NA
10 62339 NA 22 54796 NA 34 52155 NA
11 61405 NA 23 54038 NA 35 52155 NA
12 60667 NA 24 53938 NA 36 52140 0

The data on the three American disembarkations can be summarized by the following stepwise function definition:

$f(t) = \left\{ \begin{array}{cl} 54000 & 0 \leq t \lt 1 \\ 0 & 1 \leq t \lt 2 \\ 6000 & 2 \leq t \lt 3 \\ 0 & 3 \leq t \lt 5 \\ 13000 & 5 \leq t \lt 6 \\ 0 & 6 \leq t \leq 36 \end{array} \right.$

Engel (1954) estimated the effectiveness coefficients for the Americans and the Japanese to be 0.0106 and 0.0577 respectively. We can take these values to simulate the model and compare its fit to the actual observed number of active troops according to the Moorehouse data. Then we will fit these parameters using simecol's fitOdeModel function and see if we obtain an even better fit. I must remark that when using simecol, it is not necessary to know in advance the values of your model's parameters, as the fitOdeModel takes care of this problem for you. After all, estimating a model's parameters is one of the main reasons why we build simulation models in the first place.

R/simecol implementation for the simulation model of the Battle of Iwo Jima

We have come to the core of this blog entry: the implementation of the simulation model for the Battle of Iwo Jima using R and simecol. First off, let me briefly explain what simecol is. It is an R package for the simulation of ecological models through R's S4 object class system using a state-space representation. What this means (the S4 object class part) is that you can work with several model objects in memory having different parameter values, etc. and simulate them, fit their parameters separately and so on, using an object-oriented methodology that makes it easy for researchers without extensive knowledge of R programming to get their simulation models up and running quickly using a general purpose programming language like R, which provides them with flexibility in modeling while at the same time access to all the powerful statistical/analytical tools available through the base R and extensive package system.

The state-space representation part mentioned in the last paragraph means that the various types of models one can implement in simecol- 1) dynamic system models using differential equations (or simecol odeModel class objects. This is the type we will deal with here), 2) cellular automata models and 3) individual agent based models - are viewed as subtypes of a more general family of models in which you have: 1) a set of variables describing the state of the model at a given time, 2) rules governing the way these state variables change over time, 3) a set of model parameters 4) a specification for the time instants at which you wish to observe the state variables, 5) externalities (called inputs), which are not part of the state variables of the system interrelated amongst each other but which affect them and 6) a specification for a solver function, which updates the state variables based on the rules that describe their change over time.

In the case of dynamic system models with differential equations, the rules describing the way the system changes over time are obviously characterized by a system of differential equations. These differential equations usually have parameters that further characterize the dynamic behavior of the system and which the researcher is usually interested in fitting from observed data. In this battle of Iwo Jima example, the parameters would be the effectiveness coefficients of the Americans and the Japanese. In our Iwo Jima model we also have the American reinforcements- those would be the inputs (or externalities) which are included in the differential equation describing how the number of American troops changes over time. Finally, the solver function is a specification of a particular numerical integrator that solves for the state variables using the differential equations, for example, 4th-order Runge-Kutta, Euler, LSODA, etc. In this sense, simecol serves as a wrapper for another package called deSolve, which includes the compiled FORTRAN numerical integration routines for fast execution in R.

Without further ado, here's the R script for the simulation of the Battle of Iwo Jima using the simecol library:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76  library(simecol) iwojima_engel <- new("odeModel", main = function (time, init, parms, ...) { x <- init p <- parms f <- approxTime1(inputs, time, rule = 2)["reinforcements"] dame <- f - p["j"] * x["jap"] djap <- - p["a"] * x["ame"] list(c(dame, djap)) }, parms = c(j=0.0577, a=0.0106), times = c(from=0, to=36, by=0.01), init = c(ame=0, jap=21500), inputs = as.matrix(data.frame( day = c(0, 0.999, 1, 1.999, 2, 2.999, 3, 4.999, 5, 5.999, 6, 36), reinforcements = c(54000, 54000, 0, 0, 6000, 6000, 0, 0, 13000, 13000, 0, 0))), solver = "lsoda" ) #copy the iwojima Engel model onto another for fitting later iwojima_fitode <- iwojima_engel #now read the actual data... obs <- read.csv2("iwo_jima.csv") #the weightdf dataframe assigns equal weight to all observations, #except for the NA values in the japanese column (those get 0 weight) weightdf <- data.frame(ame=rep(1, nrow(obs)), jap = rep(0, nrow=(obs))) weightdf[1,"jap"] <- 1 weightdf[nrow(obs),"jap"] <- 1 #return the ssq of the Engel model (a measure of goodness of fit) ssqOdeModel(parms(iwojima_engel), iwojima_engel, obstime = obs$time, yobs=obs[2:3], weights=weightdf) #and now we fit the model parameters with simecol's fitOdeModel #including our weightdf as weights makes the PORT routine faster result_fit <- fitOdeModel(iwojima_engel, obstime=obs$time, yobs=obs[2:3], fn=ssqOdeModel, weights=weightdf, method="PORT", scale=c(1/0.1, 1/0.01), lower=c(j=0,a=0)) result_fit #we update the new model parameters parms(iwojima_fitode) <- result_fit$par ssqOdeModel(parms(iwojima_fitode), iwojima_fitode, obstime = obs$time, yobs=obs[2:3], weights=weightdf) #finally we simulate and plot the graphs iwojima_engel <- sim(iwojima_engel) iwojima_fitode <- sim(iwojima_fitode) svg("iwojima_engel.svg") matplot(x=iwojima_engel@out, main = "Model1 : Engel(1954)", xlab = "Day", ylab = "Number of active troops", type = "l", lty = c("solid", "solid"), col = c("blue","red")) points(obs$time,obs$ame,col="blue",pch=19) points(obs$time[c(1,36)],obs$jap[c(1,36)],col="red",pch=19) graphics.off() svg("iwojima_fitode.svg") matplot(x=iwojima_fitode@out, main = "Model2 : fitOdeModel", xlab = "Day", ylab = "Number of active troops", type = "l", lty = c("solid", "solid"), col = c("blue","red")) points(obs$time,obs$ame,col="blue",pch=19) points(obs$time[c(1,36)],obs$jap[c(1,36)],col="red",pch=19) graphics.off()

Let's go over the code above. First, in line 1 we load the simecol library. In lines 2-20, we define an odeModel class object called "iwojima_engel". Notice how the "main" slot serves to specify the system of differential equations as a function returning the differentials on each state variable as a list. The other simecol methods will "know" that these differentials will correspond to the "ame" and "jap" variables because they are given as a list in the same order as we specify them in the "init" slot on line 13. Notice that the f function used to represent the American reinforcements is given as a function of the "inputs" slot defined in lines 14-18 (we will later explain what that approxTime1 function does). The "times" slot in line 12 is given in very small increments of 0.01 days so that the solver function (specified as "lsoda" in line 19) can do the numerical integration in small steps.

In line 24 we read in the Moorehouse data stored in a csv file (if you want to do this simulation exercise yourself, just take the Moorehouse data from the table above and put it in a csv file in the same directory as the R script (the separation character is a semicolon). Lines 27 to 29 simply create a data frame of the same dimensions as the "obs" data frame with 1 for valid values and 0 for NA values on that data frame. This simply serves to assign equal weights to each observed non-NA value so as to make the parameter fitting faster later on.

Lines 31-35 simply invoke the ssqOdeModel function on the iwojima_engel object to compute a measure of goodness of fit (consult the help page for ssqOdeModel to see how it is calculated). Lines 38-46 do the actual fitting of the iwojima_engel model object, producing a better-fitting set of parameters (remember, the American and Japanese effectiveness parameters). To do this, the fitOdeModel invokes a non-linear optimization routine using the PORT algorithm to obtain the set of parameter values that minimize the ssqOdeModel measure. In fact, this produces a better fit of the model (indicated by a lower ssqOdeModel value).

In line 48 we simply feed the new parameters into the iwojima_fitode, which was simply a copy of the iwojima_engel object and finally, in lines 55-76 we plot both model objects. Notice how the svg graphics device is used instead of pdf or png graphics. The svg graphics device produce better looking, non pixelating graphics that look great no matter how much you zoom in the page on your browser.

Finally, for comparison, we include a little table with the main results used to compare the different parameter values (Engel's estimations and the simecol estimations):

Engel (1954)fitOdeModel
Japanese Effectiveness0.05770000.05540003
American Effectiveness0.01060000.01080927
ssqOdeModel0.31676290.05121009

Interpretation of the simulation results and some what-if questions

Before I go on with the most fun part of modeling (playing with your models and interpreting them), I must clarify that simecol is a package originally intended for ecological modeling. It has also been used for engineering and economics modeling. However, because it is so versatile, I have used it for a military-related application such as simulating a battle using a Lanchester model, which is not an application for the package that its creators originally intended nor support. I do not mean by this that my view on this matter is different from the authors of simecol. In fact, I can see their wisdom in not supporting applications of simecol that they're afraid of.

What motivated me to present this model of the Battle of Iwo Jima are mainly educational reasons. First of all, the topic of historic and real battles is usually interesting for the young audiences to whom I've given classes on modeling and simulation. For me as a teacher, this serves as a sugar-coating to make the mathematical modeling and computational implementation topic-pills go down easier. Another important reason for me as a teacher is that these type of models lend themselves to interesting interpretations and discussions in class about politics and history. It gives us, as we shall see, a scientific framework for discussing about history and other social topics. To quote one of the authors on the bibliography I consulted:

Lanchester models provide an excellent example of the strengths (and weaknesses) of simple mathematical modeling. Further, as we have seen, the basic model leads to many other ‘What if...?’ questions which can be easily investigated. Many more such questions can be asked, and of course once one begins numerical simulations the possibilities are endless. To model warfare can seem more politically challenging (not to say incorrect) than to use ecology or epidemiology, but a little understanding of how military planners arrive at their tactical conclusions can also strip away mystique and (for this author, at least) expose some of the subject’s limitations! (MacKay, 2005, p. 7)

Some possible 'what-if' questions or questions for interesting discussion:
• Given the same effectiveness of the American and Japanese troops as those found in your model, with how many troops would the Japanese have won the Battle of Iwo Jima if the Americans had disembarked the same amount of troops on the island? Would it have been possible for the Japanese to amass this number of troops given the space and logistic limitations of Iwo Jima?
• If the Japanese soldiers were 5 times more effective or lethal than the Americans, why did more Japanese die in Iwo Jima?
• How many soldiers would the Japanese have needed to make the battle linger for, say, one year? How many American casualties would this have brought about? Would this have changed the outcome of the war?
• Lanchester models are based on conventional warfare. Nowadays, this conventional warfare scenario is not as generalized as before. For example, it is said that in Venezuela there is a non-conventional war being waged, although there are no battles in the classical sense involving regular military forces. There are civilian casualties, such as those caused by one of the highest crime rates in the world. And now, there is an added element of extreme scarcity of foods and medical supplies which cause civilian casualties as well. How would you as a modern-day Lanchester propose a differential equation model for this war scenario? What state variables and parameters would you define?
As you can see, the subject lends itself to some very interesting discussions in class. I might add that mathematical modeling of this sort gives us a fresh new perspective on social issues and gives us the foundations for discussing about these issues in a more scientific and less dogmatic way. I think having this sort of discussion integrated into our educational system is much needed today, particularly in my country.

As a closing remark for the readers of this blog, let me say that I will post more about simecol and other related packages, such as FME (with more added functionalities for parameter fitting). And no, the other posts won't be about military/warfare models! Stay tuned...

Acknowledgements

I wish to thank, first and foremost, you, dear reader, for putting up with such a long post. My thanks go out to the creators of the simecol package. Dr. Thomas Petzoldt's kind help on the R-sig-dyn-mod mailing list has been invaluable in helping me come to grips with this powerful R modeling tool. I would also like to thank James Cagney for his historyanimated media on the Battle of Iwo Jima. It helped me to understand the military and historical background of the battle. Make sure to check out his historyanimated site! Finally, I would like to thank Professor Christine Lind for providing me with the Moorehouse data without which this post would not have been possible.

Notes

1. There are some excellent and thought provoking articles on the educational application of System Dynamics to school children without any Calculus background by the System Dynamics in Education Project, led by Jay Forrester himself. In a 1992 article, Forrester criticizes pre-college education as "poorly serving the needs of society", because it does not take into account the complex interactions between people and phenomena. The system dynamics view, Forrester argued, would empower schoolchildren to deal with complexity. Implementation of these ideas in the classroom entails the use of software like STELLA II or Vensim (both commercial packages with GUI's available both for Macintosh and Windows). It would be awesome if there was a such an application for R using the simecol and FME packages as backends for the actual simulation and fitting of SD models, but including a GUI with which schoolchildren could construct such models from Forrester diagrams... Oh! I'm daydreaming again.
2. There is an excellent series of animated presentations available at the Historyanimated site (see Cagney, 2008). I contacted the author himself, James Cagney, in 2011 for permission to use his material in class and for some class notes I was preparing and thinking about publishing on the web. He told me to add the following interesting note: "The animation for The Battle of Gettysburg (civilwaranimated) is used to train smoke jumpers in the western US (men who parachute from airplanes to fight forest fires) - they tell me that fighting a fire is just like fighting a battle.". So there it is, James!
3. Iwo Jima, or Iō-tō ("sulphur island" in japanese) was first visited by a Westerner in 1543, a Spaniard by the name of Bernardo de la Torre, who called the island "Isla de Sufre" or Sulphur Island (Sufre was the archaic Spanish spelling for azufre or sulphur). It is an appropriate name, as the southernmost tip of the island hosts a volcanic vent (hence the sulphur) that produces the island's most prominent geographic feature: Mount Suribachi. The fine volcanic ash sands of its beaches would prove a major nuisance for the americans as they tried to disembark amphibious vehicles during their invasion of the island. On account of this sulphur, no other scenario of any World War II battle could better resemble hell and in fact, ancient Japanese legend has it that the island was inhabited by demons.
4. A man descended from 5 generations of samurai, Koribayashi was also a veteran of the Manchurian and Chinese campaigns. He had previously served as military attaché in Washington during the 1920's, where he acquired extensive knowledge of Americans. About the latter, he wrote: "The U.S.A. is the last country in the world against whom the Japanese should wage war" (Anonymous, 2006).
5. At this point I would digress in class and talk to my students about how this Pulitzer-winning photograph was an artistic marvel. I would also mention to them that in Venezuela, we also have our own "Joe Rosenthal". His name was Héctor Rondon and his famous picture of a chaplain helping a wounded Venezuelan soldier in a street corner, amidst friendly and unfriendly crossfire, also won the Pulitzer Prize in 1963 and was featured in LIFE Magazine.

 The events surrounding this "Pietá"-like picture were those of the Porteñazo, a military coup led by communist rebels backed by Cuba against the democratically elected president Romulo Betancourt in 1962. It is ironic that while today, the Chavista propaganda talks about US imperialism and intervention in our country before the so-called Socialist Revolution, the Cuban regime had tried on several ocassions to export their revolution to Venezuela (El Porteñazo was one such incident in which over 100 Venezuelan soldiers died). That was until Hugo Chavez came along and peacefully handed control of this country to the Cuban Castro regime without a single shot being fired. At any rate I leave you a small print of the photograph for your viewing.
6. On the subject of Japanese holdouts, there is the noteworthy case of Hiroo Onoda, who did not surrender in 1945 and spent 30 years in the jungles of the Philipines (if you didn't know about Onoda, look him up. It's one hell of a story). An english progressive rock band called Camel made a concept album called Nude inspired upon the true story of Hiroo Onoda (the name Nude derives from his family name Onoda).

Bibliographical References

• ANONYMOUS. (2006). "Iwo Jima el Test Supremo". Post in online forum. Retrieved March 2011, from http://www.forosegundaguerra.com/viewtopic.php?p=18492
• Battle of Iwo Jima. (2016, August 2). In Wikipedia, The Free Encyclopedia. Retrieved August 5, 2016, from https://en.wikipedia.org/wiki/Battle_of_Iwo_Jima
• CAGNEY , J. (2008). PacificWarAnimated.com - Historical Animation of the Battle of Iwo Jima. Nonprofit History Animated Foundation. Oregon, US. Retrieved March 2011 from http://pacificwaranimated.com/Iwo.html
• ENGEL, J. H. (1954). "A Verification of Lanchester’s Law". Journal of the Operations Research Society of America, 2(2), pp. pp. 163–171.
• Iwo Jima. (2016, July 31). In Wikipedia, The Free Encyclopedia. Retrieved August 5, 2016, from https://en.wikipedia.org/wiki/Iwo_Jima
• LIND, C. (2009). Verifying Lanchester's Combat Model - Battle of Iwo Jima.
• MACKAY, N. (2005). "Lanchester Combat Models". Retrieved March, 2011 from http://arxiv.org/pdf/math/0606300.
• PETZOLDT, T. and RINKE , K. (2007). "simecol: An Object-Oriented Framework for Ecological Modeling in R". Journal of Statistical Software, 22(9), pp. 1–31. ISSN 1548-7660. http://www.jstatsoft.org/v22/i09.
• R DEVELOPMENT CORE TEAM (2009). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. http://www.R-project.org.

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