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Schedule session 5

Join a track by clicking the track title. Clicking any paper will show the abstract, and links to paper and video.

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232: Pairwise Supervised Hashing with Bernoulli Variational Auto-Encoder and Self-Control Gradient Estimator

Track 1: Deep generative models Siamak Zamani Dadaneh, Shahin Boluki, Mingzhang Yin, Mingyuan Zhou, Xiaoning Qian
[Paper] [Video (inline)] [Video (new tab)]

Semantic hashing has become a crucial component of fast similarity search in many large-scale information retrieval systems, in particular, for text data. Variational auto-encoders (VAEs) with binary latent variables as hashing codes provide state-of-the-art performance in terms of precision for document retrieval. We propose a pairwise loss function with discrete latent VAE to reward within-class similarity and between-class dissimilarity for supervised hashing. Instead of solving the optimization for training relying on existing biased gradient estimators, an unbiased, low-variance gradient estimator, which evaluates the non-differentiable loss function over two correlated sets of binary hashing codes to control the gradient variance, is adopted to optimize the hashing function to achieve superior performance compared to the state-of-the-arts, as demonstrated by our comprehensive experiments.

565: Locally Masked Convolution for Autoregressive Models

Track 1: Deep generative models Ajay Jain, Pieter Abbeel, Deepak Pathak
[Paper] [Video (inline)] [Video (new tab)]

High-dimensional generative models have many applications including image compression, multimedia generation, anomaly detection and data completion. State-of-the-art estimators for natural images are autoregressive, decomposing the joint distribution over pixels into a product of conditionals parameterized by a deep neural network, e.g. a convolutional neural network such as the PixelCNN. However, PixelCNNs only model a single decomposition of the joint, and only a single generation order is efficient. For tasks such as image completion, these models are unable to use much of the observed context. To generate data in arbitrary orders, we introduce LMConv: a simple modification to the standard 2D convolution that allows arbitrary masks to be applied to the weights at each location in the image. Using LMConv, we learn an ensemble of distribution estimators that share parameters but differ in generation order, achieving improved performance on whole-image density estimation (2.89 bpd on unconditional CIFAR10), as well as globally coherent image completions. Code is available at https://ajayjain.github.io/lmconv.

140: Walking on Two Legs: Learning Image Segmentation with Noisy Labels

Track 1: Deep generative models Guohua Cheng, Hongli Ji, Yan Tian
[Paper] [Video (inline)] [Video (new tab)]

Image segmentation automatically segments a target object in an image and has recently achieved prominent progress due to the development of deep convolutional neural networks (DCNNs). However, the quality of manual labels plays an essential role in the segmentation accuracy, while in practice it could vary a lot and in turn could substantially mislead the training process and limit the effectiveness. In this paper, we propose a novel label refinement and sample reweighting method, and a novel generative adversarial network (GAN) is introduced to fuse these two models into an integrated framework. We evaluate our approach on the publicly available datasets, and the results show our approach to be competitive when compared with other state-of-the-art approaches dealing with the noisy labels in image segmentation.

366: Adversarial Learning for 3D Matching

Track 1: Deep generative models Wei Xing, Brian Ziebart
[Paper] [Video (inline)] [Video (new tab)]

Structured prediction of objects in spaces that are inherently difficult to search or compactly characterize is a particularly challenging task. For example, though bipartite matchings in two dimensions can be tractably optimized and learned, the higher-dimensional generalization—3D matchings—are NP-hard to optimally obtain and the set of potential solutions cannot be compactly characterized. Though approximation is therefore necessary, prevalent structured prediction methods inherit the weaknesses they possess in the two-dimensional setting either suffering from inconsistency or intractability—even when the approximations are sufficient. In this paper, we explore extending an adversarial approach to learning bipartite matchings that avoids these weaknesses to the three dimensional setting. We assess the benefits compared to margin-based methods on a three-frame tracking problem.

451: Batch norm with entropic regularization turns deterministic autoencoders into generative models

Track 1: Deep generative models Amur Ghose, Abdullah Rashwan, Pascal Poupart
[Paper] [Video (inline)] [Video (new tab)]

The variational autoencoder is a well defined deep generative model that utilizes an encoder-decoder framework where an encoding neural network outputs a non-deterministic code for reconstructing an input. The encoder achieves this by sampling from a distribution for every input, instead of outputting a deterministic code per input. The great advantage of this process is that it allows the use of the network as a generative model for sampling from the data distribution beyond provided samples for training. We show in this work that utilizing batch normalization as a source for non-determinism suffices to turn deterministic autoencoders into generative models on par with variational ones, so long as we add a suitable entropic regularization to the training objective.

517: Learning to learn generative programs with Memoised Wake-Sleep

Track 1: Deep generative models Luke Hewitt, Tuan Anh Le, Joshua Tenenbaum
[Paper] [Video (inline)] [Video (new tab)]

We study a class of neuro-symbolic generative models in which neural networks are used both for inference and as priors over symbolic, data-generating programs. As generative models, these programs capture compositional structures in a naturally explainable form. To tackle the challenge of performing program induction as an ‘inner-loop’ to learning, we propose the Memoised Wake-Sleep (MWS) algorithm, which extends Wake Sleep by explicitly storing and reusing the best programs discovered by the inference network throughout training. We use MWS to learn accurate, explainable models in three challenging domains: stroke-based character modelling, cellular automata, and few-shot learning in a novel dataset of real-world string concepts.

218: Relaxed Multivariate Bernoulli Distribution and Its Applications to Deep Generative Models

Track 1: Deep generative models Xi Wang, Junming Yin
[Paper] [Video (inline)] [Video (new tab)]

Recent advances in variational auto-encoder (VAE) have demonstrated the possibility of approximating the intractable posterior distribution with a variational distribution parameterized by a neural network. To optimize the variational objective of VAE, the reparameterization trick is commonly applied to obtain a low-variance estimator of the gradient. The main idea of the trick is to express the variational distribution as a differentiable function of parameters and a random variable with a fixed distribution. To extend the reparameterization trick to inference involving discrete latent variables, a common approach is to use a continuous relaxation of the categorical distribution as the approximate posterior. However, when applying continuous relaxation to the multivariate cases, multiple variables are typically assumed to be independent, making it suboptimal in applications where modeling dependency is crucial to the overall performance. In this work, we propose a multivariate generalization of the Relaxed Bernoulli distribution, which can be reparameterized and can capture the correlation between variables via a Gaussian copula. We demonstrate its effectiveness in two tasks: density estimation with Bernoulli VAE and semi-supervised multi-label classification.

272: Semi-supervised Sequential Generative Models

Track 1: Deep generative models Michael Teng, Tuan Anh Le, Adam Scibior, Frank Wood
[Paper] [Video (inline)] [Video (new tab)]

We introduce a novel objective for training deep generative time-series models with discrete latent variables for which supervision is only sparsely available. This instance of semi-supervised learning is challenging for existing methods, because the exponential number of possible discrete latent configurations results in high variance gradient estimators. We first overcome this problem by extending the standard semi-supervised generative modeling objective with reweighted wake-sleep. However, we find that this approach still suffers when the frequency of available labels varies between training sequences. Finally, we introduce a unified objective inspired by teacher-forcing and show that this approach is robust to variable length supervision. We call the resulting method caffeinated wake-sleep (CWS) to emphasize its additional dependence on real data. We demonstrate its effectiveness with experiments on MNIST, handwriting, and fruit fly trajectory data.

543: Stable Policy Optimization via Off-Policy Divergence Regularization

Track 2: Reinforcement learning Ahmed Touati, Amy Zhang, Joelle Pineau, Pascal Vincent
[Paper] [Video (inline)] [Video (new tab)]

Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO) are among the most successful policy gradient approaches in deep reinforcement learning (RL). While these methods achieve state-of-the-art performance across a wide range of challenging tasks, there is room for improvement in the stabilization of the policy learning and how the off-policy data are used. In this paper we revisit the theoretical foundations of these algorithms and propose a new algorithm which stabilizes the policy improvement through a proximity term that constrains the discounted state-action visitation distribution induced by consecutive policies to be close to one another. This proximity term, expressed in terms of the divergence between the visitation distributions, is learned in an off-policy and adversarial manner. We empirically show that our proposed method can have a beneficial effect on stability and improve final performance in benchmark high-dimensional control tasks.

105: Exploration Analysis in Finite-Horizon Turn-based Stochastic Games

Track 2: Reinforcement learning Jialian Li, Yichi Zhou, Tongzheng Ren, Jun Zhu
[Paper] [Video (inline)] [Video (new tab)]

Exploration and exploitation trade-off is one of the key concerns in reinforcement learning. Previous work on one-player Markov Decision Processes has reached near-optimal results for both PAC and high probability regret guarantees. However, such an analysis is lacking for the more complex stochastic games with multi-players, where all players aim to find an approximate Nash Equilibrium. In this work, we address the exploration issue for the $N$-player finite-horizon turn-based stochastic games (FTSG). We propose a framework, \textit{Upper Bounding the Values for Players} (UBVP), to guide exploration in FTSGs. UBVP leverages the key insight that players choose the optimal policy conditioning on the policies of the others simultaneously; thus players can explore \textit{in the face of uncertainty} and get close to the Nash Equilibrium. Based on UBVP, we present two provable algorithms. One is \textit{Uniform}-PAC with a sample complexity of $\tilde{O}(1/\epsilon^2)$ to get an $\epsilon$-Nash Equilibrium for arbitrary $\epsilon>0$, and the other has a cumulative exploitability of $\tilde{O}(\sqrt{T})$ with high probability.

71: Regret Bounds for Decentralized Learning in Cooperative Multi-Agent Dynamical Systems

Track 2: Reinforcement learning Seyed Mohammad Asghari, Yi Ouyang, Ashutosh Nayyar
[Paper] [Video (inline)] [Video (new tab)]

Regret analysis is challenging in Multi-Agent Reinforcement Learning (MARL) primarily due to the dynamical environments and the decentralized information among agents. We attempt to solve this challenge in the context of decentralized learning in multi-agent linear-quadratic (LQ) dynamical systems. We begin with a simple setup consisting of two agents and two dynamically decoupled stochastic linear systems, each system controlled by an agent. The systems are coupled through a quadratic cost function. When both systems' dynamics are unknown and there is no communication among the agents, we show that no learning policy can generate sub-linear in $T$ regret, where $T$ is the time horizon. When only one system's dynamics are unknown and there is one-directional communication from the agent controlling the unknown system to the other agent, we propose a MARL algorithm based on the construction of an auxiliary single-agent LQ problem. The auxiliary single-agent problem in the proposed MARL algorithm serves as an implicit coordination mechanism among the two learning agents. This allows the agents to achieve a regret within $O(\sqrt{T})$ of the regret of the auxiliary single-agent problem. Consequently, using existing results for single-agent LQ regret, our algorithm provides a $\tilde{O}(\sqrt{T})$ regret bound. (Here $\tilde{O}(\cdot)$ hides constants and logarithmic factors). Our numerical experiments indicate that this bound is matched in practice. From the two-agent problem, we extend our results to multi-agent LQ systems with certain communication patterns which appear in vehicle platoon control.

17: Finite-sample Analysis of Greedy-GQ with Linear Function Approximation under Markovian Noise

Track 2: Reinforcement learning Yue Wang, Shaofeng Zou
[Paper] [Video (inline)] [Video (new tab)]

Greedy-GQ is an off-policy two timescale algorithm for optimal control in reinforcement learning. This paper develops the first finite-sample analysis for the Greedy-GQ algorithm with linear function approximation under Markovian noise. Our finite-sample analysis provides theoretical justification for choosing stepsizes for this two timescale algorithm for faster convergence in practice, and suggests a trade-off between the convergence rate and the quality of the obtained policy. Our paper extends the finite-sample analyses of two timescale reinforcement learning algorithms from policy evaluation to optimal control, which is of more practical interest. Specifically, in contrast to existing finite-sample analyses for two timescale methods, e.g., GTD, GTD2 and TDC, where their objective functions are convex, the objective function of the Greedy-GQ algorithm is non-convex. Moreover, the Greedy-GQ algorithm is also not a linear two-timescale stochastic approximation algorithm. Our techniques in this paper provide a general framework for finite-sample analysis of non-convex value-based reinforcement learning algorithms for optimal control.

355: No-regret Exploration in Contextual Reinforcement Learning

Track 2: Reinforcement learning Aditya Modi, Ambuj Tewari
[Paper] [Video (inline)] [Video (new tab)]

We consider the recently proposed reinforcement learning (RL) framework of Contextual Markov Decision Processes (CMDP), where the agent interacts with a (potentially adversarial) sequence of episodic tabular MDPs. In addition, a context vector determining the MDP parameters is available to the agent at the start of each episode, thereby allowing it to learn a context-dependent near-optimal policy. In this paper, we propose a no-regret online RL algorithm in the setting where the MDP parameters are obtained from the context using generalized linear mappings (GLMs). We propose and analyze optimistic and randomized exploration methods which make (time and space) efficient online updates. The GLM based model subsumes previous work in this area and also improves previous known bounds in the special case where the contextual mapping is linear. In addition, we demonstrate a generic template to derive confidence sets using an online learning oracle and give a lower bound for the setting.

424: Dueling Posterior Sampling for Preference-Based Reinforcement Learning

Track 2: Reinforcement learning Ellen Novoseller, Yibing Wei, Yanan Sui, Yisong Yue, Joel Burdick
[Paper] [Video (inline)] [Video (new tab)]

In preference-based reinforcement learning (RL), an agent interacts with the environment while receiving preferences instead of absolute feedback. While there is increasing research activity in preference-based RL, the design of formal frameworks that admit tractable theoretical analysis remains an open challenge. Building upon ideas from preference-based bandit learning and posterior sampling in RL, we present DUELING POSTERIOR SAMPLING (DPS), which employs preference-based posterior sampling to learn both the system dynamics and the underlying utility function that governs the preference feedback. As preference feedback is provided on trajectories rather than individual state-action pairs, we develop a Bayesian approach for the credit assignment problem, translating preferences to a posterior distribution over state-action reward models. We prove an asymptotic Bayesian no-regret rate for DPS with a Bayesian linear regression credit assignment model. This is the first regret guarantee for preference-based RL to our knowledge. We also discuss possible avenues for extending the proof methodology to other credit assignment models. Finally, we evaluate the approach empirically, showing competitive performance against existing baselines.

500: Provably Efficient Third-Person Imitation from Offline Observation

Track 2: Reinforcement learning Aaron Zweig, Joan Bruna
[Paper] [Video (inline)] [Video (new tab)]

Domain adaptation in imitation learning represents an essential step towards improving generalizability. However, even in the restricted setting of third-person imitation where transfer is between isomorphic Markov Decision Processes, there are no strong guarantees on the performance of transferred policies. We present problem-dependent, statistical learning guarantees for third-person imitation from observation in an offline setting, and a lower bound on performance in the online setting.

509: Improved Vector Pruning in Exact Algorithms for Solving POMDPs

Track 2: Reinforcement learning Eric Hansen, Thomas Bowman
[Paper] [Video (inline)] [Video (new tab)]

Exact dynamic programming algorithms for solving partially observable Markov decision processes (POMDPs) rely on a subroutine that removes, or “prunes,” dominated vectors from vector sets that represent piecewise-linear and convex value functions. The subroutine solves many linear programs, where the size of the linear programs is proportional to both the number of undominated vectors in the set and their dimension, which severely limits scalability. Recent work improves the performance of this subroutine by limiting the number of constraints in the linear programs it solves by incrementally generating relevant constraints. In this paper, we show how to similarly limit the number of variables. By reducing the size of the linear programs in both ways, we further improve the performance of exact algorithms for POMDPs, especially in solving problems with larger state spaces.

282: Slice Sampling for General Completely Random Measures

Track 3: Structure learning Peiyuan Zhu, Alexandre Bouchard-Cote, Trevor Campbell
[Paper] [Video (inline)] [Video (new tab)]

Completely random measures provide a principled approach to creating flexible unsupervised models, where the number of latent features is infinite and the number of features that influence the data grows with the size of the data set. Due to the infinity the latent features, posterior inference requires either marginalization---resulting in dependence structures that prevent efficient computation via parallelization and conjugacy---or finite truncation, which arbitrarily limits the flexibility of the model. In this paper we present a novel Markov chain Monte Carlo algorithm for posterior inference that adaptively sets the truncation level using auxiliary slice variables, enabling efficient, parallelized computation without sacrificing flexibility. In contrast to past work that achieved this on a model-by-model basis, we provide a general recipe that is applicable to the broad class of completely random measure-based priors. The efficacy of the proposed algorithm is evaluated on several popular nonparametric models, demonstrating a higher effective sample size per second compared to algorithms using marginalization as well as a higher predictive performance compared to models employing fixed truncations.

433: Permutation-Based Causal Structure Learning with Unknown Intervention Targets

Track 3: Structure learning Chandler Squires, Yuhao Wang, Caroline Uhler
[Paper] [Video (inline)] [Video (new tab)]

We consider the problem of estimating causal DAG models from a mix of observational and interventional data, when the intervention targets are partially or completely unknown. This problem is highly relevant for example in genomics, since gene knockout technologies are known to have off-target effects. We characterize the interventional Markov equivalence class of DAGs that can be identified from interventional data with unknown intervention targets. In addition, we propose a provably consistent algorithm for learning the interventional Markov equivalence class from such data. The proposed algorithm greedily searches over the space of permutations to minimize a novel score function. The algorithm is nonparametric, which is particularly important for applications to genomics, where the relationships between variables are often non-linear and the distribution non-Gaussian. We demonstrate the performance of our algorithm on synthetic and biological datasets. Links to an implementation of our algorithm and to a reproducible code base for our experiments can be found at https://uhlerlab.github.io/causaldag/utigsp.

253: Anchored Causal Inference in the Presence of Measurement Error

Track 3: Structure learning Basil Saeed, Anastasiya Belyaeva, Yuhao Wang, Caroline Uhler
[Paper] [Video (inline)] [Video (new tab)]

We consider the problem of learning a causal graph in the presence of measurement error. This setting is for example common in genomics, where gene expression is corrupted through the measurement process. We develop a provably consistent procedure for estimating the causal structure in a linear Gaussian structural equation model from corrupted observations on its nodes, under a variety of measurement error models. We provide an estimator based on the method-of-moments, which can be used in conjunction with constraint-based causal structure discovery algorithms. We prove asymptotic consistency of the procedure and also discuss finite-sample considerations. We demonstrate our method’s performance through simulations and on real data, where we recover the underlying gene regulatory network from zero-inflated single-cell RNA-seq data.

474: Model-Augmented Conditional Mutual Information Estimation for Feature Selection

Track 3: Structure learning Alan Yang, AmirEmad Ghassami, Maxim Raginsky, Negar Kiyavash, Elyse Rosenbaum
[Paper] [Video (inline)] [Video (new tab)]

Markov blanket feature selection, while theoretically optimal, is generally challenging to implement. This is due to the shortcomings of existing approaches to conditional independence (CI) testing, which tend to struggle either with the curse of dimensionality or computational complexity. We propose a novel two-step approach which facilitates Markov blanket feature selection in high dimensions. First, neural networks are used to map features to low-dimensional representations. In the second step, CI testing is performed by applying the $k$-NN conditional mutual information estimator to the learned feature maps. The mappings are designed to ensure that mapped samples both preserve information and share similar information about the target variable if and only if they are close in Euclidean distance. We show that these properties boost the performance of the $k$-NN estimator in the second step. The performance of the proposed method is evaluated on both synthetic and real data.

446: Learning LWF Chain Graphs: A Markov Blanket Discovery Approach

Track 3: Structure learning Mohammad Ali Javidian, Marco Valtorta, Pooyan Jamshidi
[Paper] [Video (inline)] [Video (new tab)]

This paper provides a graphical characterization of Markov blankets in chain graphs (CGs) under the Lauritzen-Wermuth-Frydenberg (LWF) interpretation. The characterization is different from the well-known one for Bayesian networks and generalizes it. We provide a novel scalable and sound algorithm for Markov blanket discovery in LWF CGs and prove that the Grow-Shrink algorithm, the IAMB algorithm, and its variants are still correct for Markov blanket discovery in LWF CGs under the same assumptions as for Bayesian networks. We provide a sound and scalable constraint-based framework for learning the structure of LWF CGs from faithful causally sufficient data and prove its correctness when the Markov blanket discovery algorithms in this paper are used. Our proposed algorithms compare positively/competitively against the state-of-the-art LCD (Learn Chain graphs via Decomposition) algorithm, depending on the algorithm that is used for Markov blanket discovery. Our proposed algorithms make a broad range of inference/learning problems computationally tractable and more reliable because they exploit locality.

311: Faster algorithms for Markov equivalence

Track 3: Structure learning Zhongyi Hu, Robin Evans
[Paper] [Video (inline)] [Video (new tab)]

Maximal ancestral graphs (MAGs) have many desirable properties; in particular they can fully describe conditional independences from directed acyclic graphs (DAGs) in the presence of latent and selection variables. However, different MAGs may encode the same conditional independences, and are said to be \emph{Markov equivalent}. Thus identifying necessary and sufficient conditions for equivalence is essential for structure learning. Several criteria for this already exist, but in this paper we give a new non-parametric characterization in terms of the heads and tails that arise in the parameterization for discrete models. We also provide a polynomial time algorithm ($O(ne^{2})$, where $n$ and $e$ are the number of vertices and edges respectively) to verify equivalence. Moreover, we extend our criterion to ADMGs and summary graphs and propose an algorithm that converts an ADMG or summary graph to an equivalent MAG in polynomial time ($O(n^{2}e)$). Hence by combining both algorithms, we can also verify equivalence between two summary graphs or ADMGs.

273: Robust contrastive learning and nonlinear ICA in the presence of outliers

Track 3: Structure learning Hiroaki Sasaki, Takashi Takenouchi, Ricardo Monti, Aapo Hyvarinen
[Paper] [Video (inline)] [Video (new tab)]

Nonlinear independent component analysis (ICA) is a general framework for unsupervised representation learning, and aimed at recovering the latent variables in data. Recent practical methods perform nonlinear ICA by solving classification problems based on logistic regression. However, it is well-known that logistic regression is vulnerable to outliers, and thus the performance can be strongly weakened by outliers. In this paper, we first theoretically analyze nonlinear ICA models in the presence of outliers. Our analysis implies that estimation in nonlinear ICA can be seriously hampered when outliers exist on the tails of the (noncontaminated) target density, which happens in a typical case of contamination by outliers. We develop two robust nonlinear ICA methods based on the $\gamma$-divergence, which is a robust alternative to the KL-divergence in logistic regression. The proposed methods are theoretically shown to have desired robustness properties in the context of nonlinear ICA. We also experimentally demonstrate that the proposed methods are very robust and outperform existing methods in the presence of outliers. Finally, the proposed method is applied to ICA-based causal discovery and shown to find a plausible causal relationship on fMRI data.

117: Statistically Efficient Greedy Equivalence Search

Track 3: Structure learning Max Chickering
[Paper] [Video (inline)] [Video (new tab)]

We establish the theoretical foundation for statistically efficient variants of the Greedy Equivalence Search algorithm. If each node in the generative structure has at most $k$ parents, we show that in the limit of large data, we can recover that structure using greedy search with operator scores that condition on at most $k$ variables. We present simple synthetic experiments that compare a backward-only variant of the new algorithm to GES using finite data, showing increasing benefit of the new algorithm as the complexity of the generative model increases.


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# How can I write a genetic programming algorithm, given that the Halting problem is unsolvable?

I am learning genetic programming and to practice I want to write a simple algorithm which evolves a program that solves a simple function (say, square root). I intend to represent programs as abstract syntax trees.

However, one of the functors is the while loop. Of course, in assesting a tree's fitness, I have to evaluate the program: but the halting problem is unsolvable. How can I tell if a given tree stops? Of course I can't, so what are some practicals ways to approach this problem?

Should I make my simple tree-language not turing complete? Or maybe give a timeout to each tree?

## 1 Answer

It is very unlikely indeed that finiteness is truly what you are looking for. Would you really be happy with a program that takes $$10^{10^{40}}$$ steps to complete?

If I am right, "non-haltingness" is not the real quality you are looking for. It is "halting within a reasonable time".

The best way to express this is not as a pure timeout, but by defining a measure of utility which declines with an increasing number of steps. That will give your evolutionary mechanism a steady "push" in the right direction. As for whether it is $$1/n$$, $$1/\sqrt n$$, $$2^{-n/c}$$, that is something you can decide.

Of course in practice a program that takes more than a certain number of steps is going to carry on going for ever. But you can encapsulate this intuition into the algorithm by saying that if, after $$n$$ steps, the overall utility has declined to a point when it is too low to be worth considering and it is never going to increase if you carry on stepping. Thus the timeout is still there, but as an implementation detail and not as a primary principle-in-itself.

format: plain


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# AmorphousLayer¶

class AmorphousLayer(material_specifications, length, melt_temperature=None, melt_time=None, quench_temperature=None, quench_rate=None, time_step=None)

Class to represent an amorphous layer material in MultilayerBuilder.

The amorphous layer is generated using a melt-quench MD procedure. First the system is melted and equilibrated at a high temperature. Then it is quenched from the melt_temperature to the quench_temperature at a rate of quench_rate. Then the system is quenched to zero Kelvin using a geometry optimization.

Parameters: material_specifications (MaterialSpecifications.) – A descriptor of a bulk material, plus the information needed to perform the calculations. length (PhysicalQuantity of type length) – The target length (thickness) of the layer. A length of zero will generate the smallest possible layer. melt_temperature (PhysicalQuantity of type temperature) – The temperature used during the melt. Default: 5000*Kelvin melt_time (PhysicalQuantity of type time) – The length of the melt MD simulation. Default: 40*picosecond quench_temperature (PhysicalQuantity of type temperature) – The target temperature during the quenching MD simulation. Default: 600*Kelvin quench_rate (PhysicalQuantity of type temperature per time) – The rate of temperature change during the quenching. This rate will be used to calculate the length of the quenching MD simulation. Default: 80*Kelvin/picosecond time_step (PhysicalQuantity of type time) – The MD time step.
length()
Returns: The target length (thickness) of the layer. PhysicalQuantity of type length
materialSpecifications()
Returns: The descriptor of a bulk material, plus the information needed to perform the calculations. MaterialSpecifications
meltTemperature()
Returns: The temperature used during melting. PhysicalQuantity of type temperature
meltTime()
Returns: The length of the melting MD simulation. PhysicalQuantity of type time
quenchRate()
Returns: The rate of temperature change during the quenching. PhysicalQuantity of type temperature per time
quenchTemperature()
Returns: The target temperature during the quenching MD simulation. PhysicalQuantity of type temperature
timeStep()
Returns: The MD time step. PhysicalQuantity of type time


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Address 100 N Main Ave, Lovington, NM 88260 (575) 374-2590 http://kannenterprisesonline.com

# monte carlo error simulation Maljamar, New Mexico

Based on these plots, Table 4 also provides the projected number of replications, R+, required to reduce the percent bias MCE to 0.05 or 0.005 for each of the four 2.5th Not the answer you're looking for? An estimate with zero error causes the weighted average to break down and must be handled separately. Monte Carlo Error Estimation for Multivariate Markov Chains.

The stratified sampling algorithm concentrates the sampling points in the regions where the variance of the function is largest thus reducing the grand variance and making the sampling more effective, as Lepage, VEGAS: An Adaptive Multi-dimensional Integration Program, Cornell preprint CLNS 80-447, March 1980 J. Hierarchical Spatio-Temporal Mapping of Disease Rates. Efron and Tibshirani 1993; Robert and Casella 2004; Givens and Hoeting 2005), in many cases little can be done to substantially reduce the time needed to run even a single iteration,

Random sampling of the integrand can occasionally produce an estimate where the error is zero, particularly if the function is constant in some regions. Third, viewed as statistical or mathematical experiments (Ripley 1987), it could be argued that to aid in the interpretation of results, simulation studies always should be accompanied by some assessment of This routines uses the VEGAS Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, particle filter), and mean field particle methods.

For those that did report R, we see wide variability in the number of replications used. Etymologically, why do "ser" and "estar" exist? It samples points from the probability distribution described by the function |f| so that the points are concentrated in the regions that make the largest contribution to the integral. The Annals of Statistics. 1986;14:1453–1462.Hauck WW, Anderson S.

Recursive stratified sampling is a generalization of one-dimensional adaptive quadratures to multi-dimensional integrals. This is in contrast to most scientific studies, in which the reporting of uncertainty (usually in the form of standard errors, p-values, and CIs) is typically insisted on. Here we consider a static simulation framework and consider uncertainty specifically related to the choice of simulation sample size, R.2.2 Illustrative ExampleTo illustrate MCE, consider a simple example in the context In brief, suppose that the target quantity has an integral representation given byφ=∫φ(x)fX(x)dx.Given a sample of R replicates generated under the design fX(·), X = {X1, X2, …, XR}, a natural

John Wiley & Sons. ^ Veach, Eric; Guibas, Leonidas J. (1995-01-01). "Optimally Combining Sampling Techniques for Monte Carlo Rendering". References R. Geyer CJ. “Practical Markov Chain Monte Carlo” (with discussion) Statistical Science. 1992;7:473–483.Givens GH, Hoeting JA. In particular, stratified sampling - dividing the region in sub-domains -, and importance sampling - sampling from non-uniform distributions - are two of such techniques.

Monte Carlo Methods in Statistical Physics. Multiple and Adaptive Importance Sampling When different proposal distributions, p n ( x ¯ ) {\displaystyle p_{n}({\overline {\mathbf {x} }})} , n = 1 , … , N , {\displaystyle n=1,\ldots The efficiency of VEGAS depends on the validity of this assumption. While other algorithms usually evaluate the integrand at a regular grid,[1] Monte Carlo randomly choose points at which the integrand is evaluated.[2] This method is particularly useful for higher-dimensional integrals.[3] There

Contents 1 Overview 1.1 Example 1.2 Wolfram Mathematica Example 2 Recursive stratified sampling 2.1 MISER Monte Carlo 3 Importance sampling 3.1 VEGAS Monte Carlo 3.2 Importance sampling algorithm 3.3 Multiple and From Table 3, we see that in addition to directly quantifying uncertainty, we also could use the results to form interval estimates. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is within those error bars. The result and its error estimate are based on a weighted average of independent samples.

The VEGAS algorithm computes a number of independent estimates of the integral internally, according to the iterations parameter described below, and returns their weighted average. Robert, CP; Casella, G (2004). New York: Chapman & Hall; 1993. In most cases the % error of mean is less than 5% but the error of std goes up to 30%.

Hammersley, D.C. VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling.[7] The integration region is divided into a number of "boxes", with each box getting a fixed Elements of Computational Statistics. These individual values and their error estimates are then combined upwards to give an overall result and an estimate of its error.

While the naive Monte Carlo works for simple examples, this is not the case in most problems. Robert C, Casella G. Each article was downloaded electronically, and a search was performed for any of the following terms: “bootstrap,” “dataset,” “Monte Carlo,” “repetition,” “replication,” “sample,” and “simulation.” In addition, when indicated by the Conditional skip instructions of the PDP-8 What's the longest concertina word you can find?

As far as I see the required number of simulations for any allowed percentage error $E$ (e.g., 5) is  n = \left\{\frac{100 \cdot z_c \cdot \text{std}(x)}{E \cdot \text{mean}(x)} \right\}^2 , ISSN1070-9908. ^ Cappé, O.; Guillin, A.; Marin, J. A. Why we don't have macroscopic fields of Higgs bosons or gluons?

Hall P. REPORTING OF SIMULATION STUDIESThe results given in Table 1 serve to illustrate two key points. The Effect of Monte Carlo Approximation on Coverage Error of Double-Bootstrap Confidence Intervals. Furthermore, the evaluation of (7) is based on a single simulation of length R, and its accuracy as an estimator of MCE relies on the availability of sufficient replications to get

Digital Signal Processing. Numerical Recipes: The Art of Scientific Computing (3rd ed.). We see that for R = 1000, the estimation of percent bias for the MLE β̂X is subject to substantial between-simulation variation; across the M simulations, point estimates φ^Rb range between Although we do not give detailed results here, we found that MCE was greater for φ^Rb when P(X = 1) = 0.1 compared to when P(X = 1) = 0.3, likely

Please try the request again. According to the central limit theorem, these values whould be normally dstributed around a mean . Here we build on both the asymptotic and resampling methods to develop a novel graphical approach for characterizing MCE, as a function of R. However, we should expect that the error decreases with the number of points , and the quantity defines by (271) does not.


text:
# Justification of algebraic manipulation of infinitesimals

As an engineering student, I regularly see people making arguments like this:

Consider a rectangle of dimensions $x\times 4x$. If we make $x$ bigger by a small quantity $dx$ then this will make $4x$ bigger by $4\cdot dx$ so the area of that $x \times 4x$ rectangle will change from $4x^2$ to $$(x+dx)(4x+4dx)=4(x^2+2x\cdot dx+(dx)^2)\approx4x^2+8x\cdot dx$$ with the final step justified because $dx$ is a 'small' quantity so $(dx)^2$ will be so small as to be ignorable in some mathematically rigorous way. Thus the change in area $dA$ would be $8x\cdot dx$.

Arguments like this are very common. Another random example would be in Wikipedia's proof of the brachistochrone problem which starts with the statement

$$ds^2=dx^2+dy^2$$

and proceeds to manipulate these infinitesimals as if they were ordinary constants or variables.

I'm wondering if there's a simple, analytically rigorous justification for all of this manipulation. While I feel perfectly comfortable with the idea of the derivative of a function (considered as a limit), I've never seen a similar, rigorous justification for the algebraic manipulation of infinitesimals and the cancellation of 'small' terms (like $(dx)^2$). Any thoughts or help would be appreciated.

Thankyou

-

I will just throw a few buzzwords at you :-)

The mathematically precise concept of "infinitesimal" is called "differential form". If we fix the Euclidean plane $\mathbb{R}^2$ and think of it as a "differentiable manifold", then every point in the plane has a tangential space that is "isomorph to" (or another copy of) $\mathbb{R}^2$. If we further fix a cartesian coordinate system with coordinates x and y, then a differential form is a gadget that assigns for every point p with coordinates $(x_p, y_p)$ to a tangent vector at that point a real number.

If you think of attaching a vector pointing upwards with the length of on, (0, 1), at every point in the plane, then in our example dx would spit out 0 at every point and dy would spit out 1 at every point.

This is the starting point for modern abstract "coordinate free" differential geometry.

From my experience, these concepts are usually not easy to understand for beginners, so don't worry if you don't understand everything on a first reading.

First note that you don't need the concept of a "differential form" to understand your first example:

Take a rectangle with side lengths a and b, then we have a function that gives the area, $$f: \mathbb{R}^2 \to \mathbb{R}$$ $$f: (a, b) \mapsto ab$$ If you increase both of the coordinates by h, then this is just a directional derivative, and since $f$ is differentiable we know that $$f(a + h, b + h) - f(a, b) = df*(h, h) + o(h^2)$$ holds. Here "dx" is just short hand notation for both "h" and "we know that f is differentiable and therefore that the remainder of the right hand side is $o(h^2)$, that is for smaller and smaller h the linear approximation gets better and better". The linear approximation is by definition given by applying the differential $df$ of $f$ to the vector $(h, h)$.

The second example is a little bit more complicated, here we'll really need the concept of "differential forms". To be mathematically precise, we'd have to write $$ds^2 := dx \otimes dx + dy \otimes dy$$ That is, the left hand side is defined by the right hand side, and the right hand side consists for a fixed point of elements of the tensor product of the cotangential space $T^*_pM$ with itself.

This means this gadget eats two tangential vectors on any tangent space and spits out a real number, and this operation is bilinear (linear in both input variables). So, if you fix a point on the plane, you get an element of the space $$T^*_pM \otimes T^*_pM$$ which has an algebraic structure that can be used.

-
Euclid's Elements in book V, definition 4 states that two magnitudes are comparable when they are capable of exceeding one another when multiplied sufficiently many times. Leibniz wrote in 1695 that his infinitesimals violate Euclid V.4. In modern terminology, this means that Leibnizian infinitesimals are "non-Archimedean" (this certainly does not mean that Leibniz was either familiar with or "anticipated" modern non-archimedean theories). At any rate the historic meaning of infinitesimal in analysis is clearly Leibniz's... –  user72694 Dec 9 '14 at 15:45
... Now differential forms a a wonderful thing but asserting that an infinitesimal "is" a differential form is a bit misleading. –  user72694 Dec 9 '14 at 15:46

You will hear a lot about differentials, exterior algebra and maybe even about nonstandard analysis. But the sad fact is: In over 300 years of calculus we have not come up with an easy answer to your question. All one can say is: If in a particular case "all of this manipulation" leads to a correct result, then there is also an "analytically rigorous justification" for it.

If done with professional expertise, dealing with "differentials" of all sorts in a lighthanded way is definitely a successful heuristic technique, especially in an environment where one doesn't care so much about $\epsilon$'s and $\delta$'s. But some care is needed: When you argue about the area under a curve $\gamma\!: y=f(x)$ you don't have to bother about the increase of $f$ in an interval of length $dx$, but if you want to derive a formula for the length of $\gamma$, then this little increase of $y$ plays a decisive rôle.

-
'All one can say is...' No, this is not all one can say. '[...] then there is also an "analytically rigorous justification" for it.' You seem to be using "rigorous" in contradistinction to "nonstandard analysis." There is nothing nonrigorous about nonstandard analysis. –  Ben Crowell Jan 9 '12 at 3:11
@Ben Crowell: Of course not. What I'm saying is that the various frameworks invented to salvage our everyday handling of "infinitesimals" are no easy answers to the question posed by the OP. –  Christian Blatter Jan 9 '12 at 9:09

There are various ways of answering your question. Tim van Beek has given one in terms of differential forms. Another nice way of looking at it is using nonstandard analysis (NSA). The nice thing about NSA, to my taste, is that it allows you say things the way that Leibniz and Gauss and Euler found it so useful to say them (i.e., with dx's and dy's), and it also allows you to be certain that what you're doing is logically rigorous. NSA doesn't have to be hard and scary. Jerome Keisler wrote a very nice freshman calculus book using NSA, now available online for free: http://www.math.wisc.edu/~keisler/calc.html My own book, using a similar approach, is here: http://www.lightandmatter.com/calc/ This is also a very nice treatment: http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm

The basic idea of NSA is that just as we expanded the integers to the rationals, and the rationals to the reals, we go one more step and expand the reals to the hyperreals. The hyperreals are a number system that includes the reals, but that also includes infinitesimals. The way you know whether you're doing something logically correct is that if you write down all of the elementary axioms of the real number system (x+y=y+x, etc.), then all of those are true in the hyperreals. "Elementary" means axioms that only say things like "for every number...," not ones that say "for every set of numbers..."

-
Thanks, I had wondered if NSA would pop up here, though I've only heard of them fleetingly. I'll finish Spivak, maybe do some conventional analysis, then get back to this :) –  tom Jan 9 '12 at 11:35

This stems from simple theorem regarding the change of variables, also called substitution, used for integration: $\int {f\left( {g\left( t \right)} \right)g'\left( t \right)dt} = \int {f\left( y \right)dy}$. If we put $g\left( t \right) = y$ in $\int {f\left( {g\left( t \right)} \right)g'\left( t \right)dt}$, we have $\int {f\left( y \right)\frac{{dy}} {{dt}}dt} = \int {f\left( y \right)dy}$ which we know is true from the theorem. So, when we integrate, it's allowed for us to think that, when $y = g\left( t \right)$, then $dy = g'\left( t \right)dt$.

In your example, you have $y\left( x \right) = 4{x^2}$, and want to calculate $y'\left( x \right)$. We have $y'\left( x \right) = \frac{{dy}}{{dx}} = 8x$ (you forgot to multiply by $x$ in your example). Now, we use abuse of notation to write $dy = 8xdx$.

Similar arguments can be made with integration of multivariate functions. Such consideration gives rise to differential forms.

-
thanks, I fixed it. So are you saying that all of the manipulation of infinitesimals I demonstrated can be justified using the substitution rule? –  tom Jan 9 '12 at 1:40
Unfortunately, no. As Tim has diligently written, when there is more than one variable involved, differential forms come into play. But for univariate functions, that's the background for such abuse of notation. Another justification for using "d" notation is <en.wikipedia.org/wiki/…; , which can be understood that it is "valid" to cancel out terms in expressions like this $\frac{{dz}} {{dy}}\frac{{dy}} {{dx}} = \frac{{dz}} {{dx}}$ –  Alen Jan 9 '12 at 1:44
That does make some sense though - when we go from $\frac{dy}{dx} = f(y)g(x)$ to $\frac{dy}{f(y)}=g(x)\cdot dx$ we are really just applying the substitution rule in diguise –  tom Jan 9 '12 at 2:36

There is a justification for this method: we choose a value for dx so that dx^2 is indiscernible whereas 8x.dx isn't - it's called de facto calculus. See my answer here - Could we assign a numerical value to an infinitesimal? This justification is relative - it would fail for larger values of dx and too much precision in measuring the area. In smooth infinitesimal analysis however this kind of discarding is generally acceptable.

-


text:
## Installation

# install from bioconductor
if (!requireNamespace("BiocManager", quietly = TRUE))
install.packages("BiocManager")
BiocManager::install("cosmosR")

# install the development version from GitHub
# install.packages("remotes")
remotes::install_github("saezlab/cosmosR")

## Introduction

COSMOS (Causal Oriented Search of Multi-Omic Space) is a method that integrates phosphoproteomics, transcriptomics, and metabolomics data sets. COSMOS leverages extensive prior knowledge of signaling pathways, metabolic networks, and gene regulation with computational methods to estimate activities of transcription factors and kinases as well as network-level causal reasoning. This pipeline can provide mechanistic explanations for experimental observations across multiple omic data sets.

library(cosmosR)

## Tutorial section: signaling to metabolism

In this part, we can set up the options for the CARNIVAL run, such as timelimit and min gap tolerance.

The user should provide a path to its CPLEX/cbc executable

You can check the CARNIVAL_options variable to see all possible options that can be adjusted

In this example, we will use the built-in solver lpSolve. User should be aware that lpSolve should ONLY be used for TESTS. To obtain meaningful results, best solver is cplex, or cbc if not possible.

CARNIVAL_options <- cosmosR::default_CARNIVAL_options()
# CARNIVAL_options$solverPath <- "~/Documents/cplex" # CARNIVAL_options$solver <- "cplex" #or cbc
CARNIVAL_options$solver <- "lpSolve" #or cbc CARNIVAL_options$timelimit <- 3600
CARNIVAL_options$mipGAP <- 0.05 CARNIVAL_options$threads <- 2

In the next section, we prepare the input to run cosmosR The signaling inputs are the result of footprint based TF and kinase activity estiamtion For more info on TF activity estiamtion from transcriptomic data, see:https://github.com/saezlab/transcriptutorial (Especially chapter 4)

Here we use of toy PKN, to see the full meta PKN, you can load it with data(meta_network).

The metabolites in the prior knowledge network are identified as XMetab_PUBCHEMidcompartment or XMetab_BIGGidcompartment or example “XMetab_6804m”. The compartment code is the BIGG model standard (r, c, e, x, m, l, n, g). Thus we will first need to map whatever identifer for metabolite the data has to the one of the network. Genes are identified as XENTREZid (in the signaling part of network) or XGene####__ENTREZid (in the reaction network part of network)

The maximum network depth will define the maximum number of step downstream of kinase/TF COSMOS will look for deregulated metabolites. Good first guess for max depth could be around 6 (here is 15 for the toy dataset)

The differential experession data is used to filter out wrong TF-target interactions in this context after a pre-optimisation.

The list of genes in the differential expression data will also be used as a reference to define which genes are expressed or not (all genes in the diff_expression_data are considered expressed, and genes that are no in diff_expression_data are removed from the network)

data(toy_network)
data(toy_signaling_input)
data(toy_metabolic_input)
data(toy_RNA)
test_for <- preprocess_COSMOS_signaling_to_metabolism(meta_network = toy_network,
signaling_data = toy_signaling_input,
metabolic_data = toy_metabolic_input,
diff_expression_data = toy_RNA,
maximum_network_depth = 15,
remove_unexpressed_nodes = TRUE,
CARNIVAL_options = CARNIVAL_options
)
## [1] "COSMOS: all 2 signaling nodes from data were found in the meta PKN"
## [1] "COSMOS: all 3 metabolic nodes from data were found in the meta PKN"
## [1] "COSMOS: 4653 of the 15919 genes in expression data were found as transcription factor target"
## [1] "COSMOS: 4653 of the 5303 transcription factor targets were found in expression data"
## [1] "COSMOS: removing unexpressed nodes from PKN..."
## [1] "COSMOS: 0 interactions removed"
## [1] "COSMOS: removing nodes that are not reachable from inputs within 15 steps"
## [1] "COSMOS: 10 from  19 interactions are removed from the PKN"
## [1] "COSMOS: 1 input/measured nodes are not in PKN any more: XMetab__190___c____ and 0 more."
## [1] "COSMOS: removing nodes that are not observable by measurements within 15 steps"
## [1] "COSMOS: 0 from  9 interactions are removed from the PKN"
## [1] "COSMOS:  1 interactions are removed from the PKN based on consistency check between TF activity and gene expression"
## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers."
## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers."
## [1] "COSMOS:  0 interactions are removed from the PKN based on consistency check between TF activity and gene expression"
## [1] "COSMOS: all 2 metabolic nodes from data were found in the meta PKN"
## [1] "COSMOS: 4653 of the 15919 genes in expression data were found as transcription factor target"
## [1] "COSMOS: 4653 of the 5303 transcription factor targets were found in expression data"

In this part, we can set up the options for the actual run, such as timelimit and min gap tolerance.

The running time should be much higher here than in pre-optimisation. You cna increase the number of threads to use if you have many available CPUs.

CARNIVAL_options$timelimit <- 14400 CARNIVAL_options$mipGAP <- 0.05
CARNIVAL_options$threads <- 2 This is where cosmosR run. test_result_for <- run_COSMOS_signaling_to_metabolism(data = test_for, CARNIVAL_options = CARNIVAL_options) ## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers." ## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers." Finally, we process the results of the first cosmosR run, to translate gene names and metabolites name. data(metabolite_to_pubchem) data(omnipath_ptm) test_result_for <- format_COSMOS_res(test_result_for, metab_mapping = metabolite_to_pubchem, measured_nodes = unique(c(names(toy_metabolic_input), names(toy_signaling_input))), omnipath_ptm = omnipath_ptm) ## Tutorial section: metabolism to signaling Before we run the metabolism to signaling part, we need to prepare again the inputs. CARNIVAL_options$timelimit <- 3600
CARNIVAL_options$mipGAP <- 0.05 CARNIVAL_options$threads <- 2

Now that the correct time is set up for the pre-optimisation run, we can prepare the inputs.

test_back <- preprocess_COSMOS_metabolism_to_signaling(meta_network = toy_network,
signaling_data = toy_signaling_input,
metabolic_data = toy_metabolic_input,
diff_expression_data = toy_RNA,
maximum_network_depth = 15,
remove_unexpressed_nodes = FALSE,
CARNIVAL_options = CARNIVAL_options

)
## [1] "COSMOS: all 2 signaling nodes from data were found in the meta PKN"
## [1] "COSMOS: all 3 metabolic nodes from data were found in the meta PKN"
## [1] "COSMOS: 4653 of the 15919 genes in expression data were found as transcription factor target"
## [1] "COSMOS: 4653 of the 5303 transcription factor targets were found in expression data"
## [1] "COSMOS: removing nodes that are not reachable from inputs within 15 steps"
## [1] "COSMOS: 1 from  19 interactions are removed from the PKN"
## [1] "COSMOS: 1 input/measured nodes are not in PKN any more: X2305 and 0 more."
## [1] "COSMOS: removing nodes that are not observable by measurements within 15 steps"
## [1] "COSMOS: 11 from  18 interactions are removed from the PKN"
## [1] "COSMOS: 2 input/measured nodes are not in PKN any more: XMetab__124886___c____, XMetab__6426851___c____ and 0 more."
## [1] "COSMOS:  0 interactions are removed from the PKN based on consistency check between TF activity and gene expression"
## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers."
## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers."
## [1] "COSMOS:  0 interactions are removed from the PKN based on consistency check between TF activity and gene expression"
## [1] "COSMOS: all 1 signaling nodes from data were found in the meta PKN"
## [1] "COSMOS: all 1 metabolic nodes from data were found in the meta PKN"
## [1] "COSMOS: 4653 of the 15919 genes in expression data were found as transcription factor target"
## [1] "COSMOS: 4653 of the 5303 transcription factor targets were found in expression data"

Then we can run cosmosR to connect metabolism to signaling. The running time here usually needs to be longer, as this problem seems to be harder to solve for CPLEX.

## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers."
## [1] "lpSolve does not scale well with large PKNs. This solver is mainly for testing purposes. To run COSMSO, we recommend using cplex, or cbc solvers."

Finally we can format the result of the backward run as well (same as for forward run)

test_result_back <- format_COSMOS_res(test_result_back,
metab_mapping = metabolite_to_pubchem,
measured_nodes = unique(c(names(toy_metabolic_input),
names(toy_signaling_input))),
omnipath_ptm = omnipath_ptm)

## Tutorial section: Merge forward and backward networks and visualise network

Here we simply take the union of forward and backward runs to create a full network solution lopping between signaling, gene-regulation and metabolism. Since there is an overlapp between the result network of forward and backward run, you may optionally want to check if there are any node sign that are incoherent in the overlapp between the two solutions.

full_sif <- as.data.frame(rbind(test_result_for[[1]], test_result_back[[1]]))
full_attributes <- as.data.frame(rbind(test_result_for[[2]], test_result_back[[2]]))

full_sif <- unique(full_sif)
full_attributes <- unique(full_attributes)

This function will generate a dynamic network plot centered on a given node of the network solution, and connecting it to measured nodes in the given range (here 5 steps).

network_plot <- display_node_neighboorhood(central_node = 'BCAT1',
sif = full_sif,
att = full_attributes,
n = 5)

network_plot

This network represent the flow of activities that can connect FOXM1 up-regulation with glutathione (CID 124886) accumulation. Here, FOXM1 can activate MYC, which in turn activate BCAT1. The activation of BCAT1 can lead to the increased production of glutamate (CID 33032), whioch in turn can be converted to glutathione GGT enzymes.

It is important to understand that each of this links are hypothetical. The come from a larger pool of potential molecular interactions present in multiple online databases and compiled in omnipath, STITCH and recon metabolic network. They exist in the literature and are interactions that are known to potentially exists in other experimental contexts. Thus, COSMOS compile all those potential interactions together and proposes a coherent set that can explain the data at hand. Here, such a set of mechanistic hypothesis is plotted as a network connecting FOXM1 and glutathione production.

Those links should however be considered only as potential mechanistic connections, and will need to be further confirmed experimentally.

sessionInfo()
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Catalina 10.15.7
##
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base
##
## other attached packages:
## [1] cosmosR_0.99.2
##
## loaded via a namespace (and not attached):
##   [1] segmented_1.3-4      Category_2.56.0      bitops_1.0-7
##   [4] fs_1.5.0             bit64_4.0.5          doParallel_1.0.16
##   [7] httr_1.4.2           rprojroot_2.0.2      tools_4.0.2
##  [10] bslib_0.2.4          utf8_1.2.1           R6_2.5.0
##  [13] KernSmooth_2.23-20   DBI_1.1.1            BiocGenerics_0.36.1
##  [16] colorspace_2.0-1     tidyselect_1.1.1     curl_4.3.1
##  [19] bit_4.0.4            compiler_4.0.2       cli_2.5.0
##  [22] textshaping_0.3.3    graph_1.68.0         Biobase_2.50.0
##  [25] CARNIVAL_1.2.0       desc_1.3.0           sass_0.3.1
##  [31] proxy_0.4-25         RBGL_1.66.0          rappdirs_0.3.3
##  [34] pkgdown_1.6.1        systemfonts_1.0.1    stringr_1.4.0
##  [37] digest_0.6.27        mixtools_1.2.0       rmarkdown_2.8
##  [40] pkgconfig_2.0.3      htmltools_0.5.1.1    bcellViper_1.26.0
##  [43] dbplyr_2.1.1         fastmap_1.1.0        htmlwidgets_1.5.3
##  [46] rlang_0.4.11         rstudioapi_0.13      RSQLite_2.2.7
##  [49] visNetwork_2.0.9     jquerylib_0.1.4      generics_0.1.0
##  [52] jsonlite_1.7.2       viper_1.24.0         dplyr_1.0.6
##  [55] RCurl_1.98-1.3       magrittr_2.0.1       Matrix_1.3-3
##  [58] Rcpp_1.0.6           munsell_0.5.0        S4Vectors_0.28.1
##  [61] fansi_0.4.2          lifecycle_1.0.0      stringi_1.5.3
##  [64] yaml_2.2.1           UniProt.ws_2.30.0    MASS_7.3-54
##  [67] BiocFileCache_1.14.0 org.Hs.eg.db_3.12.0  grid_4.0.2
##  [70] blob_1.2.1           parallel_4.0.2       crayon_1.4.1
##  [73] lattice_0.20-44      splines_4.0.2        annotate_1.68.0
##  [76] hms_1.0.0            knitr_1.33           dorothea_1.2.2
##  [79] pillar_1.6.0         igraph_1.2.6         lpSolve_5.6.15
##  [82] codetools_0.2-18     stats4_4.0.2         XML_3.99-0.6
##  [85] glue_1.4.2           evaluate_0.14        vctrs_0.3.8
##  [88] foreach_1.5.1        gtable_0.3.0         purrr_0.3.4
##  [91] kernlab_0.9-29       assertthat_0.2.1     cachem_1.0.4
##  [94] ggplot2_3.3.3        xfun_0.22            xtable_1.8-4
##  [97] e1071_1.7-6          ragg_1.1.2           class_7.3-19
## [100] survival_3.2-11      tibble_3.1.1         iterators_1.0.13
## [103] AnnotationDbi_1.52.0 memoise_2.0.0        IRanges_2.24.1
## [106] ellipsis_0.3.2       GSEABase_1.52.1


text:
# Analytic countably splitting families

@article{Spinas2004AnalyticCS,
title={Analytic countably splitting families},
author={Otmar Spinas},
journal={Journal of Symbolic Logic},
year={2004},
volume={69},
pages={101 - 117}
}
• O. Spinas
• Published 1 March 2004
• Mathematics
• Journal of Symbolic Logic
Abstract A family A ⊆ (ω) is called countably splitting if for every countable F ⊆ [ω]ω, some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an Fσ splitting family that does not contain a closed splitting family.
18 Citations
Perfect set theorems
We study splitting, innitely often equal (ioe) and rening families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set
Hypergraphs and proper forcing
It turns out that many of the quotient posets are proper and for this broad class of posets, most fusion arguments and iteration preservation arguments can be replaced with simple combinatorial considerations concerning the hypergraphs.
On splitting trees
• Mathematics
• 2020
We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and
Ramsey type properties of ideals
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We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then
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Preface 1. Introduction 2. Background facts 3. Analytic equivalence relations and models of set theory 4. Classes of equivalence relations 5. Games and the Silver property 6. The game ideals 7.
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vol


text:
Electromagnetics - Method of Images ?

1. Jul 11, 2006

s0undmind

Electromagnetics - Method of Images ???

Hello all, I am in the process of preparing for a entrance examination. In the course of studying I came across this problem that I just can not figure out. Here it is:

A perfectly conducting plane is located in free space at x=4, and a uniform inifinte line charge of 40nC/m lies along the line x=6, y=3 (the intersection of these planes). Let V=0 at the conducting plane. At P(7,-1,5) find (a) V (potential at P); (b) E (electric field at P).

Answer: -316V ; -45.4ax V/m (ax is the unit vector in the x direction)

I have the answers as you can see but I have no idea on how to arrive to them. I have tried for hours. I would appreciate any help or comments.
THank you.

2. Jul 12, 2006

Staff: Mentor

Are you sure about the answer to (b)? It sure seems like there would be a y component to the E-field at (7,-1,5).

What equation do you use to calculate the Electric field due to a charge distribution? Once you have the E field, how do you calculate the voltage V?

3. Jul 15, 2006

s0undmind

Thank you for your reply. Well, I started to think about it again and I agree that there should be a y component. I guess it is a typo in the text especially since I was able to calculate a 45.4ax component (corresponding to the egative of x component in the answer) and a 99ay component. I added the electric field contributions from the uniform charge density(at x=6,y=3,z) and its image (at x=2,y=3,z) and obtained this answer.

For the Voltage, I performed an ugly line integral from a point on the conducting surface. I chose a point lying on a normal line from the conducting surface to P(7,-1,5) so that only x would vary in the integral.
After some manipulations and using the wolfram integrator I arrived at the answer of 343.5V, which is close but does not agree with the answer in the text and the sign is wrong. I wonder if there's an easier way to solve this problem. If you have any hints I would like to hear to them.

Thank you.

4. Aug 25, 2006

leright

I know this thread is old, but I would like to say that this question is poorly explained. A voltage is measured between two points. A voltage is simply a path integral of the E-field between two points (for electrostatic conditions the path taken to get from one point to the other does not matter since the field is conservative...the only thing that matters is the point you start at and the point you end at....YES, this is important). The problem, especially if it is an entrance exam problem, should have specified TWO points that you are finding the voltage between, and at what point you are starting at.

Now, if only one point is specified in the problem and you are to find the voltage there, the other point is taken to be infinity. This is known as an absolute potential. However, you still need to know if you are starting at infinity and moving a test charge to the point, or if you are starting at the point and moving the test charge to infinity. The "convention", which is not clearly explained, is that when finding absolute potentials you start at the point and move the positive test charge to infinity.

However, absolute potentials are a little bit ugly when dealing with the theoretical construct of the infinite plane, since the E-field due to an infinite plane of charge is constant everywhere in space, which would yield an absolute potential of -infinity.

EDIT: I am an idiot. It says V=0 at the infinte plane!!

Last edited: Aug 25, 2006


text:
The Fibonacci sequence is a sequence where the next term is the sum of the previous two terms. A Fibonacci number, Fibonacci sequence or Fibonacci series are a mathematical term which follow a integer sequence. When you divide any number in the sequence by the number right before it ��� Similar to all sequences, the Fibonacci sequence can also be evaluated with the help of a finite number ��� $\endgroup$ ��� Charles Jun 27 '14 at 13:31 His method is right, but he started at the wrong place. to calculate the first 12 values in the fibonacci number sequence 1 1 2 3 5 8 13' 'fibonacci series assembly language program april 30th, 2018 - fibonacci series assembly language program pdf free download here lab 6 fibonacci numbers ��� It really prints me the indices but without 0s. $\endgroup$ ��� ShreevatsaR Jun 27 '14 at 7:52 $\begingroup$ @ShreevatsaR: That's standard, just take the base-$\varphi$ logarithm of the number times $\sqrt5$ and round. Find the sum of the first thirty terms of the sequence: 1, 5, 6, 11, 17, 28 if the 30th term is 2888956 and the 31st term is 4674429 I figured the pattern was tn = tn-1 + tn-2 so the first term would be a, then b, a+b, a+2b basicly the Fibonacci Sequence. We can find the 20th Fibonacci number by calculating the Fibonacci sequence out to the 20th term, but that would... See full answer below. There are an infinite number of Fibonacci numbers. If n = 1, then it should return 1. I modified the above recursive approach to output the number of calls to the fib function for each number, and here are some values: #1: 1 #10: 67 #20: 8361 #30: 1028457 #40: 126491971 python program for nth fibonacci number; fibonacci with recursion; fibonnaci sequence recurisve java; nth fibonacci number python; Get the 30th number of Fibonacci sequence. Fibonacci rounding formula proof. Must use a recursive function to implement it. 3. The first two terms of the Fibonacci sequence are 0 followed by 1. Fibonacci sequence formula; Golden ratio convergence; Fibonacci sequence table On my machine, the above code starts going really slow at around the 30th fibonacci number, which is still only 6 digits long. The Fibonacci Numbers The numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ... Each Fibonacci number is the sum of the previous two Fibonacci numbers! It's the sum of the 29th and the 30th fibonacci numbers. F 0 = 0 F 1 = 1 F n = F n-1 + F n-2, if n>1 . Derivation of Binet's formula, which is a closed form solution for the Fibonacci numbers. Formula using fibonacci numbers. Method 1 ( Use recursion ) A simple method that is a direct recursive implementation mathematical recurrence relation given above. - There should be 5 columns of numbers per row, as shown in the sample canvas above. For n = 9 Output:34. Fibonacci numbers, the elements of the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, ���, each of which, after the second, is the sum of the two previous numbers. You may hear the term ���the Golden Number��� or the ���Golden Ratio��� associated with the Fibonacci sequence, and this refers to the ratio between the Fibonacci numbers . r = common ratio. 2. The Fibonacci sequence is a sequence F n of natural numbers defined recursively: . This free number sequence calculator can determine the terms (as well as the sum of all terms) of an arithmetic, geometric, or Fibonacci sequence. For n > 1, it should return F n-1 + F n-2. n = number of the term. How is the Binet's formula for Fibonacci reversed in order to find the index for a given Fibonacci number? Task. Solutions can be iterative or recursive (though recursive solutions are generally considered too slow and are mostly used as an exercise in recursion). 1000-digit Fibonacci number Published on Friday, 30th August 2002, 06:00 pm; Solved by 154878; Difficulty rating: 5%. So it have to print: 7 (fib(7) = 13) 0 (14 is not a fib number) 0 (9 is not a fib number) 5 (fib(5) = 5) The constraints are: 0 < a,b < 100 count of the numbers < 10000 and each number should be in the range [0, 1000000] I did the output. The Fibonacci sequence is defined by the recurrence relation: F n = F n���1 + F n���2, where F 1 = 1 and F 2 = 1. Prove the sum of the even Fibonacci numbers. Problem 25. In general, the n th term is given by f(n-1)+f(n-2) To understand this sequence, you might find it useful to read the Fibonacci Sequence tutorial over here. The Number Of Elementary''Fibonacci Series Algorithm and Flowchart Code with C April 27th, 2018 - Fibonacci Series algorithm and flowchart which can be used write and each subsequent number is the sum of in this series the n th term is the 30th Number in the Fibonacci Number Sequence = 514229 . ... 30th Fibonacci Number 31st Fibonacci Number 32nd Fibonacci Number 33rd Fibonacci Number 34th Fibonacci Number 35th Fibonacci Number 36th Fibonacci Number 37th Fibonacci Number 38th Fibonacci Number 39th Fibonacci Number The arithmetic equations are written on specific notations, for deep learning & understanding of scientific notation you can use Scientific Notation Calculator. Hot Network Questions The 20th Fibonacci number is 6,765. Enter value of n:20 20th number in the fibonacci series: 6765 ----- Enter value of n:10 10th number in the fibonacci series: 55 ----- Enter value of n:30 30th number in the fibonacci series: 832040 ----- Enter value of n:40 40th number in the fibonacci series: 102334155 ----- Enter value of n:45 45th number in the fibonacci series: 1134903170 What is Arithmetic Sequence Calculator? A textSize of 20 is adequate. Following are different methods to get the nth Fibonacci number. $\begingroup$ I think the part "compute the index of the Fibonacci number" is what the question is about. The Fibonacci sequence is a simple, yet complete sequence, i.e all positive integers in the sequence can be computed as a sum of Fibonacci numbers with any integer being used once at most. Explore many other math calculators, as well as hundreds of other calculators addressing health, fitness, finance, math, and more. The second Fibonacci number, F(1), is 1. So now that we have a little background on what a Fibonacci number is, let's work through it and try to see if 30 is a Fibonacci number. In this case this is the regular fibonacci sequence. Write a function to generate the n th Fibonacci number. 1. Hence the first 12 terms will be: F 1 = 1 You note an interesting pattern concerning the $30n$th Fibonacci number: they are all divisible by 31. Dennis S. has given you the wrong answer. The next number is found by adding up the two numbers before it. The Fibonacci sequence is one where a number is found by adding up the two numbers before it. The first Fibonacci number, F(0), is zero. Pastebin is a website where you can store text online for a set period of time. Write a function int fib(int n) that returns F n.For example, if n = 0, then fib() should return 0. When a Fibonacci number is calculated, the program should draw the number on the screen using global x and y coordinates that are adjusted each time. Let n any positive integer. Binet's formula to golden ratio. If F n is what we use to describe the nth Fibonacci number, then F n = F n���1 + F n���2 This can continue on forever. Pastebin.com is the number one paste tool since 2002. These numbers were first noted by the medieval Italian mathematician Leonardo Pisano (���Fibonacci���) in his Liber abaci (1202; ���Book of the fibonacci recursive java; how to find nth fibonacci number in python; code for fibonacci series using recursion Beware! - Do not worry about numbers being too big past the 30th Fibonacci number. Fibonacci sequence is a sequence of numbers, where each number is the sum of the 2 previous numbers, except the first two numbers that are 0 and 1. The Fibonacci sequence is named after Italian mathematician Fibonacci. 0. Fibonacci Numbers & Sequence. - There should be 5 columns of numbers per row, as shown in Fibonacci. 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