Abstract

Sage is a large free open source software package aimed at all areas of mathematical computation. Hundreds of people have contributed to the project, which has steadily grown in popularity since 2005. This paper describes the motivation for starting Sage and the history of the project.

Introduction

The goal of the Sage project (http://www.sagemath.org) is to create a viable free open source alternative to Magma, Maple™, Mathematica®, and MATLAB®, which are the most popular non-free closed source mathematical software systems. Magma is (by far) the most advanced non-free system for structured abstract algebraic computation, Mathematica and Maple are popular and highly developed systems that shine at symbolic manipulation, and MATLAB is the most popular system for applied numerical mathematics. Together there are over 3,000 employees working at the companies that produce the four Ma's listed above, which take in over a hundred million dollars of revenue annually.

By a viable free alternative to the Ma's, we mean a system that will have the important mathematical features of each Ma, with comparable speed. It will have 2d and 3d graphics, an interactive graphical user interface, and documentation, including books, papers, school and college curriculum materials, etc. A single alternative to all of the Ma's is not necessarily a drop-in replacement for any of the Ma's; in particular, it need not run programs written in the custom languages of those systems. Thus an alternative may be philosophically different than the open source system Octave, which understands the MATLAB source language and attempts to implement the entire MATLAB library.

Thus far, I have discussed the types of causes that will be identified, how they can be represented as logical formulas, and how the definitions hold up to common counterexamples. This chapter addresses how these relationships can be inferred from a set of data. I begin by examining the set of hypotheses to be tested, the types of data one may make inferences from, and how to determine whether formulas are satisfied directly in this data (without first inferring a model). Next, I discuss how to calculate the causal significance measure introduced in the previous chapter (ϵavg) in data, and how to determine which values of this measure are statistically significant. I then address inference of relationships and their timing without prior knowledge of either. The chapter concludes by examining theoretical issues including the computational complexity of the testing procedures.

Testing Prima Facie Causality

Chapter 4 introduced a measure for causal significance and showed how probabilistic causal relationships can be represented using probabilistic temporal logic formulas. This representation allows efficient testing of arbitrarily complex relationships. In this chapter, I adapt standard PCTL model checking procedures to validate formulas directly in a set of time series data without first inferring a model (as this can be computationally complex or infeasible in many cases).

Why did Alice develop heart disease in her fifties? What led to the volatility in the U.S. stock market in August 2011? Who shot John F. Kennedy? The inference method described so far aims to find causal relationships that hold in general, while these questions seek causal explanations for one-time events. We do not want to know what causes heart disease, stock market crashes, or death by shooting in general but rather aim to determine why each of these particular events happened. This is a challenging problem, as we need to make such determinations with incomplete and often conflicting information. Few algorithmic methods have been developed to automate this process, yet this may have wide applications to situations with continuous monitoring, such as in intensive care units. Physicians there are overwhelmed with information and need to distinguish between factors causing a particular patient's current symptoms and side effects of their underlying illness to determine the best course of treatment.

This chapter begins in section 6.1 with a discussion of the distinction between type and token causality, and review of methods for token-level reasoning. In section 6.2, I introduce a new approach that links the type-level theory developed in earlier chapters with token-level observation sequences and develops methods for ranking explanations with incomplete and uncertain information. Finally, this is illustrated through worked out examples in section 6.3 and analysis of test cases that have proven difficult for prior approaches in section 6.4.

Philosophical Foundations of Causality

When discussing causality and causal inference we must first distinguish between the thing itself and how to recognize it. Most scientific work on causality involves developing methods for providing evidence for causal relationships, while work in philosophy addresses what it means for something to be a cause. This philosophical work is not immediately applicable to practical problems, but it provides a necessary starting point for work by computer scientists, epidemiologists, and economists. This section introduces readers not familiar with the philosophical literature to how philosophers have conceptualized causality and why this problem is still unsolved after centuries of work. I begin with a review of the primary ways philosophers have addressed causality leading up to more recent probabilistic methods. The review is not an unbiased survey of causality, but rather a discussion of its philosophical foundations through the lens of researchers aiming to build inference methods upon them. As a result, I omit large bodies of work such as process-based theories (Dowe, 2000; Salmon, 1994) and mechanistic models (Glennan, 1996; Machamer et al., 2000) because knowledge of these is not required to understand the later sections. I also raise concerns (such as computational complexity) that differ from those of philosophers but are important when translating these methods to practice.

While Aristotle is often credited with the first formal theory of causality in his Physics and Metaphysics, the most influential modern discussion of causality comes from David Hume in the 18th century. Hume attempted to define both what a cause is and what is meant by the term; as well as how we can come to possess causal knowledge and what is needed to infer it from observations. The core of Hume’s work is arguing that we come to know of causal relationships by inferring them from observations, so they may also be subjective due to beliefs and perception.

Whether we want to know the cause of a stock's price movements (in order to trade on this information), the key phrases that can alter public opinion of a candidate (in order to optimize a politician's speeches), or which genes work together to regulate a disease causing process (in order to intervene and disrupt it), many goals center on finding and using causes. Causes tell us not only that two phenomena are related, but how they are related. They allow us to make robust predictions about the future, explain the relationship between and occurrence of events, and develop effective policies for intervention.

While predictions are often made successfully on the basis of associations alone, these relationships can be unstable. If we do not know why the resulting models work, we cannot foresee when they will stop working. Lung cancer rates in an area may be correlated with match sales if many smokers use matches to light their cigarettes, but match sales may also be influenced by blackouts and seasonal trends (with many purchases around holidays or in winter). A spike in match sales due to a blackout will not result in the predicted spike in lung cancer rates, but without knowledge of the underlying causes we would not be able to anticipate that failure. Models based on associations can also lead to redundancies, since multiple effects of the true cause may be included as they are correlated with its occurrence.

Thus far, I have evaluated the approach developed here conceptually, but the goal is to apply the methods to actual data. Before applying a new approach to a new domain, though, it must first be evaluated on datasets where the true relationships are known. This chapter discusses two types of applications: validation on simulated neuronal and financial time series (to determine how well the algorithms can recover known causes) and experimentation on financial time series (to discover novel relationships).

Simulated Neural Spike Trains

We begin our study of applications with synthetically generated neural spike trains. The underlying relationships here are simple (one neuron causing another to fire in some predefined window of time), but the data allow validation of the algorithms for inferring relationships and their timing, and comparison against other methods. There has been much recent work on determining the connectivity between neurons by applying causal inference methods to spike train measurements (Brown et al., 2004; Hesse et al., 2003; Kamiński et al., 2001) but timing information is a central part of the causal relationships, so it will be useful to compare the approach to others that include this information to varying extents. I begin with a comparison where all algorithms are provided with the known times before examining how well the approach can recover these timings without such prior knowledge.

Synthetic MEA data

The data were created to mimic multi-neuronal electrode array (MEA) experiments, in which neuron firings may be tracked over a period of time. Data was generated for five different structures, with neurons denoted by the 26 characters of the English alphabet. Each dataset contained 100,000 firings generated using one of the five structures plus a degree of noise (this is a parameter that was varied).

Preliminaries

The first few chapters of the book reviewed causality (highlighting some primary approaches to reasoning about and inferring it), probability, and logic, so that readers without expertise in these areas could follow the later discussions. The remainder of the book is devoted to developing a new approach that builds on probabilistic causality and temporal logic to infer complex causal relationships from data and explain the occurrence of actual events (called token causality, and the subject of chapter 6). The first task is to determine exactly what causes will be inferred and how these fit in with other theories of causality and causal inference. This chapter will focus on conceptual differences, while chapter 7 contains experimental comparisons against other inference methods. When discussing causality or causal inference, it is important to be precise about the meaning ascribed to the term “causal.” Many fields (including epidemiology, biology, economics, and politics) have developed their own criteria and conventions for what evidence is needed to substantiate a causal relationship. It is common to draw causal conclusions in biology from few experiments where a gene is suppressed (knocked-out) and one tests whether a given observable trait (phenotype) is present in the absence of the knocked-out gene. When the trait is absent the usual explanation is that the gene causes it, but this does not mean it is the sole cause (it may be only one of a set of necessary conditions) nor does it mean that the presence of the trait indicates noncausality.

Probability

Whether we want to determine the likelihood of a stock market crash or if people with a given gene have a higher risk of a disease, we need to understand the details of how to calculate and assess probabilities. But first, what exactly are probabilities and where do they come from? There are two primary views. The frequentist view says that probabilities relate to the proportion of occurrences in a series of events. For example, the probability of a coin coming up heads being 1/2 means that with a large number of coin flips, half should be heads and half tails. The probability then corresponds to how often something will occur. However, we also discuss the probability of events that may only happen once. We may want to know the probability that a recession will end if a policy is enacted, or the chances of a federal interest rate change on a particular date. In the frequentist case, we can get close to inferring the true probability by doing a large number of tests, but when an event may only occur once, we must instead rely on background knowledge and belief. There is another interpretation of probability, referred to as the Bayesian or subjectivist view. Here the probabilities correspond to degrees of belief in the outcome occurring. In this case, one must have what is called a prior, on which the belief is based.

We are frequently faced with the problem of determining whether a result is significant. People in a region may seem to have a high rate of cancer, but is it out of the ordinary? Are children much more likely to develop autism after being vaccinated? Which genes are differentially expressed in tumor and normal cells? In most cases, we end up with a numerical result, and must determine a threshold at which to call it significant. As we will see shortly, this becomes more complicated when we test many hypotheses at once, as it is then likelier that we will observe something that seems significant by chance. This chapter reviews the basic concepts in and approaches to evaluating statistical tests. We begin with the simplest case of a single result before discussing the modifications needed when doing many tests at once.

Preliminaries

Say we want to determine whether or not a coin is fair, so we flip it 10 times. Our assumption is that the coin is fair, meaning that the probability of heads (H) (or tails (T)) on any given flip is 1/2. However, the sequence we observe is 9 heads and 1 tail. We then want to determine how likely it is that this would occur given our initial hypothesis (called the null hypothesis) that the coin is fair. The basic concept is that we attempt to determine whether the results conform to the null hypothesis, usually denoted H0, that there is no difference, or whether the observations do deviate significantly, and might be more plausibly explained by an alternative hypothesis, usually denoted H1.

At the core of many disciplines – including biomedicine, finance, and the social sciences – is the search for causes. To predict future events, understand the connection between phenomena, explain why things happen, and intervene to alter outcomes, researchers must determine the causal relationships governing the behavior of the systems they study. Automating this process has been a difficult pursuit for many reasons, from insufficient data and computing power to the more fundamental question of what causality is and how it can be inferred from observational data alone.

However, many of the previous barriers to inferring complex causal relationships are falling. Through technological advances enabling interrogation of the activities of single cells, the increasing adoption of electronic health records, and the prevalence of sites like Twitter that broadcast the thoughts and actions of millions of users, we now face a flood of data. As predicted by Moore's law, computers have also become faster and cheaper, making it possible to analyze this newly generated information. These datasets are too large for manual analysis, making automated inference not just a possibility, but a necessity. Medical doctors now have patients who log their own vital statistics and symptoms between visits and must integrate this data (that captures critical moments between appointments and admissions) with the patient's history and their own background knowledge. Stock traders are confronted with worldwide financial, political, and other events (reported at a rate far faster than one could read), and must extract the pertinent information and reconcile it with what is known about how markets behave in response to news.