Public Articles

Pascal's triangle

Pascal’s triangle is filled with patterns that can solve
many mathematical problems. One of the neat things that the triangle can do is help with
binomial expressions. First let’s take a look at binomial expressions. Binomial expressions relate to the sum or difference of two
terms such as:

\(x+2\)

\(3x+5\)

\(c-d\)

2 things you need to do so your resolutions make you feel like a queen

2016 has just started. We made New Years' resolutions. Some of us have sticked to your plans for now. And this makes us feel good.

Maybe you have not made any comittments because you think this is a stupid habit. Maybe you do not take these words as serious commitments?

Or maybe you have made plans but you are already desperate because your goals are so huge and far away.

Or you were lazy during the first weeks of January and now it feels like you can never make your resolutions a habit.

Stirling Numbers

Stirling numbers of the second kind in discrete mathematics
are used to show numerous combinatoric properties like partitioning a set, (a
number of ways to write an integer of a set), and forming a recurrence relation.

Blog Post 2

1 $\underline{\text{Pascal's Triangle}}$ In discrete mathematics, Pascal’s triangle is the arrangement of binomial coefficients in such a way that a triangle is formed. Although such a pattern was studied centuries before his time, we refer to Pascal’s triangle in relation to Blaise Pascal, a French mathematician. The triangle was originally developed by the ancient Chinese, but Pascal was the first person to discover the importance of all of the patterns that occur within it. His work allegedly stemmed from the popularity of gambling. After considering a question asked to him about gambling with dice, Pascal’s *Arithmetical Triangle* resulted.

The triangle is created by starting at the top, row 0, with the number 1. Each row afterwards begins and ends with 1 and the pattern follows that as you move through the row, you add the number above and to the left with the number above and to the right for any given position. A portion of Pascal’s triangle is shown below.

Blog Post 1

1 $\underline{\text{Binomial Coefficient}}$ In mathematics, the binomial coefficient is written as ${n \choose k}$ and can be pronounced as “*n* choose *k*.” Alternatively, binomial coefficients are also sometimes given the notation *C*(*n*, *k*). In this case, the *C* stands for the word “choices” or “combination” (Benjamin, 2009, p. 8). This is because there are ${n \choose k}$ ways of choosing *k* elements from a set containing a number of *n* elements. For example, we can consider the set *A* = {1, 2, 3, 4}. If we wish to know how many subsets of 2 can be created using this set, we are essentially asking how many ways there are of choosing 2 elements from a set with 4 total elements. Therefore, we can identify that *k* = 2 and *n* = 4. Hence, we have ${4 \choose 2}$. To calculate such a problem, we typically would want to write out by hand all the possible combinations. Doing so, one would find that there are six pairs of size two subsets, namely {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, and {3, 4}. However, it becomes clear to see that when we are dealing with large sets of values, this work can become tedious. Therefore, it is convenient to utilize the following formula:

\[{n \choose k} = \frac{n!}{(n-k)! \cdot k!} \]

A Laurent Series Tranform for Integer Sequences via Generalised Continued Fractions

We investigate a function defined by a generalised continued fraction and find it to be closely related to a generating function of series OEIS:A000698. We then alter the function to contain the set of primes rather than the set of real numbers. This continued fraction function also has a similar form with coefficients 2, 6, 48, 594, 10212, 230796, ⋯. We also transform the sequence 1, 1, 1, 1, 1⋯ and gain the Catalan numbers as coefficients of the corresponding Laurent Series. We then provide examples which transform into various OEIS sequences.

Define the function \begin{equation}
f(x)=\frac{1}{x+\frac{2}{x+\frac{3}{x+\cdots}}} = \underset{k=1}{\overset{\infty}{\mathrm \large K \normalsize}} \frac{k}{x}
\end{equation} evaluating this function until convergence for 16 decimal places, gives something that looks like 1/*x*, however, after diagnosing the coefficients of the Laurent series, by subtracting likely integer terms we very easily find \begin{equation}
\lim_{x\to \infty}f(x)=\frac{1}{x}-\frac{2}{x^3}+\frac{10}{x^5}-\frac{74}{x^7}+\frac{706}{x^9}-\frac{8162}{x^{11}}+\frac{110410}{x^{13}}-\cdots,
\end{equation} with a search on OEIS giving sequence A000698 which we believe to be plausible. We can easily imagine a similar function \begin{equation}
g(x)=\frac{2}{x+\frac{3}{x+\frac{5}{x+\frac{7}{x+\cdots}}}} = \underset{k=1}{\overset{\infty}{\mathrm \large K \normalsize}} \frac{p_k}{x}
\end{equation} where the top row of numbers are the prime numbers, *p*_{k}. Using the first 9999 primes in the continued fraction, this also converges to 16 decimal places for large enough *x*, and appears to be described by an integral coefficient Laurent series \begin{equation}
g(x)=\frac{2}{x}-\frac{6}{x^3}+\frac{48}{x^5}-\frac{594}{x^7}+\frac{10212}{x^9}-\frac{230796}{x^{11}}+\frac{6569268}{x^{13}}-\cdots
\end{equation}

That would allow conjecture for a very interesting relationship \begin{equation}
\underset{i=1}{\overset{\infty}{\mathrm \large K \normalsize}} \frac{p_i}{x} = \frac{2}{x} - \frac{6}{x^3} + \frac{48}{x^5} -\frac{594}{x^7} + \frac{10520}{x^9} -\cdots
\end{equation} where *p*_{1} = 2,*p*_{2} = 3 and so one for primes, and the capital *K* notation is that for the continued fraction.

A further assessment of many integer sequences inside the continued fraction should be made, and a check of corresponding Laurent series undertaken. It may be common place to find integral coefficients.

The first function investigated in this document is also of interest, in OEIS the Laurent coefficient sequence A000698 is commented “Number of nonisomorphic unlabeled connected Feynman diagrams of order 2n-2 for the electron propagator of quantum electrodynamics (QED), including vanishing diagrams.” The continued fraction clearly has the potential to capture the recursive aspect of the theory, and this may be why the series align.

Many thanks to OEIS for providing their invaluable service. Thanks to Wolfram|Alpha for plots as below, and Mathematica for calculations. This work was undertaken in my spare time while being funded by the EPSRC.

Pascal's Triangle

Pascal's Triangle was made famous by a French mathematician named Blaise Pascal. He did not necessarily invent this triangle of numbers, but he did contribute a lot when it comes to its uses.
In fact, Indian, Iranian, and Chinese mathematicians discovered this well before Pascal was putting it to use.
Ok, before going further let's explain and show how to create Pascal's Triangle.

Ozone Paper Moved Offline

and 2 collaborators

We develop a quantitative method for determining Stratosphere to Troposphere Transport events (STTs) and a minimum bound for this transported ozone quantity using ozonesondes over Melbourne, Macquarie Island, and Davis.

Binomial Coefficients

Binomial coefficients in discrete mathematics are denoted \({n \choose k}\), where we say \(n\) choose \(k.\) The variable, \(n,\) is usually known as the upper index while the variable, \(k,\) is known as the lower index. Binomial coefficients are the same amount of combinations of \(k\) items that can be chosen from a set of \(n\) items where order does not matter. Therefore, \({n \choose k}\) gives the \(k\) subsets that are possible out of \(n\) total items. When looking further into binomial coefficients, we notice that the values are non-negative.

Open Science as a Service: Status and future potential from a German non-university research institution perspective

and 1 collaborator

Authorea was used to write this article for the BOBCATSSS 2016 symposium in Lyon, France (January 27-29 2016).

2016HCT Prelim.

Personal news and content curation is an exciting NLP application. Systems providing this service are often characterised by a collaborative approach that combines human and machine intelligence. As the scope of the problem increases however, so too does the importance of automation. To this end we propose a novel method for scoring news articles and other related content. It is natural to view this problem in a learning-to-rank framework. The training phase of our model first makes use of a pairwise transform. This alters the problem from the ranking of a whole corpus to many individual pairwise comparisons (is article 'a' better than article 'b'). This transformed set is then used to determine the optimal weights in a logistic regression model. These can then be used directly to classify the non-transformed test set. We also perform a comprehensive review and selection process on a large range of candidate features. Our final features involve measures of centrality, informativeness, complexity and within-group similarity.

Microbial Eukaryote Proposal

and 2 collaborators

#Project Summary (1 page)

Due: January 25, 2016 at 5 PM (local time)

DEB - Biodiversity: Discovery & Analysis Cluster

Solicitation: http://www.nsf.gov/pubs/2015/nsf15609/nsf15609.htm

Cluster description: http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=503666&org=DEB&from=home

##Overview

The broad goal with this proposal is to increase the overall knowledge of the true diversity of microbial eukaryotes by identifying and culturing microeukaryotes from seagrass beds.

Microorganisms, and specifically marine microbial eukaryotes, represent an underexplored area of diversity. Microbial eukaryotes are known to be important on a number of trophic levels in the marine system **CITE**, and microbial eukaryotes found in seagrass beds likely contribute to their tremendous biodiversity and roles as important players in nutrient cycling and carbon sequestration in the oceans. We will use a combination of sequencing and culturing techniques to (1) characterize microeukaryotes in a global census of the seagrass *Zostera marina*, (2) Explore microbial eukaryotic diversity across the Order Alismatales, including the 3 separate lineages of seagrasses and their freshwater and brackish relatives, and (3) Create a publicly available culture collection of microbial eukaryotes from *Zostera marina* samples from Bodega Bay, CA.

##Intellectual Merit

Microorganisms comprise the majority of diversity on Earth. Traditionally classified using morphological approaches, the advent of sequence data has dramatically altered our views of microbial evolution and diversity. Specifically, high throughput sequencing technologies have enabled us to explore multiple genes and genomes from microorganisms, giving us insight into genome complexity and function in these unseen organisms. As a result microbial ecologists are finding themselves in uncharted territory as they analyze large data sets full of "unclassified" organisms, and it now clear that microorganisms are much more diverse than previously thought.

Although certain pathogenic microeukaryotes have been studied in great detail (ex. *giardia*, see \cite{Adam_2001}) for review, environmental microeukaryotes, specifically marine microeukatyores, are grossly uncharacterized despite their important functional roles in their ecosystems \cite{Caron_2008}. Novel marine microeukaryotic lineages have previously been found at all phylogenetic scales \cite{Massana_2008}; however, many of these novel organisms are still a mystery to us as they have yet to be cultured. It is estimated that the total diversity of microbial eukaryotes is much higher than what we currently have in culture \cite{Mora_2011} \cite{Pawlowski_2012}.

Seagrasses are a unique system in which to explore marine microbial eukaryotic diversity. These important marine angiosperms provide habitat and food to many rare and endemic species, and contain tremendous levels of biodiversity that has currently only been characterized at the macrobe level \cite{Orth_2006}. Seagrasses are known to be important contributors to biogeochemical processes within the ocean and are one of the largest carbon sinks on earth, sequestering carbon 35X faster than Tropical Rainforests \cite{Mcleod_2011}.

Given their importance in the complex marine food web and their contributions to nutrient cycling within the oceans, we hypothesize that seagrass-associated marine microbial eukaryotes are important to both the high levels of macrobe biodiversity within seagrass beds and to their role in nutrient cycling and carbon sequestration in the ocean ecosystem.

We propose to perform a global census of microbial eukaryotes found in association with the leaves, roots, and sediment of the seagrass *Zostera marina*. We will then expand our investigation to census the microbial eukaryotes found in association with plants across the Order Alismatales, which includes three independent lineages of seagrasses. Concurrently with the afformentioned censuses, we will establish a culture collection of microbial eukaryotes found associated with *Zostera marina* from Bodega Bay, California. We are uniquely positioned to be successful at the proposed research; using funds provided by the Gordon and Betty Moore Foundation, we have already established a program to explore bacterial diversity within seagrass beds, and have completed the majority of field work and formed ongoing collaborations with other seagrass researchers from both the Zostera Experimental Network (ZEN) and other research institutions.

##Broader Impacts

The project we propose here is a global interdisciplinary collaboration that will result in increased knowledge of the biodiversity of an understudied group of organisms from an important marine ecosystem. The preposed project is the first to explore seagrass-associated microbial eukaryotes using both sequence and culture based methods, and will generate large amounts of publicly available sequence data and numerous new entries of novel marine organisms to culture collections.

The project we are proposing will include a large outreach component both at the local level (undergraduate researchers, high school students) and the global level (website, collaborators). Undergraduates and local high school students will be intimately involved in creating the culture collection and our progress will be transparently available on our lab website.

Sample Blog Post Math 381

A graph is a structure used in discrete math that is used to show the relationship between various objects. The objects form the vertices of the graph. Two vertices are connected by a line if they satisfy a certain relationship. We call these lines edges. By formalizing the way we connect the dots we are able to rigorously prove mathematical claims.

A quick introduction to version control with Git and GitHub

Many scientists write code as part of their research. Just as experiments are logged in laboratory notebooks, it is important to document the code you use for analysis. However, a few key problems can arise when iteratively developing code that make it difficult to document and track which code version was used to create each result. First, you often need to experiment with new ideas, such as adding new features to a script or increasing the speed of a slow step, but you do not want to risk breaking the currently working code. One often utilized solution is to make a copy of the script before making new edits. However, this can quickly become a problem because it clutters your filesystem with uninformative filenames, e.g. `analysis.sh`

, `analysis_02.sh`

, `analysis_03.sh`

, etc. It is difficult to remember the differences between the versions of the files, and more importantly which version you used to produce specific results, especially if you return to the code months later. Second, you will likely share your code with multiple lab mates or collaborators and they may have suggestions on how to improve it. If you email the code to multiple people, you will have to manually incorporate all the changes each of them sends.

Fortunately, software engineers have already developed software to manage these issues: version control. A version control system (VCS) allows you to track the iterative changes you make to your code. Thus you can experiment with new ideas but always have the option to revert to a specific past version of the code you used to generate particular results. Furthermore, you can record messages as you save each successive version so that you (or anyone else) reviewing the development history of the code is able to understand the rationale for the given edits. Also, it facilitates collaboration. Using a VCS, your collaborators can make and save changes to the code, and you can automatically incorporate these changes to the main code base. The collaborative aspect is enhanced with the emergence of websites that host version controlled code.

In this quick guide, we introduce you to one VCS, Git (git-scm.com), and one online hosting site, GitHub (github.com), both of which are currently popular among scientists and programmers in general. More importantly, we hope to convince you that although mastering a given VCS takes time, you can already achieve great benefits by getting started using a few simple commands. Furthermore, not only does using a VCS solve many common problems when writing code, it can also improve the scientific process. By tracking your code development with a VCS and hosting it online, you are performing science that is more transparent, reproducible, and open to collaboration \cite{23448176, 24415924}. There is no reason this framework needs to be limited only to code; a VCS is well-suited for tracking any plain-text files: manuscripts, electronic lab notebooks, protocols, etc.

ProCS15: A DFT-based chemical shift predictor for backbone and C\(\beta\) atoms in proteins

We present ProCS15: A program that computes the isotropic chemical shielding values of backbone and C*β* atoms given a protein structure in less than a second. ProCS15 is based on around 2.35 million OPBE/6-31G(d,p)//PM6 calculations on tripeptides and small structural models of hydrogen-bonding. The ProCS15-predicted chemical shielding values are compared to experimentally measured chemical shifts for Ubiquitin and the third IgG-binding domain of Protein G through linear regression and yield RMSD values below 2.2, 0.7, and 4.8 ppm for carbon, hydrogen, and nitrogen atoms respectively. These RMSD values are very similar to corresponding RMSD values computed using OPBE/6-31G(d,p) for the entire structure for each protein. The maximum RMSD values can be reduced by using NMR-derived structural ensembles of Ubiquitin. For example, for the largest ensemble the largest RMSD values are 1.7, 0.5, and 3.5 ppm for carbon, hydrogen, and nitrogen. The corresponding RMSD values predicted by several empirical chemical shift predictors range between 0.7 - 1.1, 0.2 - 0.4, and 1.8 - 2.8 ppm for carbon, hydrogen, and nitrogen atoms, respectively.

ProCS15: A DFT-based chemical shift predictor for backbone and C\(\beta\) atoms in proteins

We present ProCS15: A program that computes the isotropic chemical shielding values of backbone and C*β* atoms given a protein structure in less than a second. ProCS15 is based on around 2.35 million OPBE/6-31G(d,p)//PM6 calculations on tripeptides and small structural models of hydrogen-bonding. The ProCS15-predicted chemical shielding values are compared to experimentally measured chemical shifts for Ubiquitin and the third IgG-binding domain of Protein G through linear regression and yield RMSD values below 2.2, 0.7, and 4.8 ppm for carbon, hydrogen, and nitrogen atoms respectively. These RMSD values are very similar to corresponding RMSD values computed using OPBE/6-31G(d,p) for the entire structure for each protein. The maximum RMSD values can be reduced by using NMR-derived structural ensembles of Ubiquitin. For example, for the largest ensemble the largest RMSD values are 1.7, 0.5, and 3.5 ppm for carbon, hydrogen, and nitrogen. The corresponding RMSD values predicted by several empirical chemical shift predictors range between 0.7 - 1.1, 0.2 - 0.4, and 1.8 - 2.8 ppm for carbon, hydrogen, and nitrogen atoms, respectively.

A quick introduction to version control with Git and GitHub

Many scientists write code as part of their research. Just as experiments are logged in laboratory notebooks, it is important to document the code you use for analysis. However, a few key problems can arise when iteratively developing code that make it difficult to document and track which code version was used to create each result. First, you often need to experiment with new ideas, such as adding new features to a script or increasing the speed of a slow step, but you do not want to risk breaking the currently working code. One often utilized solution is to make a copy of the script before making new edits. However, this can quickly become a problem because it clutters your filesystem with uninformative filenames, e.g. `analysis.sh`

, `analysis_02.sh`

, `analysis_03.sh`

, etc. It is difficult to remember the differences between the versions of the files, and more importantly which version you used to produce specific results, especially if you return to the code months later. Second, you will likely share your code with multiple lab mates or collaborators and they may have suggestions on how to improve it. If you email the code to multiple people, you will have to manually incorporate all the changes each of them sends.

Fortunately, software engineers have already developed software to manage these issues: version control. A version control system (VCS) allows you to track the iterative changes you make to your code. Thus you can experiment with new ideas but always have the option to revert to a specific past version of the code you used to generate particular results. Furthermore, you can record messages as you save each successive version so that you (or anyone else) reviewing the development history of the code is able to understand the rationale for the given edits. Also, it facilitates collaboration. Using a VCS, your collaborators can make and save changes to the code, and you can automatically incorporate these changes to the main code base. The collaborative aspect is enhanced with the emergence of websites that host version controlled code.

In this quick guide, we introduce you to one VCS, Git (git-scm.com), and one online hosting site, GitHub (github.com), both of which are currently popular among scientists and programmers in general. More importantly, we hope to convince you that although mastering a given VCS takes time, you can already achieve great benefits by getting started using a few simple commands. Furthermore, not only does using a VCS solve many common problems when writing code, it can also improve the scientific process. By tracking your code development with a VCS and hosting it online, you are performing science that is more transparent, reproducible, and open to collaboration \cite{23448176, 24415924}. There is no reason this framework needs to be limited only to code; a VCS is well-suited for tracking any plain-text files: manuscripts, electronic lab notebooks, protocols, etc.

THE PREDICTION OF OUTCOMES RELATED TO THE USE OF NEW DRUGS IN THE REAL WORLD THROUGH ARTIFICIAL ADAPTIVE SYSTEMS.

THE PREDICTION OF OUTCOMES RELATED TO THE USE OF NEW DRUGS IN THE REAL WORLD THROUGH ARTIFICIAL ADAPTIVE SYSTEMS.

Enzo Grossi & Massimo Buscema

Semeion Research Centre

Research Centre of Sciences of Communication

Via Sersale 117, Rome, 00128, Italy

INTRODUCTION

In this brief essay we will focus three main problems related to the use of new medications in the real world: 1) the prediction of drug response in individual patients; 2) the prediction of rare unwanted events after introduction of the new drug on the market; 3) the passage from preclinical phase to Phase I in human beings. The first problem is specifically felt by medical doctors who are asked to treat their patients as individuals rather than as statistics, but we have to note that, with the advent of extremely costly new drugs, also health authorities or private insurance organizations are looking for potent tools to personalize treatment plans. The second problem is typically sensed by drug agencies which sometimes are forced to withdraw marketing authorization as a bunch of deaths drug related creates rumors and disappointments at media level, while the third problem is specifically felt by Pharmaceutical Companies and Institutional Review Boards releasing the clearance for first in man trials.

1.Prediction of drug response in individual patient

Making predictions for specific outcomes (diagnosis, risk assessment, prognosis) represents a fascinating aspect of medical science. Different statistical approaches have been proposed to define models to identify factors that are predictive for the outcome of interest. Studies have been performed to define the clinical and biological characteristics that could be helpful in predicting who will benefit from an antiobesity drug for example, but results have been limited (1).

Traditional statistical approaches encounter problems when the data show big variability and not easily normalized for inherent nonlinearity. More-advanced analysis techniques, such as dynamic mathematical models, can be useful because they are particularly suitable for solving nonlinear problems frequently associated with complex biological systems.

Use of ANNs in biological systems has been proposed for different purposes, including studies on deoxyribonucleic acid sequencing (2) and protein structure (3).

ANNs have been used in different clinical settings to predict the effectiveness of instrumental evaluation (echocardiography, brain single photon emission computed tomography, lung scintigram, prostate biopsy) in increasing diagnostic sensitivity and specificity and in laboratory medicine in general (4). Also, they have proven effective in identifying gastro-oesophageal refux patients on the sole basis of clinical data (5). But the most promising application of ANNs relates to prediction of possible clinical outcomes with specific therapy. ANNs have proven effective in detecting responsiveness to methadone treatments of drug addicts (6), to pharmacological treatment in Alzheimer disease (7), to clozapine in schizophrenic patients (8) and in various fields of psychiatric research (9).

The use of ANNs for predictive modelling in obesity dates back to a decade ago, where it was proposed to model the waist-hip ratio from 13 other health parameters (10). Later, it has been proposed as a tool for body composition research (11).

One of the main factors preventing a more efficient use of new pharmacological treatments for chronic diseases like for example hypertension, cancer, Alzheimer disease or obesity is represented by the difficulty of predicting “a priori” the chance of response of the single patient to a specific drug. A major methodological setback in drawing inferences and making predictions from data collected in the real world setting, such as observational studies, is that variability in the underlying biological substrates of the studied population and the quality and content of medical intervention influence outcomes. Because there is no reason to believe that these, like other health factors, work together in a linear manner, the traditional statistical methods, based on the generalized linear model, have limited value in predicting outcomes such as responsiveness to a particular drug.

Most studies have shown that up to 50% of patients treated with new molecules given in monotherapy or as an adjunct to standard treatments may show an unsatisfactory response. As a matter of fact, when time comes for the physician to decide about type of treatment, there is very little evidence that can help her/him in drug treatment choice. Take for example obesity. Here only scanty data are available on predictive factors to the specific treatment, and attempts at developing models for predicting response to the drug by using traditional techniques of multiple regression have showed an unsatisfactory predictive capacity (i.e. inferior to 80% of total variance). (12, 13). A possible explanation could be that obesity is a so-called complex disease, where different factors interact with multiple interactions among variables, positive and negative feedback loops, and non-linear system dynamics. Another good example is Alzheimer Disease.

Clinical trials have established the efficacy of cholinesterase inhibitor drugs (ChEI), such as tacrine,

(14) donepezil, (15) and rivastigmine (16) based on improvement in cognitive aspects and in overall functioning using the Alzheimer’s Disease Scale—Cognitive subscale (ADAS-Cog) and the

Clinician’s Interviewed Based Impression of Change (CIBIC) , respectively. Although the mean score of treated patients in both scales was significantly higher than the placebo group, many subjects under active treatment showed little or no improvement (nonresponders).

However it is not possible to estimate which patients are likely to respond to pharmacological therapy with ChEI. This prediction would be an important decision-making factor in improving the use of healthcare resources.

A major methodological setback in drawing inferences and making predictions from data collected in the real world setting, such as observational studies, is that variability in the underlying biological substrates of the studied population and the quality and content of medical intervention

influence outcomes. Because there is no reason, a priori, to believe that these, like other health factors, work together in a linear manner, the traditional statistical methods, based on the generalized linear model, have limited value in predicting outcomes such as responsiveness to

a particular drug.

A possible alternative approach to the solution of the problem is represented by the use of Neural Networks. Artificial Neural Networks (ANN) represent computerized algorithms resembling interactive processes of human brain. They allow to study very complex non-linear phenomena like biological systems. Like the brain, ANNs recognize patterns, manage data, and, most significantly, learn. These statistical-mathematical tools can determine the existence of a correlation between series of data and a particular outcome and when “trained” can predict output data once given the input. They work well in pattern recognition and discrimination tasks.

Although ANNs have been applied to various areas of medical research, they have not been employed in obesity clinical pharmacology.

ANN proved to be a useful method to discriminate between responders and non-responders, better than traditional statistical methods in our three experimental studies carried out with donepezil in Alzheimer disease and with sibutramine in obesity and with infliximab in Crohn disease.

In a paper published in 2002 (7) we have evaluated the accuracy of artificial neural networks compared with discriminant analysis in classifying positive and negative response to the cholinesterase inhibitor donepezil in a opportunistic group of 61 old patients of both genders affected by Alzheimer’s disease (AD) patients in real world setting along three months follow-up.

Accuracy in detecting subjects sensitive (responders) or not (nonresponders) to therapy was based on the standard FDA criterion standard for evaluation of efficacy: the scores of Alzheimer’s Disease Assessment Scale—Cognitive portion and Clinician’s Interview Based Impression of Change—plus scales. In this study ANNs were more effective in discriminating between responders and nonresponders than other advanced statistical methods, particularly linear discriminant analysis. The total accuracy in predicting the outcome was 92.59%.

In a second study we evaluated the use of artificial neural networks in predicting response to infliximab treatment in patients with Crohn's disease(18).

In this pilot study , different ANN models were applied to a data sheet with demographic and clinical data from 76 patients with steroid resistant/dependant or fistulizing CD treated with Infliximab to compare accuracy in classifying responder and non responder subjects with that of linear discriminant analysis.

Eighty one outpatients with CD (31 men, 50 women; mean age± standard deviation 39.9 ± 15 range: 12-81 ) partecipating to an Italian Multicentric Study (17) , were enrolled in the study. All patients were treated, between April 1999 and December 2003, with a dose of Infliximab 5 mg/kg of body weight for luminal refractory (CDAI > 220–400) (43 patients), fistulizing CD (19 patients) or both of them (14 patients) .

The final data sheet consisted of 45 independent variables related to the anagraphic and anamnestic data (sex, age at diagnosis, age at infusion , smoking habit, previous Crohn’s related abdominal surgery [ileal or ileo-cecal resections] and concomitant treatments including immunomodulators and corticosteroids ) and to clinical aspects ( location of disease, perianal disease, type of fistulas, extraintestinal manifestations, clinical activity at the first infusion [CDAI], indication for treatment). Smokers were defined as those smoking a minimum of 5 cigarettes per day for at least 6 months before their first dose of Infliximab. Non smokers were defined as those who had never smoked before, those who had quit smoking at least 6 months before their first dose of Infliximab, or those who smoked fewer than 5 cigarettes per day. Concomitant immunosuppressive use was defined as initiation of methotrexate before their first Infliximab infusion or initiation of 6-mercaptopurine (6-MP) or azathioprine more than 3 months before their first Infliximab infusion.

Assessment of response was determined by clinical evaluation 12 weeks after the first infusion for all patients. Determination of response in patients with inflammatory CD was based on the Crohn’s Disease Activity Index (CDAI). For clear - cut estimate clinical response was evaluated as complete response or partial / no response.

Complete response was defined as (a) clinical remission (CDAI < 150) in luminal refractory disease and (b) temporary closure of all draining fistulas at consecutive visits in the case of enterocutaneous and perianal fistulas; entero-enteric fistulas were evaluated by small bowel barium enema and vaginal vescical fistula by lack of drainage at consecutive visits . For patients with both indications the outcome was evaluated independently for each indication.

Two different experiments were planned following an identical research protocol. The first one included all 45 independent variables including frequency and intensity Crohn disease symptoms, plus numerous other social and demographic characteristics, clinical features and history. In the second experiment the IS system coupled to the T&T system automatically selected the most relevant variables and therefore 22 variables were included in the model.

Discriminant analysis was also performed on the same data sets to evaluate the predictive performance of this advanced statistical method by a statistician blinded to ANN results. Different models were assessed to optimise the predictive ability. In each experiment the sample was randomly divided into two sub-samples, one for the training phase and the other for the testing phase, with the same record distributions used for ANN validation.

ANNs reached an overall accuracy rate of 88% while LDA performance was only of 72%.

Finally in a third study we evaluated the performance of ANN in predicting response to Warfarin(17).

A total of 377 patients were included in the analysis. The most frequent clinical indication for anticoagulation was atrial fibrillation (69%); other indications included heart valve prosthesis (10%) and pulmonary embolism (8%). The large majority of patients, 325, 86%) were on concurrent drug treatment: on average, they were taking 3 (IQR 1-4) medications potentially interacting with warfarin. The median weekly maintenance dose (WMD) of warfarin was 22.5 mg (IQR 16.3-28.8mg). Thirteen patients whose INR values were not within thetherapeutic range were erroneously included in the analysis: their median weekly maintenance dose was 21.4 mg (IQR 12.2-30.0 mg), the INR was higher than 3.0 (INR 3.7 and 4.3) in 2, and lower than 2.0 in 11 (median INR 1.5, IQR 1.5-1.7).

Demographic, clinical and genetic data (CYP2C9 and VKORC1 polymorphisms) were used. The final prediction model was based on 23 variables selected by TWIST® system within a bipartite division of the data set (training and testing) protocol.

TWIST system is based on a population of n ANNs, managed by an evolutionary system able to extracts from the global dataset the best training and testing sets and to evaluate the relevance of the different variables of the dataset in a sophisticated way, slecting the most relevant for the problem on study.

The ANN algorithm reached high accuracy, with an average absolute error of 5.7 mg of the warfarin maintenance dose. In the subset of patients requiring ≤21 mg and 21-49 mg (45 and 51% of the cohort, respectively) the absolute error was 3.86 mg and 5.45 with a high percentage of subjects being correctly identified (71 and 73%, respectively). This performance is higher than those obtained in different studies carried out with traditional statistical techniques. In conclusion ANN appears to be a promising tool for vitamin K antagonist maintenance dose prediction.

1.1 Selection of informative variables: how evolutionary algorithms work

To include only the most informative of the available variables we used a genetic algorithm, called the Genetic Doping Algorithm [19], which uses the principles of evolution to optimize the training and testing sets and to select the minimum number of variables capturing the maximum amount of available information in the data. Contrary to statistical linear models using indicator variables, TWIST does not require the omission of a reference category. This is due to the focus of the artificial neural network on prediction rather than estimation. If some of the indicator variables can completely account for the predictive ability of the others, those will be excluded by the algorithm during the selection process. The method is called the TWIST protocol and has been previously applied successfully in similar problems [20,21]. The advantages of the approach are the sub-setting of the data in two representative sets for training and testing, which is problematic in small datasets, and the use of a combination of criteria to determine the fit of the model. TWIST is comprised of two systems, the T&T for resampling of the data and the IS for feature selection, both using artificial neural networks (ANNs). The T&T system splits the data into training and testing sets in such a way that each subset is statistically representative of the full sample. This non-random selection of subsets is crucial when small samples are considered and the selection of non-characteristic and extreme subsets is likely. The IS system uses the training and testing subsets produced to identify a vector of 0s and 1s, describing the absence or presence of an indicator variable, that is able to optimize the categorization of the individuals in cases and controls compared to their observed status. For this, a population of vectors, with each vector a combination of the indicator variables, is allowed to “evolve” through a number of generations in order optimize the prediction of target variable, as a natural population evolves to optimize fitness under a specific set of environmental conditions. The vectors with the best predictive ability are overrepresented in the next generation while a smaller number of sub-optimal vectors are maintained to give rise to the following generation. Some instability, in the form of low predictive ability vectors, is introduced in the process to avoid the problem of finding a solution which is optimal under a narrow set of conditions, also known as a local optimum. This step ensures that the attributes do not include redundant information or noise variables that will decrease the accuracy of the map and increase both the computing time and the amount of examples necessary during learning. In addition, feature selection permits the easier interpretation of the graph of relationships between the variables.

2. Prediction of rare unwanted events

Drug-induced injuries are a growing concern for health authorities. The world population is continuously growing older because of an increased life expectancy and is thus using more and more drugs, whether prescription or over-the-counter drugs.

Therefore, chances of drug-induced injuries are rising. Over the years, a number of postmarketing labelling changes or drug withdrawals from the market due to postmarketing discoveries have occurred. Even the best planned and carefully designed clinical studies have limitations. To detect all potential adverse drug reactions, you need quite a large number of subjects exposed to the drug and the number of subjects participating in the clinical studies might not be large enough to detect especially rare adverse drug reactions.

To minimise the risk of postmarketing discoveries such as unrecognised adverse drug reactions, certain risk factors, e.g. laboratory or ECG abnormalities, are subject of increased regulatory review.

The most frequent cause of safety-related withdrawal of medications (e.g. bromfenac, troglitazone) from the market and for FDA non-approval is the drug-induced liver injury (DILI). Different degrees of liver enzyme elevations after drug intake can result in hepatotoxicity, which can be fatal due to the irreversible damage to the liver. Since animal models cannot always predict human toxicity, drug-induced hepatotoxicity is often detected after market approval. In the United States, DILI is contributing to more than 50% of acute liver failure cases (data from WM Lee and colleagues from the Acute Liver Failure Study Group).

The second leading cause for withdrawing approved drugs from the market is QT interval prolongation, which can be measured during electrocardiogram (ECG). Some non-cardiovascular drugs (e.g. terfenadine) have the potential to delay cardiac repolarisation and to induce potentially fatal ventricular tachyarrhythmias such as Torsades de Pointes.

Drug toxicity is also a common cause of acute or chronic kidney injury and can be minimised or prevented by vigilance and early treatment. NSAIDs, aminoglycosides, and calcineurin inhibitors are for example some drugs that are known to induce kidney dysfunction. Most events are reversible, with kidney function returning to normal when the drug is discontinued.

Consequently, the pharmaceutical industry has a strong interest to identify drugs bearing the risk of causing adverse drug reactions as early as possible in order to improve the drug development programme.

A patient developing a severe side effect to a particular medication can be considered an outlier. Suppose that you are deriving probabilities of future occurrences of severe side effects from the data collected in large clinical trials carried out before the commercialization of your product. These trials provide healthy authorities that your product is effective and safe and so that its deserves a registration or marketing authorization.

Now, say that you estimate that an event happens every 1,000 patients treated. You will need a lot more data than 1,000 patients to ascertain its frequency, say 3,000 patients. Now, what if the event happens once every 5,000 patients? The estimation of this probability requires some larger number, 15,000 or more. The smaller the probability, the more observations you need, and the greater the estimation error for a set number of observations. Therefore, to estimate a rare event you need a sample that is larger and larger in inverse proportion to the occurrence of the event.

If small probability events carry large impacts ( in this example death), and (at the same time) these small probability events are more difficult to compute from past data itself, then our empirical knowledge about the potential contribution—or role—of rare events (probability × consequence) is inversely proportional to their impact. The future challenge in this particular setting will be to derive from the limited amount of information available in the pre registration phase of drug development subtle, weak but true signals that something will go bad in the future after the marketing approval, when the new drug will be exposed in the real world to hundreds of thousand of subjects, a twofold increase in order of magnitude in comparison with pre-registration experience. These patients of the real world will be very different from patients encountered in the phase 3 clinical trial, generally speaking “clean patients” i.e. no concomitant disease, few concomitant treatments, age not beyond a certain value, good compliance, an so on. On the other hand in the post marketing phase the new drug will be exposed to “dirty patients” i.e subjects with substantial co-morbidity, many concomitant treatments, extreme age, poor compliance ( which mean also taking by mistake or intentionally excess of drug in the attempt to compensate for missed assumptions of the drug). Some artificial adaptive systems based on a new mathematics, could be able to learn from a large phase 3 study the hidden links among rare events and a particular profile of a patients even if no patients with such a particular profile actually exists in the data set.

There are basically two possibilities: the first is to use an “associative memory” or autoassociative artificial neural network able to navigate in the hypersurface of a dataset in search of rare occurrence linked to a particular assembly of variables; the second is to use a pseudo-inverse function coupled with an evolutionary algorithms able to repopulate a specific probability density function distribution with virtual records enabling the search of rare events, not available in the original data set.

2.1 Autoassociative artificial neural networks.

The NR-NN is a new recurrent network provided with a new powerful algorithm (“Re-Entry” by the Semeion Research Centre), that can dynamically adapt its trajectory to answer according to the different questions, during the recall phase.

This new artificial neural network, developing an associative memory, can identify the best possible connection between variables and can generate alternative data scenarios to follow the dynamic effects. During the training phase, the algorithm optimizes the weight of all the possible interconnections between variables in order to minimize the error. The training phase is followed by a rigorous validation protocol which foresees the correct reconstruction of variables that are randomly deleted by each record.

During the querying phase of the database, the NR-NN can answer the following questions:

• prototypical question (the characteristic prototype of a patient with a particular side effect or without a particular side effect),

• virtual question (the prototypical profile of a patient having a side effect with specific characteristics, even if no subject with these variables is actually present in the data set),

These special dynamics of NR-NN allow us to distinguish 3 types of variables:

• Discriminant variables: variables that are "switched on" only for a specific prototype;

• Indifferent variables: variables that are "switched off" for both prototypes;

• Metastable variables: variables that are "switched on" for both prototypes; in other words they act in opposite ways according to the context of the other variables. Metastable variables are specific of non-linear systems.

The possibility to simulate rather than carrying out in reality a post marketing surveillance study in uncertain situations could help in optimizing decisions and save lives if the drug could cause severe side effects in rare patients.

Actually there are very few known computer aided simulators for clinical trials, for example the simulation method and the simulator proposed by Pharsight Corporation, Mountain View, California, USA. The basic features of the said known simulator is disclosed in “ Case Study in the use of Bayesian hierarchical modelling and simulation for design and analysis of a clinical trial, by William R. Gillespie, Bayesian CTS example at FDA/Industry workshop September 2003. The method on which the known simulator operates is a well known statistical algorithm known as “Montecarlo Algorithm”.

This simulator however is not constructed in order to simulate the results or the trend of the results of a phase four clinical trial. Thus the prediction cannot be seen as very reliable. The method furthermore is more oriented on better planning the trials relatively to the kind of individuals and the way the trials has to be carried out in order to have bigger chances of success.

From theoretical point of view the use of Artificial Adaptive Systems should offer the possibility to infer results that are likely to be obtained in a advanced phase of marketing from the analysis of the data assembly related to the pre registrative phases. In other words the aim is to simulate and predict the results of post marketing surveillance from the data of phase 3, or better from an observational open study carried out in a large sample of the recipient population supposed to be exposed to the new drug after the commercialisation. The only requirement is that this observational study would consists in a assembly of a population not excessively “clean” but corresponding at least in part to the mix of variables encountered in the real world. The key point is to establish the “implicit function” relating the input variables or independent variables, to the dependent variable ( specific outcome of a subject).

It is interesting to note that at variance with classical statistics, which act on a particular data set with a vertical horizon, AAS tend to act with a horizontal approach. ( see figure 1).

When the implicit function is established, for example with autoassociative neural networks, is possible to navigate on the hyper surface of the data set asking questions. For example one could ask which is the prototype of a subject having a specific negative outcome, and how this profile depend from the presence of the new drug.

Let’s consider the problem of drug-induced hepatotoxicity. During the Phase 3 clinical trials of a new drug the hepatic function has been closely monitored ad slight to moderate elevation of hepatic enzymes has been recorded in a small proportion of patients; for example 2% of the exposed population. Liver enzyme levels can range in a certain interval from zero to upper normal range till 10 times of the normal range in case massive hepatic necrosis.

Let’s take ALT: normal range is 1-20 units; a very high value is 400 units.

In the trial 2% of patients showed elevation of ALT till 50 units; none of them had severe necrosis with very high ALT values.

To adapt the collected data to neural network processing scale the ALT values are scaled from 0 (equal to 1 ALT unit) to 1 ( equal to ALT 400 units).

So a subject with ALT= 20 will be coded as 0.05; while a subject with ALT = 40 will be coded as 0.1.

After the training phase with the auto-associative neural network, we ask the network which is the prototype of a patient with a very high ALT value, by setting the value of ALT in our scaled data set at 1.0 ( 400 units). During the query when ALT input is set on, the network will activate all its units in a dynamic, competitive and cooperative process at the same time..

This external activation of ALT variable of the original dataset generates a process. Each step of this process is a state of a dynamical system. At each state, each variable will take a specific value, generated by the previous negotiation of that variable with all the others. The process will terminate when the system reaches its natural attractor. At this point, all the states of the process represent the prototype of the patients with strong elevation of ALT values.

This prototype can be used to monitor patients candidate to receive the drug after commercialisation whose profile is close to the prototype.

With the same approach we can generate prototype of patients having other severe reactions, as defined by dramatic changes of specific biomarkers.

2.2 Pseudo-inverse function and evolutionary algorithms

A method for generating new records using an evolutionary algorithm (close to but different from a genetic algorithm) has been recently developed at Semeion Institute. This method, called Pseudo-Inverse Function (in short P-I Function), is able to generate new (virtual) data from a small set of observed data. P-I Function can be of aid when practical constraints limit the number of cases collected during clinical trials, or in case of a population that shows some potentially interesting safety risk traits, but whose small size can seriously affect the reliability of estimates, or in case of secondary analysis on small samples.

The applicative ground is given by research design with one or more dependent and a set of independent variables. The estimation of new cases takes place according to the maximization of a fitness function and outcomes a number as large as needed of ‘virtual’ cases, which reproduce the statistical traits of the original population. The algorithm used by P-I Function is known as Genetic Doping Algorithm (GenD), designed and implemented by Semeion Research Centre; among its features there is an innovative crossover procedure, which tends to select individuals with average fitness values, rather than those who show best values at each ‘generation’.

A particularly thorough research design has been put on: (1) the observed sample is half-split to obtain a training and a testing set, which are analysed by means of a back propagation neural network; (2) testing is performed to find out how good the parameter estimates are; (3) a 10% sample is randomly extracted from the training set and used as a reduced training set; (4) on this narrow basis, GenD calculates the pseudo-inverse of the estimated parameter matrix; (5) ‘virtual’ data are tested against the testing data set (which has never been used for training).

The algorithm has been validated on a particularly difficult data set composed by only 44 respondents, randomly sampled from a broader data set taken from the General Social Survey 2002. The major result is that networks trained on the ‘virtual’ resample show a model fit as good as the one of the observed data, though ‘virtual’ and observed data differ on some features. It can be seen that GenD ‘refills’ the joint distribution of the independent variables, conditioned by the dependent one.

This approach could be very interesting to expand the subset of patients who in the course of a clinical Phase 3 trial suffer for severe side effects. In the new virtual population thanks to the higher number of records it would be more appropriate to test statistical assumptions and derive predictive models.

3.Prediction of drug toxicity dose-related in the passage from preclinical phase to Phase I in human beings.

The recent disaster occurred in France where 5 healthy volunteers suffered from extremely severe Central nervous system side effects following the administration of the highest dosage of a new chemical entity in the frame of dose-escalation Phase I clinical trial underlines the complexity to transfer data coming from animal studies to man choosing a dose range of medication with an acceptable risk of toxicity.

Phase I studies are designed to test safety and tolerability of a drug, as well as how, and how fast, the chemical is processed by the human body. Most of these studies are carried out by specialized research contract companies; the subjects are usually healthy volunteers who receive modest financial compensation.

Serious incidents in phase I studies are rare, but they can never be completely excluded because a drug's behavior in animals isn't always a good predictor of its effects in humans. The last publicly known similar incident occurred in 2006, when six men in the United Kingdom suffered severe organ dysfunction after receiving small doses of a monoclonal antibody named TGN1412.

It is clear that differences in metabolic pathways in animal and human species, genetic predisposition, non linear pharmacokinetics of certain drugs, and other biological factor can interact in a complex way giving rise to unexpected catastrophic event non commonly foreseeable with traditional mathematical approaches.

These phenomena have to do with chaos theory(22).

A major breakthrough of the twentieth century, which has been facilitated by computer science, has been the recognition that simple rules not always lead to “ stable order,” but in many circumstances instead lead to an apparent disorder characterized by marked instability and unpredictable variation for reasons intrinsic to the rules themselves.

The phenomenon of rules causing emerging disorder, counter-intuitive to many people, is the environment currently being explored as “self-organization,” “fractals,” i .e. (a fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole, a property called self-similarity), “non-linear dynamical systems” and “chaos.”

Chaos theory, also called nonlinear systems theory, provides new insights into processes previously thought to be unpredictable and random. It also provides a new set of tools that can be used to analyze physiological and clinical data like for example unexpected reaction to pharmaceutical agents, electric signals coming from the heart or from the brain etc.

Most of the reasoning in the section 2 of this essay applyies as well to this specific problem.

in addition to autoassociative neural networksm evolutionary systems like those explained in section 1, when fed with adequate amount of information of different types, related to the pharmacological experiments performed in the preclinical development of a new drug can easily exploit a waste range of combinatorial occurrence and propose the safest protocol to be applied in healthy volunteers who for the first time will experiment the new compound.

Evolutionary computation, offers practical advantages to the researcher facing difficult optimization problems. These advantages are multifold, including the simplicity of the approach, its robust response to changing circumstance, its flexibility, and many other facets. The evolutionary algorithm can be applied to problems where heuristic solutions are not available or generally lead to unsatisfactory results.

Conclusions

New mathematical laws coupled by computer programming allow today to perform predictions which till few years ago were considered impossible. The prediction of response to a specific treatment in individual patients with supervised artificial neural networks has been documented and validated extensively in the literature. Less experience exists in predicting major unwanted effects of a new drugs after the commercialisation or predicting unexpected toxicity in Phase I clinical trials. In doing this task a close cooperation between regulatory agency scientific team and bio mathematicians experts in the use of new generations artificial adaptive system has to be strongly recommended.

.

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Welcome to Authorea!

Hey, welcome. Double click anywhere on the text to start writing. In addition to simple text you can also add text formatted in **boldface**, *italic*, and yes, math too: *E* = *m**c*^{2}! Add images by drag’n’drop or click on the “Insert Figure” button.

Authenticating Route Transitions in an SPA: What to do About the Developer Console

In a single page app, all of the decisions about what view/subview to render occurs on the client. This means that ideally the client would be able to authenticate the currently logged in user on transitions to sensitive pages and access its data without going back to the server. This means that special care needs to be made to protect our application from a malicious user interacting with the developer console present in all modern browsers. One possible security vulnerability is the escalation of a globally stored user user role. This would cause the hacker to view a part of the website that they were forbidden to.

Growth of 48 Built Environment Bacterial Isolates on Board the International Space Station (ISS)

and 6 collaborators

**Abstract**

Background: While significant attention has been paid to the potential risk of pathogenic microbes aboard crewed spacecraft, much less has focused on the non-pathogenic microbes in these habitats. Preliminary work has demonstrated that the interior of the International Space Station (ISS) has a microbial community resembling those of built environments on earth. Here we report results of sending 48 bacterial strains, collected from built environments on earth, for a growth experiment on the ISS. This project was a component of Project MERCCURI (Microbial Ecology Research Combining Citizen and University Researchers on ISS).

Results: Of the 48 strains sent to the ISS, 45 of them showed similar growth in space and on earth. The vast majority of species tested in this experiment have also been found in culture-independent surveys of the ISS. Only one bacterial strain that avoided contamination showed significantly different growth in space. *Bacillus safensis* JPL-MERTA-8-2 grew 60% better in space than on earth.

Conclusions: The majority of bacteria tested were not affected by conditions aboard the ISS in this experiment (e.g., microgravity, cosmic radiation). Further work on *Bacillus safensis* could lead to interesting insights on why this bacteria grew so much better in space.

UZFor2015 - Timing Analysis Report

The data has been taken from \cite{Potter_2011}. All timing stamps are in BJD using the TDB time scale. No further transformations needed. A total of 42 timing measurements exist. However, Potter et al. have not included data points from Dai et al. (2010). See Potter et al. (2011) text for details. In this analysis I will consider the full set of timings as a start and as presented in Potter et al. (2011).

In general I am using IDL for the timing analysis. The cycle or ephemeris numbers have been obtained from IDL> ROUND((BJDMIN-TZERO)/PERIOD) where BJDMIN are all 42 timing measurements, TZERO is an arbitrary timing measurement that defines the CYCLE=*E*=0 and PERIOD is the binary orbital period (0.087865425 days) and was taken from \cite{Potter_2011}, Table 2. In this work I will use TZERO=BJD 2,450,021.779388. It is a bit different from the TZERO used in \cite{Potter_2011} in order to introduce a bit variation and also because I think the center of mass of the data points is as chosen by me.

As a first step I used IDL’s LINFIT code to fit a straight line with the MEASURE_ERROR keyword set to an array holding the timing measurements errors (Table 2, 3rd column, Potter et al. 2011). This way the square of deviations are weighted with 1/*σ*^{2} where *σ* is the standard timing error for each timing measurement. This is standard procedure and was also used in Potter et al. (2011). The average or mean timing error for the 42 measurements is 6.0 seconds (the standard deviation is also 6.0 seconds) with 0.74 seconds as the smallest and 17 seconds as the largest error. Also I have rescaled the timing measurements by subtracting the first timing measurement from all the others. Rescaling introduces nothing spooky to the analysis and has the advantage to avoid dynamic range problems. This is in particular needed for a later analysis when using MPFIT. Using LINFIT the resulting reduced *χ*^{2} value was 95.22 (*χ*^{2} = 3808.82 with (42-2) degrees of freedom) with the ephemeris (or computed timings) given as \begin{equation}
T(E) = BJD~2450021.77890(6) + E \times 0.0878654291(1)
\end{equation} The corresponding root-mean-square (RMS) scatter of the data around the best-fit line is 27.5 seconds and the corresponding standard deviation is 27.7 seconds. As expected they should both be similar. To measure scatter of data around any best-fit model, I will use the RMS quantity. The RMS scatter is 5 times the average timing error and could be indicative of a systematic process.

As a test the CURVEFIT routine has been used in a similar manner. The resulting reduced chi2 was also 95.22 matching and confirming the result from the previous section. The /NODERIVATIVE keyword does not change anything and expressions for the partial derivative has been included. The RMS also agrees with the results obtained from LINFIT. However, the formal 1*σ* uncertainties in the best-fit parameters (TZERO and PERIOD) are one magnitude smaller compared to the equivalent values obtained from LINFIT. The data and the best-fit line (obtained from LINFIT) is shown in Fig. [linearfit] with the residuals plotted in Fig. [linearfit_res]. There is absolutely no difference when using the results from CURVEFIT.

After fitting a straight line and visually inspecting the residual plots I cannot see any convincing trend that should justify a quadratic ephemeris (linear + a quadratic term). What I see is a sinusoidal variation around the best-fit line. Relative to the linear line the first timing measurement arrives 20s earlier than expected. Then the trend goes down and increases again to 40s at E=0, then decreases again to a minimum to around 20s and increases again thereafter. There is no obvious quadratic trend from looking at the residuals in Fig. [linearfit_res].

Although there is no obvious reason to include a quadratic term I will nevertheless consider a quadratic model. I will do this by again using IDL’s CURVEFIT procedure and the MPFIT package (also IDL) which is a more sophisticated fitting tool utilizing the Levenberg-Marquardt least-squares minimization algorithm developed by Marwardt.

The results from CURVEFIT are surprising. The best-fit *χ*^{2} value was 3718.89 yielding a reduced *χ*^{2} of 95.36 with (42-3 DoF). The RMS scatter of the residuals around the quadratic model fit was 31 seconds. This means that the fit became worse compared to a linear ephemeris model. The resulting residual plot is shown in Fig. [quadfit_res]. The corresponding best-fit parameters along with formal uncertainties for a quadratic ephemeris are \begin{eqnarray}
T(E) &=& T + P \times E + A \times E^2 \\
&=& 24550021.778895(6) + 0.0878654269(3) \times E + 4.3(5)\times 10^{-14} \times E^2
\end{eqnarray}

I have also used MPFIT to fit a quadratic ephemeris to the Potter et al. (2011) timing data. The resulting *χ*^{2} is 3718.94 with (42-3) degrees of freedom yielding a reduced *χ*^{2} of 95.36. This is identical to the results obtained with CURVEFIT and thus confirmed independently. This is really surprising. The RMS scatter of data around the quadratic ephemeris is around 31 seconds. I will not state the best-fit values for the three model parameters (and their uncertainties) as obtained from MPFIT.

Based on the above result I cannot see that the residuals relative to a linear ephemeris allow the inclusion of a secular term accounting for a quadratic ephemeris. The *χ*^{2} increases with an extra parameter which is not what is expected. I will continue now and fit a 1- and 2-companion model.

We have considered a linear + 1-LTT model (excluding secular changes as described in a quadratic ephemeris). We have again used MPFIT for this task. The model is taken from Irwin (19??). We considered 10^{7} initial guesses. The initial guess for the reference epoch and binary period were taken from the best-fit obtained from a linear ephemeris model. Inital guesses for the semi-amplitude of the light-time orbit were taken from an estimate of the amplitude as shown in Fig. 2. Initial guesses for the eccentricity covered the interval [0,0.9995]. Initial guess for the argument of pericenter covered the interval [0,360] degrees. Initial guess for the orbital period was also estimated from Fig. 2. Initial guess for the time of pericenter passage were obtained from T0 and the orbital period of the light-time orbit. Initial guesses were drawn at random. The methodology follows the same techniques as described in Hinse et al. (2012). Best-fit parameters were obtained from the best-fit solution covariance matrix as returned by MPFIT. Parameters errors should be considered as formal. The best-fit had a *χ*^{2} = 185.2 with (42-7) degrees of freedom resulting in a reduced *χ*_{ν}^{2} = 5.3. The corresponding RMS scatter of data points around the best-fit is 15.7 seconds. The best-fit parameters are listed in Table [BestFitParamsLinPlus1LTT] and shown in Fig. [BestFitModel_LinPlus1LTT]. Recalling the average timing error (of 42 timing measurements) to be 6 seconds, that means that the RMS residuals are on a 2.6*σ* level.

T_{0} (BJD) |
2, 450, 021.77924 ± 3 × 10^{−5} |

P_{0} (days) |
0.0878654289 ± 2 × 10^{−10} |

asinI (AU) |
0.00043 ± 2 × 10^{−5} |

e |
0.65 ± 0.03 |

ω (radians) |
6.89 ± 0.04 |

T_{p} (BJD) |
2, 408, 616.0 ± 50 |

P (days) |
6020 ± 35 |

RMS (seconds) | 15.7 |

\label{BestFitParamsLinPlus1LTT}

At the present stage some inconsistencies were discovered in the reported timing uncertainties as listed in Table 1 in Potter et al. (2011). For example the timing uncertainty reported by \cite{Warren_1995} is 0.000023 days, while Potter et al. (2011) reports 0.00003 and 0.00004 days. Furthermore, after scrutinizing the literature we found that several timing measurements were omitted in Potter et al. (2011). We tested for the possibility that Potter et al. (2011) adopts timing uncertainties from the spread of data around a best-fit linear regression. However, that seems not the case: As a test, we used the five timing measurements from \cite{Beuermann1988} as listed in Table 1 in Potter et al. (2011). We fitted a linear straight line using CURVEFIT as implemented in IDL and found a scatter of 0.00004 to 0.00005 days depending on the metric used to measure scatter around the best-fit. The quoted uncertainties in Potter et al. (2011) are smaller by at least a factor of two. We conclude that Potter et al. (2011) must be in error when quoting timing uncertainties in their Table 1. Similar mistakes when quoting timing uncertainties apply to data listed in \cite{Ramsay1994}. Furthermore, after scrutinizing the literature for timing measurements of UZ For we found several timing measurements that were omitted in Potter et al. (2011). For example six eclipse timings were reported by \cite{BaileyCropper_1991} with a uniform uncertainty of 0.00006 days. However, Potter et al. (2011) only reports three of the six timings. Furthermore, a total of five new timings were reported by \cite{Ramsay1994}, but only one were listed in Potter et al. (2011). We can not come up with a good explanation why those extra timing measurements should be omitted or discarded. All of the new data points have been presented in the original works alongside with data points used in the analysis of Potter et al. (2011).

In this research we make use of all timing measurements that have been obtained with reasonable accuracy. We have therefore recompiled all available timing measurements from the literature. We list them in Table [NewTimingData]. The original HJD(UTC) time stamps from the literature were converted to the BJD(TDB) system using the on-line time utilities^{1} \citep{Eastman_2010}. Not all sources of timing measurements provide explicit information of the the time standard used. In that case we assume that HJD time stamps are valid in the UTC standard. This assumption is to some extend justified since the first timing measurement was taken in august 1983. At that time the UTC time standard for astronomical observations was widespread. All new measurements presented in \cite{Potter_2011} were taken directly from their Table 1. Some remarks are at place. By finding additional timing measurements (otherwise omitted in Potter et al. 2011) in the literature we decided to follow a different approach to estimate timing uncertainties. For measurements that were taken over a short time period one can determine a best-fit line and estimate timing uncertainties from the data scatter. The underlying assumption in this method is that no significant astrophysical signal (interaction between binary components or additional bodies) is contained in the timing measurements over a few consecutive observing nights. Therefore, the scatter around a linear ephemeris should be a reasonable measure of how well timings were measured. In other words, only a first-order effect due to a linear ephemeris is observed. Higher-order eclipse timing variation effects are negligible for data sets obtained during a few consecutive nights. The advantage is that for a given data set the same telescope/instrument were used as well as weather conditions were likely not to have changed much from night to night. Furthermore, most likely the same technique was applied to infer the individual time stamps of a given data set. In Table [NewTimingData] we list the original quoted uncertainties presented in the literature as *σ*_{lit}. We also list the uncertainty obtained from the scatter of the data around a best-fit linear regression line. The corresponding reduced *χ*^{2} statistic for each fit is also tabulated in the third column. From the reduced *χ*^{2} for each data set one can scale the corresponding uncertainties such that *χ*_{ν}^{2} = 1 is enforced \citep{Bevington2003Book}. This step is only permitted if a high confidence in the applied model is justified. We think that this is the case when time stamps have been obtained over a short time interval. However, ultimately the timing uncertainty depends on the sampling of the eclipse event at a sufficiently high signal-to-noise ratio. The \cite{Imamura_1998} data set was split in two since those time stamps were obtained from two observing runs each lasting for a few days. Furthermore, we have calculated three data scatter metrics around the best-fit line: a) the root-mean-square, b) the standard deviation and c) the standard deviation as given by \cite{Bevington2003Book} and defined as \begin{equation}
\sigma^2 = \frac{1}{N-2} \sum_{i=1}^{N}(y_{i} - a - bx_{i})^2
\label{BevEq6p15}
\end{equation} where *N* is the number of data points, *a*, *b* the two parameters for a linear line and (*x*_{i}, *y*_{i}) is a given timing measurement at a given epoch. We have tested the dependence of scatter on the weight used and found no difference in the scatter metrics when applying a weight of one for all measurements. Finally some additional details need to be mentioned. We only inferred new timing uncertainties for data sets with more than two measurements. For a given data set we used the published ephemeris (orbital period) to calculate the eclipse epochs. For the time stamps presented in \cite{BaileyCropper_1991} no ephemeris was stated. We therefore, used their eclipse cycles for the independent variable to calculate a best-fit line. The reference epoch in each fit was placed to be in or near the middle of the data set. Two data points were discarded in the present analysis. We removed one time stamp from \cite{Ferrario_1989} due to a too high timing uncertainty. Another time stamp was removed from the new data presented in Potter et al. (2011), namely the time stamp BJD(TDB) 2,454,857.36480850. This eclipse is duplicated as it was observed also with the much larger SALT/BVIT instrument resulting in a lower timing error. We therefore use only the SALT/BVIT measurement in the present analysis which makes use of a total of 54 timing stamps. The average or mean timing error for the 54 measurements is 5.7 seconds (the standard deviation is 6.5 seconds) with 0.33 seconds as the smallest and 26.5 seconds as the largest error. Also we have rescaled the timing measurements by subtracting the first time stamp from all the others. Rescaling introduces nothing spooky to the analysis and has the advantage to avoid dynamic range problems when carrying out the process of least-squares minimization. The total baseline of the data set spans 27 years.

BJD(TDB) | σ_{lit} |
χ_{ν}^{2} |
σ_{lit, scaled} |
σ_{RMS} |
STD | Eq. [BevEq6p15] | Remarks |
---|---|---|---|---|---|---|---|

2455506.427034 | 0.0000100 | – | – | – | – | – | HIPPO/1.9m, \cite{Potter_2011} |

2455478.485831 | 0.0000100 | – | – | – | – | – | HIPPO/1.9m, \cite{Potter_2011} |

2455450.544621 | 0.0000100 | – | – | – | – | – | HIPPO/1.9m, \cite{Potter_2011} |

2454857.364805 | 0.0000086 | – | – | – | – | – | SALT/BVIT, \cite{Potter_2011} |

2454417.334722 | 0.0000086 | – | – | – | – | – | SALT/SALTICAM, \cite{Potter_2011} |

2453408.288086 | 0.0000086 | 0.198 | 3.83E-6 | 0.0000070 | 0.0000070 | 0.0000100 | UCTPOL/1.9m, \cite{Potter_2011} |

2453407.321574 | 0.0000100 | 0.198 | 4.45E-6 | 0.0000070 | 0.0000070 | 0.0000100 | UCTPOL/1.9m, \cite{Potter_2011} |

2453405.300663 | 0.0000350 | 0.198 | 1.56E-5 | 0.0000070 | 0.0000070 | 0.0000100 | UCTPOL/1.9m, \cite{Potter_2011} |

2453404.334042 | 0.0000600 | – | – | – | – | – | SWIFT, \cite{Potter_2011} |

2452494.839196 | 0.0000870 | – | – | – | – | – | XMM OM, \cite{Potter_2011} |

2452494.575626 | 0.0000350 | – | – | – | – | – | UCTPOL/1.9m, \cite{Potter_2011} |

2452493.609058 | 0.0000700 | – | – | – | – | – | UCTPOL/1.9m, \cite{Potter_2011} |

2451821.702394 | 0.0000100 | – | – | – | – | – | WHT/S-Cam, \cite{de_Bruijne_2002} |

2451528.495434 | 0.0000200 | 0.134 | 7.32E-6 | 0.0000040 | 0.0000050 | 0.0000070 | WHT/S-Cam, \cite{Perryman_2001} |

2451528.407579 | 0.0000200 | 0.134 | 7.32E-6 | 0.0000040 | 0.0000050 | 0.0000070 | WHT/S-Cam, \cite{Perryman_2001} |

2451522.432730 | 0.0000200 | 0.134 | 7.32E-6 | 0.0000040 | 0.0000050 | 0.0000070 | WHT/S-Cam, \cite{Perryman_2001} |

2450021.779400 | 0.0000600 | 2.237 | 8.97E-5 | 0.0000500 | 0.0000600 | 0.0000900 | CTIO 1m/photometer, set II, \cite{Imamura_1998} |

2450021.691660 | 0.0000600 | 2.237 | 8.97E-5 | 0.0000500 | 0.0000600 | 0.0000900 | CTIO 1m/photometer, set II, \cite{Imamura_1998} |

2450018.704120 | 0.0000600 | 2.237 | 8.97E-5 | 0.0000500 | 0.0000600 | 0.0000900 | CTIO 1m/photometer, set II, \cite{Imamura_1998} |

2449755.634995 | 0.0000600 | 0.427 | 3.92E-5 | 0.0000200 | 0.0000300 | 0.0000300 | CTIO 1m/photometer, set I, \cite{Imamura_1998} |

2449755.547165 | 0.0000600 | 0.427 | 3.92E-5 | 0.0000200 | 0.0000300 | 0.0000300 | CTIO 1m/photometer, set I, \cite{Imamura_1998} |

2449753.614046 | 0.0000600 | 0.427 | 3.92E-5 | 0.0000200 | 0.0000300 | 0.0000300 | CTIO 1m/photometer, set I, \cite{Imamura_1998} |

2449752.647586 | 0.0000600 | 0.427 | 3.92E-5 | 0.0000200 | 0.0000300 | 0.0000300 | CTIO 1m/photometer, set I, \cite{Imamura_1998} |

2449733.405017 | 0.0000400 | – | – | – | – | – | EUVE, \cite{Potter_2011} |

2449310.332595 | 0.0000230 | – | – | – | – | – | EUVE, \cite{Warren_1995} |

2449276.680076 | 0.0000230 | – | – | – | – | – | EUVE, \cite{Warren_1995} |

2448784.721419 | 0.0000300 | – | – | – | – | – | HST, \cite{Potter_2011} |

2448483.606635 | 0.0000200 | 4.413 | 4.20E-5 | 0.0000300 | 0.0000400 | 0.0000400 | ROSAT, \cite{Ramsay1994} |

2448483.430915 | 0.0000200 | 4.413 | 4.20E-5 | 0.0000300 | 0.0000400 | 0.0000400 | ROSAT, \cite{Ramsay1994} |

2448483.343045 | 0.0000200 | 4.413 | 4.20E-5 | 0.0000300 | 0.0000400 | 0.0000400 | ROSAT, \cite{Ramsay1994} |

2448482.903785 | 0.0000200 | 4.413 | 4.20E-5 | 0.0000300 | 0.0000400 | 0.0000400 | ROSAT, \cite{Ramsay1994} |

2448482.727955 | 0.0000200 | 4.413 | 4.20E-5 | 0.0000300 | 0.0000400 | 0.0000400 | ROSAT, \cite{Ramsay1994} |

2447829.184858 | 0.0000600 | 0.120 | 2.08E-5 | 0.0000170 | 0.0000190 | 0.0000200 | AAT, \cite{BaileyCropper_1991} |

2447829.096998 | 0.0000600 | 0.120 | 2.08E-5 | 0.0000170 | 0.0000190 | 0.0000200 | AAT, \cite{BaileyCropper_1991} |

2447829.009088 | 0.0000600 | 0.120 | 2.08E-5 | 0.0000170 | 0.0000190 | 0.0000200 | AAT, \cite{BaileyCropper_1991} |

2447828.130518 | 0.0000600 | 0.120 | 2.08E-5 | 0.0000170 | 0.0000190 | 0.0000200 | AAT, \cite{BaileyCropper_1991} |

2447828.042638 | 0.0000600 | 0.120 | 2.08E-5 | 0.0000170 | 0.0000190 | 0.0000200 | AAT, \cite{BaileyCropper_1991} |

2447827.954778 | 0.0000600 | 0.120 | 2.08E-5 | 0.0000170 | 0.0000190 | 0.0000200 | AAT, \cite{BaileyCropper_1991} |

2447437.920514 | 0.0000300 | – | – | – | – | – | 2.3m Steward obs., \cite{Allen_1989} |

2447128.809635 | 0.0009000 | 0.059 | 2.18E-4 | 0.0002000 | 0.0002000 | 0.0002000 | 2.3m Steward obs., \cite{Berriman_1988} |

2447128.722035 | 0.0009000 | 0.059 | 2.18E-4 | 0.0002000 | 0.0002000 | 0.0002000 | 2.3m Steward obs., \cite{Berriman_1988} |

2447127.843835 | 0.0009000 | 0.059 | 2.18E-4 | 0.0002000 | 0.0002000 | 0.0002000 | 2.3m Steward obs., \cite{Berriman_1988} |

2447127.755635 | 0.0009000 | 0.059 | 2.18E-4 | 0.0002000 | 0.0002000 | 0.0002000 | 2.3m Steward obs., \cite{Berriman_1988} |

2447145.064339 | 0.0000600 | 1.046 | 6.14E-5 | 0.0002000 | 0.0002000 | 0.0003000 | AAT, \cite{Ferrario_1989} |

2447127.227739 | 0.0003000 | 1.046 | 3.07E-4 | 0.0002000 | 0.0002000 | 0.0003000 | AAT, \cite{Ferrario_1989} |

2447127.139439 | 0.0003000 | 1.046 | 3.07E-4 | 0.0002000 | 0.0002000 | 0.0003000 | AAT, \cite{Ferrario_1989} |

2447097.792555 | 0.0002500 | 0.069 | 6.58E-5 | 0.0000600 | 0.0000500 | 0.0000700 | ESO/MPI 2.2m, \cite{Beuermann1988} |

2447094.717355 | 0.0002300 | 0.069 | 6.05E-5 | 0.0000600 | 0.0000500 | 0.0000700 | ESO/MPI 2.2m, \cite{Beuermann1988} |

2447091.554235 | 0.0002300 | 0.069 | 6.05E-5 | 0.0000600 | 0.0000500 | 0.0000700 | ESO/MPI 2.2m, \cite{Beuermann1988} |

2447090.587785 | 0.0001200 | 0.069 | 3.16E-5 | 0.0000600 | 0.0000500 | 0.0000700 | ESO/MPI 2.2m, \cite{Beuermann1988} |

2447089.709005 | 0.0003000 | 0.069 | 7.89E-5 | 0.0000600 | 0.0000500 | 0.0000700 | ESO/MPI 2.2m, \cite{Beuermann1988} |

2447088.742545 | 0.0003000 | 0.069 | 7.89E-5 | 0.0000600 | 0.0000500 | 0.0000700 | ESO/MPI 2.2m, \cite{Beuermann1988} |

2446446.973823 | 0.0001600 | – | – | – | – | – | EXOSAT, \cite{Osborne_1988} |

2445567.177636 | 0.0001600 | – | – | – | – | – | EXOSAT, \cite{Osborne_1988} |

\label{NewTimingData}

In this work we are not using the F-test as a statistical tool to perform model selection. The F-test is based on the assumption that uncertainties are Gaussian. This assumption might be violated if the data is affected by time-correlated red noise due to atmospheric effects and/or additional astrophysical effects that influence the shape of the eclipse profile. There exist no studies in the literature that has addressed this question and therefore we judge that the outcome of an F-test is unreliable.

In the following we will consider the newly compiled data set with timing uncertainties obtained from rescaling the published uncertainties in order to ensure *χ*^{2} = 1 over short time intervals. We have determined the following linear ephemeris using MPFIT. We followed the monte-carlo approach and determined a best-fit model by generating 10 million random initial guesses. We used best-fit parameters from LINFIT to obtain a first estimate of the initial epoch and period. Then initial guesses were drawn from a Gaussian distribution centered at the LINFIT values with standard deviation given by five times the formal LINFIT uncertainties. The linear ephemeris is shown in Fig. [Linearfit_NEW]. The resulting reduced *χ*^{2} value was 162.5 (*χ*^{2} = 8448.6 with (54-2) degrees of freedom) with the ephemeris (or computed timings) given as \begin{equation}
T(E) = BJD_{TDB}~2,450,018.703604(3) + E \times 0.08786542817(9)
\end{equation} Residuals are shown in Fig. [Linfit_NEW_Res] and displays a systematic variation. The corresponding RMS scatter of the data around the best-fit line is 28.9 seconds. The scatter is 5 times the average timing error and could be indicative of a systematic process of astrophysical origin.

We have also considered a quadratic model to the new data set. However, and judged by eye from Fig. [Linfit_NEW_Res], there is no obvious upward or downward parabolic trend in the data. Nevertheless we added a quadratic term and generated 10 million initial guesses to find a best-fit model. The resulting reduced *χ*^{2} value increased to 165.7 with 54-3 degrees of freedom. We therefore, decide to not consider a quadratic ephemeris in our further analysis.

Using scaled uncertainties we have considered a linear + 1-LTT model. We have again used MPFIT. The model is taken from Irwin (19??) and described in Hinse et al. (2012). We considered 10^{7} initial guesses. The initial guess for the reference epoch and binary period were taken from the best-fit obtained from a linear ephemeris model. Inital guesses for the semi-amplitude of the light-time orbit were taken from an estimate of the amplitude as shown in Fig. 2. Initial guesses for the eccentricity covered the interval [0,1]. Initial guess for the argument of pericentre covered the interval [0,360] degrees. Initial guess for the orbital period was also estimated from Fig. [Linfit_NEW_Res]. Initial guess for the time of pericentre passage were obtained from T0 and the orbital period of the light-time orbit. Initial guesses were drawn at random. The methodology follows the same techniques as described in Hinse et al. (2012). Best-fit parameters were obtained from the best-fit solution covariance matrix as returned by MPFIT. Parameters errors should be considered as formal [OFF-THE-RECORD: FINAL ERRORS WILL USE BOOTSTRAP TECHNIQUE]. The best-fit had a *χ*^{2} = 717.6 with 47 degrees of freedom resulting in a reduced *χ*_{ν}^{2} = 15.3. The corresponding RMS scatter of data points around the best-fit is 20.0 seconds. The best-fit parameters are listed in Table [BestFitParamsLinPlus1LTT_New_AllData] and shown in Fig. [BestFitModel_LinPlus1LTT_New_AllData]. Recalling the average timing error to be 6 seconds, that means that the RMS residuals are on a 3.3*σ* level indicating a significant signal of some origin. However, upon close inspection of Fig. [BestFitModel_LinPlus1LTT_New_AllData] the origin of the large scatter is mainly due to data obtained by \cite{Beuermann1988}, \cite{Berriman_1988}, \cite{Ferrario_1989} and a single point from \cite{Allen_1989} located between cycle number -27,000 and -35,000. In the following we investigate the effect of the resulting model when removing those data points.

T_{0} (BJD) |
2, 450, 021.77919 ± 3 × 10^{−5} |

P_{0} (days) |
0.0878654283 ± 1 × 10^{−10} |

asinI (AU) |
0.00048 ± 3 × 10^{−5} |

e |
0.76 ± 0.03 |

ω (radians) |
3.84 ± 0.04 |

T_{p} (BJD) |
2, 461, 743.0 ± 53 |

P (days) |
5964 ± 25 |

RMS (seconds) | 20.0 |

χ^{2} |
717.6 |

red. χ^{2} |
15.3 |

\label{BestFitParamsLinPlus1LTT_New_AllData}

To start we have removed a total of eight points: three points from \cite{Ferrario_1989}, four points from \cite{Berriman_1988} and a single point from \cite{Allen_1989}. The average deviation of those points from our best-fit model (Fig. [BestFitModel_LinPlus1LTT_New_AllData] and Table [BestFitParamsLinPlus1LTT_New_AllData]) was around 35 seconds. The minimum timing uncertainty is 0.33 seconds. The maximum timing uncertainty is 13.8 seconds. The mean of the timing uncertainty is 3.7 seconds. This data set is very similar to the data set investigated by Potter et al. (2011). Our new model now had a *χ*^{2} = 467.1 and a reduced *χ*^{2} = 12 with 39 DoF resulting in a RMS scatter of 13 seconds. We show the resulting best-fit parameters in Fig. [BestFitModel_LinPlus1LTT_RedDataSet1] and Table [BestFitParamsLinPlus1LTT_RedDataSet1]. As a result we first note that the removal of eight data points did not change significantly the model. This points towards that those discarded points do not contribute significantly to constrain the model during the fitting process. Further we note that our model is significantly different from the first elliptical term model presented in Potter et al. (2011). The most striking difference is dominantly seen in the eccentricity parameter. While they found a near-circular model we find a highly eccentric solution. Next we continue our analysis by removing an additional six data points.

T_{0} (BJD) |
2, 450, 021.69149 ± 4 × 10^{−5} |

P_{0} (days) |
0.0878654287 ± 1 × 10^{−10} |

asinI (AU) |
0.00047 ± 3 × 10^{−5} |

e |
0.73 ± 0.04 |

ω (radians) |
0.74 ± 0.03 |

T_{p} (BJD) |
2455832.0 ± 28 |

P (days) |
6012 ± 23 |

RMS (seconds) | 13.0 |

χ^{2} |
467.1 |

red. χ^{2} |
12.0 |

\label{BestFitParamsLinPlus1LTT_RedDataSet1}

In this section we investigate the effects by removing a total of 14 data points. Six from \cite{Beuermann1988}, three from \cite{Ferrario_1989}, four points from \cite{Berriman_1988} and a single point from \cite{Allen_1989}. The minimum timing uncertainty is 0.33 seconds. The maximum uncertainty is 13.8 seconds and the mean is 3.5 seconds. The resulting best-fit model is shown in Fig. [BestFitModel_LinPlus1LTT_RedDataSet2] with best-fit parameters listed in Table [BestFitParamsLinPlus1LTT_RedDataSet2]. We note that the resulting best-fit model has not changed significantly. Also the RMS scatter is comparable with the mean timing uncertainty. From this we can conclude that the timing errors should be scaled with $\sqrt{\chi^2}$ if the model is the correct description of the signal.

T_{0} (BJD) |
2, 450, 021.69150 ± 3 × 10^{−5} |

P_{0} (days) |
0.0878654279 ± 1 × 10^{−10} |

asinI (AU) |
0.00049 ± 3 × 10^{−5} |

e |
0.79 ± 0.03 |

ω (radians) |
6.91 ± 0.03 |

T_{p} (BJD) |
2467502 ± 57 |

P (days) |
5901 ± 20 |

RMS (seconds) | 4.4 |

χ^{2} |
161.0 |

red. χ^{2} |
4.9 |

\label{BestFitParamsLinPlus1LTT_RedDataSet2}

Finally, we have also discarded the two first timing measurements from \cite{Osborne_1988}. The mean timing uncertainty is 3 seconds. Again we found a best-fit model as shown in Fig. [BestFitModel_LinPlus1LTT_RedDataSet3] with best-fit parameters listed in Table [BestFitParamsLinPlus1LTT_RedDataSet3]. Also in this case the model did not change much compared to previous investigations. This points towards that the data taken at earlier epochs (discarded) does not play an important role to constrain the model. The RMS scatter of 4 seconds is comparable with the mean uncertainty and does not point towards a signal that could be due to an additional companion.

Based on rescaled timing uncertainties we find: We find no qualitative (visual inspection of residuals) and quantitative (increased chi2) justification for including a quadratic term in any model. We find that certain data points can be discarded without significantly affecting the best-fit model as obtained when all data were included. Therefore those data points do no play a significant role to constrain the model. We find that there is no significant evidence for a 2nd companion when only considering timing data of good quality.

T_{0} (BJD) |
2, 450, 021.69149 ± 4 × 10^{−5} |

P_{0} (days) |
0.0878654279 ± 1 × 10^{−10} |

asinI (AU) |
0.00050 ± 5 × 10^{−5} |

e |
0.79 ± 0.05 |

ω (radians) |
5.66 ± 0.05 |

T_{p} (BJD) |
2467498 ± 70 |

P (days) |
5900 ± 23 |

RMS (seconds) | 4.0 |

χ^{2} |
160.0 |

red. χ^{2} |
5.2 |

\label{BestFitParamsLinPlus1LTT_RedDataSet3}

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