# Installing a Native Ubuntu Terminal on Windows 10

Modern clusters and cloud platforms requires users to login through SSH on Unix terminals in order to have access to its functionalities. Such terminals also give its users access to a wide range of commands and programs such as awk and Vim which are not available on Windows. Linux and Mac users have such terminals installed by default, but Windows users normally have to install Cygwin or use a browser-based terminal, both with limited capabilities such as the inability to install new packages.

Fortunately, Microsoft has established a partnership with Canonical (Ubuntu’s parent company) which brought part of the Linux kernel to Windows 10, allowing users to install Ubuntu’s terminal on Windows through official means without the need for compatibility layers. Using Ubuntu’s terminal on Windows has the advantages of being able to use apt-get and dpkg to install new packages, which was not possible with Cygwin, and of running Python and C/C++ codes faster. Here are the steps to install Ubuntu terminal on Windows 10:

1. On Windows Settings, click on “Update & Security.” There, click on “For Developers” close to the bottom on left pane and turn on the option “Developer Mode.”
2. Windows Settings, click on “Apps” -> “Programs and Features” (right pane) -> “Turn Windows features on or off”  (left pane) and check “Windows Subsystem for Linux.”

2. Open Windows PowerShell as administrator, type the following line and press enter:
Enable-WindowsOptionalFeature -Online -FeatureName Microsoft-Windows-Subsystem-Linux

1. Open Microsoft Store (Windows’ app store), look for “Ubuntu Terminal,” and install it.
2. Now you should have it installed and a shortcut on your quick-start bar.
3. Open the Ubuntu Terminal and type:
sudo apt-get update


In order to install programs such as the Intel compiler and profiler (free for students), pip, Vim, GNUPlot or the most recent version of GCC, just type:

sudo apt-get install program_to_be_installed


If the package you installed has graphical components, such GNUPlot and Python/Matplotlib, you will need to install a program on Windows to display the graphical components from the Ubuntu terminal. One such option is Xming. To use Xming, follow the following steps:

1. Install Xming from here.
2. Run it. Click “next” until you can click on “Finish.” This process will have to be repeated every time you open the terminal.
3. Open Ubuntu terminal
4. Type the following and press enter:
echo "export DISPLAY=localhost:0.0" >> .bashrc

1. Close and re-open the terminal
2. In order to make sure you can run graphic applications, run the following two commands:
sudo apt-get install x11-apps
xeyes


# Preparing Data for a Time Series Analysis

This semester, I had the opportunity to take Dr. Scott Steinschneider’s new class, Hydrologic Engineering in a Changing Climate, that is offered here at Cornell. In this class, we covered time series analysis, extreme value modeling, and trend tests. I chose to do a final project which focused on using a time series approach to forecast electricity demand in California. As I worked through my project, it became apparent to me that data are rarely in the form where a time series model can be applied directly. Consequently, multiple transformations are usually necessary before a model can be fit to the data set. In this post, I outline some of the inherent characteristics of data that might warrant a transformation as well as the steps that can be taken to address these problems.

Given a data set that you would like to fit any Box-Jenkins model to, you should ask yourself the following two questions?

1. Is normality a reasonable assumption for the residuals?
2. Are the data stationary?

### Normality

Normality can be checked before fitting the model because if the original data are not normal, then there is a good chance that the residuals won’t be as well. If you fit a histogram to the data and it looks like Figure 1, you probably need to apply some form of normalizing transformation.

###### Figure 1: Histogram of Monthly Flow

Two traditional transformations that you can try are a log transform or a Box-Cox transform, shown in the following two equations, where xt is the original data point.

Log Transform:

Box-Cox Transform:

Sometimes a log transformation can be too drastic and skew the data the opposite way. The Box-Cox transform is effectively a less intense transformation that one can try if the log transform is not suitable. Note that when λ=0, the Box-Cox transform reduces to a simple log transform.

The powerTransform function in the R package, car, can be used to find a lambda that will maximize normality.

### Stationarity

For a data set to exhibit stationarity, the following three principles must be true for us to be confident that our model will represent our data well:

For some lag term, s,

1. E[xt]=E[xt+s] (The mean of the data set does not change with time)
2. Var[xt]=Var[xt+s] (The variance of the data set does not change with time)
3. Cov[xt,xt+s]= γ (The covariance between data points is some constant value,γ)

Outlined below are some of the characteristics of a data set that can cause a violation of one or more of these principles.

Seasonality

Seasonality in data can exist if a time series pattern repeats over a fixed and known period. Figure 2 shows monthly inflow into the Schoharie Creek Reservoir. Periodicity is apparent, but it isn’t until we look at the autocorrelation function (ACF) of the data, shown in Figure 3, that we see that there is a clear repetition occurring every 12 months.

###### Figure 3: ACF of Monthly Inflow

One effective way to get rid of this monthly seasonality is to use the following de-seasonalizing equation:

The seasonality is removed from each data point by subtracting the corresponding monthly mean (xmt) and dividing by the month’s standard deviation ( smt). This equation can also be used to account for daily or yearly seasonality as well.

Differencing is another way to address seasonality in data. A seasonal difference is the difference between an observation and the corresponding observation from the previous year.

Where m=12 for monthly data, m=4 for quarterly data, and so on 1.

Trend

A trend, shown in the first panel of Figure 4, is a clear violation of the first requirement for stationarity. There are a couple options that one can implement to deal with trends: differencing and model fitting.

###### Figure 4: De-trending process1

From the above figures, it is clear that differencing can be used to account for seasonality but can also be used to dampen a trend. A first difference is performed by subtracting the value of the current observation from the one in the time step before. It can be applied as follows:

If the transformed data is plotted and still has a trend, a second difference can further be applied.

It is important to note the distinction between seasonal and first differences. Seasonal differencing is the difference from one year to the next, while first differencing is the difference between one observation and the next. Seasonal and trend differencing can both be applied, but sometimes, if seasonal differencing is performed first, it will remove the need for further differencing1.

In Figure 4, note how a log transform, seasonal differencing, and second differencing is necessary to ultimately remove the trend.

###### Figure 5: Modeling Fitting with Ordinary Least Squares2

If a monotonic trend is observed, such as the one in Figure 5, a model fitting can be performed. In this example, a linear model is fit to the trend by choosing coefficients that minimize the sum of squares. This model is then subtracted from the original data to give residuals. The goal is for the resulting residuals to be stationary. Note that a polynomial model can also be fit to the trend if appropriate2 .

Heteroscedasticity

Heteroscedasticity describes the phenomenon when the data do not exhibit a constant variance. This is a violation of the second principle. Heteroscedasticity tends to appear in financial time series (i.e. prices of stocks and bonds) which can be very volatile, but it appears less so in hydrological data3. I did not have to address heteroscedasticity in the electricity load data for my project, and some statisticians suggest that one doesn’t have to deal with it unless it is very severe as weak heteroscedasticity tends be taken care of with normalization and de-seasonalization.

One way to check for heteroscedasticity in a time series is with the McLeod-Li test for conditional heteroscedasticity. If heteroscedasticity is present, consider using an ARCH/GARCH model, if an AR or ARMA model can be fit to the data, respectively, or a hybrid ARCH-ARIMA model if the latter models are not appropriate.

### Choosing a Time Series Model

Once the necessary transformations have been performed, you are ready to fit a time series model to your data. R has a some useful packages for this: forecast and stats. Some helpful functions in these packages include:

auto.arima (forecast) – This function tells you what model is the best fit for your data, the coefficients for the lag terms, and variance of errors (along with other useful information).

arima.sim (stats) – This function allows you to simulate a set of data from your time series model.

predict (stats) – This function will provide a prediction for n time steps into the future based on the chosen time series model. Keep in mind it is best when used to predict just the next few time step.

Finally, remember that back-transformations must be performed on all simulations or predictions to get them into back into the original space.

*For a really helpful explanation of different time series notation, check this previous post.

### References

*All information or figures not specifically cited came from class notes and homework from Dr. Scott Steinschneider’s class

(1) Stationarity and Differencing: https://www.otexts.org/fpp/8/1

(2) Removal of Trend and Seasonality, UC Berkeley: https://www.stat.berkeley.edu/~gido/Removal%20of%20Trend%20and%20Seasonality.pdf

(3) Heteroscedasticity: http://www.math.canterbury.ac.nz/~m.reale/econ324/Topic2.pdf

# Nondimensionalization of differential equations – an example using the Lotka-Volterra system of equations

I decided to write about nondimensionalization today since it’s something I only came across recently and found very exciting. It’s apparently a trivial process for a lot of people, but it wasn’t something I was taught how to do during my education so I thought I’d put a short guide out on the interwebs for other people. Nondimensionalization (also referred to as rescaling) refers to the process of transforming an equation to a dimensionless form by rescaling its variables. There are a couple benefits to this:

• All variables and parameters in the new system are unitless, i.e. scales and units not an issue in the new system and the dynamics of systems operating on different time and/or spatial scales can be compared;
• The number of model parameters is reduced to a smaller set of fundamental parameters that govern the dynamics of the system; which also means
• The new model is simpler and easier to analyze, and
• The computational time becomes shorter.

I will now present an example using a predator-prey system of equations. In my last blogpost, I used the Lotka-Volterra system of equations for describing predator-prey interactions. Towards the end of that post I talked about the logistic Lotka-Volterra system, which is in the following form:Where x is prey abundance, y is predator abundance, b is the prey growth rate, d is the predator death rate, c is the rate with which consumed prey is converted to predator abundance, a is the rate with which prey is killed by a predator per unit of time, and K is the carrying capacity of the prey given its environmental conditions.

The first step is to define the original model variables as products of new dimensionless variables (e.g. x*) and scaling parameters (e.g. X), carrying the same units as the original variable.The rescaled models are then substituted in the original model:
Carrying out all cancellations and obvious simplifications:Our task now is to define the rescaling parameters X, Y, and T to simplify our model – remember they have to have the same units as our original parameters.

 Variable/parameter Unit x mass 1 y mass 1 t time b 1/time d 1/time a 1/(mass∙time) 2 c mass/mass 3 K mass

There’s no single correct way of going about doing this, but using the units for guidance and trying to be smart we can simplify the structure of our model. For example, setting X=K will remove that term from the prey equation (notice that this way X has the same unit as our original x variable).

The choice of Y is not very obvious so let’s look at T first. We could go with both T=1/b or T=1/d. Unit-wise they both work but one would serve to eliminate a parameter from the first equation and the other from the second. The decision here depends on what dynamics we’re most interested in, so for the purposes of demonstration here, let’s go with T=1/b.

We’re now left with defining Y, which only appears in the second term of the first equation. Looking at that term, the obvious substitution is Y=b/a, resulting in this set of equations:

Our system of equations is still not dimensionless, as we still have the model parameters to worry about. We can now define aggregate parameters using the original parameters in such a way that they will not carry any units and they will further simplify our model.

By setting p1=caK/b and p2=d/b we can transform our system to:a system of equations with no units and just two parameters.

1 Prey and predator abundance don’t have to necessarily be measured using mass units, it could be volume, density or something else. The units for parameters a, c, K would change equivalently and the rescaling still holds.

2 This is the death rate per encounter with predator per time t.

3 This is the converted predator (mass) per prey (mass) consumed.

# An Introduction to Copulas

Modeling multivariate probability distributions can be difficult when the marginal probability density functions of the component random variables are different. Copulas are a useful tool to model dependence between random variables with any marginal distributions. This post will introduce the idea of a copula, run through the basic math that underlies its composition and discuss some common copulas in use. Through researching for this post I found several comprehensive coding examples in Matlab, Python and R, so instead of creating my own I’m focusing on the a theoretical introduction to copulas in this post and will link to the coding tutorials at the end.

### What is a copula?

The word copula is derived from the Latin noun for a link or a tie (as is the English word “couple”) and its purpose is to describe the dependence structure between two variables. Sklar’s theorem states that “Any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the two variables” [1].

To understand how a copula can describe the dependence function between random variables its helpful to first review some simple statistics.

The above statements can be summarized as saying that the values of the CDF of any marginal distribution are uniformly distributed on the interval [0,1] ie. if you make a random draw from any distribution, you have the same probability of drawing the largest value (U=1) of that distribution as the smallest possible value (U=0) or the median value (U=.5).

So what does this have to do with copulas? Great question, a copula is actually a joint distribution of the CDFs of the random variables it is modeling. Put formally:

A k dimensional copula is a function $c:[0,1]^k \rightarrow [0,1]$ and is a CDF with uniform marginals [1].

So now that we’ve defined what a copula is, lets take a look at the form of some commonly used ones.

### The Gaussian Copula

The Gaussian takes the form:

Where:

Φ_R is the joint standard normal CDF with:

ρ_(i,j) is the correlation between random variables X_i and X_j.

Φ^-1 is the inverse standard normal CDF.

It’s important to note that the Pearson Correlation coefficient is a bad choice for ρ row here, a rank based correlation such as Spearman’s ρ or Kendall’s τ are better options since they are scale invariant and do not require linearity.

### Issues with Tail Dependency

The Gaussian copula is a helpful tool and relatively easy to fit, even for relatively large numbers of RVs with different marginal distributions. Gaussian copulas however, do not do a good job capturing tail dependence and can cause one to underestimate risk of simultaneously being in the tails of each distribution. The failure of the Gaussian copula to capture tail dependence has been blamed for contributing the the 2008 financial crisis after it was widely used by investment firms on Wall Street (this is actually a really interesting story, for more details check out this article from the financial times) .

Tail dependency can be quantified by the coefficients of upper and lower tail dependence (λ_u and λ_l) defined as:

### The Student t Copula

Like the student t distribution, the student t copula has a similar shape to the Gaussian copula, but with fatter tails, thus it can do a slightly better job capturing tail dependence.

Where:

t_ν,Σ is the joint student t CDF, Σ is covariance matrix (again don’t use Pearson correlation coefficient), ν is the degrees of freedom and t^-1_ν is the inverse student t CDF.

### Archimedean Copulas

Archimedian copulas are a family of copulas with the following form:

ψ(u|θ) is called the generator function and θ is the parameters for the copula.

3 common Archimedean Copulas are:

• Gumbel: which is good at modeling upper tail dependence
• Clayton: which is good at modeling lower tail dependence
• Frank: has lighter tails and more density in the middle

It’s important to note that these copulas are usually employed for bivariate cases, for more than two variables, the Gaussian or Student t copulas are usually used.

A comparison of the shape of the copulas above can be found in Figure 1.

Image source: wikipedia.org

### Coding Copulas

There are numerous packages for modeling copulas in Matlab, Python and R.

In Matlab, the Statistics and Machine learning Toolbox has some helpful functions. You can find some well narrated examples of copulas here. There’s also the Multivariate Copula Analysis Toolbox from UC Irvine.

In Python, the copulalib package can be used to model the Clayton, Frank and Gumbel copulas. The statsmodels  package also has copulas built in. I found this post on the copulalib package, it has an attached Jupyter notebook with nice coding examples and figures. Here’s a post on statsmodels copula implementation, along with example Jupyter notebook.

Finally, here’s an example of coding copulas in R using the copulas library.

### References and Acknowledgments

1. Rüschendorf, L. (2013). Mathematical risk analysis dependence, risk bounds, optimal allocations and portfolios. Berlin: Springer.

I’d like to note that the majority of the content in the post came from Scott Steinschneider’s excellent course, BEE 6940: Multivariate Analysis, at Cornell.

# Simple tricks to make your C/C++ code run faster

Us, engineers with no formal computer science training, for a myriad of good reasons tend to favor friendly programming languages such as Python over evil C/C++. However, when our application is performance sensitive and we cannot wait for results sitting in our chairs for as long as a Python code would require us to, we are sometimes forced to us C/C++. Why then not making the most of it when it this is the case?

Here are some simple tips to make your C/C++ code run even faster, and how to get some advice about further performance improvements. The last idea (data locality) is transferable to Python and other languages.

Most important trick

Improve your algorithm. Thinking if there is a simpler way of doing what you coded may reduce your algorithm’s complexity (say, from say n3 to n*log(n)), which would:

• yield great speed-up when you have lots of data or needs to run a computation several times in a row, and
• make you look smarter.

Compiler flags

First and foremost, the those who know what they are doing — compiler developers — do the work for you by calling the compiler with the appropriate flags. There are an incredible amount of flags you can call for that, but I would say that the ones you should have on whenever possible are -O3 and –march=native.

The optimization flags (-O1 to -O3, the latter more aggressive than the former) will perform a series of modification on your code behind the scenes to speed it up, some times by more than an order of magnitude. The issue is that this modifications may eventually make your code behave differently than you expected, so it’s always good to do a few smaller runs with -O0 and -O3 and compare their results before getting into production mode.

The –march=native flag will make the compiler fine tune your code to the processor it is being compiled on (conversely, –march=haswell would fine tune it to haswell architectures, for example). This is great if you are only going to run your code on your own machine or in another machine known to have a similar architecture, but if you try to run the binary on a machine with a different architecture, specially if it is an older one, you may end up with illegal instruction errors.

Restricted pointer array

When declaring a pointer array which you are sure will not be subject to pointer aliasing — namely there will be no other pointer pointing to the same memory address –, you can declare that pointer as a restrict pointer, as below:

• GCC: double* __restrict__ my_pointer_array
• Intel compiler: double* restrict my_pointer_array

This will let the compiler know that it can change order of certain operations involving my_pointer_array to make your code faster without changing some read/write order that may change your results. If you want to use the restricted qualifier with the intel compiler the flag -restrict must be passed as an argument to the compiler.

Aligned pointer array

By aligning an array, you are making sure the data is going to lie in the ideal location in memory for the processor to fetch it and perform the calculations it needs based on that array. To help your compiler optimize your data alignment, you need to (1) align your array when it is declared by a specific number of bytes and (2) tell your the compiler the array is aligned when using it to perform calculations — the compiler has no idea whether or not arrays received as arguments in function are aligned. Below are examples in C, although the same ideas apply to C++ as well.

GCC

#include <stdio.h>
#include <omp.h>

#define SIZE_ARRAY 100000
#define ALIGN 64

void sum(double *__restrict__ a, double *__restrict__ b, double *__restrict__ c, int n) {
a = (double*) __builtin_assume_aligned(a, ALIGN);
b = (double*) __builtin_assume_aligned(b, ALIGN);
c = (double*) __builtin_assume_aligned(c, ALIGN);
for (int i = 0; i < n; ++i)
c[i] = a[i] + b[i];
}

int main(void) {

double a[SIZE_ARRAY] __attribute__((aligned(ALIGN )));
double b[SIZE_ARRAY] __attribute__((aligned(ALIGN )));
double c[SIZE_ARRAY] __attribute__((aligned(ALIGN )));

for (int i = 0; i < SIZE_ARRAY; ++i) {
a[i] = 5.;
b[i] = 2.;
}
double start_time = omp_get_wtime();
sum(a, b, c, SIZE_ARRAY);
double time = omp_get_wtime() - start_time;
printf("%0.6fs", time);
}


Intel compiler

#include <stdio.h>
#include <omp.h>

#define SIZE_ARRAY 100000
#define ALIGN 64

void sum(double* restrict a, double* restrict b, double* restrict c, int n) {
__assume_aligned(a, ALIGN);
__assume_aligned(b, ALIGN);
__assume_aligned(c, ALIGN);
for (int i = 0; i < n; ++i)
c[i] = a[i] + b[i];
}

int main(void) {

__declspec(align(ALIGN )) float a[SIZE_ARRAY];
__declspec(align(ALIGN )) float b[SIZE_ARRAY];
__declspec(align(ALIGN )) float c[SIZE_ARRAY];

for (int i = 0; i < SIZE_ARRAY; ++i) {
a[i] = 5.;
b[i] = 2.;
}
double start_time = omp_get_wtime();
sum(a, b, c, SIZE_ARRAY);
double time = omp_get_wtime() - start_time;

printf("%0.6fs", time);
}


Edit: In a comment to this post, Krister Walfridsson not only caught an issue with my GCC code, for which mention I thank him, but he also shows the differences in machine code generated with and without alignment.

Data Locality

Computers are physical things, which means that data is physically stored and needs to be physically moved around in memory and between cache and processor in order to be used in calculations. This means that, if your data is stored all over the place in memory — e.g. in multiple pointer arrays in different parts of memory –, the processor will have to reach out to several parts of memory to fetch all your data before performing any computations. By having the data intelligently laid out in memory you ensure all data required for each computation is stored close to each other and in cache at the same time, which becomes even more important if your code uses too much data to fit in the cache at once.

In order to making your processor’s life easier, it is a good idea to ensure that all data required for a calculation step is close together. For example, if a given computation required three arrays of fixed sizes, it is always a good idea to merge them into one long array, as in the example below for the Intel compiler.

#include <stdio.h>
#include <omp.h>

#define SIZE_ARRAY 100000
#define ALIGN 64

void sum(double* restrict a, double* restrict b, double* restrict c, int n) {
__assume_aligned(a, ALIGN);
__assume_aligned(b, ALIGN);
__assume_aligned(c, ALIGN);
for (int i = 0; i < n; ++i)
c[i] = a[i] + b[i];
}

int main(void) {

__declspec(align(ALIGN )) float abc[3 * SIZE_ARRAY];

for (int i = 0; i < 2 * SIZE_ARRAY; ++i) {
a[i] = 5.;
b[i] = 2.;
}
double start_time = omp_get_wtime();
sum(abc, abc + SIZE_ARRAY, abc + 2 * ARRAY, SIZE_ARRAY);
double time = omp_get_wtime() - start_time;

printf("%0.6fs", time);
}


or even, since c[i] depends only on b[i] and a[i], we can have the values of a, b and c intercalated to assure that all computations will be performed on data that is right next to each other in memory:

#include <stdio.h>
#include <omp.h>

#define SIZE_ARRAY 100000
#define ALIGN 64
#define STRIDE 3

void sum(double* restrict abc, int n, int stride) {
__assume_aligned(abc, ALIGN);

for (int i = 0; i < n; i += stride)
abc[i+2] = abc[i] + abc[i+1];
}

int main(void) {
__declspec(align(ALIGN )) double abc[3 * SIZE_ARRAY];

for (int i = 0; i < 3 * SIZE_ARRAY; i += STRIDE) {
abc[i] = 5.;
abc[i+1] = 2.;
}
double start_time = omp_get_wtime();
sum(abc, SIZE_ARRAY, STRIDE );
double time = omp_get_wtime() - start_time;

printf("%0.6fs", time);
}


Conclusion

According a class project in which we had to write C code to perform matrix multiplication, the improvements suggested should may improve the performance of your code by 5 or 10 times. Also, the idea of data locality can be transferred to other languages, such as Python.

# Exploring the stability of systems of ordinary differential equations – an example using the Lotka-Volterra system of equations

Stability when dealing with dynamical systems is important
because we generally like the systems we make decisions on to be predictable.
As such, we’d like to know whether a small change in initial conditions could
lead to similar behavior. Do our solutions all tend to the same point? Would
slightly different initial conditions lead to the same or to a completely
different point for our systems.

This blogpost will consider the stability of dynamical systems of the form:

The equilibria of which are denoted by x* and y*,
respectively.

I will use the example of the Lotka-Volterra system of
equations, which is the most widely known method of modelling many predator-prey/parasite-host interactions encountered in natural systems. The Lotka-Volterra predator-prey equations were discovered independently by both Alfred Lotka and Vito Volterra in 1925-26. Volterra got to these equations while trying to explain why, immediately after WWI, the number of predatory fish was much larger than before the war.

The system is described by the following equations:

Where a, b, c, d > 0 are the parameters describing the
growth, death, and predation of the fish.

In the absence of predators, the prey population (x) grows
exponentially with an intrinsic rate of growth b.

Total predation is proportional to the abundance of prey and
the abundance of predators, at a constant predation rate a.

New predator abundance is proportional to the total
predation (axy) at a constant conversion rate c.

In the absence of prey, the predator population decreases at
a mortality rate d.

The system demonstrates an oscillating behavior, as
presented in the following figure for parameters a=1, b=1, c=2, d=1.

Volterra’s explanation for the rise in the numbers of
predatory fish was that fishing reduces the rate of increase of the prey
numbers and thus increases the rate of decrease of the predator. Fishing does
not change the interaction coefficients. So, the number of predators is
decreased by fishing and the number of prey increases as a consequence. Without
any fishing activity (during the war), the number of predators increased which
also led to a decrease in the number of prey fish.

To determine the stability of a system of this form, we
first need to estimate its equilibria, i.e. the values of x and y for which:

An obvious equilibrium exists at x=0 and y=0, which kinda

We’ll first look at a system that’s still alive, i.e x>0
and y>0:

And

Looking at these expressions for the equilibria we can also
see that the isoclines for zero growth for each of the species are straight
lines given by b/a for the prey and d/ca for the predator, one horizontal and
one vertical in the (x,y) plane.

In dynamical systems, the behavior of the system near an
equilibrium relates to the eigenvalues of the Jacobian (J) of F(x,y) at the equilibrium.
If the eigenvalues all have real parts that are negative, then the equilibrium
is considered to be a stable node; if the eigenvalues all have real parts that
are positive, then the equilibrium is considered to be an unstable node. In the
case of complex eigenvalues, the equilibrium is considered a focus point and
its stability is determined by the sign of the real part of the eigenvalue.

I found the following graphic from scholarpedia to be a
useful illustration of these categorizations.

So we can now evaluate the stability of our equilibria.
First we calculate the Jacobian of our system and then plug in our estimated
equilibrium.

To find the eigenvalues of this matrix we need to find the
values of λ that satisfy: det⁡(J-λI)=0  where I is
the identity matrix and det denotes the determinant.

Our eigenvalues are therefore complex with their real parts equal to 0. The equilibrium is therefore a focus point, right between instability and asymptotic stability. What this means for the points that start out near the equilibrium is that they tend to both converge towards the equilibrium and away from it. The solutions of this system are therefore periodic, oscillating around the equilibrium point, with a period , with no trend either towards the
equilibrium or away from it.

One can arrive at the same conclusion by looking at the
trace (τ) of the Jacobian and its determinant (Δ).

The trace is exactly zero and the determinant is positive
(both d,b>0) which puts the system right in between stability and
instability.

Now let’s look into the equilibrium where x*=0 and y*=0, aka
the total death.

Both b and d are positive real numbers which means that the
eigenvalues will always have real values of different signs. This makes the
(0,0) an unstable saddle point. This is important because if the equilibrium of
total death were a stable point, initial low population levels would tend to
converge towards their extinction. The fact that this equilibrium is unstable
means that the dynamics of the system make it difficult to achieve total death
and that prey and predator populations could be infinitesimally close to zero
and still recover.

Now consider a system where we’ve somehow killed all the
predators (y=0). The prey would continue to grow exponentially with a growth
rate b. This is generally unrealistic for real-life systems because it assumed
infinite resources for the prey. A more realistic model would consider the prey
to exhibit a logistic growth, with a carrying capacity K. The carrying capacity of a biological species is the maximum population size of the species that can be sustained indefinitely given the necessary resources.

The model therefore becomes:

Where a, b, c, d, K > 0.

To check for this system’s stability we have to go through
the same exercise.

The predator equation has remained the same so:

For zero prey growth:

Calculating the eigenvalues becomes a tedious exercise at
this point and the time of writing is 07:35PM on a Friday. I’d rather apply a
small trick instead and use the isoclines to derive the stability of the system. The isocline for the predator zero-growth has remained the same (d/ca), which is a straight line (vertical on the (x,y) vector plane we draw before). The isocline for the prey’s zero-growth has changed to:

Which is again a straight line with a slope of –b/aK, i.e.,
it’s decreasing when moving from left to right (when the prey is increasing). Now looking at the signs in the Jacobian of the first system:

We see no self-dependence for each of the two species (the
two 0), we see that as the predator increases the prey decreases (-) and that
as the prey increases the predator increases too (+).

For our logistic growth the signs in the Jacobian change to:

Because now there’s a negative self-dependence for the prey-as its numbers increase its rate of growth decreases. This makes the trace (τ) of the Jacobian negative and the determinant positive, which implies that our system is now a stable system. Plotting the exact same dynamic system but now including a carrying capacity, we can see how the two populations converge to specific numbers.

This semester I’m taking my first official CS class here at Cornell, CS 5220 Applications of Parallel Computers taught by Dave Bindel (for those of you in the Reed group or at Cornell, I would definitely recommend taking this class, which is offered every other year, once you have some experience coding in C or C++). In addition to the core material we’ve been learning in the class, I’ve been learning a lot by examining the structure and syntax of the code snippets written by the instructor and TA that are offered as starting points and examples for our class assignments. One element in particular  that stood out to me from our first assignment was the many function calls made through the makefile. This post will first take a closer look into the typical composition of a makefile and then examine how we can harness the structure of a makefile to help improve workflow on complicated projects.

## Dissecting a typical makefile

On the most basic level, a makefile simply consists of series of rules that each have an associated set of actions. Makefiles are how you use the “make” utility, a software package installed on all linux systems. Make has its own syntax similar to bash but with some distinct idiosyncrasies. For example, make allows you to store a snippet of code in whats called a “macro” (these are pretty much analogous to variables in most other languages).  A macro to store the flags you would like to run with your compiler could be defined like this:

CFLAGS  = -g -Wall

To referenced the CFLAGS macro, use a dollar sign and brackets, like this:

 $(CFLAGS) There are a series of “special” predifined macros that can be used in any makefile which are fairly common, you can find them here. Now that we’ve discussed makefile syntax, lets take a look at how rules are structured within a makefile. A rule specified by a makefile has the following shape: target: prerequisites recipe ... ... The target is usually the name of the file that is generated by a program, for example an executable or object file. A prerequisite is the specified input used to create the target (which can often depend on several files). The recipe is the action that make carries out for the intended target (note: this must be indented at every line). For example, a rule to build an executable called myProg from a c file called myProg.c using the gcc compiler with flags defined in CFLAGS might look like this: myProg: myProg.c gcc$(CFLAGS) $< ## Make the makefile do the work The most common rules within makefiles call the compiler to build code (hence the name “makefile”) and many basic makefiles are used for this sole purpose. However, a rule simply sends a series commands specified by its recipe to the command line and a rule can actually specify any action or series of actions that you want. A ubiquitous example of a rule that specifies an action is “clean”, which may be implemented like this: clean: rm -rf *.o$(PROGRAM)

Where PROGRAM is a macro containing the names of the executable compiled by the makefile.

In this example, the rule’s target is an action rather than an output file. You can call “clean” by simply typing “make clean” into the command line and you will remove all .o files and the executable PROGRAM from your working directory.

Utilizing this application of rules, we can now have our makefile do a lot of our command line work for us. For example, we could create a rule called “run” which submits a series of pbs jobs into a cluster.

run:
qsub job-$*.pbs We can then enter “make run” into the command line to execute this rule, which will submit the .pbs jobs for us (note that this will not perform any of the other rules defined in the makefile). Using this rule may be handy if we have a large number of jobs to submit. Alternatively we could make a rule called “plot” which calls a plotting function from python: plot: python plotter.py$(PLOTFILES)

Where PLOTFILES is a macro containing the name of files to be plotted and plotter.py is a python function that takes the file names as input.

Those are just two examples (loosely based on a makefile given in CS 5220) of how you can use a makefile to do your command line work for you, but the possibilities are endless!! Ok, maybe that’s getting a bit carried away, but I do find this functionality to be a simple and elegant way to improve the efficiency of your workflow on complex projects.

For some background on makefiles, I’d recommend taking a look a Billy’s post from last year. I also found the GNU make user manual helpful as well as this tutorial from Swarthmore that has some nice example makefiles.