\documentclass[12pt]{article}
\usepackage{graphicx,amssymb,amsmath}
\usepackage{geometry}
\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{lem}{Lemma}
\newtheorem{cor}{Corollary}
\geometry{letterpaper,portrait,margin=1in}
%\setlength{\topmargin}{-0.5in}
%\setlength{\left}{0.8in}
%\setlength{\right}{0.8in}
%\setlength{\textheight}{9in}
\begin{document}
\centerline{MONT105Q -- Mathematical Journeys}
\centerline{Information on Final Projects}
\centerline{March 7, 2016}
\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus, one of the assignments for the seminar this semester will 
be a final project.  You will be working on this project in {\it teams of 2 or 3} and the goals
will be to prepare a roughly 10 page research paper and an oral presentation to the class on your 
project.  The presentations will be given the final class meetings of the semester -- 
May 2, 4, 6, and 9.
\vskip 10pt
\noindent
{\it Schedule and Deadlines}
\begin{itemize}
\item  By Friday, March 18 -- inform me which general topic you want to work on and who you 
will be working with.  If your proposed topic has several different directions
that might be pursued (see the descriptions below), please give an indication of which aspect
you would like to work on.  If you need assistance in forming ``teams,'' I will
be happy to help with that.   Ideally, each group will work on a different project, although in 
some cases, the topics are large enough that if more than one group wants to try
that, there will be ways to ``split up'' the topic into several parts.  See me to discuss the 
possibilities.
\item  Friday, April 15 -- Each group will submit, 
{\it by email}, a bibliography of sources to be used for your
project.  You should identify {\it at least six articles, books, or 
web sites}, that will be relevant.
For each of your sources, write up a short paragraph giving a rough description
of how that source relates to your main topic, what kind of information you 
will take from it, and how you will be using it (including a preliminary
estimate of how reliable you think the information there is).  
The project descriptions below contain some first places to look, but 
you should plan to spend some time searching for additional sources of information.
\emph{Your final project papers may also refer to additional sources if you find that is
necessary.}   
\item  During the week of April 18, each team
will meet with me during office hours (or at another time
if that is not convenient) for a progress report and a chance
to look at any questions that have come up as you have started to
work on the project.   I will be happy to discuss any
aspect of the project at other times too, of course.
\item  We will decide which groups present which day when we get closer to the dates.
\item  All final project papers will be due no later than 5:00pm on 
\emph{Monday, May 9}.  (This is an absolutely firm deadline.)
\end{itemize}

\vskip10pt
\noindent
{\it Other Information}
\vskip 10pt
\begin{itemize}
\item  Ms. Barbara Merolli, our
Science Librarian, will be visiting our class after Easter Break 
to introduce herself and give some introductory information about using 
the library resources to identify sources for your project.  She will be more 
than happy to assist you in the important process of assembling the resources 
you will use for your project. 
\item  The presentations can be done with PowerPoint.
I will be happy to help out with technical details if needed.
\item  I will ask each group to do a ``dry run'' of their
presentation with me at least one day before you go in front of the class.
The purpose of this is to give you some feedback about what
is working and what is not, and to give you some practice to minimize
the effect of ``nerves'' when the time comes for the real thing.
\end{itemize}

\vskip 10pt
\noindent
{\it The Final Project Report}
\vskip 10pt
This writing assignment will be somewhat different from the papers that we have done 
previously in this class.  You should think of the written portion as the final write-up 
of the investigations you did on the data sets you looked at.  So you should probably 
{\it not} try to start writing until most or all of the mathematical/statistical
work has been done and you have thought over what the results said carefully.  The 
evaluation of your project reports will be based on how well you have addressed the 
following guidelines and expectations:

\begin{itemize}
\item  Distill your investigations into a central
argument.  A good ``technical report'' of this kind should be
more than just a compilation of all the different things you 
did.  It should be {\it well organized} and the writing 
should give the reader a clear indication where you are heading
with your central argument at all times. 
\item A typical outline/breakdown 
into sections might look like this:  
\vskip 10pt
\begin{itemize}
\item {\it Introduction} -- Catch the reader's 
attention, explain the significance of the problem or topic.
Say what you will do in general terms, without going into all the 
details from the start.  
\item {\it Methodology} -- Say where the data came from
(and what the quantities are and what units they are measured in), describe 
the statistical methods you used and how you applied them.
\item  {\it Results} -- Say how what you found addresses
the central argument.  In discussing the results of statistical tests,
it is considered good form to report the $p$-value returned by the test
(the smallest $\alpha$ for which the null hypothesis would be rejected) 
as an indication of the strength of the conclusion.  However, it is not
necessary to reproduce the calculation of the test statistic, and the 
mechanics of carrying out the test, etc.  
Also, don't go overboard by including every single table and graph you generate.  
Be selective -- one well-chosen graph can be just as informative as several similar 
ones!  On the other hand, {\it don't ``cherry-pick'' your results} -- be honest 
and include findings that might point to a limitation in the methods
you used, or a shortcoming of your main argument.
\item {\it Discussion} -- Don't overstate the 
importance of your findings, and (again) be honest if there are limitations.
Give proper credit to sources you consulted that contributed
to your ideas about the problem you studied.  (In a more formal article,
it would be expected that another section reviewing
the most relevant contributions of previous work on related subjects 
would be included -- that kind of full literature review is {\it not expected 
for this assignment}.) Discuss, if possible, how your results could be extended 
or generalized.
\item {\it References} -- Include all books, articles, 
websites you used in the preparation of the work.  For the websites,
give the full URL, and the date you consulted.  
\end{itemize}
\item  Be clear, concise, and correct in your writing. 
Aim for {\it no typos, misspellings, or grammatical problems}.  But even
more importantly, each paragraph should have a clearly evident purpose
in relation to your main argument.  
\item  Use graphs, tables, of data sparingly in the main 
text.  (If you want to include more of these, that can be done in an additional
Appendix section at the end.) 
\item  Proofread your work carefully and have an ``impartial''
reader or readers look at it and give you comments.  This can be one of the 
other teams or me.  Be prepared and willing to {\it revise} your work
based on the comments you get.  Of course, this means that the writing
must not be put off until the evening of May 8(!)  Be sure you 
get started early enough so that the input can be put to productive use.   
\end{itemize}

\vskip 10pt
\noindent
{\it Ideas for Project Topics}
\vskip 10pt
\noindent
The first group of project ideas use statistics
to understand trends and patterns in data {\it that change in some way over time}.
The technical name for this sort of data set is a {\it time series}. For example, a time 
series might represent average yearly temperature or precipitation at a particular location, 
or the batting average of a baseball player over the years of his career, or the concentration 
of a chemical in a water source measured in successive weeks.

\vskip 10pt 
\noindent
{\it Statistical Background}
\vskip 10pt
We can think of breaking a time series up into several components:
\vskip 10pt
\centerline{Time Series = Trend + Cycle + Residual,}
\vskip 10pt
\noindent
where the Trend might be an upward or downward movement, the Cycle
might represent a regular, repeating changing pattern
(e.g. the normal temperature changes due to the change of the
seasons), and the Residual represents random variation.
One of the most important questions
to ask in dealing with time series data is whether there is a way
to identify whether there is some increasing or decreasing
long-term trend involved in a given time series.

There are many different methods that statisticians use to address questions 
of this type.  \emph{For examples and further discussion, it will be 
valuable to start out by identifying and studying a general discussion
of this topic that is not too technical (i.e. geared toward people
who actually need to perform such analyses rather than toward statisticians
who are interested in extending these methods or developing more powerful ones.}

One basic method would be to use a {\it regression} of the data versus time.  The
slope of the regression line can then be used to decide if there is an upward or 
downward trend over time.  However, if the trend is not linear, or if some of the necessary
conditions (``football-shaped scatter plot'' or homoscedasticity,
normal distribution of the residual term, etc.) for the regression analysis are not 
met, then this approach is not especially appropriate.

There is an alternative class of methods called {\it nonparametric
statistical methods}, including in particular a method called
the {\it Mann-Kendall test for trends}.  These do not require
any of the assumptions needed for regression.   As a result, the
Mann-Kendall test has been very widely applied to study time
series arising in areas such as pollution control, climate science,
and other areas.  The Mann-Kendall
test works like this.  Call the time series $x_i$, for $i = 1, \ldots, n$.
\vskip 10pt
\begin{itemize}
\item  Essentially the only assumptions necessary (for the
basic version) are (1) if there is a trend, then it is ``monotone''
(either increasing for the whole time period, or decreasing for the
whole time period), (2) there is no nonzero periodic Cycle term, and
(3) the Residual term is purely random (not ``autocorrelated'').
\item  The test proceeds by computing for each time
$i$ and all later times $j > i$ the {\it sign} of the difference
$x_j - x_i$ ($+1$ if $x_j > x_i$, $-1$ if $x_j < x_i$, and $0$ if
$x_j = x_i$).  Let $S$ be the {\it sum} of all of these signs.
\item  Then a quantity called the {\it variance}, $V$, is computed by the following
formula:
$$V = \frac{1}{18}\left(n(n-1)(2n + 5) - \sum_{k=1}^g t_k(t_k-1)(2t_k+5)\right),$$
where $g$ is the number of different groups of ``ties'' in the time series
data, and $t_k$ is the number of terms $x_i$ in the $k$th group of ``ties''
for $1 \le k \le g$.
(If there are no duplicate values in the time series, then this last term
is zero.)
\item  Then the statistic
$$Z = \begin{cases}
               \frac{S - 1}{\sqrt{V}} & {\rm if}\ S > 0\\
                        &   \\
                     0 & {\rm if}\  S = 0\\
                        &   \cr
              \frac{S + 1}{\sqrt{V}} & {\rm if}\ S < 0
\end{cases}$$
is computed and this is used to infer the presence or absence of trends
as follows.
\item  Provided that $n > 10$, $Z$
is approximately {\it normally distributed}, so we can set up
confidence intervals or hypothesis tests with rejection regions specified
by the percentage points of the standard normal curve.
\item  For instance, for a (upper-tail) test of the alternative hypothesis
$H_a$:  There is an increasing trend in the data, versus the null
hypothesis $H_0$: there is no upward trend, at the $\alpha = 5\%$
(Type I error probability) level, we would reject $H_0$ if $Z \ge 1.645$,
and not reject $H_0$ if $Z < 1.645$.  As usual with hypothesis testing,
the rationale for why the test is set up this way is that the chance
that $Z$ has a value greater than $1.645$ is only $5\%$ if the null hypothesis
is true.  There are corresponding two-tail tests as well, where the alternative
hypothesis would be that there is some trend (either upward or downward).
\item  For smaller length time series, $n\le 10$, there are
tables available in many books and online that replace the standard normal table
for setting up the corresponding rejection regions.
\end{itemize}

\noindent
Among the advantages of the Mann-Kendall test are that it applies very widely,
its conclusions are not greatly affected by gross errors or outliers (because
it is only whether an increase or a decrease has occurred from one time period
to another that matters, not the {\it magnitude} of the change), and the computations
can be carried out by hand if necessary.  One disadvantage is
that it does not apply (and generates misleading results) when an upward or downward
trend is combined with a cyclic (seasonal) variation.  There are ``souped-up''
versions that deal with seasonal variations as well, though (``Seasonal Mann-Kendall''
tests).  Other ``corrections'' have been devised to deal with cases where the
Residual term is not purely random.
\vskip 10pt
\noindent
{\it A Mann-Kendall Example ("By Hand")}
\vskip 10pt

    Say we have a time series with successive values:

    $$12.9, 14.8, 13.2, 15.7, 13.8, 15.6, 12.9, 11.0, 17.3, 15.9, 19.0, 18.3, 20.1, 21.7$$

    The Mann-Kendall $S$  is computed by comparing each value \emph{to all later values}
(not any earlier ones)
    and recording a $1$ if the later value is larger, a $-1$ if the later value
    is smaller, and a $0$ if the later value is the same:  Here we get the following
\[
\begin{array}{rrrrrrrrrrrrrr}
\hline
         &14.8& 13.2& 15.7& 13.8& 15.6& 12.9& 11.0& 17.3& 15.9& 19.0& 18.3& 20.1& 21.7\\  
\hline      
    12.9&   1&  1&  1&  1&  1&  0&  -1&  1&  1&  1&  1&  1&  1\\
    14.8&    & -1&  1& -1&  1& -1&  -1&  1&  1&  1&  1&  1&  1\\
    13.2&    &   &  1&  1&  1& -1&  -1&  1&  1&  1&  1&  1&  1\\
    15.7&    &   &   & -1& -1& -1&  -1&  1&  1&  1&  1&  1&  1\\
    13.8&    &   &   &   &  1& -1&  -1&  1&  1&  1&  1&  1&  1\\
    15.6&    &   &   &   &   & -1&  -1&  1&  1&  1&  1&  1&  1\\
    12.9&    &   &   &   &   &   &  -1&  1&  1&  1&  1&  1&  1\\
    11.0&    &   &   &   &   &   &    &  1&  1&  1&  1&  1&  1\\
    17.3&    &   &   &   &   &   &    &   & -1&  1&  1&  1&  1\\
    15.9&    &   &   &   &   &   &    &   &   &  1&  1&  1&  1\\
    19.0&    &   &   &   &   &   &    &   &   &   & -1&  1&  1\\
    18.3&    &   &   &   &   &   &    &   &   &   &   &  1&  1\\
    20.1&    &   &   &   &   &   &    &   &   &   &   &   &  1
\end{array}
\]
\noindent
    Counting up, there are $91$ pairs, $18$ $-1$'s, $1$ $0$, and $72$ $1$'s.  Hence 

                 $$S = -18 + 0 + 72 = 54$$

    The variance is computed like this (there is one pair of equal values):

                 $$V = (14)(13)(33)/18 - (2)(1)(9)/18$$

    Then the Mann-Kendall statistic is

                 $$Z = \frac{S - 1}{\sqrt{V}} \doteq 2.9058 \quad {\rm (approximately)}$$

\vskip 10pt
\noindent
    \emph{Interpretation}:  There appears to be an upward trend
    in the data and the Mann-Kendall test confirms this with a pretty 
    small $p$-value.  The probability of observing this large a value of $Z$
    if there were no trend would be about  $p = .0018$ (using the standard
    normal table).  
\vskip 10pt
As indicated above, it is certainly {\it possible} to compute
the Mann-Kendall $Z$-statistic by hand if necessary.  However,
as you can probably guess, for long time series this can be somewhat
tedious and prone to computational errors if you are not careful.
For that reason, it is much more common to perform the calculations in
software.  Many of the major commercial and research statistical
software packages contain commands or have add-on packages to do this
computation.  There is also an Excel template spreadsheet
called MAKESENS (developed in Finland for environmental applications)
that is set up to perform these calculations.  This can be freely
downloaded from various sources on the web, including:
\vskip 10pt
\centerline{\tt http://en.ilmatieteenlaitos.fi/makesens}
\vskip 10pt
\noindent
There is a users' manual for use of this spreadsheet available here too.
\vskip 10pt
\noindent
{\bf Important Notes:}
\begin{itemize}  
\item If you use this spreadsheet, you will need to modify the 
Annual Data input page to accept different input time series -- the Calculate 
Trend Statistics ``button'' should then compute everything you need.  
\item Because this spreadsheet involves
macros -- small programs embedded in the cells of the spreadsheet --
you will probably need to override security features of Windows, 
your antivirus software, etc. when you download it to get a working version.  
See me early if you want to use
this so that we can get the technical issues resolved.
\end{itemize}

\vskip 10pt
\noindent
{\it Ideas for project topics related to trend analysis}
\vskip 10pt
\noindent
{\it Topic 1.  Implementing the Mann-Kendall and Seasonal Mann-Kendall Tests.}
\vskip 10pt
Even though there are available Mann-Kendall spreadsheets and packages
available for general use like the MAKESENS Excel template mentioned above, 
I am still a firm believer that when you are
learning a new computational process, then it can be very valuable to ``convince'' a computer
to do it for you ( ;) ) by programming the process (either in a spreadsheet,
or in some other sort of programming environment).  For this project,
the goal would be to develop your own Mann-Kendall and
Seasonal Mann-Kendall procedures and test them thoroughly on
various inputs.  Everyone else would be using the Mann-Kendall
test more or less as a statistical ``black box'' -- your presentation
would involve digging a bit deeper into exactly how the tests work and
explaining some of the fine points.  Note:  If you choose to work on
this one, you should have a fairly high level of skill and experience
in either spreadsheet macro programming, or in programming in some
other environment (e.g. C++, Maple, etc.)  If you have {\it never}
done anything like this, it would be better to consider a different
topic.
\vskip 10pt
\noindent
{\it Topic 2.  Analyzing Trends in Climate Data}
\vskip 10pt
There are a large collection of interesting
data sets related to measurements of levels of various ``greenhouse''
gasses such as $CO_2$, $NO_2$ etc. in the atmosphere, temperatures,
rainfall amounts, snow cover levels and durations, and many
other things available for download from the web site of the Carbon
Dioxide Information Analysis Center (CDIAC) at Oak Ridge National Labs,
\vskip 10pt
\centerline{\tt http://cdiac.ornl.gov/}
\vskip 10pt
\noindent
and the National Atmospheric and Oceanographic Administration (NOAA) 
National Centers for Environmental Information (NCEI):
\vskip 10pt
\centerline{\tt http://www.ncdc.noaa.gov/climate-information/statistical-weather-and-climate-information}
\vskip 10pt
\noindent
Some of these data sets are really \emph{too simple} for the kind of
trend analysis provided by regression or Mann-Kendall.  For instance
if you look at the time series of atmospheric $CO_2$ levels measured
at various locations on the Earth, there is a clear monotonically
increasing trend (superimposed on a seasonal variation that is tied
to the growth cycles of plants) that requires no statistical analysis whatsoever(!)
On the other hand, whether there are trends for some of the other trace
gas amounts (e.g. carbon monoxide, chloroform, carbon tetrachloride,
chlorofluorocarbons, etc.) is far less obvious.   One very good project
here would be to ``pick your favorite trace gas,'' look at the various
data sets available here and perform trend analyses, comparing results
from different locations.    ({\it Note:} There is certainly enough to do for
2 or 3 teams to work in this general area.)

One interesting question is how precipitation amounts have changed
at the locations where temperatures have increased over the past
50 or so years.  Do you find significant associations?  ({\it Note:}  Again
by looking at different geographical areas, 2 or three teams could
work on different aspects of this.)
\vskip 10pt
\noindent
{\it Topic 3.  Analysis of Trends in Baseball Statistics}
\vskip 10pt
This is literally a {\it huge} topic, as you can probably
guess.  Two or three groups could easily find plenty to
do on various aspects of this.  I recommend using the
\vskip 10pt
\centerline{\tt http://www.baseball-reference.com}
\vskip 10pt
\noindent
web site as your primary source -- this has team and individual stats for
almost the whole history of professional baseball(!)  Among the
kinds of questions that might be interesting to look at are:
Are there typical patterns for statistics like yearly home run production,
or on-base percentage, or slugging percentage, or OPS (on-base plus
slugging) for individual batters, or statistics such as
earned run average, strikeout rate per nine innings, etc. for pitchers over their
careers?  Does this make it easier to understand some personnel
decisions that team managements make?  On the other hand, you could
ask how the game itself has changed over time by looking at aggregate
(average) home runs, slugging percentage, earned run average, etc.
over some range of years.   Then again, you could look at how the
use of particular tactics like the base stealing, sacrifice bunts,
etc. has changed over time.

Another sort of question:  Are professional baseball players getting
better?  Are they getting more consistent?  Is there more or less variation
in the level of achievement of all batters now versus 50 years ago?
Similarly, you could ask about the variation in pitching statistics!
I'm purposely going to leave this one very open-ended because
I'm very interested in seeing what you will come up with!
\vskip 10pt
\noindent
The next topics don't necessarily involve trend analysis, but they could
be approached with some of the same ideas about time series data described
above.
\vskip 10pt 
\noindent
{\it Topic 4.  The ``Hot Hand Phenomenon'' -- Fallacy or Reality?}
\vskip 10pt
In 1985, Thomas Gilovich, Robert Vallone and Amos Tversky published an article [GVT]
called ``The Hot Hand in Basketball: On the Misperception of Random Sequences.''
Contrary to the testimony of many basketball players and sports fans that 
there were times when players were ``in the flow'' or ``hot'' and were both
more likely than not to make their next shots and aware that they were playing
at a higher level, the authors of this article claimed to show via a statistical 
analysis that the data just didn't support these perceptions.  In fact, they said, the outcomes 
of previous shots seemed to have no effect at all on whether subsequent shots were made
or not.  Charles Wheelan mentions this study on pages 102-103 of \emph{Naked Statistics}
as an example of when we misinterpret independent events as being 
\emph{not independent}.  This article attracted a lot of attention and has generated
a fairly substantial literature about the ``hot hand'' phenomenon from the psychological 
point of view.  

However, in September 2015, Joshua Miller and Adam Sanjurjo released a preprint
``Surprised by the Gambler's and Hot Hand Fallacies? A Truth in
the Law of Small Numbers''  in which they claimed to have identified a flaw in the [GVT] 
analysis.  When that flaw is corrected, they claimed that the data show there really can 
be ``hot'' (or ``cold'') hands in this sort of situation.  Slightly later, 
Brett Green and Jeffrey Zwiebel, of the Stanford University School of Business
released another preprint called ``The Hot Hand Fallacy: Cognitive Mistakes or Equilibrium 
Adjustments? Evidence from Baseball'' that claimed to find results similar to Miller and 
Sanjurjo's in baseball data.  
A good place to start looking at all of this is an article on {\tt slate.com} by 
Jordan Ellenberg describing this rethinking of the [GVT] results.  

I think it's
probably safe to say that the jury is really still out on all of this, but it's
an interesting question for any sports fan, and a really strong indication of 
the fact that the application of mathematics and statistics to real world
questions is more like the rest of science (where new insights can lead to 
radical rethinking of our basic understanding of how the world works)
than a realm of pure mathematical, Euclidean certainty.  

There are several possible projects here:
\vskip 10pt
\begin{itemize}
\item A (much) more theoretical topic would be to look closely at the original arguments and
data in [GVT] and try to understand why Gilovich, Vallone and Tversky came
to their original conclusion. Then try to understand the criticism that 
Miller and Sanjurjo raised and why they think [GVT] ``got it wrong.''
There's some pretty sophisticated mathematics here, but
it will be interesting to work through this if you're up for a challenge.  
\item It would also be possible to look at new or different data (different
sports, for instance) to see whether
the hypothesis of the ``hot hand'' seems to hold up there.  You should 
try to follow the sorts of analysis that [GVT] and Miller and Sanjurjo did
here.  
\item It would also be possible to design a questionnaire and carry out
a survey of students at the college to see if they have heard
of this new research, and determine whether they believe in and/or
think they have observed or experienced the ``hot hand.''  
Then you would need to analyze the data from that survey.  For instance, it 
would be interesting to know whether there is a difference in perceptions between
varsity athletes and the general student body, whether there are differences in 
perception between different sports, etc.  
{\bf Important Note:}  If you want to try this, there is a procedure you will need 
to follow to get approval from 
the Human Subjects Committee for the research, and this will take time,
so talk to me soon to get the process started.  
\end{itemize}

\vskip 10pt
\noindent
{\it Topic 5.  A Topic of Your Choice}
\vskip 10pt
If there is another topic you would prefer to work on, I am open
to suggestions.  This could be a similar application of trend
analysis to a different subject.  Or, you might want to try working
on something completely different (something possibly related to one of 
the readings this semester).  If you want to propose a topic
of your own, you {\it must get my approval} before starting to 
work.  For the March 26 deadline above, write up a short
description of the topic or questions you want to look at and
how you want to try to address them.  I will let you know 
as soon as possible whether you have my approval.
\end{document}
