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\hypertarget{project-1-modeling-populations}{%

\section{Project 1: Modeling

Populations}\label{project-1-modeling-populations}}

Alex Wenstrup and Jasmine Kamdar

September 27, 2018

\hypertarget{questions}{%

\subsection{Questions:}\label{questions}}

\begin{itemize}

Why was there a population rise in Germany between 1950-1970?

\begin{itemize}

\tightlist

\item

What was the effect on the population of Portugal (one of the 7

locations where migrant workers were coming to Germany from) during

this time?

\end{itemize}

\end{itemize}

\hypertarget{methodology}{%

\subsection{Methodology:}\label{methodology}}

\begin{itemize}

Overlay a generic logistic growth model on top of the German

population in the specified date range

Calculate and plot residuals to identify where the greatest deviations

from logistic growth occurred

Research significant events in Germany within the specified date range

Add data and model for other affected population as a way to validate

our results

Adjust models for both populations to account for found events

Plot residuals for new models, compare to residuals for old model, and

evaluate effectiveness of new model

\end{itemize}

\hypertarget{note-this-report-includes-all-processes-done-for-the-portuguese-population-inline-with-those-done-for-the-german-population.-that-is-not-indicative-of-our-actual-order-of-operations-rather-it-is-done-for-continuity-in-this-report.}{%

\subsubsection{Note: This report includes all processes done for the

report.}\label{note-this-report-includes-all-processes-done-for-the-portuguese-population-inline-with-those-done-for-the-german-population.-that-is-not-indicative-of-our-actual-order-of-operations-rather-it-is-done-for-continuity-in-this-report.}}

\hypertarget{step-1-find-and-import-data}{%

\subsection{Step 1: Find and Import

data}\label{step-1-find-and-import-data}}

We obtained our data from gapminder.com, an established resource and

tool for tracking world demographic data. After downloading their world

population data as an excel spreadsheet, we removed unnecesary rows and

columns so that our CSV file contained only data relevant to our model.

Source: https://www.gapminder.org/data/

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\hypertarget{step-2-overlay-a-generic-growth-model}{%

\subsection{Step 2: Overlay a generic growth

model}\label{step-2-overlay-a-generic-growth-model}}

We had to pick some generic growth model from which to calculate the

largest deviations from expected population. For this purpose, we chose

a simple quadratic growth model because of its face validity, easy

workability, and general good fit.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

Here we create an update function that models each year's population

growth as a function of current population, an assumed maximum

population, and some groth constant aG or aP.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

Here we create our run\_simulation functions, which iterate the our

update function through the timeframe for which we have data, and return

the results.

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{ \hspace*{\fill} \\}

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{ \hspace*{\fill} \\}

Note that these models were not created by linearizing our data and

fitting a least squares regression line. We eyeballed these models,

because their exact precision was not important.

\hypertarget{step-3-plot-residuals-to-identify-large-deviations}{%

\subsection{Step 3: Plot residuals to identify large

deviations}\label{step-3-plot-residuals-to-identify-large-deviations}}

At this point, it was already fairly evident where the largest

deviations were from our model, but a relative residual plot would serve

two purposes: - It made it even easier to see deviations from expected

growth - It gave us a means to compare our adjusted model to our generic

model

To calculate a relative residual, we divided the residual from our model

by the true population at the time.

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\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_25_0.png}

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{ \hspace*{\fill} \\}

\hypertarget{step-4-research}{%

\subsection{Step 4: Research!}\label{step-4-research}}

Now that we have identified 1970, and the buildup to that date, as a

significant deviation from expected population, we knew exactly where to

look for population events. A quick google search for `german population

boom 1970' led us to this study: http://countrystudies.us/germany/84.htm

which we then verified with this New York Times article:

https://www.nytimes.com/1984/08/19/magazine/germany-s-guest-workers.html

\hypertarget{gastarbeiter}{%

\subsubsection{Gastarbeiter}\label{gastarbeiter}}

We found that due to Germany's booming economy in the 1950's, many

workers were brought in to work in a variety of industries, including

infrastructure. Between 1955 and 1973, seven countries, including

Portugal, entered agreements with Germany which made it easier for

workers to travel to Germany for work. These workers, called

gastarbeiter, were originally meant to stay only for 3 years, though

many of them stayed in Germany permanently.

\hypertarget{step-5-a-new-model}{%

\subsection{Step 5: A New Model}\label{step-5-a-new-model}}

Knowing the Gastarbeiter migration event, we created a model for each of

the countries with the expectation of population growth and decline. The

first step for creating this new model was coding a function of the rate

of migrants flowing into Germany.

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{ \hspace*{\fill} \\}

This function starts off at 1960 rather than the actual start date of

1955 because we predicted that the program did not get popular enough to

significantly affect the population. The shape of the graph is then a

parabolic curve because the popularity would highten in the middle years

of the program and then slowly decrease to 0 as the German economy

suffered in the early 70's.

We made a new system to account for new variables.

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To create our new model for Germany, we added our travel\_popularity

function to our generic quadratic model. We also accounted for 1/3 of

migrants leaving Germany after 3 years.

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{ \hspace*{\fill} \\}

To create our new model for Portugal we subtracted people from the

generic model according to the travel\_popularity fuction. We also added

back the 1/3 of the workers who returned to Portugal. We also accounted

for the fact that Portugal was one of seven countries from which

migrants went to Germany by multiplying the function by 1/5. This

assumes that 1/5 of the migrants in Germany came from Portugal.

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To see how our new model compared with our generic model, we made two

residual plots that noted the difference between the our models and the

actual data.

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\hypertarget{interpretation}{%

\subsection{Interpretation:}\label{interpretation}}

\begin{itemize}

The quadratic growth model roughly approximated the population curve,

though there were clearly variations in the population that were not

accounted for in that model

This growth model, along with its residual plot, showed us when

significant deviations in populations occurred

The new model almost perfectly accounted for the change in Germany's

population, and accounted for over half of the population decrease in

Portugal

Our migrant model was successful because in the time range 1955-1973,

the residuals were significantly lower than those of a generic

logistic model

\end{itemize}

A few causes of error include:

\begin{itemize}

We were unsure how many migrants workers stayed in Germany

We were unsure how the popularity of migration changed with time

We were unsure what percentage of the migrant workers in Germany came

from Portugal

\end{itemize}

\hypertarget{sources}{%

\subsection{Sources}\label{sources}}

\begin{itemize}

https://www.gapminder.org/data/

http://countrystudies.us/germany/84.htm

https://www.nytimes.com/1984/08/19/magazine/germany-s-guest-workers.html

\end{itemize}

% Add a bibliography block to the postdoc

\end{document}