Category: Blog

• Life and Logic: “Hegel’s Concept of Life” by Karen Ng

  Georg Wilhelm Friedrich Hegel is one of philosophy’s giants, and his influence on the science of logic and self-consciousness can’t be ignored. Philosopher Karen Ng puts Hegel’s thought and arguments into words in Hegel’s Concept of Life. Reason comes from life in itself, Ng explains.

  Following and responding to Immanuel Kant’s writing, Hegel describes a type of internal purposiveness around which self-consciousness, freedom, and logic develop. Hegel derives a purposiveness from Kant’s third Critique of Judgment. Nature itself has a purposiveness, and, from this, judgement attains its power.

  For a thing generated either by art, or by nature, …Art is the principle in a thing other than that which is generated, nature is a principle in the thing itself.

  Aristotle, Metaphysics

  Hegel cites Kant’s use of Aristotle’s understanding of nature in distinguishing between external and internal purposiveness. While the external purposiveness uses artifact creation and instrumental action, the internal type uses organic production and life the same way Aristotle differentiated between art and nature. This is pertinent for understanding Hegel’s philosophical method in the Differenzschrift (1801) and Phenomenology of Spirit (1807). In those texts, Hegel cites Fichte and Schelling in arguing against Kant that internal purposiveness is part of the activity of cognition. Ng offers her own interpretation, too. Hegel’s critique of Fichte’s idealism as “subjective” rests on Fichte’s inability to conceive of nature as internally purposive and living. From there, the cognition relates to the self and the world.

  Ng interprets Hegel’s Science of Logic in a nuanced fashion that Hegel’s Subjective Logic are part of Hegel’s version of a critique of judgement. One can understand life as making intelligibility possible. Hegel’s theory of judgement is made up of reflective and teleological judgements such that a species or kind creates the objective context for predication. “Objective universality” is the context needed for predication, particularly the normative predicates ascription to the subject. Life is, then, something original of judgement, and presupposes the actualization of self-conscious cognition.

  April 7, 2020

  Philosophy

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• Books about Nothing: On the death of the novel

  What use do books have nowadays? In The Decline of the Novel, author Joseph Bottum paints a grim portrait. The novel is dead, and, if not, dying. Fiction is no longer about grappling with reality.

  For almost three hundred years, the novel was a major art form, perhaps the major art form, of the modern world—the device by which . . . we tried to explain ourselves to ourselves.

  Joseph Bottum

  Novels used to emerge from storytelling. They were a more mature form of them that would let the reader take a look inside someone else. Yet, they have met their fate. Long gone are the days of Austen, the Brontës, Dickens, Kafka, or anyone else we can remember. Now, novels are about letting readers find something else to divert their attention or entertain themselves. Bottum delivers this story of stories through examples and illustrations of these changes in reading. He supports his arguments through an exploration of how the role art plays has changed over time.

  The Catholic author Bottum began his research on the historic trends of Protestantism losing cultural significance in American and European life. He believes the first novel, Cervantes’ 1605 Catholic Don Quixote, and how the form of the novel began were meant to reveal the “thick self in a thin universe.” For the first 300 years of its life, the novel kept this purpose. In the midst of the Reformation, the “Protestantism of the air” set the scene for the writing of the time, even for non-Protestants. And it caught on by readers and scholars of religious groups, especially Protestants.

  The modern novel…came into being to present the Protestant story of the individual soul as it strove to understand its salvation and achieve its sanctification, illustrated by the parallel journey of the new-style characters, with their well-finished interiors, as they wandered through their adventure in the exterior world.

  Joseph Bottum

  Society began embracing the idea that they were pushing forward humanity somewhere. With progress of all forms, the novel promised something more than detailed stories of modern selves. They would create stories of the crises of modern selves with the urgency and relevance for the readers of the time period. In some cases, they offered solutions to the problems of the self. Like a remedy for the soul, they could enchant the reader who hungered for stories to understand the world around them. In the reduction of existence to science and technology, government and bureaucracy, and commerce and economics, novels provided meaning in reality. They gave purpose to the natural and physical world in which there was none, making them supernatural and metaphysical.

  With this power, the novel was religious, Bottum argues. At the time, the secular realm consisted of the social norms that built civilization enforced by power. Uncovering this social value was, then, a religious act.

  But those were the days of the past. The novel’s slow and steady decline reflects society’s inability to address the issues in the supernatural and metaphysical realm. “The decline of the novel’s prestige reflects and confirms…a new crisis born of the culture’s increasing failure of intellectual nerve and terminal doubt about its own progress,” Bottum argues. With modernity of all aspects of society, “the thick inner world of the self increasingly came to seem ill-matched with the impoverished outer world, stripped of all the old enchantment that had made exterior objects seem meaningful and important. . . . This is what we mean by the crisis of the self: Why does anything matter, what could be important, if meaning is invented, coming from the self rather than to the self?”

  The dying began with the final decade of the twentieth century. Protestantism lost traction in Western civilization. “Of the authors who have published novels since the early 1990s, none are mandatory reading,” Bottum writes. How true this is depends on who you ask. It’s definitely possible that the novels over the past century don’t conform to Bottum’s view. Writers who are amazing with portraying characterization, dialogue, plot, and other features can still fall far from this purpose of a novel. The novel’s purpose in creating a meaning that transcends words themselves doesn’t follow from those aspects of writing. It’s something else. Many books over the past century start off well with a lot of potential, but don’t seem to reach this stage of a novel’s purpose. They, instead, leave the reader wanting more, circling around to how things started in the beginning, or use some other method of avoiding this transcendence. A lot of writers tend to shy away from metaphysical purpose and stick with themes that there’s no meaning in their work, that it was never meant to provide that to begin with, or some similar postmodern theme. We’ve come to associate these secular searches for truth by avoiding what Bottum has described as the purpose of the novel.

  In thrusting the novel to the limits of postmodernism, one can ask “Where do we go from here?” Hungarian novelist László Krasznahorkai has tried pushing through this nihilism in his writing like The Melancholy of Resistance. His apocalyptic godless story has a description of a human body decomposing. In graphic detail of the chemical, it creates a redundancy of the form. Bottum argues that the American writer Tom Wolfe has a metaphysical component from the absence of a moral framework. Instead of having the ideal conception of ethics from which to measure distance of events and actions, Wolfe’s writing “needs a greater thickness than the world seems to possess,” Bottum says. “What he discovers instead is the culture’s failure of nerve, and it ruins the attempt to go where he wants to go,” Bottum writes. “The ending of a Tom Wolfe novel is usually a disaster, or at least a minor fall, because the resources necessary to conclude a story of justification and sanctification simply do not exist for him.” The American George Saunders and French Michel Houellebecq have established themselves as post-postmodernists to this end. Beyond the boundaries of postmodernism, they’ve written stories that capture what Bottum has intended. They give life to the novel in a sort of resurrection. They provide a philosophy with aesthetics, metaphysics, and other characteristics that Bottum argues novels have lost.

  Krasznahorkai and Houellebecq perceive Bottum’s problem with the death of the novel. They both attempt to provide solutions with a metaphysics of the imagination for the empty space in their work. Houellebecq creates worlds that use transcendence in fitting ways even when in the lost abyss. Krasznahorkai avoids the problem by making a testament to modernity in ways other writers don’t.

  Works across popular biography, New Journalism, graphic novels, and genre fiction have explored new forms of novelism. Though these aren’t novels, we can turn to them in determining things that novels have missed. They don’t quite capture the movements that Dickens or Austen once created. Bottum believes this “signals…an end of confidence, about the past values and future goals of what conceived itself as Western culture.”

  Bottum is cautious in arguing that, though the novel has died, the writers before our time did not have more genius than current ones do. He mentions Naipaul, Vargas Llosa, Byatt, Pynchon, Roth, DeLillo, Coetzee, Robinson, Amis, Rushdie, McCarthy, Murakami, Eugenides as examples who are talented writers, yet show something different than what Defoe, Dickens, Austen, Faulkner, Hemingway, Steinbeck, or Mann did in their work.

  All this is testimony, I think, to the current problem of culture’s lack of belief in itself, derived from the fading of a temporal horizon….Without a sense of the old goals and reasons…all that remains are the crimes the culture committed in the past to get where it is now. Uncompensated by achievement, unexplained by purpose, these unameliorated sins must seem overwhelming: the very definition of the culture.

  Where do we go? “Why, indeed, should we write or even read book-length fiction for insight into the directions of the culture and the self?” Bottum asks.

  There aren’t many people nowadays who believe reading novels is essential to being part of the public sphere like reading the news or searching for a peek into the human condition. While Bottum’s book still suffers from issues in the way its constructed such as how it centers on essays that don’t resonate as well as they could, it still reveals this truth about our novel-reading habits. One may argue about why or how these changes have occurred. In the age of information technology, our communication has become more superficial and simplistic. Fiction no longer carries the mysterious aura of power it once did. Freudian psychoanalysis has made the human person itself instrument such that everything we do becomes mere mechanical processes. The novelist, in these dimensions, doesn’t have much to work with. There’s no transcendence.

  It’s also worth emphasizing the times of religious authority in many areas of society no longer holds the same water given the advances in communication and culture over the centuries since Don Quixote. As the novel gained its own aesthetic and culture significance, it had already begun losing the curiosity of the elusive human condition.

  Writing itself has changed, too. It’s become a form of seeking status. Nowadays writing is more about changing the world rather than investigating it, and many people are more concerned with the prestige and power that comes along with it rather than the long, arduous craft of becoming a better writer.

  The media’s portrayal of reality, through all these trends, becomes more supra-fictional. The phrase “truth is stranger than fiction” may resonate with many readers nowadays. Some areas of fiction like crime and horror still try their best to catch our attention, though.

  Overall, the story of the novel meeting its demise presents these postmodern and post-postmodern issues that many of us experience whenever we open up a book. When its story ends, we’ll see if a new one begins.

  March 14, 2020

  Education, Philosophy

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• Where do numbers come from? Philosophers have sought answers.

  

  The answer may lie in Irish mathematician Paolo Zellini’s recent book The Mathematics of the Gods and the Algorithms of Men: A Cultural History. The philosophical debate determines to answer the question if numbers are discovered like a diamond in a cave or invented like a new phone. Whether numbers are real or fake, it doesn’t make a difference to most people, even those who use mathematics in their everyday lives. An engineer needs to know if the physics of a bridge are sturdy enough, but probably doesn’t need to know whether those physics were invented or discovered. Still, understanding that it’s not relevant to most issues means that you can appreciate a greater philosophical inquiry by approaching the problem. Figuring it out for what it is presents those new methods of reasoning. When there’s nothing practical to gain, then the real learning begins.

  So where did mathematics come from? How did we start using numbers to count things? Zellini says that, historically, “2 apples” came before the number 2 did. We saw many things in front of us and used numbers to count them. Enumeration itself was meant to give reality to these things. Mathematics and numbers were powerful, and this attribution gave them their power. Philosophers who wrote about divinity believed numbers created this reality through divine powers, as Zellini explains in The Mathematics of the Gods and the Algorithms of Men.

  So if math was from the Gods, were algorithms from the men? In some way. The debates throughout the 1800s and 1900s lead to the theories of computer science in solving algorithms and difficult math problems. The ways numbers behaved in different calculations were the basis for questions of how things can change or not. Einstein’s theories of relativity and developments in the creation of computers took advantage of these methods of thinking. There, the foundation of mathematics in science and technology is apparent. But Zellini takes things a step further. Math not only showed how important calculations are to society, but dictated fundemental searches for what is real.

  Numbers have a reality. This isn’t the same reality as the difference between real and imaginary numbers (such as the imaginary i unit). It’s a reality of how these numbers came about. They tell us what is and isn’t. Zellini writes their “calculability,” or this mathematical practicality, determines this. These theoretical questions of what kinds of math problems can be solved or how algorithms behave speak to how a system of rules for numbers may work. Zellini is very careful not to draw too many conclusions that math is the sole method of understanding reality or that these revelations will change every field of research that uses numbers. Instead, he presents more of a guide for how the amount of money you have in your pocket or temperature forecast tomorrow are real enough for the purposes they serve, even if other numbers aren’t as real.

  Zellini’s writing is still insightful and relevant, though. Numbers are different from what they enumerate. The power of hundreds of thousands of voters supporting one candidate over the other relies on calculations in an increasingly data-driven world. The models built upon machine learning and statistics depend upon all sources of information. This data comes from a small part of our experience, though. The algorithms and computers that control the analysis, prediction, and other methods create the reality that can dominate the experience they claim to represent. As we rely increasingly on forecasts and cost-benefit models of risks, we, in many ways, find ourselves turning back to the philosophical power of numbers. Back to the big questions of what a 50% win chance in an election means adds up, Zellini reveals.

  It’s disappointing, then, that Zellini’s appeal to philosophy depends so much on ancient mathematics that don’t flow so well with the philosophy itself. Making strong references to Heidegger and Nietzsche and a rambling explanation from classical philosophy are fine, but the work still falls short. It stays too well within the intellectual landscape of dead white men in a way that it doesn’t represent numbers, calculation, and algorithms as well as it could. The connections between mathematics and philosophy are still weak. Zellini even makes incorrect historical claims about the cultural history of math and philosophy.

  I’m sure there are better stories of the history of mathematics and philosophy such historian David Wootton’s The Invention of Science: A New History of the Scientific Revolution. Still, Zellini’s explanation of the power of numbers is difficult to ignore in today’s issues of population and economics.

  March 9, 2020

  Philosophy, Science

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• Why life might be worth living, according to philosopher William James

  If you ever find yourself in doubt of yourself and other things in your life, remember to remain cognizant in evaluating things. Questioning whether life is worth living is only a part of many larger questions that many people face at some point or another. Whether you find satisfying answers can be difficult, though. Turning to philosophy can provide answers, with some effort at least.

  “Is Life Worth Living?” A bold title for the 1896 lecture of philosopher and psychologist William James. And what better way to begin such a than with an 1881 self-help book of a similar title. James himself had been through the existential dilemma. He would ask, was life worth living?

  The short answer is that it depends on the liver. Satisfied? If not, there is a more elaborate response. Philosopher John Kaag’s new book Sick Souls, Healthy Minds: How William James Can Save Your Life explores the Father of American Psychology’s personal journey in figuring out if life is worth living.

  James would wake up each day “with a horrible dread at the pit of [his] stomach,” contemplating suicide in his early 20s and wondering “how other people could live, how I myself had ever lived, so unconscious of that pit of insecurity beneath the surface of life.” Through an arduous journey of figuring out what made life meaningful and worth living, the philosopher ends up conceding to “our usual refined optimisms and intellectual and moral consolations” and live as though life were worth living.

  After Kaag witnessed a suicide by jumping off of the William James Hall at Harvard University in 2014, the philosopher began questioning why it had happened. Sick Souls, Healthy Minds aims to remedy those actions by offering James as a friend in those trying times of misery. Kaag shares own difficult time at age 30 as he was researching William James at Harvard University while going through a divorce and dealing with the death of his alcoholic father. Like his previous book on Nietzsche, Kaag searches for practical wisdom by combining his autobiographical experience alongside the famous philosopher. I still found myself believing that, though Kaag himself went through a tremendous amount of stress, his own story still pales in comparison to James’ style and work.

  James’ research in studying philosophy and psychology alongside one another, radical empiricism, pragmatism, “anti-intellectualism” (to be clarified later) and overall revolutionary role in the theory of emotion that still resounds to this day make his life and rumination on its meaning much more impactful. His own life, from going on a scientific journey through the Amazon, studying medicine, and pondering life’s purpose, especially in light of On the Origin of Species, published in 1859, lead him to think humans were merely animals in a deterministic world of cause-and-effect. Choice, like free will, was only an illusion. This lead to his diary entries in 1870, in which he assumed free will was no illusion, and, out of his own free will, he would believe in free will. He wrote he would, “accumulate grain on grain of willful [sic] choice like a very miser” through making habits. After reading French philosopher Charles Renouvier’s, he came to believe these thoughts and kept them close in everything he did.

  James’ pragmatism, that truth is not statically there to be perceived or discovered but is, in many cases, what we create in the stride of living, we can jump across the abyss that Nietzsche warned about staring into by jumping across it. James would write about a type of “anti-intellectualism” against the idea that the minds have “a world complete in itself” and need simply to find this world while having no power to re-determine its already-given character. These gave the psychologist-philosopher a type of deterministic that James would use to describe a type of “rich and active commerce” between minds and reality.

  When new ideas join older ones, they “marry” one another, James described. You can form beliefs as hypotheses, and their values depend on how they relate to you. This hypothesis of life makes life valuable.

  But Kaag also warns the prideful dangers of pragmatism, even if his explanations are a bit indulgent. Kaag’s doubts crept up on him during his first wedding, but his mother suggested to continue with the wedding as planned. He realized he could determine the truth that his marriage would be a happy one, but he also couldn’t the same way he could. It seemed as though James’ free will wouldn’t have helped.

  James’ other work reflects the groundbreaking discoveries in psychology and cognitive science while creating the Department of Experimental Psychology at Harvard. James believed emotions are “constituted by, and made up of, those bodily changes which we ordinarily call their expression or consequence.” Being sad is not the cause of crying, but is what it feels like to cry in this sort of “biofeedback” in which we figure out our own emotions. This means, according to James, that whistling a happy tune could prevent yourself from feeling sad. The psychologist-philosopher mocked the cognitivist idea that emotions could simply be states of mind which cause us to have visceral reactions. Without the fiery passion of anger within your heart or heavy weight of mourning at a funeral, an emotion would only be “feelingless cognition.”

  If, as Nietzsche said, every great philosophy is “a confession on the part of its author and a kind of involuntary and unconscious memoir,” then the emphasis should be on “involuntary and unconscious.” Maybe, in philosophizing, the personal should let themselves feel what they feel.

  March 5, 2020

  Philosophy

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• Neurons that work together, explained

  A theoretical physicist can sit at a computer with a pen and paper may not seem like a likely candidate to understanding how the brain works, but, according to physicists who study statistics and algebra, they can figure out revolutionary theories about how the nervous system works. When I met Princeton theoretical physicist William Bialek in 2013 during my undergraduate years at Indiana University-Bloomington, I asked him about the “magic” of physics and how scientist can capture abstract ways of thinking and apply them to how neurons in the brain work. Bialek’s book “Spikes: Exploring the Neural Code,” one of my inspirations to step into neuroscience research, and his work alongside other researchers in physics and mathematics can answer key questions in neuroscience.

  Pairwise Interactions

  Often in neuroscience we are confronted with a small sample measurement of a few neurons from a large population. Although many have assumed, few have actually asked: What are we missing here? What does recording a few neurons really tell you about the entire network? Correlations of neurons dominated large networks of neurons. Using Ising models from statistical physics, the researchers of Schneidman et al. 2006 looked at large networks and their ability to correct for errors in representing sensory data. They argue that correlations are due to pairwise, but not 3-wise interactions between neurons, although some might argue that closer inspection reveals otherwise. Pairwise interactions are how neurons forms pairs among themselves to act together. Their pairwise maximum entropy approach can capture the activity of RGB neurons effectively.

  Using an elegant preparation retina on a micro electrode array (MEA) viewing defined scenes/stimuli, the researchers showed that statistical physics models that assume pairwise correlations, but disregard any higher order phenomena, perform very well in modeling the data. This indicates a certain redundancy exists in the neural code. The results are also replicated with cultured cortical neurons on a MEA. They noted a dominance of pairwise interactions. This would imply that learning rules depending on pairwise correlations could, on their own, create nearly optimal internal models describing how the retina computes codewords. The brain could, then, assess new events for their degree of surprise with reasonable accuracy. The central nervous system alone could learn the maximum entropy model from the data provided by the retina alone, but the conditionally independent model is not biologically realistic in this sense. Although the pairwise correlations are small and weak and the multi-neuron deviations from independence are large, the maximum entropy model consistent with the pairwise correlations captures almost all of the structure in the distribution of responses from the full population of neurons. The weak pairwise correlations imply strongly correlated states. 

  

  If you modeled the cells independent from one another, they would form the Poisson distribution. The actual distribution is almost exponential, so this doesn’t fit well. For example, the probability of K = 10 neurons spiking together is ~10⁵ x larger than expected in the independent model. For this model, the specific response patterns across the population of neurons show that the N-letter binaries (patterns of 0s and 1s) differ greatly from the experimental results. These discrepancies show the failure of independent coding. The difference between prediction and empirical observation is anti-correlated in clusters of spikes. 

  

  Instead, a group of neurons comes to a decision through pairwise correlations. These rates are predicted with >10% accuracy. The rates scatter between predictions and observations is confined largely to rare events for which the measurement of rates is itself uncertain.

  

  The Jensen–Shannon divergence measures similarity between two probability distributions. This metric can be used to measure mutual information of a random variable to an associated mixture distribution, as the researchers did. In previous work, the researchers had used the same principle to a joint distribution and the product of its two marginal distributions and measure how reliably you can decide if a given response comes from the joint distribution or the product distribution. 

  

  The fractions of full network correlations in 10-cell groups the maximum entropy model of second order plotted as a function of the full network correlation, measured by the multi-information I_N. The ratio is larger when I_N itself is larger, so that the pairwise model is more effective in describing populations of cells with stronger correlations, and the ability of this model to capture ~90% of the multi-information holds independent of many details. 

  The Maximum Entropy Method

  

  The most general description of the population activity of n neurons, which uses all possible correlation functions among cells, can be written using the maximum entropy principle as shown in the equation above for a probability p̂, Lagrange multipliers h_i, and J_ij, Z as the normalization constant, and the other variables representing each individual event probability. This method also uses Laplace’s principle of insufficient reason, which states that two events are to be assigned equal probabilities if there is no reason to think otherwise, and Jayne’s principle of maximum entropy, the idea that distributions are determined so as to maximize the entropy (as a measure of uncertainty) in a way consistent with given measurements.

  For N neurons, the maximum entropy distributions with Kth-order correlations (K=1, 2, …N) can account for the interactions. Entropy difference (multi-information)  I_N = S₁ – S_N measures the total amount of correlation in the network, independent of whether it arises from pairwise, triplet or more-complex correlations. They found this across organisms, network sizes, appropriate bin sizes, Each entropy value S_K decreases monotonically toward the true entropy S : S₁ ≥ S₂ ≥,… ≥ S_N. The contribution of the Kth-order correlation is I_(K) = S_K-1 – S_K and is always positive. More correlation always decreases entropy. 

  In a physical system, the maximum entropy distribution is the Boltzmann distribution, and the behavior of the system depends on the temperature, T. For the network of neurons, there is no real temperature, but the statistical mechanics of the Ising model predicts that when all pairs of elements interact, increasing the number of elements while fixing the typical strength of interactions is equivalent to lowering the temperature, T, in a physical system of fixed size, N. This mapping predicts that correlations will be even more important in larger groups of neurons.

  The active neurons are those that send an action potential down the axon in any given time window, and the inactive ones are those that do not. Because the neural activity at any one time is modelled by independent bits, Hopfield suggested that a dynamical Ising model would provide a first approximation to a neural network which is capable of learning.

  The researchers looked for maximum entropy distribution consistent with experimental findings. Ising models with pairwise interactions are the least structured, or maximum-entropy, probability distributions that exactly reproduce measured pairwise correlations between spins. Schneidman and the researchers used such models to describe the correlated spiking activity of populations of neurons in the salamander retina subjected to naturalistic stimuli. They showed that for groups of N≈10 neurons (which can be fully sampled during a typical experiment) these models with O(N²) tunable parameters provide a good description of the full distribution over 2^N possible states. 

  They found the maximum entropy model of second order captures over 95% of the multi-information in experiments on cultured networks of cortical neurons. There would be implications for learning rules could be enough to generate nearly optimal internal models for the distribution of “codewords” in the retinal vocabulary and let the brain accurately evaluate new events for their degree of surprise.

  Accounting for Cell Bias

  

  The researchers noted they needed to account for the pairwise interactions and cell bias values. Interactions have different signs, the researchers showed that frustration would prevent the system from freezing into a single state in about 40% of all triplets. With enough minimum energy patterns, the system has a representational capacity, and the network can identify the whole pattern uniquely just as Hopfield models of associative memory do. The system would have a holographic or error-correcting property, so that an observer who has access only to a fraction of the neurons would nonetheless be able to reconstruct the activity of the whole population.

  

  The pairwise correlation model also uncovers subtle biases in decision making. It will tell you about how they influence each other, on average. Pairwise maximum entropy models reveal that the code relies on strongly correlated network states and shows distributed error-correcting structure.

  To figure out if the pairwise correlations are an effective description of the system, you need to determine if the reduction in entropy from the correlations captures all or most of the multi-information I_N. The researchers conclude that, even if the pairwise correlations are small and the multi-neuron deviations from independence are large, the maximum entropy model consistent with the pairwise correlations captures almost all of the structure in the distribution of responses from the full population of neurons. This means the weak pairwise correlations imply strongly correlated states. 

  Other Effects

  

  Intrinsic bias dominates small groups of cells, but, in large groups, almost all of the ~N² pairs of cells are significantly interacting (top). This shifts the balance so that the typical values of the intrinsic bias are reduced while the effective field contributed by other cells increases (bottom). In the Ising model, if all pairs of cells interact significantly with one another, you can limit the typical size of interactions by showing how J_ij changes with increasing N. There were no signs of significant changes in J with growing N with the values the researchers tested. 

  Extrapolation

  

  For weak correlations, you can solve the Ising model in perturbation theory to show that the multi-information I_N is the sum of mutual information terms between all pairs of cells, and I_N ~ N(N – 1) (left). This is in agreement with the empirically estimated I_N up to N = 15, the largest value for which direct sampling of the data provides a good estimate. Monte Carlo simulations of the maximum entropy models suggest that this agreement extends up to the full population of N = 40 neurons in their experiment (G. Tkačik, E.S., R.S., M.J.B. and W.B., unpublished data). The potential for extrapolation to larger networks of neurons can be shown through the error-correction that comes about (right). The error-correction emerges when figuring out how N-cell activity can predict (N+1)-cell activity. Uncertainty decreases by the number of cells. In a 40-cell population, three cells with spiking probability have an near-perfect linear encoding of the number of spikes generated by other cells in the network. Through these methods of becoming more and more accurate and robust, they showed findings that are similar to how single pyramidal cell spiking correlates with more collective responses. 

  Challenges to the Model

  The case of two correlated neurons has proven to be particularly challenging, because the Fokker–Planck equations are analytically tractable only in the linear regime of correlation strengths (r ≈ 0) and only for a limited set of current correlation functions. Some analytical results for the spike cross-correlation function have been obtained using advanced approximation techniques for the probability density and expressed as an infinite sum of implicit functions (Moreno-Bote and Parga, 2004, 2006). Similarly, the correlation coefficient of two weakly correlated leaky-integrate-and-fire neurons has been obtained for identical neurons in the limit of large time bins. 

  Correlations between neurons can occur at various timescales. It’s possible, by integrating the cross-correlation function (xcorr in Matlab, correlate in numpy) between two neurons, to read off the timescale of the correlation (Bair, Zohary and Newsome 2001). This can help to distinguish correlations due to monosynaptic or disynaptic connections, which are visible at short timescales, with correlations due to slow drift in oscillations, up-down states, attention, etc., which occur at much longer timescales. Correlations depend on physical distance on the cortical map as well as tuning distance between two neurons (Smith and Kohn, 2008).

  Decoding techniques of Ising model can be applied to simulated neural ensemble responses from a mouse visual cortex model with an improvement in decoder performance for a model with heterogeneous as opposed to homogeneous neural tuning and response properties. Their results demonstrate the practicality of using the Ising model to read out, or decode, spatial patterns of activity comprised of many hundreds of neurons (Schaub et al. 2011).

  Discussion

  The research seems to reflect general trends of “the whole is greater than the sum of its parts” or even “less is more,” both concepts in science and philosophy that date back centuries. I even emailed Elad Schneidman a few days ago about this, and he responded, “I think that this idea must predate the ancient greeks ;-)”.

  Their work used the application of the maximum entropy formalism of Schneidman et al. 2003, to ganglion cells. The same way a group of neurons behaves differently than the sum (or combination) of each independent neuron gives the research leverage and potential for these systems-like problems of neurocomputation and emergent phenomena. 

  The work in deriving an Ising model (or using a maximum entropy method) from statistical mechanics shows the importance of a priori proof work in using equations and theories to deduce “what follows from what.” It’s a great example of using the principles and methods of abstraction that mathematicians and physicists use in solving problems in biology and neuroscience. In my own writing, I’ve described this sort of attention to abstract models and ideas as relevant to biology in a previous blogpost.

  In this paper, the researchers very well theorized which shortcomings and limitations their model would have and addressed them appropriately by fitting their model to experimental work. As a result, their research testifies to the power of computational and theoretical research in both describing and explaining empirical phenomena. 

  Recreating the Results

  

  Related Research

  In that same year, Tkačik and other researchers would use the same recordings and use Monte-Carlo-based methods to construct the appropriate Ising model for the complete 40-neuron dataset. They showed that pairwise interactions still account for the observed higher-order correlations and argue why the effects of three-body interactions should be suppressed. 

  They examined the thermodynamic properties of Ising models of various sizes derived from the data to suggest a statistical ensemble from which the observed networks could have been drawn and, consequently, to create synthetic networks of 120 neurons. They found that with increasing size the networks operate closer to a critical point and start exhibiting collective behaviors reminiscent of spin glasses. They examined more closely the appearance of multiple single-spin-flip stable states.

  The method of using a maximum entropy model is equivalent to the method of Roudi et al. 2009, where they described a method of normalizing the the Kullback–Leibler divergence D_KL(P, P˜) (for P˜ approximation to distribution P, with the distance from the independent maximum entropy fit. The quality of the pairwise model comes from normalizing this by the corresponding distance of the distribution P from an independent maximum entropy fit D_KL(P, P₁), where P₁ is the highest entropy distribution consistent with the mean firing rates of the cells (equivalently, the product of single-cell marginal firing probabilities): Δ = 1 – D_KL(P, P˜)/D_KL(P, P₁) where Δ = 1 means the pairwise model perfectly fits the additional information left out by the independent model, and Δ = 0 means the pairwise model doesn’t improve at all compared to the independent model. 

  In 2014, Tkačik and the researchers from Schneidman et al. 2006 published “Searching for Collective Behavior in a Large Network of Sensory Neurons” with K-pairwise models, more specialized variations of the pairwise models to estimate entropy, classify activity patterns, show that the neural codeword ensembles are extremely inhomogeneous, and demonstrate that the state of individual neurons is highly predictable from the rest of the population, which would allow for error correction. 

  Barreiro et al. 2014 found that, over a broad range of stimuli, output spiking patterns are surprisingly well-captured by the pairwise model. They studied an analytically tractable simplification of the retinal ganglion cell mode, and found that in the simplified model, bimodal input signals produce larger deviations from pairwise predictions than unimodal inputs. The characteristic light filtering properties of the upstream retinal ganglion cell circuitry would suppress bimodality in light stimuli, thus removing a powerful source of higher-order interactions. The researchers said this gave a novel explanation for the surprising empirical success of pairwise models.

  Ostojic et al. 2009 studied how functional interactions would depend on biophysical parameters and network activity that variations in the background noise changed the amplitude of the cross-correlation function as strongly as variations of synaptic strength. They found that the postsynaptic neuron spiking regularity has a pronounced influence on cross-correlation function amplitude. This suggests an efficient and flexible mechanism for modulating functional interactions.

  In 1995, Mainen & Sejnowski showed that single neurons have very reliable responses to current injections. Nevertheless, cortical neurons seem to have Poisson or supra-Poisson variability. It’s possible to find a bound on decodability using the Fisher information matrix (Sompolinsky & Seung 1993). Under the assumption of independent Poisson variability, it is possible to derive a simple scheme for ML decoding that can be implemented in neuronal populations (Jazayeri & Movshon 2006).

  The accumulation of noise sources and various other mechanisms cause cortical neuronal populations to be correlated. This poses challenges for decoding. You can get a little more juice out of decoding algorithms by considering pairwise correlations (Pillow et al. 2008).

  References

  Bair, W. “Correlated firing in macaque visual area MT: time scales and relationship to behavior.” (2001). Journal of Neuroscience. 

  Barreiro, et al. “When do microcircuits produce beyond-pairwise correlations?” (2014). Frontiers. 

  Bialek, William and Rangnathan, Ramek. “Rediscovering the power of pairwise interactions.” (2018). Arxiv. 

  Hopfield, J.J. “Neural networks and physical systems with emergent collective computational abilities.” (1982). Proc. Natl Acad. Sci. USA 79, 2554–-2558. 

  Jazayeri, M, Movshon, A. “Optimal representation of sensory information by neural populations.” (2006). Nature Neuroscience. 

  Mainen, ZF, Sejnowski, TJ. “Reliability of spike timing in neocortical neurons.” (1955). Science. 

  Moreno-Bote, R., and Parga, N. “Role of synaptic filtering on the firing response of simple model neurons.” (2004). Phys. Rev. Lett. 92, 028102.

  Moreno-Bote, R., and Parga, N. “Auto- and crosscorrelograms for the spike response of leaky integrate-and-fire neurons with slow synapses.” (2006). Phys. Rev. Lett. 96, 028101.

  Ostojic, et al. “How Connectivity, Background Activity, and Synaptic Properties Shape the Cross-Correlation between Spike Trains.” (2009). The Journal of Neuroscience. 

  Pillow, Jonathan, et al. “Spatio-temporal correlations and visual signalling in a complete neuronal population.” (2008). Nature. 

  Roudi, et al. “Pairwise Maximum Entropy Models for Studying Large Biological Systems: When They Can Work and When They Can’t.” (2009). PLoS Computational Biology. 

  Schaub, Michael and Schultz, Simon. “The Ising decoder: reading out the activity of large neural ensembles.” (2011). Journal of Computational Neuroscience. 

  Schneidman et al. “Network Information and Connected Correlations.” (2003). Physical Review Letters. 

  Schneidman et al. “Weak pairwise correlations imply strongly correlated network states in a neural population.” (2006). Nature. 

  Seung, HS, Sompolinsky, H. “Simple models for reading neuronal population codes.” (1993). PNAS. 

  Shlens, Jonathan, et al. “The structure of multi-neuron firing patterns in primate retina” (2006). The Journal of Neuroscience 26.32: 8254-8266.

  Smith, Matthew, and Kohn, Adam. “Spatial and Temporal Scales of Neuronal Correlation in Primary Visual Cortex.” (2008). Journal of Neuroscience. 

  Tkačik, Gašper et al.  “Ising models for networks of real neurons.” (2006). arXiv.org:q-bio.NC/0611072. 

  Tkačik, Gašper et al. “Searching for Collective Behavior in a Large Network of Sensory Neurons.” (2014). PLoS Comput Biol. 

  February 9, 2020

  Science

  ---

• The Emergent Beauty of Mathematics

  

  Like a flower in full bloom, nature reveals its patterns shaped by mathematics. Or as particles collide with one another, they create snowflake-like fractals through emergence. How do fish swarm in schools or consciousness come about from the brain? Simulations can provide answers.

  Through code, you can simulate how living cells or physical particles would interact with one another. Using equations that govern the behavior of how cells act when they meet one another or how they would grow and evolve into larger structures. In the gif above, you can use diffusion-limited aggregation to create Brownian trees. These are the structures that emerge when particles move randomly with respect to one another. Particles in fluid (like dropping color dye into water) take these patterns when you look at them under a microscope. As the particles collide and form trees, they create shapes and patterns like water crystals on glass. These visuals can give you a way of appreciating how beautiful mathematics is. The way mathematical theory can borrow from nature and how biological communities of living organisms themselves depend on physical laws shows how such an interdisciplinary approach provides a way to bridge different disciplines.

  

  In the code, the particles are set to move with random velocities in two dimensions and, if they collide with the tree (a central particle at the beginning), they form parts of the tree. As the tree grows bigger over time, it takes the shapes of branches the same way neurons in the brain form trees that send signals between one another. These fractals, in their uniqueness, give them a kind of mathematical beauty.

  

  Flashing lights coming and going away like stars shining in the sky are more than just randomness. These models of cellular interactions are known as cellular automaton. The gif above shows an example of Conway’s game of life, a simulation of how living cells interact with one another.

  These cells “live” and “die” according to four simple rules: (1) live cells with fewer than two live neighbors die, as if by underpopulation, (2) live cells with two or three live neighbors live on to the next generation, (3) live cells with more than three live neighbors die, as if by overpopulation and (4) dead cells with exactly three live neighbors become live cells, as if by reproduction.

  

  Through these rules, specific shapes emerge such as “gliders” or “knightships” you can further describe with rules and equations. You’ll find natural versions of cells obeying rules like the colorful patterns on a seashell. Complex structures emerging from more basic, fundamental sets of rules unite these observations. While the beauty of these structures becomes more and more apparent from the patterns between different disciplines, searching for these patterns in other contexts can be more difficult such as human behavior.

  Recent writing like Genesis: The Deep Origin of Societies by biologist E.O. Wilson take on the debate over how altruism in humans evolved. While the shape of snowflakes can emerge from the interactions between water molecules, humans getting along with one another seems far more complicated and higher-level. Though you can find similar cooperation in ants and termites creating societies, how did science let this happen?

  Biologists have answered that organisms choose to mate with individuals and increase the survival chances of themselves and their offspring while passing on their genes. Though they’ve argued this for decades, Wilson offers a contrary point of view. In groups, selfish organisms defeat altruistic ones, but altruistic groups beat selfish groups overall. This group selection drives the emergence of altruism. Through these arguments, both sides have appealed to the mathematics of nature, showing its growing importance in recognizing the patterns of life.

  Wilson clarifies that data analysis and mathematical modeling should come second to the biology itself. Becoming experts on organisms themselves should be a priority. Regardless of what form it takes, the beauty is still there, even if it’s below the surface.

  December 22, 2019

  Science

  ---

• How to Create Interactive Network Graphs (from Twitter or elsewhere) with Python

  In this post, a gentle introduction to different Python packages will let you create network graphs users can interact with. Taking a few steps into graph theory, you can apply these methods to anything from links between the severity of terrorist attacks or the prices of taxi cabs. In this tutorial, you can use information from Twitter to make graphs anyone can appreciate.

  The code for steps 1 and 2 can be found on GitHub here, and the code for the rest, here.

  Table of contents:

  1. Get Started
  2. Extract Tweets and Followers
  3. Process the Data
  4. Create the Graph
  5. Evaluate the Graph
  6. Plot the Map

  1. Get Started

  

  Make sure you’re familiar with using a command line interface such as Terminal and you can download the necessary Python packages (chart-studio, matplotlib, networkx, pandas, plotly and python-twitter). You can use Anaconda to download them. This tutorial will introduce parts of the script you can run from the command line to extract tweets and visualize them.

  If you don’t have a Twitter developer account, you’ll need to login here and get one. Then create an app and find your keys and secret codes for the consumer and access tokens. This lets you extract information from Twitter.

  2. Extract Tweets and Followers

  

  To extract Tweets, run the script below. In this example, the tweets of the UCSC Science Communication class of 2020 are analyzed (in screennames) so their Twitter handles are used. Replace the variables currently defined as None below with them. Keep these keys and codes safe and don’t share them with others. Set datadir to the output directory to store the data.

  The code begins with import statements to use the required packages including json and os, which should come installed with Python.

  import json
  import os
  import pickle
  import twitter 
  
  screennames = ["science_ari", "shussainather", "laragstreiff",                  "scatter_cushion", "jessekathan", "jackjlee",                 "erinmalsbury", "joetting13", "jonathanwosen",                 "heysmartash"] 
  
  CONSUMER_KEY = None
  CONSUMER_SECRET = None
  ACCESS_TOKEN_KEY = None
  ACCESS_TOKEN_SECRET = None
  
  datadir = "data/twitter"

  Extract the information we need. This code goes through each screen name and accesses their tweet and follower information. It then saves the data of both of them to output JSON and pickle files.

  t = twitter.Api(consumer_key = CONSUMER_KEY,
                   consumer_secret = CONSUMER_SECRET,
                   access_token_key = ACCESS_TOKEN_KEY,
                   access_token_secret = ACCESS_TOKEN_SECRET)
   for sn in screennames:
       """
       For each user, get the followers and tweets and save them
       to output pickle and JSON files.
       """
       fo = datadir + "/" + sn + ".followers.pickle"
       # Get the follower information.
       fof = t.GetFollowers(screen_name = sn)
       with open(fo, "w") as fofpickle:
           pickle.dump(fof, fofpickle, protocol = 2)
       with open(fo, "r") as fofpickle:
           with open(fo.replace(".pickle", ".json"), "w") as fofjson:
               fofdata = pickle.load(fofpickle)
               json.dump(fofdata, fofjson)  # Get the user's timeline with the 500 most recent tweets.
      timeline = t.GetUserTimeline(screen_name=sn, count=500)
      tweets = [i.AsDict() for i in timeline]
      with open(datadir + "/" + sn + ".tweets.json", "w") as tweetsjson:
           json.dump(tweets, tweetsjson) # Store the informtion in a JSON. 

  This should extract the follower and tweets and save them to pickle and JSON files in the datadir.

  3. Process the Data

  Now that you have an input JSON file of tweets, you can set it to the tweetsjson variable in the code below to read it as a pandas DataFrame.

  For the rest of the tutorial, start a new script for convenience.

  import json 
  import matplotlib.pyplot as plt 
  import networkx as nx 
  import numpy as np 
  import pandas as pd 
  import re 
  from plotly.offline import iplot, plot
  from operator import itemgetter 

  Use pandas to import the JSON file as a pandas DataFrame.

  df = pd.read_json(tweetsjson) 

  Set tfinal as the final DataFrame to make.

  tfinal = pd.DataFrame(columns = ["created_at", "id", "in_reply_to_screen_name",                                       "in_reply_to_status_id", "in_reply_to_user_id",                                        "retweeted_id", "retweeted_screen_name", "user_mentions_screen_name",                                       "user_mentions_id", "text", "user_id", "screen_name", "followers_count"])

  Then, extract the columns you’re interested in and add them to tfinal.

  eqcol = ["created_at", "id", "text"]
  tfinal[eqcol] = df[eqcol]
  tfinal = filldf(tfinal)
  tfinal = tfinal.where((pd.notnull(tfinal)), None)

  Use the following functions to extract information from them. Each function extracts information form the input df DataFrame and adds it to the tfinal one.

  First, get the basic information: screen name, user ID and how many followers.

  def getbasics(tfinal):
       """
       Get the basic information about the user.
       """
       tfinal["screen_name"] = df["user"].apply(lambda x: x["screen_name"])
       tfinal["user_id"] = df["user"].apply(lambda x: x["id"])
       tfinal["followers_count"] = df["user"].apply(lambda x: x["followers_count"])
       return tfinal

  Then, get information on which tweets have been retweeted.

  def getretweets(tfinal):
       """
       Get retweets.
       """
       # Inside the tag "retweeted_status" will find "user" and will get "screen name" and "id". 
       tfinal["retweeted_screen_name"] = df["retweeted_status"].apply(lambda x: x["user"]["screen_name"] if x is not np.nan else np.nan)
       tfinal["retweeted_id"] = df["retweeted_status"].apply(lambda x: x["user"]["id_str"] if x is not np.nan else np.nan)
       return tfinal

  Figure out which tweets are replies and to who they are replying.

  def getinreply(tfinal):
       """
       Get reply info.
       """
       # Just copy the "in_reply" columns to the new DataFrame.
       tfinal["in_reply_to_screen_name"] = df["in_reply_to_screen_name"]
       tfinal["in_reply_to_status_id"] = df["in_reply_to_status_id"]
       tfinal["in_reply_to_user_id"]= df["in_reply_to_user_id"]
       return tfinal

  The following function runs each of these functions to get the information into tfinal.

  def filldf(tfinal):
       """
       Put it all together.
       """
       getbasics(tfinal)
       getretweets(tfinal)
       getinreply(tfinal)
       return tfinal

  You’ll use this getinteractions() function in the next step when creating the graph. This takes the actual information from the tfinal DataFrame and puts it into the format that a graph can use.

  def getinteractions(row):
       """
       Get the interactions between different users.
       """
       # From every row of the original DataFrame.
       # First we obtain the "user_id" and "screen_name".
       user = row["user_id"], row["screen_name"]
       # Be careful if there is no user id.
       if user[0] is None:
           return (None, None), []

  For the remainder of the for loop, get the information if it’s there.

      # The interactions are going to be a set of tuples.
      interactions = set()
  
      # Add all interactions. 
      # First, we add the interactions corresponding to replies adding 
      # the id and screen_name.
      interactions.add((row["in_reply_to_user_id"], 
      row["in_reply_to_screen_name"]))
      # After that, we add the interactions with retweets.
      interactions.add((row["retweeted_id"], 
      row["retweeted_screen_name"]))
      # And later, the interactions with user mentions.
      interactions.add((row["user_mentions_id"], 
      row["user_mentions_screen_name"]))
  
      # Discard if user id is in interactions.
      interactions.discard((row["user_id"], row["screen_name"]))
      # Discard all not existing values.
      interactions.discard((None, None))
      # Return user and interactions.
      return user, interactions

  4. Create the Graph

  Initialize the graph with networkx.

  graph = nx.Graph()

  Loop through the tfinal DataFrame and get the interaction information. Use the getinteractions function to get each user and interaction involved with each tweet.

  for index, tweet in tfinal.iterrows():
       user, interactions = getinteractions(tweet)
       user_id, user_name = user
       tweet_id = tweet["id"]
       for interaction in interactions:
           int_id, int_name = interaction
           graph.add_edge(user_id, int_id, tweet_id=tweet_id)
           graph.node[user_id]["name"] = user_name
           graph.node[int_id]["name"] = int_name

  5. Evaluate the Graph

  In the field of social network analysis (SNA), researchers use measurements of nodes and edges to tell what graphs re like. This lets you separate the signal from noise when looking at network graphs.

  First, look at the degrees and edges of the graph. The print statements should print out the information about these measurements.

  degrees = [val for (node, val) in graph.degree()]
  print("The maximum degree of the graph is " + str(np.max(degrees))) 
  print("The minimum degree of the graph is " + str(np.min(degrees)))
  print("There are " + str(graph.number_of_nodes()) + " nodes and " + str(graph.number_of_edges()) + " edges present in the graph")
  print("The average degree of the nodes in the graph is " + str(np.mean(degrees))) 

  Are all the nodes connected?

  if nx.is_connected(graph):
       print("The graph is connected")
   else:
       print("The graph is not connected")
   print("There are " + str(nx.number_connected_components(graph)) + " connected in the graph.")

  Information about the largest subgraph can tell you what sort of tweets represent the majority.

  largestsubgraph = max(nx.connected_component_subgraphs(graph), key=len)
   print("There are " + str(largestsubgraph.number_of_nodes()) + " nodes and " + str(largestsubgraph.number_of_edges()) + " edges present in the largest component of the graph.")

  The clustering coefficient tells you how close together the nodes congregate using the density of the connections surrounding a node. If many nodes are connected in a small area, there will be a high clustering coefficient.

  print("The average clustering coefficient is " + str(nx.average_clustering(largestsubgraph)) + " in the largest subgraph")
  print("The transitivity of the largest subgraph is " + str(nx.transitivity(largestsubgraph)))
  print("The diameter of our graph is " + str(nx.diameter(largestsubgraph)))
  print("The average distance between any two nodes is " + str(nx.average_shortest_path_length(largestsubgraph)))

  Centrality tells you how many direct, “one step,” connections each node has to other nodes in the network, and there are two ways to measure it. “Betweenness centrality” represents which nodes act as “bridges” between nodes in a network by finding the shortest paths and counting how many times each node falls on one. “Closeness centrality,” instead, scores each node based on the sum of the shortest paths.

  graphcentrality = nx.degree_centrality(largestsubgraph)
  maxde = max(graphcentrality.items(), key=itemgetter(1))
  graphcloseness = nx.closeness_centrality(largestsubgraph)
  graphbetweenness = nx.betweenness_centrality(largestsubgraph, normalized=True, endpoints=False)
  maxclo = max(graphcloseness.items(), key=itemgetter(1))
  maxbet = max(graphbetweenness.items(), key=itemgetter(1))

  print("The node with ID " + str(maxde[0]) + " has a degree centrality of " + str(maxde[1]) + " which is the max of the graph.")
  print("The node with ID " + str(maxclo[0]) + " has a closeness centrality of " + str(maxclo[1]) + " which is the max of the graph.")
  print("The node with ID " + str(maxbet[0]) + " has a betweenness centrality of " + str(maxbet[1]) + " which is the max of the graph.")

  6. Plot the Map

  Get the edges and store them in lists Xe and Ye in the x- and y-directions.

  Xe=[]
  Ye=[]
  for e in G.edges():
      Xe.extend([pos[e[0]][0], pos[e[1]][0], None])
      Ye.extend([pos[e[0]][1], pos[e[1]][1], None])

  Define the Plotly “trace” for nodes and edges. Plotly uses these traces as a way of storing the graph data right before it’s plotted.

  trace_nodes = dict(type="scatter",
                   x=Xn, 
                   y=Yn,
                   mode="markers",
                   marker=dict(size=28, color="rgb(0,240,0)"),
                   text=labels,
                   hoverinfo="text")
  
  trace_edges = dict(type="scatter",                  
                   mode="lines",                  
                   x=Xe,                  
                   y=Ye,                 
                   line=dict(width=1, color="rgb(25,25,25)"),                                         hoverinfo="none")

  Plot the graph with the Fruchterman-Reingold layout algorithm. This image shows an example of a graph plotted with this algorithm, designed to provide clear, explicit ways the nodes are connected.

  

  pos = nx.fruchterman_reingold_layout(G)

  Use the axis and layout variables to customize what appears on the graph. Using the showline=False, option, you will hide the axis line, grid, tick labels and title of the graph. Then the fig variable creates the actual figure.

  axis = dict(showline=False, 
            zeroline=False,
            showgrid=False,
            showticklabels=False,
            title=""
            )
  
  layout = dict(title= "My Graph",  
               font= dict(family="Balto"),
               width=600,
               height=600,
               autosize=False,
               showlegend=False,
               xaxis=axis,
               yaxis=axis,
               margin=dict(
               l=40,
               r=40,
               b=85,
               t=100,
               pad=0,   
       ),
       hovermode="closest",
       plot_bgcolor="#EFECEA", # Set background color.           
       )
  
   fig = dict(data=[trace_edges, trace_nodes], layout=layout)

  Annotate with the information you want others to on each node. Use the labels variable to list (with the same length as pos) what should appear as an annotation.

  labels = range(len(pos))

  def make_annotations(pos, anno_text, font_size=14, font_color="rgb(10,10,10)"):
       L=len(pos)
       if len(anno_text)!=L:
           raise ValueError("The lists pos and text must have the same len")
       annotations = []
       for k in range(L):
           annotations.append(dict(text=anno_text[k], 
                                   x=pos[k][0], 
                                   y=pos[k][1]+0.075,#this additional value is chosen by trial and error
                                   xref="x1", yref="y1",
                                   font=dict(color= font_color, size=font_size),
                                   showarrow=False)
                             )
       return annotations  
  fig["layout"].update(annotations=make_annotations(pos, labels))

  Finally, plot.

  iplot(fig)

  

  December 21, 2019

  Education, Science

  ---

• Make a word cloud in a single line of Python

  

  This is a concise way to make a word cloud using Python. It can teach you basics of coding while creating a nice graphic.

  It’s actually four lines of code, but making the word cloud only takes one line, the final one.

  import nltk
  from wordcloud import WordCloud
  nltk.download("stopwords")
  WordCloud(background_color="white", max_words=5000, contour_width=3, contour_color="steelblue").generate_from_text(" ".join([r for r in open("mobydick.txt", "r").read().split() if r not in set(nltk.corpus.stopwords.words("english"))])).to_file("wordcloud.png")

  Just tell me what to do now!

  The first two lines lines specify the required packages you must download with these links: nltk and wordcloud. You may also try these links: nltk and wordcloud to download them. The third line downloads the stop words (common words like “the”, “a” and “in”) that you don’t want in your word cloud.

  The fourth line is complicated. Calling the WordCloud() method, you can specify the background color, contour color and other options (found here). generate_from_text() takes a string of words to put in the word cloud.

  The " ".join() creates this string of words separated by spaces from a list of words. The for loop in the square brackets[] creates this list of each word from the input file (in this case, mobydick.txt) with the r variable letting you use each word one at a time in the list.

  The input file is open(), read() and split() into its words under the condition (using if) they aren’t in nltk.corpus.stopwords.words("english"). Finally, to_file() saves the image as wordcloud.png.

  How to use this code

  In the code, change "mobydick.txt" to the name of your text file (keep the quotation marks). Save the code in a file makewordcloud.py in the text file’s directory, and use a command line interface (such as Terminal) to navigate to the directory.

  Run your script using python makewordcloud.py, and check out your wordcloud.png!

  December 17, 2019

  Science

  ---

• Global Mapping of Critical Minerals

  The periodic table below illustrates the global abundance of critical minerals in the Earth’s crust in parts per million (ppm). Hover over each element to view! Lanthanides and actinides are omitted due to lack of available data.

  Data is obtained from the USGS handbook “Critical Mineral Resources of the United States— Economic and Environmental Geology and Prospects for Future Supply.” The code used is found here.

  Bokeh Plot https://cdn.pydata.org/bokeh/release/bokeh-1.4.0.min.js Bokeh.set_log_level(“info”);

  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  Because these minerals tend to concentrate in specific countries like niobium in Brazil or antimony in China and remain central to many areas of society such as national defense or engineering, governments like the US have come forward with listing these minerals as “critical.”

  

  The abundance across different continents is shown in the map above.

  You can find gallium, the most abundant of the critical minerals, in place of aluminum and zinc, elements smaller than gallium. Processing bauxite ore or sphalerite ore (from the sediment-hosted, Mississippi Valley-type and volcanogenic massive sulfide) of zinc yield gallium. The US meets its gallium needs through primary, recycled and refined forms of the element.

  German and indium have uses in electronics, flat-panel display screens, light-emitting diodes (LEDs) and solar power arrays. China, Belgium, Canada, Japan and South Korea are the main producers of indium while germanium production can be more complicated. In many cases, countries import primary germanium from other ones, such as Canada importing from the US or Finland from the Democratic Republic of the Congo, to recover them.

  Rechargable battery cathodes and jet aircraft turbine engines make use of cobalt. While the element is the central atom in vitamin B12, excess and overexposure can cause lung and heart dysfunction and dermatitis.

  As one of only three countries that processes beryllium into products, the US doesn’t put much time or money into exploring for new deposits within its own borders because a single producer dominates the domestic berllyium market. Beryllium finds uses in magnetic resonance imaging (MRI) and medical lasers.

  December 15, 2019

  Education, Science

  ---

• A Deep Learning Overview with Python

  This course proposes a quick introduction to deep learning and two of its major networks, convolutional neural networks (CNNs) and recurrent neural networks (RNNs). The purpose is to give an intuitive sense of how to implement deep learning approaches for various tasks. To use this iPython notebook, run the python code in separate files for each cell. The content below each cell of this notebook is the output for running those cells.

  Simple perceptron¶

  In [1]:

  import numpy as np
  
  # sigmoid function
  def sigmoid(x,deriv=False):
      if(deriv==True):
          return x*(1-x)
      return 1/(1+np.exp(-x))
      
  # input dataset
  X = np.array([[0,0,1],
                [0,1,1],
                [1,0,1],
                [1,1,1]])
      
  # output dataset            
  y = np.array([[0,0,1,1]]).T
  
  # seed random numbers to make calculation
  # deterministic (just a good practice)
  np.random.seed(1)
  
  # initialize weights randomly with mean 0
  syn0 = 2*np.random.random((3,1)) - 1
  
  for j in range(100000):
  
      # forward propagation
      l0 = X
      l1 = sigmoid(np.dot(l0,syn0))
  
      # how much did we miss?
      l1_error = y - l1
      if (j% 10000) == 0:
          print("Error:" + str(np.mean(np.abs(l1_error))))
  
      # multiply how much we missed by the 
      # slope of the sigmoid at the values in l1
      l1_delta = l1_error * sigmoid(l1,True)
  
      # update weights
      syn0 += np.dot(l0.T,l1_delta)
  
  print()
  print("Prediction after Training:")
  print(l1)
  

  Error:0.517208275438
  Error:0.00795484506673
  Error:0.0055978239634
  Error:0.00456086918013
  Error:0.00394482243339
  Error:0.00352530883742
  Error:0.00321610234673
  Error:0.00297605968522
  Error:0.00278274003022
  Error:0.0026227273927
  
  Prediction after Training:
  [[ 0.00301758]
   [ 0.00246109]
   [ 0.99799161]
   [ 0.99753723]]
  

  What is the loss function here? How is it calculated?

  Any idea how it would perform on non-linearly separable data? How could we test it?

  Multilayer perceptron¶

  Let’s use the fact that the sigmoid is differenciable (while the step function we saw in the slides is not). This allows us to add more layers (hence more modelling power).

  In [2]:

  import numpy as np
  
  def sigmoid(x,deriv=False):
  	if(deriv==True):
  	    return x*(1-x)
  
  	return 1/(1+np.exp(-x))
      
  X = np.array([[0,0,1],
                [0,1,1],
                [1,0,1],
                [1,1,1]])
                  
  y = np.array([[0],
  			  [1],
  			  [1],
  			  [0]])
  
  np.random.seed(1)
  
  # randomly initialize our weights with mean 0
  syn0 = 2*np.random.random((3,4)) - 1
  syn1 = 2*np.random.random((4,1)) - 1
  
  for j in range(100000):
  
  	# Feed forward through layers 0, 1, and 2
      l0 = X
      l1 = sigmoid(np.dot(l0,syn0))
      l2 = sigmoid(np.dot(l1,syn1))
  
      # how much did we miss the target value?
      l2_error = y - l2
      
      if (j% 10000) == 0:
          print("Error:" + str(np.mean(np.abs(l2_error))))
          
      # in what direction is the target value?
      # were we really sure? if so, don't change too much.
      l2_delta = l2_error*sigmoid(l2,deriv=True)
  
      # how much did each l1 value contribute to the l2 error (according to the weights)?
      l1_error = l2_delta.dot(syn1.T)
      
      # in what direction is the target l1?
      # were we really sure? if so, don't change too much.
      l1_delta = l1_error * sigmoid(l1,deriv=True)
  
      syn1 += l1.T.dot(l2_delta)
      syn0 += l0.T.dot(l1_delta)
      
  print()
  print(l2)
  

  Error:0.496410031903
  Error:0.00858452565325
  Error:0.00578945986251
  Error:0.00462917677677
  Error:0.00395876528027
  Error:0.00351012256786
  Error:0.00318350238587
  Error:0.00293230634228
  Error:0.00273150641821
  Error:0.00256631724004
  
  [[ 0.00199094]
   [ 0.99751458]
   [ 0.99771098]
   [ 0.00294418]]
  

  Setting up the environment¶

  We have done toy examples for feedforward networks. Things quickly become complicated, so let’s go deeper by relying on high-level frameworks: TensorFlow and Keras. Most technicalities are thus avoided so that you can directly play with networks.

  In [ ]:

  !conda install tensorflow keras
  

  In [3]:

  import tensorflow as tf
  import keras
  

  /Users/syedather/.local/lib/python3.6/site-packages/matplotlib/__init__.py:1067: UserWarning: Duplicate key in file "/Users/syedather/.matplotlib/matplotlibrc", line #2
    (fname, cnt))
  Using TensorFlow backend.
  

  In [4]:

  hello = tf.constant('Hello, TensorFlow!')
  sess = tf.Session()
  print(sess.run(hello))
  

  b'Hello, TensorFlow!'
  

  CNNs¶

  We are going to use the MNIST dataset for our first task. The code below loads the dataset and shows one training example and its label.

  In [5]:

  from __future__ import print_function
  import keras
  from keras.datasets import mnist
  from keras.models import Sequential
  from keras.layers import Dense, Dropout, Flatten
  from keras.layers import Conv2D, MaxPooling2D
  from keras import backend as K
  from pylab import *
  
  # the data, split between train and test sets
  (x_train, y_train), (x_test, y_test) = mnist.load_data()
  
  print("The first training instance is labeled as: "+str(y_train[0]))
  

  The first training instance is labeled as: 5
  

  In [6]:

  figure(1)
  imshow(x_train[0], interpolation='nearest')
  

  Out[6]:

  <matplotlib.image.AxesImage at 0x1259b2320>

  Now study the following code. What is the network we use? How many layers? What hyper parameters?

  In [7]:

  # Setup some hyper parameters
  batch_size = 128
  num_classes = 10
  epochs = 15
  
  # input image dimensions
  img_rows, img_cols = 28, 28
  
  # This is some technicality regarding Keras' dataset
  if K.image_data_format() == 'channels_first':
      x_train = x_train.reshape(x_train.shape[0], 1, img_rows, img_cols)
      x_test = x_test.reshape(x_test.shape[0], 1, img_rows, img_cols)
      input_shape = (1, img_rows, img_cols)
  else:
      x_train = x_train.reshape(x_train.shape[0], img_rows, img_cols, 1)
      x_test = x_test.reshape(x_test.shape[0], img_rows, img_cols, 1)
      input_shape = (img_rows, img_cols, 1)
  
  # We convert the matrices to floats as we will use real numbers
  x_train = x_train.astype('float32')[:1000]
  x_test = x_test.astype('float32')[:200]
  x_train /= 255
  x_test /= 255
  print('x_train shape:', x_train.shape)
  print(x_train.shape[0], 'train samples')
  print(x_test.shape[0], 'test samples')
  
  # convert class vectors to binary class matrices
  y_train = keras.utils.to_categorical(y_train, num_classes)[:1000]
  y_test = keras.utils.to_categorical(y_test, num_classes)[:200]
  
  
  # Build network
  model = Sequential()
  model.add(Conv2D(32, kernel_size=(3, 3),
                   activation='relu',
                   input_shape=input_shape))
  model.add(Conv2D(64, (3, 3), activation='relu'))
  model.add(MaxPooling2D(pool_size=(2, 2)))
  # model.add(Dropout(0.25))
  model.add(Flatten())
  model.add(Dense(128, activation='relu'))
  # model.add(Dropout(0.5))
  model.add(Dense(num_classes, activation='softmax'))
  
  model.compile(loss=keras.losses.categorical_crossentropy,
                optimizer=keras.optimizers.Adam(),
                metrics=['accuracy'])
  
  # Train
  model.fit(x_train, y_train,
            batch_size=batch_size,
            epochs=epochs,
            verbose=1,
            validation_data=(x_test, y_test))
  
  # Evaluate on test data
  score = model.evaluate(x_test, y_test, verbose=0)
  print()
  print('Test loss:', score[0])
  print('Test accuracy:', score[1])
  
  # Evaluate on training data
  score = model.evaluate(x_train, y_train, verbose=0)
  print()
  print('Train loss:', score[0])
  print('Train accuracy:', score[1])
  

  x_train shape: (1000, 28, 28, 1)
  1000 train samples
  200 test samples
  Train on 1000 samples, validate on 200 samples
  Epoch 1/15
  1000/1000 [==============================] - 4s 4ms/step - loss: 1.7244 - acc: 0.5660 - val_loss: 0.9116 - val_acc: 0.7900
  Epoch 2/15
  1000/1000 [==============================] - 4s 4ms/step - loss: 0.5967 - acc: 0.8320 - val_loss: 0.5148 - val_acc: 0.8100
  Epoch 3/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.4394 - acc: 0.8670 - val_loss: 0.3056 - val_acc: 0.8600
  Epoch 4/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.3296 - acc: 0.9050 - val_loss: 0.3263 - val_acc: 0.9000
  Epoch 5/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.2205 - acc: 0.9360 - val_loss: 0.2092 - val_acc: 0.9200
  Epoch 6/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.1684 - acc: 0.9560 - val_loss: 0.1870 - val_acc: 0.9450
  Epoch 7/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.1325 - acc: 0.9690 - val_loss: 0.1597 - val_acc: 0.9350
  Epoch 8/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.0990 - acc: 0.9740 - val_loss: 0.1617 - val_acc: 0.9400
  Epoch 9/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.0636 - acc: 0.9840 - val_loss: 0.1434 - val_acc: 0.9450
  Epoch 10/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.0393 - acc: 0.9960 - val_loss: 0.1545 - val_acc: 0.9400
  Epoch 11/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.0267 - acc: 0.9950 - val_loss: 0.1444 - val_acc: 0.9400
  Epoch 12/15
  1000/1000 [==============================] - 4s 4ms/step - loss: 0.0158 - acc: 1.0000 - val_loss: 0.1642 - val_acc: 0.9350
  Epoch 13/15
  1000/1000 [==============================] - 3s 3ms/step - loss: 0.0090 - acc: 1.0000 - val_loss: 0.1475 - val_acc: 0.9450
  Epoch 14/15
  1000/1000 [==============================] - 4s 4ms/step - loss: 0.0057 - acc: 1.0000 - val_loss: 0.1556 - val_acc: 0.9350
  Epoch 15/15
  1000/1000 [==============================] - 4s 4ms/step - loss: 0.0041 - acc: 1.0000 - val_loss: 0.1651 - val_acc: 0.9350
  
  Test loss: 0.165074422359
  Test accuracy: 0.935
  
  Train loss: 0.00311407446489
  Train accuracy: 1.0
  

  Is there anything wrong here?

  How do you think a linear classifier performs?

  In [8]:

  # Setup some hyper parameters
  batch_size = 128
  num_classes = 10
  epochs = 15
  
  # input image dimensions
  img_rows, img_cols = 28, 28
  
  # This is some technicality regarding Keras' dataset
  if K.image_data_format() == 'channels_first':
      x_train = x_train.reshape(x_train.shape[0], 1, img_rows, img_cols)
      x_test = x_test.reshape(x_test.shape[0], 1, img_rows, img_cols)
      input_shape = (1, img_rows, img_cols)
  else:
      x_train = x_train.reshape(x_train.shape[0], img_rows, img_cols, 1)
      x_test = x_test.reshape(x_test.shape[0], img_rows, img_cols, 1)
      input_shape = (img_rows, img_cols, 1)
  
  # We convert the matrices to floats as we will use real numbers
  x_train = x_train.astype('float32')[:1000]
  x_test = x_test.astype('float32')[:200]
  x_train /= 255
  x_test /= 255
  print('x_train shape:', x_train.shape)
  print(x_train.shape[0], 'train samples')
  print(x_test.shape[0], 'test samples')
  
  # convert class vectors to binary class matrices
  y_train = keras.utils.to_categorical(y_train, num_classes)[:1000]
  y_test = keras.utils.to_categorical(y_test, num_classes)[:200]
  
  
  # Build network
  model = Sequential()
  model.add(Conv2D(32, kernel_size=(3, 3),
                   activation='relu',
                   input_shape=input_shape))
  model.add(Conv2D(64, (3, 3), activation='relu'))
  model.add(MaxPooling2D(pool_size=(2, 2)))
  model.add(Dropout(0.25))
  model.add(Flatten())
  model.add(Dense(128, activation='relu'))
  model.add(Dropout(0.5))
  model.add(Dense(num_classes, activation='softmax'))
  
  model.compile(loss=keras.losses.categorical_crossentropy,
                optimizer=keras.optimizers.Adam(),
                metrics=['accuracy'])
  
  # Train
  model.fit(x_train, y_train,
            batch_size=batch_size,
            epochs=epochs,
            verbose=1,
            validation_data=(x_test, y_test))
  
  # Evaluate on test data
  score = model.evaluate(x_test, y_test, verbose=0)
  print()
  print('Test loss:', score[0])
  print('Test accuracy:', score[1])
  
  # Evaluate on training data
  score = model.evaluate(x_train, y_train, verbose=0)
  print()
  print('Train loss:', score[0])
  print('Train accuracy:', score[1])
  

  x_train shape: (1000, 28, 28, 1)
  1000 train samples
  200 test samples
  

  ---------------------------------------------------------------------------
  ValueError                                Traceback (most recent call last)
  <ipython-input-8-a1470fe28059> in <module>()
       53           epochs=epochs,
       54           verbose=1,
  ---> 55           validation_data=(x_test, y_test))
       56 
       57 # Evaluate on test data
  
  ~/anaconda3/lib/python3.6/site-packages/keras/models.py in fit(self, x, y, batch_size, epochs, verbose, callbacks, validation_split, validation_data, shuffle, class_weight, sample_weight, initial_epoch, steps_per_epoch, validation_steps, **kwargs)
      961                               initial_epoch=initial_epoch,
      962                               steps_per_epoch=steps_per_epoch,
  --> 963                               validation_steps=validation_steps)
      964 
      965     def evaluate(self, x=None, y=None,
  
  ~/anaconda3/lib/python3.6/site-packages/keras/engine/training.py in fit(self, x, y, batch_size, epochs, verbose, callbacks, validation_split, validation_data, shuffle, class_weight, sample_weight, initial_epoch, steps_per_epoch, validation_steps, **kwargs)
     1628             sample_weight=sample_weight,
     1629             class_weight=class_weight,
  -> 1630             batch_size=batch_size)
     1631         # Prepare validation data.
     1632         do_validation = False
  
  ~/anaconda3/lib/python3.6/site-packages/keras/engine/training.py in _standardize_user_data(self, x, y, sample_weight, class_weight, check_array_lengths, batch_size)
     1478                                     output_shapes,
     1479                                     check_batch_axis=False,
  -> 1480                                     exception_prefix='target')
     1481         sample_weights = _standardize_sample_weights(sample_weight,
     1482                                                      self._feed_output_names)
  
  ~/anaconda3/lib/python3.6/site-packages/keras/engine/training.py in _standardize_input_data(data, names, shapes, check_batch_axis, exception_prefix)
      111                         ': expected ' + names[i] + ' to have ' +
      112                         str(len(shape)) + ' dimensions, but got array '
  --> 113                         'with shape ' + str(data_shape))
      114                 if not check_batch_axis:
      115                     data_shape = data_shape[1:]
  
  ValueError: Error when checking target: expected dense_4 to have 2 dimensions, but got array with shape (1000, 10, 10)

  Let’s use this model to predict a value for the first training instance we vizualized.

  In [ ]:

  print(model.predict(np.expand_dims(x_train[0], axis=0)))
  

  Is the model correct here? What is the output of the network?

  RNNs¶

  We will now switch to RNNs. These require more resources, so we can’t do the fanciest applications during the workshop. We will do some sentiment classification of movie reviews.

  In [9]:

  from __future__ import print_function
  import numpy as np
  import keras
  from keras.preprocessing import sequence
  from keras.models import Sequential
  from keras.layers import Dense, Dropout, Embedding, LSTM, Bidirectional
  from keras.datasets import imdb
  
  # Number of considered words, based on frequencies
  max_features = 20000
  # cut texts after this number of words
  maxlen = 100
  batch_size = 32
  
  print('Loading data...')
  (x_train, y_train), (x_test, y_test) = keras.datasets.imdb.load_data(num_words=max_features, index_from=3)
  
  # This is just for pretty printing the sentences...
  word_to_id = keras.datasets.imdb.get_word_index()
  word_to_id = {k:(v+3) for k,v in word_to_id.items()}
  word_to_id["<PAD>"] = 0
  word_to_id["<START>"] = 1
  word_to_id["<UNK>"] = 2
  id_to_word = {value:key for key,value in word_to_id.items()}
  
  print("Here's the input for the first training instance:")
  print(' '.join(id_to_word[id] for id in x_train[0] ))
  

  Loading data...
  Downloading data from https://s3.amazonaws.com/text-datasets/imdb.npz
  17465344/17464789 [==============================] - 2s 0us/step
  Downloading data from https://s3.amazonaws.com/text-datasets/imdb_word_index.json
  1646592/1641221 [==============================] - 0s 0us/step
  Here's the input for the first training instance:
  <START> this film was just brilliant casting location scenery story direction everyone's really suited the part they played and you could just imagine being there robert <UNK> is an amazing actor and now the same being director <UNK> father came from the same scottish island as myself so i loved the fact there was a real connection with this film the witty remarks throughout the film were great it was just brilliant so much that i bought the film as soon as it was released for retail and would recommend it to everyone to watch and the fly fishing was amazing really cried at the end it was so sad and you know what they say if you cry at a film it must have been good and this definitely was also congratulations to the two little boy's that played the <UNK> of norman and paul they were just brilliant children are often left out of the praising list i think because the stars that play them all grown up are such a big profile for the whole film but these children are amazing and should be praised for what they have done don't you think the whole story was so lovely because it was true and was someone's life after all that was shared with us all
  

  What do you think about this text? Is it a positive or negative review?

  In [10]:

  print("Here are the dataset shapes")
  print(len(x_train), 'train sequences')
  print(len(x_test), 'test sequences')
  
  print("And the input for the first instance is represented as:")
  print(x_train[0])
  

  Here are the dataset shapes
  25000 train sequences
  25000 test sequences
  And the input for the first instance is represented as:
  [1, 14, 22, 16, 43, 530, 973, 1622, 1385, 65, 458, 4468, 66, 3941, 4, 173, 36, 256, 5, 25, 100, 43, 838, 112, 50, 670, 2, 9, 35, 480, 284, 5, 150, 4, 172, 112, 167, 2, 336, 385, 39, 4, 172, 4536, 1111, 17, 546, 38, 13, 447, 4, 192, 50, 16, 6, 147, 2025, 19, 14, 22, 4, 1920, 4613, 469, 4, 22, 71, 87, 12, 16, 43, 530, 38, 76, 15, 13, 1247, 4, 22, 17, 515, 17, 12, 16, 626, 18, 19193, 5, 62, 386, 12, 8, 316, 8, 106, 5, 4, 2223, 5244, 16, 480, 66, 3785, 33, 4, 130, 12, 16, 38, 619, 5, 25, 124, 51, 36, 135, 48, 25, 1415, 33, 6, 22, 12, 215, 28, 77, 52, 5, 14, 407, 16, 82, 10311, 8, 4, 107, 117, 5952, 15, 256, 4, 2, 7, 3766, 5, 723, 36, 71, 43, 530, 476, 26, 400, 317, 46, 7, 4, 12118, 1029, 13, 104, 88, 4, 381, 15, 297, 98, 32, 2071, 56, 26, 141, 6, 194, 7486, 18, 4, 226, 22, 21, 134, 476, 26, 480, 5, 144, 30, 5535, 18, 51, 36, 28, 224, 92, 25, 104, 4, 226, 65, 16, 38, 1334, 88, 12, 16, 283, 5, 16, 4472, 113, 103, 32, 15, 16, 5345, 19, 178, 32]
  

  What do these numbers represent? Is there any limitation you can imagine coming from this?

  In [11]:

  print('Pad sequences (samples x time)')
  x_train = sequence.pad_sequences(x_train, maxlen=maxlen)[:5000]
  x_test = sequence.pad_sequences(x_test, maxlen=maxlen)[:5000]
  print('x_train shape:', x_train.shape)
  print('x_test shape:', x_test.shape)
  y_train = np.array(y_train)[:5000]
  y_test = np.array(y_test)[:5000]
  
  model = Sequential()
  model.add(Embedding(max_features, 128, input_length=maxlen))
  model.add(Bidirectional(LSTM(64)))
  model.add(Dropout(0.5))
  model.add(Dense(1, activation='sigmoid'))
  
  model.compile('adam', 'binary_crossentropy', metrics=['accuracy'])
  
  print('Train...')
  model.fit(x_train, y_train,
            batch_size=batch_size,
            epochs=4,
            validation_data=[x_test, y_test])
  

  Pad sequences (samples x time)
  x_train shape: (5000, 100)
  x_test shape: (5000, 100)
  Train...
  Train on 5000 samples, validate on 5000 samples
  Epoch 1/4
  5000/5000 [==============================] - 54s 11ms/step - loss: 0.6032 - acc: 0.6570 - val_loss: 0.4283 - val_acc: 0.8056
  Epoch 2/4
  5000/5000 [==============================] - 54s 11ms/step - loss: 0.2761 - acc: 0.8918 - val_loss: 0.4403 - val_acc: 0.7948
  Epoch 3/4
  5000/5000 [==============================] - 61s 12ms/step - loss: 0.1101 - acc: 0.9670 - val_loss: 0.6366 - val_acc: 0.8026
  Epoch 4/4
  5000/5000 [==============================] - 56s 11ms/step - loss: 0.0478 - acc: 0.9868 - val_loss: 0.6637 - val_acc: 0.7954
  

  Out[11]:

  <keras.callbacks.History at 0x1392d76d8>

  In [12]:

  print("The neural net predicts that the first instance sentiment is:")
  print(model.predict(np.expand_dims(x_train[0], axis=0)))
  

  The neural net predicts that the first instance sentiment is:
  [[ 0.99445081]]
  

  Remarks? Comments?

  How do the training scores compare to the test scores? How can we improve this? What are the current limitations?

  This RNN use case takes more time to train but it is definitely more impressive. We will model the language, by training on a novel. For each (set of) word(s) in the novel, the objective is to predict the following word. This can be done on any text, and we don’t need annotated data – the text itself is enough.

  Have a look at the following piece of code and try to understand what it does. Then, run it and see the network generating text! At first, the output is not meaningful, but it becomes so over time. This is the magic I was referring to.

  Beware: this will take longer to run on a CPU. A GPU is recommended, but you can still try to run it for a while to see the predictions evolve. On my laptop, an epoch takes 6mins so the full training takes 6hrs. About 20 epochs are required for the generated text to be somewhat meaningful.

  Note, however, that although this seems long, training actual deep learning models for concrete tasks takes days, even on multiple GPUs. This is mostly because of the data size and the much deeper networks.

  In [ ]:

  from __future__ import print_function
  from keras.callbacks import LambdaCallback
  from keras.models import Sequential
  from keras.layers import Dense, Activation
  from keras.layers import LSTM
  from keras.optimizers import RMSprop
  from keras.utils.data_utils import get_file
  import numpy as np
  import random
  import sys
  import io
  
  # We load a text from Nietzsche
  path = get_file('nietzsche.txt', origin='https://s3.amazonaws.com/text-datasets/nietzsche.txt')
  with io.open(path, encoding='utf-8') as f:
      text = f.read().lower()
  print('corpus length:', len(text))
  
  # We create dictionaries of character > index and the other way around
  chars = sorted(list(set(text)))
  print('total chars:', len(chars))
  char_indices = dict((c, i) for i, c in enumerate(chars))
  indices_char = dict((i, c) for i, c in enumerate(chars))
  
  # cut the text in semi-redundant sequences of maxlen characters
  maxlen = 40
  step = 3
  sentences = []
  next_chars = []
  for i in range(0, len(text) - maxlen, step):
      sentences.append(text[i: i + maxlen])
      next_chars.append(text[i + maxlen])
  print('nb sequences:', len(sentences))
  
  print('Vectorization...')
  x = np.zeros((len(sentences), maxlen, len(chars)), dtype=np.bool)
  y = np.zeros((len(sentences), len(chars)), dtype=np.bool)
  for i, sentence in enumerate(sentences):
      for t, char in enumerate(sentence):
          x[i, t, char_indices[char]] = 1
      y[i, char_indices[next_chars[i]]] = 1
  
  
  # build the model: a single LSTM
  print('Build model...')
  model = Sequential()
  model.add(LSTM(128, input_shape=(maxlen, len(chars))))
  model.add(Dense(len(chars)))
  model.add(Activation('softmax'))
  
  optimizer = RMSprop(lr=0.01)
  model.compile(loss='categorical_crossentropy', optimizer=optimizer)
  
  
  def sample(preds, temperature=1.0):
      # helper function to sample an index from a probability array
      preds = np.asarray(preds).astype('float64')
      preds = np.log(preds) / temperature
      exp_preds = np.exp(preds)
      preds = exp_preds / np.sum(exp_preds)
      probas = np.random.multinomial(1, preds, 1)
      return np.argmax(probas)
  
  
  def on_epoch_end(epoch, logs):
      # Function invoked at end of each epoch. Prints generated text.
      print()
      print('----- Generating text after Epoch: %d' % epoch)
  
      start_index = random.randint(0, len(text) - maxlen - 1)
      for diversity in [0.2, 0.5, 1.0, 1.2]:
          print('----- diversity:', diversity)
  
          generated = ''
          sentence = text[start_index: start_index + maxlen]
          generated += sentence
          print('----- Generating with seed: "' + sentence + '"')
          sys.stdout.write(generated)
  
          for i in range(400):
              x_pred = np.zeros((1, maxlen, len(chars)))
              for t, char in enumerate(sentence):
                  x_pred[0, t, char_indices[char]] = 1.
  
              preds = model.predict(x_pred, verbose=0)[0]
              next_index = sample(preds, diversity)
              next_char = indices_char[next_index]
  
              generated += next_char
              sentence = sentence[1:] + next_char
  
              sys.stdout.write(next_char)
              sys.stdout.flush()
          print()
  
  print_callback = LambdaCallback(on_epoch_end=on_epoch_end)
  
  model.fit(x, y,
            batch_size=128,
            epochs=60,
            callbacks=[print_callback])
  

  Downloading data from https://s3.amazonaws.com/text-datasets/nietzsche.txt
  606208/600901 [==============================] - 0s 0us/step
  corpus length: 600893
  total chars: 57
  nb sequences: 200285
  Vectorization...
  Build model...
  Epoch 1/60
  200285/200285 [==============================] - 281s 1ms/step - loss: 1.9553
  
  ----- Generating text after Epoch: 0
  ----- diversity: 0.2
  ----- Generating with seed: "to
  agree with many people. "good" is no "
  to
  agree with many people. "good" is no and it is the the of the same the of the sention of the strenge of the most the self-our of the inderent that the sensive indeed the one of the constitute of the most of the semple of the desire of the sensive of the most of the semple of the sempathy of the one of the into the every to a soul of the some of the persent the free of the semple of the most of the sention of the of the spiritual the 
  ----- diversity: 0.5
  ----- Generating with seed: "to
  agree with many people. "good" is no "
  to
  agree with many people. "good" is no may a suptimes and also orage mankind the one of indeed of one streng the possible the sensition and the inderenation of a sul the in a sould be the orting a solitiarity of religions in a man of such and a scient, in every of and the self-to and of a revilued it is the most in the indeed, and it is assual that the ord of the of the distiture in its all the manter of the soul permans the decours of
  ----- diversity: 1.0
  ----- Generating with seed: "to
  agree with many people. "good" is no "
  to
  agree with many people. "good" is no causest and hew the fown of every groktulr
  destined a the art it noteriness of one it all and
  and cothinded of that rendercaterfroe to doe," in the pational the is the onl yutre
  allor upitsoon,--one
  viburan mused a "master in the that niver if
  a pridicle quesiles of
  the shoold enss nowxing to
  feef ma.t--wute disequerly that then her rewadd finale the eeblive alse rusurefver" a selovery catte he re
  ----- diversity: 1.2
  ----- Generating with seed: "to
  agree with many people. "good" is no "
  to
  agree with many people. "good" is no likeurenes, it is novamentstisuser'stone, indos paces. fund, wethel feel the
  que let doee new eveny that is that the catel. thotgy is
  within ceoks of theregeritades) and itwas brutmes ageteron
  clyrelogilabl freephi; its. by an? andaver happ
  one of his absuman artificss? itself old a
  ooker himsood and bus hray
  fined in smuch is sudtirers of rerarder from and
  afutty
  mest utfered with to "bewnook one
  Epoch 2/60
   81664/200285 [===========>..................] - ETA: 2:37 - loss: 1.6395

  December 14, 2019

  Science

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