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# Cellular Automata/Examples of Plankton and Fish Dynamics

## Definition of prey-predator systems

Prey-predator systems are a well known biological example of excitable media. This example is based upon an article (see references) about plankton and fish dynamics. The dynamics are given as a PDE system. The purpose of this example is to observe pattern formation and other dynamic properties of prey-predator systems using both finite difference equations and cellular automata.

### PDE describing prey-predator systems

We define the prey-predator system using the a system of general reaction-diffusion differential equations.

${\displaystyle u_{t}(\mathbf {r} ,t)=D_{u}\nabla ^{2}u(\mathbf {r} ,t)+f(u,v)}$
${\displaystyle v_{t}(\mathbf {r} ,t)=D_{v}\nabla ^{2}v(\mathbf {r} ,t)+g(u,v)}$

The first equation describes the prey population ${\displaystyle u(\mathbf {r} ,t)}$  and the seccond equation describes the predator population ${\displaystyle v(\mathbf {r} ,t)}$ . Nonlinear functions ${\displaystyle f(u,v)}$  and ${\displaystyle g(u,v)}$  describe local prey-predator dynamics, without them the system is decomposed into two separate diffusion equations.

We are interested in prey-predator dynamics of aquatic systems, where phytoplankton (small plants, food producers) is the prey and zooplankton (small animals) is the predator. Since we observe only slowly moving plankton the diffusion coefficient ${\displaystyle D_{u}=D_{v}=D}$  depends mainly on marine turbulence. The two nonlinear functions describing prey-predator dynamics are

${\displaystyle f(u,v)=P(u)-E(u,v)\,}$
${\displaystyle g(u,v)=\kappa E(u,v)-\mu v\,}$

${\displaystyle P(u)}$  describes local growth and mortality of the prey, ${\displaystyle E(u,v)}$  describes the predation, ${\displaystyle \kappa }$  is food utilization and ${\displaystyle \mu }$  is the mortality rate of the predator. The selection of functions ${\displaystyle P(u)}$ , ${\displaystyle E(u,v)}$  and parameters ${\displaystyle \kappa }$ , ${\displaystyle \mu }$  depends on the observed population. Here a simple example is used, for a detailed description of the equations see the references.

${\displaystyle u_{t}(\mathbf {r} ,t)=D\nabla ^{2}u(\mathbf {r} ,t)+{\frac {\alpha }{b}}u(b-u)-\gamma {\frac {u}{u+H}}v}$
${\displaystyle v_{t}(\mathbf {r} ,t)=D\nabla ^{2}v(\mathbf {r} ,t)+\kappa \gamma {\frac {u}{u+H}}v-\mu v}$

${\displaystyle \alpha }$  is the maximal growth rate of the prey, ${\displaystyle b}$  is the carrying capacity of the prey population, and H is the half-saturation abundance of prey.

### Algebraic analysis

The first step is to introduce dimensionless variables

${\displaystyle {\tilde {u}}={\frac {u}{b}}\quad {\tilde {v}}={\frac {v\gamma }{b\alpha }}\quad {\tilde {t}}=\alpha {t}\quad {\tilde {\mathbf {r} }}=\mathbf {r} \left({\frac {\alpha }{D}}\right)^{\frac {1}{2}}}$

and the new dimensionless parameters are

${\displaystyle h={\frac {H}{b}}\quad m={\frac {\mu }{\alpha }}\quad k={\frac {\kappa \gamma }{\alpha }}}$

the PDE system with only dimensionless quantities is (tildes are omitted)

${\displaystyle u_{t}=\nabla ^{2}u+u(1-u)-{\frac {u}{u+h}}v}$
${\displaystyle v_{t}=\nabla ^{2}v+k{\frac {u}{u+h}}v-mv}$

### Local Dynamics

Local dynamics cycle (starting in (u*,v*))

The local dynamics of the system are defined by the nonlinear reaction part of equations

${\displaystyle u_{t}=u(1-u)-{\frac {u}{u+h}}v\qquad v_{t}=v+k{\frac {u}{u+h}}v-mv}$

We first observe stationary conditions ${\displaystyle u_{t}=v_{t}=0\,}$  There are three stationary states ${\displaystyle (u,v)}$

• total extinction ${\displaystyle (0,0)}$  is a sedal point for ${\displaystyle k,m,h>0}$
• predator extinction ${\displaystyle (1,0)}$  is a sedal point if ${\displaystyle (u_{*},v_{*})}$  is in the biologically meaningful region ${\displaystyle u,v\geq 0}$  and a stable point otherwise
• and coexistence ${\displaystyle (u_{*},v_{*})}$  can be a sedal or stationary point
${\displaystyle u_{*}={\frac {rh}{1-r}}\quad r=m/k\quad v_{*}=(1-u_{*})(h+u_{*})}$

Although the reaction dynamics depend on three parameters ${\displaystyle h}$  is the most important. If

${\displaystyle h<{\frac {1-r}{r}}}$

than the population is in a biologically meaningful region ${\displaystyle u_{*},v_{*}\geq 0}$ . If

${\displaystyle h<{\frac {1-r}{1+r}}}$

than there is a stable limit cycle surrounding the steady state. There is a figure of such a cycle where the path begins near from the stationary point ${\displaystyle (u_{*},v_{*}+\Delta {v})}$  and the parameters are ${\displaystyle k=0.2}$ , ${\displaystyle r=0.3}$  and ${\displaystyle h=0.4}$ .

## Numerical simulation using finite difference equations

A direct approach to simulate prey-predator dynamics in the plankton example is to discretize the reaction-diffusion differential equation, and to solve it numerically. The dynamics is first observed in an one dimensional space and than on two dimensions. In both cases using a zero-flux boundary ${\displaystyle \nabla u=\nabla v=0}$ .

The PDE discretization method used is FTCS, which is the simplest. It is capable of producing complex behavior we would like to observe, but it has stability and convergence problems discussed below.

### One dimension

The simulation is first performed in an one dimensional space of ${\displaystyle 4000}$  elements. The discretization parameters are

${\displaystyle \Delta x=1\quad \Delta t=0.2\quad d=0.2}$

The prey-predator system parameters are

${\displaystyle D=1\quad k=2.0\quad r=0.4\quad h=0.3}$

For initial conditions the uniform stationary point ${\displaystyle (u_{*},v_{*})}$  is used

${\displaystyle u(x,0)=u_{*}\,}$
${\displaystyle v(x,0)=v_{*}+\epsilon x+\delta \,}$

a gradual increase of predator population from the left to the right is added to ignite the spatial dynamics, the parameters are

${\displaystyle \epsilon =10^{-6}\quad \delta =-1.5\cdot 10^{-3}}$

There are two figures showing the system at times 0, 640, 2600. At a certain point in time and space the species distribution becomes chaotic. The chaotic part shows a fast growth.

Spatial distributions of populations in the 1D case at time t=640

Spatial distributions of populations in the 1D case at time t=2640

The system is not converging by decreasing the time difference ${\displaystyle \Delta {t}\rightarrow 0}$ . The source of the problem can be seen by observing the local dynamics of ${\displaystyle u(t)}$  and ${\displaystyle v(t)}$  inside the chaotic region ${\displaystyle x=1500}$  from time ${\displaystyle t=0}$  to ${\displaystyle t=640}$  using two different time differences ${\displaystyle \Delta {t}=0.20}$  and ${\displaystyle \Delta {t}=0.02}$ . With decreasing the time difference the influence of diffusion grows over the influence of local dynamics. On the above figure the effect would be seen as a slower spreading of the chaotic region.

Local dynamics (u(t),v(t)) observed at x=1500 t=0-640 with Δt=0.20

Local dynamics (u(t),v(t)) observed at x=1500 t=0-640 with Δt=0.02

### Two dimensions

The simulation is then performed in a two-dimensional space of ${\displaystyle 576\times 384}$  elements. The discretization parameters are

${\displaystyle \Delta x=\Delta y=2\quad \Delta t=0.5\quad d=0.125}$

The prey-predator system parameters are

${\displaystyle D=1\quad k=2.0\quad r=0.4\quad h=0.3}$

For initial conditions the uniform stationary point ${\displaystyle (u_{*},v_{*})}$  is used

${\displaystyle u(x,0)=u_{*}+\epsilon _{1}(x-0.1y-144).*(x-0.1y-432)\,}$
${\displaystyle v(x,0)=v_{*}+\epsilon _{2}(x-450)-\epsilon _{3}(y-150)\,}$

a gradual increase of predator population is added to ignite the spatial dynamics, the parameters are

${\displaystyle \epsilon _{1}=2\cdot 10^{-7}\quad \epsilon _{2}=3\cdot 10^{-5}\quad \epsilon _{3}=1.2\cdot 10^{-4}}$

2D dynamics are observed on time points ${\displaystyle t=\{0,100,150,200,300,400,1000\}}$ . Spiral patterns can be seen arising from the initial gradient, at the end patterns brake and the system ends in chaos.

t=0

t=100

t=150

t=200

t=300

t=400

t=1000

Another problem arises because the FTCS method is not unconditionally stable. Some results may become negative, and the functions describing local dynamics diverge very fast in the negative quadrants.

## References

1. Alexander B. Medvinsky, Sergei V. Petrovskii, Irene A. Tikhonova, Horst Malchow, Bai-Lian Li, Spatiotemporal Complexity of Plankton and Fish Dynamics, SIAM Review, Volume 44, Number 3, pp. 311-370
1. Marcus R. Garvie, Finite difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB PDF


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# HIDDEN
# Clear previously defined variables
%reset -f

import os
os.chdir(os.path.expanduser('~/notebooks/02'))

# HIDDEN
import warnings
# Ignore numpy dtype warnings. These warnings are caused by an interaction
# between numpy and Cython and can be safely ignored.
# Reference: https://stackoverflow.com/a/40846742
warnings.filterwarnings("ignore", message="numpy.dtype size changed")
warnings.filterwarnings("ignore", message="numpy.ufunc size changed")

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
%matplotlib inline
import ipywidgets as widgets
from ipywidgets import interact, interactive, fixed, interact_manual
import nbinteract as nbi

sns.set()
sns.set_context('talk')
np.set_printoptions(threshold=20, precision=2, suppress=True)
pd.options.display.max_rows = 7
pd.options.display.max_columns = 8
pd.set_option('precision', 2)
# This option stops scientific notation for pandas
# pd.set_option('display.float_format', '{:.2f}'.format)

## SRS vs. "Big Data"¶

As we have previously mentioned, it is tempting to do away with our long-winded bias concerns by using huge amounts of data. It is true that collecting a census will by definition produce unbiased estimations. Perhaps bias will no longer be a problem if we simply collect many data points.

Suppose we are pollsters in 2012 trying to predict the popular vote of the US presidential election, where Barack Obama ran against Mitt Romney. Since today we know the exact outcome of the popular vote, we can compare the predictions of a SRS to the predictions of a large non-random dataset, often called administrative datasets since they are often collected as part of some administrative work.

We will compare a SRS of size 400 to a non-random sample of size 60,000,000. Our non-random sample is nearly 150,000 times larger than our SRS! Since there were about 120,000,000 voters in 2012, we can think of our non-random sample as a survey where half of all voters in the US responded (no actual poll has ever surveyed more than 10,000,000 voters).

# HIDDEN
total = 129085410
obama_true_count = 65915795
romney_true_count = 60933504
obama_true = obama_true_count / total
romney_true = romney_true_count / total

# 1 percent off
obama_big = obama_true - 0.01
romney_big = romney_true + 0.01

Here's a plot comparing the proportions of the non-random sample to the true proportions. The bars labeled truth show the true proportions of votes that each candidate received. The bars labeled big show the proportions from our dataset of 60,000,000 voters.

# HIDDEN
pd.DataFrame({
'truth': [obama_true, romney_true],
'big': [obama_big, romney_big],
}, index=['Obama', 'Romney'], columns=['truth', 'big']).plot.bar()
plt.title('Truth compared to a big non-random dataset')
plt.xlabel('Candidate')
plt.ylabel('Proportion of popular vote')
plt.ylim(0, 0.75);

We can see that our large dataset is just slightly biased towards the Republican candidate, just as the Gallup Poll was in 1948. To see this effects of this bias, we simulate taking simple random samples of size 400 from the population and large non-random samples of size 60,000,000. We compute the proportion of votes for Obama in each sample and plot the distribution of proportions.

srs_size = 400
big_size = 60000000
replications = 10000

def resample(size, prop, replications):
return np.random.binomial(n=size, p=prop, size=replications) / size

srs_simulations = resample(srs_size, obama_true, replications)
big_simulations = resample(big_size, obama_big, replications)

Now, we plot the simulation results and overlay a red line indicating the true proportion of voters that voted for Obama.

bins = bins=np.arange(0.47, 0.55, 0.005)
plt.hist(srs_simulations, bins=bins, alpha=0.7, normed=True, label='srs')
plt.hist(big_simulations, bins=bins, alpha=0.7, normed=True, label='big')

plt.title('Proportion of Obama Voters for SRS and Big Data')
plt.xlabel('Proportion')
plt.ylabel('Percent per unit')
plt.xlim(0.47, 0.55)
plt.ylim(0, 50)
plt.axvline(x=obama_true, color='r', label='truth')
plt.legend();

As you can see, the SRS distribution is spread out but centered around the true population proportion of Obama voters. The distribution created by the large non-random sample, on the other hand, is very narrow but not a single simulated sample produces the true population proportion. If we attempt to create confidence intervals using the non-random sample, none of them will contain the true population proportion. To make matters worse, the confidence interval will be extremely narrow because the sample is so large. We will be very sure of an ultimately incorrect estimation.

In fact, when our sampling method is biased our estimations will often first become worse as we collect more data since we will be more certain about an incorrect result. In order to make accurate estimations using an even slightly biased sampling method, the sample must be nearly as large as the population itself, a typically impractical requirement. The quality of the data matters much more than its size.

## Takeaways¶

Before accepting the results of a data analysis, it pays to carefully inspect the quality of the data. In particular, we must ask the following questions:

1. Is the data a census (does it include the entire population of interest)? If so, we can just compute properties of the population directly without having to use inference.
2. If the data is a sample, how was the sample collected? To properly conduct inference, the sample should have been collected according to a completely described probability sampling method.
3. What changes were made to the data before producing results? Do any of these changes affect the quality of the data?

For the curious reader interested in a deeper comparison between random and large non-random samples, we suggest watching this lecture by the statistician Xiao-Li Meng.

format: plain


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# Bayes factor

In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing.

Given a model selection problem in which we have to choose between two models M1 and M2, on the basis of a data vector x. The Bayes factor K is given by

[itex]K = \frac{p(x|M_1)}{p(x|M_2)}.[itex]

This is similar to a likelihood-ratio test, but instead of maximising the likelihood Bayesians average it over the parameters. Generally, the models M1 and M2 will be parametrised by vectors of parameters θ1 and θ2; thus K is given by

[itex]K = \frac{p(x|M_1)}{p(x|M_2)} = \frac{\int \,p(\theta_1|M_1)p(x|\theta_1, M_1)d\theta_1}{\int \,p(\theta_2|M_2)p(x|\theta_2, M_2)d\theta_2}.[itex]

A value of K > 1 means that the data indicate that M1 is more likely than M2 and vice versa. Note that classical hypothesis testing gives one hypothesis (or model) preferred status (the 'null hypothesis'), and only considers evidence against it. Harold Jeffreys gave a scale for interpretation of K:

K Strength of evidence
< 1 Negative (supports M2)
1 to 3 Barely worth mentioning
3 to 12 Positive
12 to 150 Strong
> 150 Very strong

Many Bayesian statisticians would use a Bayes factor as part of making a choice, but would also combine it with their estimates of the prior probability of each of the models and the loss functions associated with making the wrong choice.

## Example

Suppose we have a random variable which produces either a success or a failure. We want to consider a model M1 where the probability of success is q=½, and another model M2 where q is completely unknown and we take a prior distribution for q which is uniform on [0,1]. We take a sample of 200, and find 115 success and 85 failures. The likelihood is:

[itex]{200 \choose 115}q^{115}(1-q)^{85}[itex]

So we have

[itex]P(X=115|M_1)={200 \choose 115}\left({1 \over 2}\right)^{200}=0.00595...[itex]

but

[itex]P(X=115|M_2)=\int_{q=0}^1 1{200 \choose 115}q^{115}(1-q)^{85}dq = {1 \over 201} = 0.00497...[itex]

The ratio is then 1.197..., which is "barely worth mentioning" even if it points very slightly towards M1.

This is not the same as a classical likelihood ratio test, which would have found the maximum likelihood estimate for q, namely 115200=0.575, and from that get a ratio of 0.1045..., and so pointing towards M2. A frequentist hypothesis test would have produced an even more dramatic result, saying that that M1 could be rejected at the 5% confidence level, since the probability of getting 115 or more successes from a sample of 200 if q=½ is 0.0200..., and as a two-tailed test of getting a figure as extreme as or more extreme than 115 is 0.0400... Note that 115 is more than two standard deviations away from 100.

M2 is a more complex model than M1 because it has a free parameter which allows it to model the data more closely. The ability of Bayes factors to take this into account is a reason why Bayesian inference has been put forward as a theoretical justification for and generalisation of Occam's razor, reducing Type I errors.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy


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Now showing items 2102-2121 of 10788

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Abstract Background High-throughput measurement of allele-specific expression (ASE) is a relatively new and exciting application area for array-based technologies. In this paper, we explore several data sets which make use ...
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The $\alpha$-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for $\alpha$ $\geq$ 1/2. In this article, we derive a necessary and sufficient algebraic condition ...
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# M. Gyulassy

The author has been a faculty member of the Columbia University physics department since $1993$. Prior to that he was a senior nuclear theorist at the Lawrence Berkeley National Laboratory. He obtained his Ph.D. in $1974$ from the University of California after immigrating to the US from Hungary in $1956$. His over $270$ publications cover topics on the theory of ultradense nuclear and QCD matter and the phenomenology of high-energy nuclear collisions.


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### Home > PC3 > Chapter 12 > Lesson 12.2.1 > Problem12-95

12-95.

Given $z_1 = 6\Big[\cos(\frac { 7 \pi } { 12 }) + i \sin(\frac { 7 \pi } { 12 })\Big]$ and $z_2 = 3 \Big[\cos(\frac { \pi } { 4 }) + i \sin (\frac { \pi } { 4 })\Big]$ compute each of the following values. Write your answers in $a + bi$ form.

1. $z_{1}z_{2}$

Given two complex numbers $z_1=r_1(\cos(a)+i\sin(a))$ and $z_2=r_2(\cos(b)+i\sin(b))$, the product of the two numbers is $z _ { 1 } z _ { 2 } = r _ { 1 } r _ { 2 } ( \operatorname { cos } ( a + b ) + i \operatorname { sin } ( a + b ) )$

1. $\frac { z 1 } { z _ { 2 } }$

Given two complex numbers $z_1=r_1(\cos(a)+i\sin(a))$ and $z_2=r_2(\cos(b)+i\sin(b))$, the quotient of the two numbers is $\frac { z _ { 1 } } { z _ { 2 } } = \frac { r _ { 1 } } { r _ { 2 } } ( \operatorname { cos } ( a - b ) + i \operatorname { sin } ( a - b ) )$


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# In which field of chemistry does the study of kinetics, pressure, and phase changes lie?

This falls under the $\text{physical chemistry}$ umbrella........


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# What is the coefficient of (a) x^10. (b) x^11. In the expansion of (2 x - 1 / x^2)^{16}? Fully...

## Question:

What is the coefficient of

(a) {eq}x^{10} {/eq}.

(b) {eq}x^{11} {/eq}.

In the expansion of {eq}\bigg(2 x - \dfrac 1 {x^2}\bigg)^{16} {/eq}? Fully evaluate your answers.

## Binomial Theorem:

The binomial theorem is used to expand the higher order polynomials. It is also used in the mass function of the binomial distribution. That is why, the name of the distribution is binomial. The combination method is involved in the theorem.

Given Information:

The binomial theorem is:

{eq}{\left( {a + b} \right)^n} = \sum\limits_{r = 0}^n {{}^n{C_r}{{\left( a \right)}^r}{{\left( b \right)}^{n - r}}} {/eq}

The following expression can be expanded using Binomial theorem:

{eq}\begin{align*} {\left( {2x - \dfrac{1}{{{x^2}}}} \right)^{16}} &= \sum\limits_{r = 0}^{16} {{}^{16}{C_r}{{\left( {2x} \right)}^r}{{\left( { - {x^{ - 2}}} \right)}^{16 - r}}} \\ &= {}^{16}{C_0}{\left( {2x} \right)^0}{\left( { - {x^{ - 2}}} \right)^{16}} + {}^{16}{C_1}\left( {2x} \right){\left( { - {x^{ - 2}}} \right)^{15}} + ............ + {}^{16}{C_{16}}{\left( {2x} \right)^{16}}{\left( { - {x^{ - 2}}} \right)^{16 - 16}}\\ &= 65536{x^{16}} - 524288{x^{13}} + 1966080{x^{10}} - 4587520{x^7} + 7454720{x^4} - 8945664x + \dfrac{{8200192}}{{{x^2}}} - \dfrac{{5857280}}{{{x^5}}} + \dfrac{{3294720}}{{{x^8}}} - \dfrac{{1464320}}{{{x^{11}}}} + \dfrac{{512512}}{{{x^{14}}}} - \dfrac{{139776}}{{{x^{17}}}} + \dfrac{{29120}}{{{x^{20}}}} - \dfrac{{4480}}{{{x^{23}}}} + \dfrac{{480}}{{{x^{26}}}} - \dfrac{{32}}{{{x^{29}}}} + \dfrac{1}{{{x^{32}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right) \end{align*} {/eq}

a)

The coefficient of {eq}{x^{10}}\,\,in\,\,the\,\,\exp ansion\,\,of\,\,{\left( {2x - \dfrac{1}{{{x^2}}}} \right)^{16}} {/eq} is 1966080. This is obtained from (1).

b)

The coefficient of {eq}{x^{11}}\,\,in\,\,the\,\,\exp ansion\,\,of\,\,{\left( {2x - \dfrac{1}{{{x^2}}}} \right)^{16}} {/eq} is 0. This is obtained from (1).


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## Part 1 of 3: Building/Loading/Scoring Against Predictive Models in R

August 31, 2011
By

In this first installment, I'm going to focus on:Building/evaluating a predictive model with partitioned dataSaving the predictive model to diskLoading the predictive model from diskScoring data against a predictive model (within R)This installment is ...

## Seriously … why don’t math classes use computers?…

August 31, 2011
By

Seriously … why don’t math classes use computers? Excel, simple Python scripts, Mathematica / Sage, everything beyond the TI-83. Kids could be creating totally sweet visuals instead of cribbing formulae. And thinking instead of copying. I can sa...

## Seriously … why don’t math classes use computers?…

August 31, 2011
By

Seriously … why don’t math classes use computers? Excel, simple Python scripts, Mathematica / Sage, everything beyond the TI-83. Kids could be creating totally sweet visuals instead of cribbing formulae. And thinking instead of copying. I can sa...

## Story of the Ljung-Box Blues: Progress Not Perfection

August 31, 2011
By

In the last post we determined that our ARIMA(2,2,2) model failed to pass the Ljung-Box test.  In todays post we seek to completely discredit the last posts claim and finally arrive at some needed closure. The Ljung-Box is first performed on the s...

## rnpn: An R interface for the National Phenology Network

August 31, 2011
By

The team at rOpenSci and I have been working on a wrapper for the USA National Phenology Network API. The following is a demo of some of the current possibilities. We will have more functions down the road. Get the publicly available code, and contribu...

## XLConnect – A platform-independent interface to Excel

August 31, 2011
By

XLConnect is a comprehensive and platform-independent R package for manipulating Microsoft Excel files from within R. XLConnect differs from other related R packages in that it is completely cross-platform and as such runs under Windows, Unix/Linux and Mac (32- and 64-bit). Moreover, it … Continue reading →

## Posts of the year

August 30, 2011
By

Like last year, here are the most popular posts since last August: Home page 92,982 In{s}a(ne)!! 6,803 “simply start over and build something better” 5,834 Julien on R shortcomings 2,373 Parallel processing of independent Metropolis-Hastings algorithms 1,455 Do we need an integrated Bayesian/likelihood inference? 1,361 Coincidence in lotteries 1,256 #2 blog for the statistics geek?! 863

## What language is R written in?

August 30, 2011
By

On of the nice things about R is that a lot if it is written in the R language. That means, as an R user, if you want to see how R calculates a certain statistic, or you want to modify an existing function for your own use, you can just look at the R code by typing the...

## The Visual Difference – R and Anscombe’s Quartet

August 30, 2011
By

I spent a chunk of today trying to get my thoughts in order for a keynote presentation at next week’s The Difference that Makes a Difference conference. The theme of my talk will be on how visualisations can be used to discover structure and pattern in data, and as in many or my other recent

## Getting Started with Latent Dirichlet Allocation using RTextTools + topicmodels

RTextTools bundles a host of functions for performing supervised learning on your data, but what about other methods like latent Dirichlet allocation? With some help from the topicmodels package, we can get started with LDA in just five steps. Text in

## Nomograms everywhere!

August 30, 2011
By

At useR!, Jonty Rougier talked about nomograms, a once popular visualisation that has fallen by the wayside with the rise of computers. I’d seen a few before, but hadn’t understood how they worked or why you’d want to use them. Anyway, since that talk I’ve been digging around in biology books from the 60s and

## R combined gps-track plot of spatial intensity

August 30, 2011
By

To get a quick impression about the temporal stay of places it is helpful to generate a plot of the trackpoints spatial density (intensity). As the 3d visualisation has both advatages and disadvantages, a combination with a 2D plot is useful to interpret the data. The data used in this example is a gps record

## Realized beta and beta equal 1

August 30, 2011
By

What does beta look like in the out-of-sample period for the portfolios generated to have beta equal to 1? In the comments Ian Priest wonders if the results in “The effect of beta equal 1″ are due to a shift in beta from the estimation period to the out-of-sample period.  (The current post will make … Continue reading...

## How Much of R is Written in R Part 2: Contributed Packages

August 29, 2011
By

So that mean old boss of mine is at it again.  This morning I came in beaming about how many people had read my post How Much of R is Written in R (thanks by the way!).  He then asks me about one little line in that post; the one about how if you looked

## Sharing live R functions with OpenCPU

August 29, 2011
By

OpenCPU is a new initiative from R user Jeroen Ooms to make innovations in statistics, visualization and data-science more widely applicable. Based on open-source principles, it's a web-based service that lets you upload data visualizations and analyses as R scripts, and allow others to run them on demand. For example, you can upload a script to visualize a company's...

## another lottery coincidence

August 29, 2011
By
$another lottery coincidence$

Once again, meaningless figures are published about a man who won the French lottery (Le Loto) for the second time. The reported probability of the event is indeed one chance out of 363 (US) trillions (i.e., billions in the metric system. or 1012)… This number is simply the square of which is the number of

## The effect of beta equal 1

August 29, 2011
By

Investment Performance Guy had a post about beta equal 1.  It made me wonder about the properties of portfolios with beta equal 1.  When I looked, I got a bigger answer than I expected. Data I have some S&P 500 data lying about from the post ‘On “Stock correlation has been rising”‘.  So laziness dictated … Continue reading...

## Comparing Two Distributions

August 29, 2011
By

Here I compare two distributions, flowering duration of indigenous and allochtonous plant species. The hypothesis is that alien compared to indigenous plant species exhibit longer flowering periods. Read more »

## R is a cool image editor #2: Dithering algorithms

August 29, 2011
By

Here I implemented in R some dithering algorithms: - Floyd-Steinberg dithering - Bill Atkinson dithering - Jarvis-Judice-Ninke dithering - Sierra 2-4a dithering - Stucki dithering - Burkes dithering - Sierra2 dithering - Sierra3 dithering For each algorithm, I wrote a 2-dimensional convolution function (a matrix passing over a matrix); it is slow because I didn't implemented any fasting tricks. It can be easily implemented in C, then used...

## Slides of 10+ talks at R Users Groups

August 29, 2011
By

Links to slides of 10+ talks at R Users Groups in Australia are provided below. Slides of the talks are downloadable at the links, including R codes if any. MelbURN: Melbourne Users of R Network: Experiences with using R in … Continue reading →

## Real-time Scoring/Optimization of Predictive Models in R

August 28, 2011
By

I'm working on a 3 part post on how to build, score and perform optimization with predictive models in R. Having done this type of work in IBM SPSS for a number of years, I wanted to replicate it in R. It's amazing how little is published on how to s...

## Ra vs. compiler package

August 28, 2011
By

R seems to have two byte code compilers: the Ra add-on module (and the accompanying "jit" package) and the "compiler" package came with the default installation. I wonder how they differentiate from each other and what the strengths and weaknesses...

## HPC for biological research

August 28, 2011
By

In early May I had the opportunity to attend a workshop on using high performance computing in R hosted at Nimbios. I’ve been meaning to write a summary of the meeting ever since but got sidetracked by various other projects. Since a collaborator recently asked for meeting notes I finally took the time to write

## Real-time data collection and analysis in class

August 28, 2011
By

As September draws nearer, my mind inevitably turns away from my lofty (and largely unmet) summer research goals, and toward teaching.  This semester I will be trying out a teaching technique using live data collection and analysis as a tool to encourage student engagement.  The idea is based on the electronic polling technology known as

## Support Vector Machine with GPU

August 27, 2011
By

Most elementary statistical inference algorithms assume that the data can be modeled by a set of linear parameters with a normally distributed noise component. A new class of algorithms called support vector machine (SVM) remove such constraint. rea...

## Some Additional Thoughts on Useless Averages

In my last post, I described three situations where the average of a sequence of numbers is not representative enough to be useful: in the presence of severe outliers, in the face of multimodal data distributions, and in the face of infinite-variance distributions.  The post generated three interesting comments that I want to respond to here.First and foremost, I...

## Forecasting In R: The Greatest Shortcut That Failed The Ljung-Box

August 27, 2011
By

Okay so you want to forecast in R, but don't want to manually find the best model and go through the drudgery of plotting and so on.  I have recently found the perfect function for you.  Its called auto.arima and it automatically fits the bes...

## SIGKDD 2011 Conference — Days 2/3/4 Summary

August 27, 2011
By

<< My review of Day 1. I am summarizing all of the days together since each talk was short, and I was too exhausted to write a post after each day. Due to the broken-up schedule of the KDD sessions, I group everything together instead of switching back and forth among a dozen different topics. By far the most enjoyable...


text:
Yess C.

# What would be the molarity (M2) of the 100.00 mL (V2) of dilute HCl if you used a 25.00 (V1) mL volumetric pipet?

Quantitative dilution of solutions from higher to lower concentrations is a critical skill in this lab course. You have already diluted HCl from higher to lower concentrations using two different approaches requiring slightly different calculations

For example one procedure could use a 100.00 mL (V2) volumetric flask and a 10.00 mL (V1) volumetric pipet or pipettor to prepare a 0.500 M (M2) solution of HCl. A 10.00 mL volumetric pipet or pipettor is used to transfer 5.00 M HCl (M1) to a 100.00 mL (V2) volumetric flask and then the flask is filled exactly to 100.00 mL with distilled water and mixed well. The same dilution can be done using 5.00 mL (V1) and a 50.00 mL (V2) volumetric flask if less solution is needed.

One way to make different concentrations of HCl (M2) in a 100.00 mL (V2) volumetric flask from a stock 5.00 M (M1) solution would be to use different volumetric pipets. What would be the molarity (M2) of the 100.00 mL (V2) of dilute HCl if you used a 25.00 (V1) mL volumetric pipet? Do not include units in your answer.


text:
# Behavioural Items with ScriptableObjects

My game has an item system that is relatively small compared to most of the RPGs, but it is pretty similar to the games like Half-Life and Counter-Strike where you have an inventory consists of several menus (categories) and a few slots in them. Whenever the player picks up or receives the weapon, from then on they can select it using their inventory which is quite simplistic, in my opinion, and I like it.

First of all, I want to indicate that I am willing to separate logic and data. For data, I am using ScriptableObjects and the basic concept behind is basically referencing ScriptableObjects to MonoBehaviours and handle the tasks in there according to the data. This can be of course made with Prefabs but the thing is, you would de unconsciously duplicating common components that an item would use and only change some of the fields in one of your MonoBehaviours that is related to items. So my approach to that is, keep the data with ScriptableObjects, create only single Prefab, then tie the related components to it and the logic MonoBehaviour which will reference the certain data.

To begin with, I want to talk about how I designed my ScriptableObjects (in other words, data structures). There is the base (abstract) class called ItemData which contains common stuff like every item will have, for example, HUD icon. Then, there are two more abstract classes which are called PassiveItemData & ActiveItemData and derived from ItemData.

public enum ActiveItemType {
Active,
Firearm,
Melee
}

public abstract class ActiveItemData : ItemData {
public Sprite wieldingSprite;

public ActiveItemType type;

public SupplyData resourceType;
}

public abstract class PassiveItemData : ItemData {

}

PassiveItemData does not have any fields at the moment, but might do in the future. For now, the only class that derives from it is this:

public class SupplyData : PassiveItemData {
[Tooltip("The maximum amount that the player can carry this type of resource at once.")]
public int max;
}

SupplyData is the data of consumable items (e.g. ammo) which have no use on their own but can be consumed by other "active" items. Active items are the items that have actual functionalities (e.g. firearm, melee weapon) and can be directly used by the players and switched to via inventory. And a typical active item would also consume a supply. In games like Half-Life, there is no such thing called "active items", whatever that is can be switched to and directly used, they are called weapons in there. The reason that I named mine version as active items is that because, in my game, there won't be only weapons that can be directly used but also items like Medical Syringe (which consumes syringes) for healing purposes. My problem is I don't know where should I put the methods that perform the tasks like fire, use, swing, etc... Because they are not very generic and may belong to the only specific type of items, which is kinda tricky to implement.

My point was to put them inside a MonoBehaviour that will be tied to the Prefab. However, it is not possible to implement it with a single MonoBehaviour since the data would reference either a firearm or melee weapon data. Say, it is a melee weapon, it'd be non-sense to keep a field like ammoInMagazine that is related to only firearms within the script. But you can never know for sure if the reference data will be firearm or melee weapon. So you have to keep it in there whatsoever. Yet, I don't want this to happen. So I used several more classes that are going to keep the data and perform the tasks accordingly that derives from MonoBehaviour. And the issue has now come to how am I going to use these MonoBehaviours? Create prefabs for each MonoBehaviour? Like Firearm.cs MonoBehaviour to Firearm Prefab and MeleeWeapon.cs MonoBehaviour to Melee Weapon Prefab? Oh, my God. Does this mean I'd have to deal with this each time I implement a new item type? No way I do that. Or maybe I do, but not sure if this is the right approach.

I also thought of doing all these without creating MonoBehaviours and Prefabs that they are going to be tied. But in this case, there are going to be other situations where you might not be able to configure the "fire point", for example.

So, people, this might be a little longer than you expected but I still don't even think I included all the details. Nevertheless, I believe you have the big picture and can suggest a thing or two. Thanks in advance.

• I think your worry about duplicating data in prefabs might be holding you back from more elegant component-based solutions. A little extra data won't cause undue hardship, and it can free you from the inheritance polymorphism knot you find yourself in. – DMGregory Apr 29 '20 at 15:29

It sounds to me like your items have enough individual variation that trying to support them off a single prefab and a rigid inheritance hierarchy of ScriptableObject types is likely to be a headache.

For cases like this, where you have many item varieties that have their own particular needs, I think prefabs are actually a great way to go. While you say:

This can be of course made with Prefabs but the thing is, you would de unconsciously duplicating common components that an item would use and only change some of the fields in one of your MonoBehaviours that is related to items

This data duplication is actually very cheap. Even over hundreds of items you're probably looking at only a matter of KiB. I'd argue that's a reasonable price to pay for the flexibility we gain.

Using a prefab for each item lets us use composition over inheritance. Special behaviours like ammunition get offloaded to a component that an item can "have" rather than getting baked into what type of item it "is", and that lets use mix and match them and create interesting variants of items just by combining components, without re-coding our inheritance hierarchy every time we want to add a new combination of behaviours.

For instance, you might make a core Item component that all your inventory interaction and item use gameplay scripts interact with. It does very little on its own - it just has slots to call into other components to customize its action:

public class Item : MonoBehaviour {

public class TargetedEvent : UnityEvent<GameObject> {}

[SerializeField]
ItemSlot _slot;
public ItemSlot slot { get { return _slot;} }

[SerializeField]
UnityEvent _onUse;

[SerializeField]
TargetedEvent _onUseOnTarget;

[SerializeField]
IUseRestriction[] _restrictions;

public bool CanUse() {
if(_onUse == null) return false;

foreach(var restriction in _restrictions)
if(restriction.CanUse() == false) return false;

return true;
}

public bool CanUseOn(GameObject target) {
if(_onUseOnTarget == null) return false;

foreach(var restriction in _restrictions)
if(restriction.CanUseOn(target) == false) return false;

return true;
}

public bool TryUse() {
if(CanUse() == false)
return false;

_onUse.Invoke();

foreach(var restriction in _restrictions) {
restriction.OnUse();
}
return true;
}

public bool TryUseOn(GameObject target) {
if(CanUseOn(target) == false)
return false;

_onUseOnTarget.Invoke(target);

foreach(var restriction in _restrictions) {
restriction.OnUse();
}
return true;
}
}

Using a UnityEvent means you can wire up any function you want as the action to perform when an item is used, so you can run wild with different item effects without modifying the core access point for interacting with items.

We've also abstracted any rule that might affect when we're allowed to use this item, making them just one or more components that we can attach to the item as needed, or even stack up in different combinations:

public interface IUseRestriction {
bool CanUse();
bool CanUseOn(GameObject target);

void OnUse();
}

Your ammunition is then just one kind of IUseRestriction component that might be attached to an object.

public class Ammunition : MonoBehaviour, IUseRestriction {

[SerializeField]
int _currentAmmo;

[SerializeField]
int _maxAmmo;

public bool CanUse() { return _currentAmmo > 0; }
public bool CanUseOn(GameObject target) { return CanUse(); }

public void OnUse() { _currentAmmo--; }

int amountLoaded = _maxAmmo - _currentAmmo;

}
}

(If you have a centralized ammo inventory on your character rather than in the item itself, you can modify this to query the character's ammo inventory instead)

Other restrictions might include Cooldown or SuperMeter or EffectiveRange tests, friendly/enemy UsableOnFaction masks, even a Consumable restriction that self-destructs the item after use. ;)

Similarly, your concept of an "Active" item can be just another component:

public class Equipable : MonoBehaviour {
public EquipmentSlot slot;
public Sprite equippedSprite;
}

Now your code can test if something is equipable just by checking:

var equipable = item.GetComponent<Equipable>();
if(equipable != null) {
// Offer/execute option to equip this item,
// using the data in the equipable instance.
}

You can do similarly for Sellable items that need a price/value, Throwable items that need to draw a range-finder when used, etc.

This lets you mix and match different item effects, restrictions, etc. in any way you want, just by combining different components on the prefab, without a care for rigid inheritance hierarchies.

Maybe most of your potions are pure inventory items that can't be equipped except to a quick-use slot. But you have an acid potion that's allowed to be equipped as a weapon. No problem! You just add the Equipable component to that single prefab, and you don't have to create a whole new branch of the potion inheritance hierarchy that duplicates the equipable behaviour from the ActiveItemData type, or a branch of the ActiveItemData type that duplicates the stacking behaviour of the PassiveItemData type. New original gameplay combinations with zero code changes. 😍

If you find you have a lot of components that need to reference the same information, and the duplication bothers you, you can still parcel up just that information to a shared ScriptableObject that they all reference, without losing the flexibility of individual prefabs. Say for instance you have 20 different varieties of potion, where you're allowed to carry a stack of 20 of each, and 12 different varieties of food, where you're allowed to carry a stack of 50 of each, and 5 varieties of gemstone, where you're allowed to carry a stack of 99 of each. Each Stackable component on these prefabs can expose a StackType enum, from which you select Potion, Food, or Gem, and you can have a separate InventoryCapacity ScriptableObject that contains a map from StackType to the max item count.


text:
# Tag Info

What you need is more mutual information rather than Shannon entropy. It is dedicated to capture the influence of one variable on another (you can think about it as a non linear version of Pearson correlations). They are closely related since the mutual information $I$ between two variables $X$ and $Y$ reads: $$I(X;Y) = H(X,Y) - H(X|Y) - H(Y|X)$$ where $H$ ...


text:
# qml.utils¶

Warning

Unless you are a PennyLane or plugin developer, you likely do not need to use these utility functions.

This module contains utilities and auxiliary functions which are shared across the PennyLane submodules.

## Functions¶

 expand(U, wires, num_wires) Expand a multi-qubit operator into a full system operator. unflatten(flat, model) Wrapper for _unflatten().

## Classes¶

 OperationRecorder() A template and quantum function inspector, allowing easy introspection of operators that have been applied without requiring a QNode. Recorder(old_context) Recorder class used by the OperationRecorder.


text:
Question

# Two dice are numbered 1,2,3,4,5,6 and 1,1,2,2,3,3 respectively. They are thrown and the sum of the numbers on them is noted. Find the probability of getting each sum from 2 to 9 separately.

HINT- In order to solve such type of question we must use formula Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$ , along with proper understanding of favourable cases and total cases.

Here we have one unbiased dice which is numbered from 1 to 6. While other dice are biased on which numbers are 1,1,2,2,3,3.

When we through both the dice then every number of each dice will make a pair with each number of other dice so total numbers of possible outcomes will be

(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)

(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)

(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)

(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)

(1,3) (2,3) (3,3) (4,3) (5,3) (6,3)

(1,3) (2,3) (3,3) (4,3) (5,3) (6,3)

So total Favourable outcomes =36

The probability of getting each sum from 2 to 9

(1) Sum of 2
favourable outcome for Sum of 2 will be (1,1,), (1,1).
No. of favourable outcomes = 2
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 2)=$\dfrac{2}{{36}}{\text{ = }}\dfrac{1}{{18}}$

(2) Sum of 3
a favourable outcome for Sum of 3 will be (1,2), (1,2), (2,1). (2,1)
No. of favourable outcomes = 4
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 3)=$\dfrac{4}{{36}}{\text{ = }}\dfrac{1}{9}$
=364=91

(3) Sum of 4
a favourable outcome for Sum of 4 will be (2,2), (2,2), (3,1). (1,3)

No. of favourable outcomes = 4
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 4)=$\dfrac{4}{{36}}{\text{ = }}\dfrac{1}{9}$

(4) Sum of 5
favourable outcome for Sum of 5 will be (1,4), (4,1,), (2,3)., (2,3) (3,2). (3,2)

No. of favourable outcomes = 6
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 5)=$\dfrac{6}{{36}}{\text{ = }}\dfrac{1}{6}$

(5) Sum of 6
favourable outcome for Sum of 6 will be (1,5), (1,5), (2,4)., (2,4) (3,3). (3,3)

No. of favourable outcomes = 6
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 6)=$\dfrac{6}{{36}}{\text{ = }}\dfrac{1}{6}$

(6) Sum of 7
favourable outcome for Sum of 7 will be (1,6), (1,6), (2,5)., (2,5) (3,4). (3,4)

No. of favourable outcomes = 6
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 7)=$\dfrac{6}{{36}}{\text{ = }}\dfrac{1}{6}$

(7) Sum of 8
favourable outcome for Sum of 8 will be (2,6), (2,6,), (3,5). (3,5)
No. of favourable outcomes = 4
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 8)=$\dfrac{4}{{36}}{\text{ = }}\dfrac{1}{9}$

(8) Sum of 9
favourable outcome for Sum of 9 will be (6,3,), (3,6).
No. of favourable outcomes = 2
total number of possible cases = 36

As we know that Probability $\left( {\text{P}} \right){\text{ = }}\dfrac{{{\text{Favourable number of cases}}}}{{{\text{Total number of cases}}}}$
Probability (Sum of 9)=$\dfrac{2}{{36}}{\text{ = }}\dfrac{1}{{18}}$

NOTE- In Such Types of Question first find out the total numbers of possible outcomes then find out the number of favourable cases, then divide them using the formula which stated above, we will get the required answer.


text:
# DEVICE Photodiode modeling - Excess Noise

#1

Hi,
I am wondering if obtaining excess noise would be possible for an avalanche photodiode using Lumerical? What are your advices and suggested references for simulating this parameter?
Thanks!

#2

I appreciate any help I can get

#3

Hi @kghaffari,

Sorry for the delay, we’ve been pretty busy recently! I’m not too familiar with this application, but from what I could read (for instance on Wikipedia), you can express the excess noise ratio as:
$$F = \kappa M + \left( 2-\frac{1}{M} \right) \left( 1-\kappa\right)$$
Where $M$ is the APD’s gain, $\kappa$ is the ratio of the hole impact ionization rate to that of electrons.

The gain can be obtained from the DEVICE simulation. You can refer to this page for a some examples of avalanche photodiode simulations.

Regarding $\kappa$, it depends on the material properties as well as the electric field in the multiplying region, so I’m not sure how it can be obtained. In DEVICE, we’re using the Selberherr model to describe the impact ionization:
$$\alpha_{n,p} = \alpha_{n,p}^{\infty}exp \left[ -\left( \frac{E_{n,p}^{crit} }{\vert E \vert } \right)^2 \right]$$
where $\alpha_{n,p}$ is the impact ionization rate for electrons or holes. As $E$ can be spatially dependent (but can be obtained from the simulation), a possibility could be to integrate $\alpha_{n,p}$ over the material area/volume and then calculate $\kappa$ from that.

I will try and see if my colleague who wrote the APD examples can have a look at your question. In the meantime, I hope this will help!

#4

Hi @gbaethge
In Selberherr model, do you have any suggestions about how to account for the non-uniformity of electric field when determining |E|? Also do you have any references for alpha_n and E_crit values used in the software (particularly for silicon)?

#5

Hi @kghaffari,

In DEVICE, you can use a monitor to record the electric field distribution. Then you could use this information and integrate $\alpha_{n,p}$ over the area of the material.

Regarding the model, there is some references here. More specifically, Selberherr, S. Analysis and Simulation of Semiconductor Devices, Springer Verlag, Vienna (1984), includes tables of coefficients for some semiconductors (including Si). That said, the data is taken from the literature, so this book will also list the various sources.

#6

This helped a lot, Thank you!


text:
# map_bfs

0th

Percentile

##### Apply a function to nodes in the order of a breath first search

These functions allow you to map over the nodes in a graph, by first performing a breath first search on the graph and then mapping over each node in the order they are visited. The mapping function will have access to the result and search statistics for all the nodes between itself and the root in the search. To map over the nodes in the reverse direction use map_bfs_back().

##### Usage
map_bfs(root, mode = "out", unreachable = FALSE, .f, ...)map_bfs_lgl(root, mode = "out", unreachable = FALSE, .f, ...)map_bfs_chr(root, mode = "out", unreachable = FALSE, .f, ...)map_bfs_int(root, mode = "out", unreachable = FALSE, .f, ...)map_bfs_dbl(root, mode = "out", unreachable = FALSE, .f, ...)
##### Arguments
root

The node to start the search from

mode

How should edges be followed? 'out' only follows outbound edges, 'in' only follows inbound edges, and 'all' follows all edges. This parameter is ignored for undirected graphs.

unreachable

Should the search jump to an unvisited node if the search is completed without visiting all nodes.

.f

A function to map over all nodes. See Details

...

Additional parameters to pass to .f

##### Details

The function provided to .f will be called with the following arguments in addition to those supplied through ...:

• graph: The full tbl_graph object

• node: The index of the node currently mapped over

• rank: The rank of the node in the search

• parent: The index of the node that led to the current node

• before: The index of the node that was visited before the current node

• after: The index of the node that was visited after the current node.

• dist: The distance of the current node from the root

• path: A table containing node, rank, parent, before, after, dist, and result columns giving the values for each node leading to the current node. The result column will contain the result of the mapping of each node in a list.

Instead of spelling out all of these in the function it is possible to simply name the ones needed and use ... to catch the rest.

##### Value

map_bfs() returns a list of the same length as the number of nodes in the graph, in the order matching the node order in the graph (that is, not in the order they are called). map_bfs_*() tries to coerce its result into a vector of the classes logical (map_bfs_lgl), character (map_bfs_chr), integer (map_bfs_int), or double (map_bfs_dbl). These functions will throw an error if they are unsuccesful, so they are type safe.

Other node map functions: map_bfs_back, map_dfs_back, map_dfs

• map_bfs
• map_bfs_lgl
• map_bfs_chr
• map_bfs_int
• map_bfs_dbl
##### Examples
# NOT RUN {
# Accumulate values along a search
create_tree(40, children = 3, directed = TRUE) %>%
mutate(value = round(runif(40)*100)) %>%
mutate(value_acc = map_bfs_dbl(node_is_root(), .f = function(node, path, ...) {
sum(.N()$value[c(node, path$node)])
}))
# }

Documentation reproduced from package tidygraph, version 1.1.2, License: MIT + file LICENSE

### Community examples

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text:
beamer problem with combining columns, listings, and overlays

I am trying to come up with a beamer frame where some listing appears at the left column on overlay 1 and some other stuff appears in its place in overlay 2, while some stuff appears in the right column for both overlays. I came up with this minimal example, which doesn't work, giving me an error

Missing } inserted.
<inserted text>
}

If I comment out the \only<1>{}, it works, but not with it and I can't figure it out. Any help would be much appreciated.

\documentclass{beamer}
\usecolortheme{whale}

\usepackage{listings}

\begin{document}
\begin{frame}[t,fragile]
\frametitle{Test}

\begin{columns}
\begin{column}{0.4\textwidth}
\only<1>{
\vspace{1cm}
\begin{lstlisting}[language=SQL,mathescape, showstringspaces=false]
SELECT ENAME
FROM PROJ, ASG, EMP
\end{lstlisting}
}

\only<2>{
There is another line of text that I want to appear in left column in the \alert{second} overlay.
}
\end{column}

\begin{column}{0.4\textwidth}
\begin{center}

Some stuff that I want to appear on the right column on both overlays.

\end{center}
\end{column}

\end{columns}

\end{frame}
\end{document}

• I should have also added that if I take out the listing and put in a regular text or something, it works. It seems to be a combination of overlays and listing (and perhaps even columns). – ozsu Jul 2 '15 at 13:40

Here a solution from this answer

\documentclass{beamer}
\usecolortheme{whale}
\usepackage{listings}

\newsavebox{\mysavebox}

\begin{document}

\begin{lrbox}{\mysavebox}
\begin{lstlisting}[language=SQL,mathescape, showstringspaces=false]
SELECT ENAME
FROM PROJ, ASG, EMP
\end{lstlisting}
\end{lrbox}

\begin{frame}[t,fragile]
\frametitle{Test}

\begin{columns}
\begin{column}{0.4\textwidth}

\only<1>{
\vspace{1cm}
\usebox{\mysavebox}
}
\only<2>{
There is another line of text that I want to appear in left column in the \alert{second} overlay.
}
\end{column}

\begin{column}{0.4\textwidth}
\begin{center}

Some stuff that I want to appear on the right column on both overlays.

\end{center}
\end{column}

\end{columns}

\end{frame}
\end{document}

• Thank you. So, the solution is to make use of \lrbox construct. I'll try that. – ozsu Jul 2 '15 at 17:34


text:
# What can you say about an event outside your Light-Cone? [closed]

I'm taking a Special Relativity course and lately I've been wondering about this thing for some time. Imagine I'm at the origin of my spacetime lattice (hope I can say this), and I've synchronized all my clocks so that when light passes they start running synchronized.

At a time T2, light has reached some clocks and they start ticking.
Let's suppose that at this time there's an event far away where the light has not reached the clocks in space. This the same situation of an event occurring at time T2 outside your light cone:

2. How much do I have to wait to see it?
3. Do we say this event is too far in space, time or spacetime?

Please address any misconception (if there is) in comment, I will be glad to know them and edit my question.

## closed as unclear what you're asking by WillO, ZeroTheHero, sammy gerbil, Jon Custer, Michael SeifertJun 9 '17 at 14:04

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• (2): Are you asking, when you see the clocks starting to tick? In $2*T_2-T_1$. You can see them. But you will see them only after the light from the clock arrived to you. If you ask, when you will see A to happen, it is independent from your other clocks. It will happen in $T_1 + \frac{dist(A)}{c}$. – peterh Jun 8 '17 at 18:59
• Well you could make any assumption about any event outside your light cone, but it doesn't mean it's going to happen or is correct ;) – Kyle Kanos Jun 9 '17 at 2:21


text:
InScida workshop organised at AIMS, Muizenburg

My current research is focused on building emulators for weak lensing. Traditional Markov Chain Monte Carlo (MCMC) methods require many forward simulations, which are generally expensive to compute, in order to obtain reliable posterior densities of not only cosmological parameters but also systematic parameters. While neural networks have been explored in the earlier days, my main focus has been on Gaussian Processes, which allow for the full propagation of the uncertainty throughout the likelihood analysis. In addition, I have also been working on the MOPED (Heavens et al. 2000) compression algorithm to show that we can use the Gaussian Process emulator combined to recover the full posterior distribution of cosmological and nuisance parameters.

On a separate note, I am also interested in Machine Learning and Deep Learning (from a probabilistic perspective). I believe uncertainty quantification is important and will be a central focus in the next decade of research in Machine Learning. Moreover, while 'curve-fitting' techniques have been very successful over the past two decades, there is still the requirement to understand and interpret these algorithms.

# Publications

Below is an updated list of publications in which I was involved:

1. Fast Approximate Model for the 3D Matter Power Spectrum
Arrykrishna Mootoovaloo, Andrew Jaffe, Alan Heavens, Florent Leclercq
Accepted for publication in NeurIPS 2021

2. Many Bayesian inference problems in cosmology involve complex models. Despite the fact that these models have been meticulously designed, they can lead to intractable likelihood and each forward simulation itself can be computationally expensive, thus making the inverse problem of learning the model parameters a challenging task. In this paper, we develop an approximate model for the 3D matter power spectrum, $P_{\delta}(k,z)$, which is a central quantity in a weak lensing analysis. An important output of this approximate model, often referred to as surrogate model or emulator, are the first and second derivatives with respect to the input cosmological parameters. Without the emulator, the calculation of the derivatives requires multiple calls of the simulator, that is, the accurate Boltzmann solver, CLASS. We illustrate the application of the emulator in the calculation of different weak lensing and intrinsic alignment power spectra and we also demonstrate its performance on a toy simulated weak lensing dataset.

3. Kernel-Based Emulator for the 3D Matter Power Spectrum from CLASS
Arrykrishna Mootoovaloo, Andrew Jaffe, Alan Heavens, Florent Leclercq
Astronomy and Computing, arXiv:2105.02256, Code, Documentation

4. The 3D matter power spectrum, $P_{\delta}(k,z)$ is a fundamental quantity in the analysis of cosmological data such as large-scale structure, 21cm observations, and weak lensing. Existing computer models (Boltzmann codes) such as CLASS can provide it at the expense of immoderate computational cost. In this paper, we propose a fast Bayesian method to generate the 3D matter power spectrum, for a given set of wavenumbers, $k$ and redshifts, $z$. Our code allows one to calculate the following quantities: the linear matter power spectrum at a given redshift (the default is set to 0); the non-linear 3D matter power spectrum with/without baryon feedback; the weak lensing power spectrum. The gradient of the 3D matter power spectrum with respect to the input cosmological parameters is also returned and this is useful for Hamiltonian Monte Carlo samplers. The derivatives are also useful for Fisher matrix calculations. In our application, the emulator is accurate when evaluated at a set of cosmological parameters, drawn from the prior, with the fractional uncertainty, $\Delta P_{\delta}/P_{\delta}$ centered on 0. It is also $\sim 300$ times faster compared to CLASS, hence making the emulator amenable to sampling cosmological and nuisance parameters in a Monte Carlo routine. In addition, once the 3D matter power spectrum is calculated, it can be used with a specific redshift distribution, $n(z)$ to calculate the weak lensing and intrinsic alignment power spectra, which can then be used to derive constraints on cosmological parameters in a weak lensing data analysis problem. The software ($\texttt{emuPK}$) can be trained with any set of points and is distributed on Github, and comes with a pre-trained set of Gaussian Process (GP) models, based on 1000 Latin Hypercube (LH) samples, which follow roughly the current priors for current weak lensing analyses.

5. Data Augmentation in a Hierarchical-Based Classification scheme for Variable Stars
Zafiirah Hosenie, Robert Lyon, Benjamin Stappers, Arrykrishna Mootoovaloo, Vanessa McBrides
Accepted for publication in NeurIPS 2020

6. The accurate automated classification of variable stars into their respective sub-types is difficult. Machine learning based solutions often fall foul of the imbalanced learning problem, which causes poor generalisation performance in practice, especially on rare variable star sub-types. We attempted to overcome such deficiencies via the development of a hierarchical machine learning classifier. This ‘algorithm-level’ approach to tackling imbalance, yielded promising results on Catalina Real-Time Survey (CRTS) data. We attempt to further improve hierarchical classification performance by applying ‘data-level’ approaches to directly augment the training data so that they better describe under-represented classes. We apply and report results for three data augmentation methods in particular: Randomly Augmented Sampled Light curves from magnitude Error (RASLE), augmenting light curves with Gaussian Process modelling (GpFit) and the Synthetic Minority Over-sampling Technique (SMOTE). When combining the ‘algorithm-level’ (i.e. the hierarchical scheme) together with the ‘data-level’ approach, we further improve variable star classification accuracy by 1-4%. We found that a higher classification rate is obtained when using GpFit in the hierarchical model.

7. Parameter Inference for Weak Lensing using Gaussian Processes and MOPED
Arrykrishna Mootoovaloo, Alan Heavens, Andrew Jaffe, Florent Leclercq
MNRAS, arXiv:2005.06551, Code

8. In this paper, we propose a Gaussian Process (GP) emulator for the calculation of a) tomographic weak lensing band-power spectra, and b) coefficients of summary data massively compressed with the MOPED algorithm. In the former case cosmological parameter inference is accelerated by a factor of $\sim 10$-$30$ compared to explicit calls to the Boltzmann solver CLASS when applied to KiDS-450 weak lensing data. Much larger gains will come with future data, where with MOPED compression, the speed up can be up to a factor of $\sim 10^3$ when the common Limber approximation is used. Furthermore, the GP opens up the possibility of dropping the Limber approximation, without which the theoretical calculations may be unfeasibly slow. A potential advantage of GPs is that an error on the emulated function can be computed and this uncertainty incorporated into the likelihood. If speed is of the essence, then the mean of the Gaussian Process can be used and the uncertainty ignored. We compute the Kullback-Leibler divergence between the emulator likelihood and the CLASS likelihood, and on the basis of this and from analysing the uncertainties on the parameters, we find that in this case, the inclusion of the GP uncertainty does not justify the extra computational expense in the test application. For future weak lensing surveys such as Euclid and Legacy Survey of Space and Time (LSST), the number of summary statistics will be large, up to $\sim 10^{4}$. The speed of MOPED is determined by the number of parameters, not the number of summary data, so the gains are very large. In the non-Limber case, the speed-up can be a factor of $\sim 10^5$, provided that a fast way to compute the theoretical MOPED coefficients is available. The GP presented here provides such a fast mechanism and enables MOPED to be employed.

9. Gaussian Processes and MOPED Compression for Weak Lensing
Arrykrishna Mootoovaloo, Alan Heavens, Andrew Jaffe, Florent Leclercq
ICLR, Video, Paper, Code

10. In this paper, we propose a Gaussian Process (GP) emulator for tomographic weak lensing band power spectra, which speeds up cosmological parameter inference by a factor of $\sim$ 12 − 30 when applied to the KiDS-450 data. The emulator is built upon 8 parameters (6 cosmological and 2 systematic) using 1000 Latin Hypercube training samples. Since the prediction from the emulator is Gaussian distributed, we can also include the theoretical uncertainty in the likelihood function using the predictive uncertainty to marginalise over the latent function. Including the GP uncertainty is more demanding since it requires an $\mathcal{O}(N^{2})$ operation, for train each band power at every step in the MCMC compared to $\mathcal{O}(N_{\textrm{train}})$ for the mean alone, where $N_{\textrm{train}}$ is the number of training points in the emulator. If we use the exact solver, that is CLASS, each likelihood computation takes ∼ 0.65 seconds while each likelihood evaluation with the mean and uncertainty of the GP is $\sim$ 0.03 and $\sim$ 0.09 seconds respectively. The algorithm can be sped further when using MOPED compression algorithm which eliminates the $\mathcal{O}(N^{3})$ cost in the likelihood evaluation, where $N$ is the number of data points. Finally, with the emulator, it is realistic to drop the Limber approximation and train on the very slow but accurate full integrals.

11. Imbalance Learning for Variable Star Classification
Zafiirah Hosenie, Robert Lyon, Benjamin Stappers, Arrykrishna Mootoovaloo, Vanessa McBride
MNRAS, arXiv:2002.12386, Code

12. The accurate automated classification of variable stars into their respective sub-types is difficult. Machine learning based solutions often fall foul of the imbalanced learning problem, which causes poor generalisation performance in practice, especially on rare variable star sub-types. In previous work, we attempted to overcome such deficiencies via the development of a hierarchical machine learning classifier. This 'algorithm-level' approach to tackling imbalance, yielded promising results on Catalina Real-Time Survey (CRTS) data, outperforming the binary and multi-class classification schemes previously applied in this area. In this work, we attempt to further improve hierarchical classification performance by applying 'data-level' approaches to directly augment the training data so that they better describe under-represented classes. We apply and report results for three data augmentation methods in particular: Randomly Augmented Sampled Light curves from magnitude Error (𝚁𝙰𝚂𝙻𝙴), augmenting light curves with Gaussian Process modelling (𝙶𝚙𝙵𝚒𝚝) and the Synthetic Minority Over-sampling Technique (𝚂𝙼𝙾𝚃𝙴). When combining the 'algorithm-level' (i.e. the hierarchical scheme) together with the 'data-level' approach, we further improve variable star classification accuracy by 1-4%. We found that a higher classification rate is obtained when using 𝙶𝚙𝙵𝚒𝚝 in the hierarchical model. Further improvement of the metric scores requires a better standard set of correctly identified variable stars and, perhaps enhanced features are needed.

13. Comparing Multi-class, Binary and Hierarchical Machine Learning Classification schemes for variable stars
Zafiirah Hosenie, Robert Lyon, Benjamin Stappers, Arrykrishna Mootoovaloo
MNRAS, arXiv:1907.08189

14. Upcoming synoptic surveys are set to generate an unprecedented amount of data. This requires an automatic framework that can quickly and efficiently provide classification labels for several new object classification challenges. Using data describing 11 types of variable stars from the Catalina Real-Time Transient Surveys (CRTS), we illustrate how to capture the most important information from computed features and describe detailed methods of how to robustly use Information Theory for feature selection and evaluation. We apply three Machine Learning (ML) algorithms and demonstrate how to optimize these classifiers via cross-validation techniques. For the CRTS dataset, we find that the Random Forest (RF) classifier performs best in terms of balanced-accuracy and geometric means. We demonstrate substantially improved classification results by converting the multi-class problem into a binary classification task, achieving a balanced-accuracy rate of $\sim$99 per cent for the classification of $\delta$-Scuti and Anomalous Cepheids (ACEP). Additionally, we describe how classification performance can be improved via converting a 'flat-multi-class' problem into a hierarchical taxonomy. We develop a new hierarchical structure and propose a new set of classification features, enabling the accurate identification of subtypes of cepheids, RR Lyrae and eclipsing binary stars in CRTS data.

15. Marginal Likelihoods from Monte Carlo Markov Chains
Alan Heavens, Yabebal Fantaye, Arrykrishna Mootoovaloo, Hans Eggers, Zafiirah Hosenie, Steve Kroon, Elena Sellentin
arXiv:1704.03472, Code

16. In this paper, we present a method for computing the marginal likelihood, also known as the model likelihood or Bayesian evidence, from Markov Chain Monte Carlo (MCMC), or other sampled posterior distributions. In order to do this, one needs to be able to estimate the density of points in parameter space, and this can be challenging in high numbers of dimensions. Here we present a Bayesian analysis, where we obtain the posterior for the marginal likelihood, using kth nearest-neighbour distances in parameter space, using the Mahalanobis distance metric, under the assumption that the points in the chain (thinned if required) are independent. We generalise the algorithm to apply to importance-sampled chains, where each point is assigned a weight. We illustrate this with an idealised posterior of known form with an analytic marginal likelihood, and show that for chains of length $∼10^{5}$ points, the technique is effective for parameter spaces with up to ∼20 dimensions. We also argue that $k=1$ is the optimal choice, and discuss failure modes for the algorithm. In a companion paper (Heavens et al. 2017) we apply the technique to the main MCMC chains from the 2015 Planck analysis of cosmic background radiation data, to infer that quantitatively the simplest 6-parameter flat $\Lambda$CDM standard model of cosmology is preferred over all extensions considered.

17. No evidence for extensions to the standard cosmological model
Alan Heavens, Yabebal Fantaye, Elena Sellentin, Hans Eggers, Zafiirah Hosenie, Steve Kroon, Arrykrishna Mootoovaloo
PRL, arXiv:1704.03467, Code

18. We compute the Bayesian Evidence for models considered in the main analysis of Planck cosmic microwave background data. By utilising carefully-defined nearest-neighbour distances in parameter space, we reuse the Monte Carlo Markov Chains already produced for parameter inference to compute Bayes factors $B$ for many different model-dataset combinations. Standard 6-parameter flat $\Lambda$CDM model is favoured over all other models considered, with curvature being mildly favoured only when CMB lensing is not included. Many alternative models are strongly disfavoured by the data, including primordial correlated isocurvature models (ln$B$=−7.8), non-zero scalar-to-tensor ratio (ln$B$=−4.3), running of the spectral index (ln$B$=−4.7), curvature (ln$B$=−3.6), non-standard numbers of neutrinos (ln$B$=−3.1), non-standard neutrino masses (ln$B$=−3.2), non-standard lensing potential (ln$B$=−4.6), evolving dark energy (ln$B$=−3.2), sterile neutrinos (ln$B$=−6.9), and extra sterile neutrinos with a non-zero scalar-to-tensor ratio (ln$B$=−10.8). Other models are less strongly disfavoured with respect to flat $\Lambda$CDM. As with all analyses based on Bayesian Evidence, the final numbers depend on the widths of the parameter priors. We adopt the priors used in the Planck analysis, while performing a prior sensitivity analysis. Our quantitative conclusion is that extensions beyond the standard cosmological model are disfavoured by Planck data. Only when newer Hubble constant measurements are included does ΛCDM become disfavoured, and only mildly, compared with a dynamical dark energy model (ln$B\sim$+2).

19. Bayes Factors via Savage-Dickey Supermodels
Arrykrishna Mootoovaloo, Bruce A. Bassett, Martin Kunz
arXiv:1609.02186

20. We outline a new method to compute the Bayes Factor for model selection which bypasses the Bayesian Evidence. Our method combines multiple models into a single, nested, Supermodel using one or more hyperparameters. Since the models are now nested the Bayes Factors between the models can be efficiently computed using the Savage-Dickey Density Ratio (SDDR). In this way model selection becomes a problem of parameter estimation. We consider two ways of constructing the supermodel in detail: one based on combined models, and a second based on combined likelihoods. We report on these two approaches for a Gaussian linear model for which the Bayesian evidence can be calculated analytically and a toy nonlinear problem. Unlike the combined model approach, where a standard Monte Carlo Markov Chain (MCMC) struggles, the combined-likelihood approach fares much better in providing a reliable estimate of the log-Bayes Factor. This scheme potentially opens the way to computationally efficient ways to compute Bayes Factors in high dimensions that exploit the good scaling properties of MCMC, as compared to methods such as nested sampling that fail for high dimensions.


text:
# zbMATH — the first resource for mathematics

On derivations in near-rings. II. (English) Zbl 0911.16026
Saad, Gerhard et al., Nearrings, nearfields and $$K$$-loops. Proceedings of the conference on nearrings and nearfields, Hamburg, Germany, July 30–August 6, 1995. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 426, 191-197 (1997).
[For part I cf. the author and G. Mason, Near-rings and near-fields, Proc. Conf., Tübingen 1985, North-Holland Math. Stud. 137, 31-35 (1987; Zbl 0619.16024).]
A near-ring $$N$$ with a derivation and such that $$xNy=\{0\}$$ implies that $$x=0$$ or $$y=0$$ and satisfying certain conditions on multiplicative ideals is a commutative ring.
For the entire collection see [Zbl 0874.00036].

##### MSC:
 16Y30 Near-rings 16W25 Derivations, actions of Lie algebras
##### Keywords:
derivations; near-rings; commutativity theorems