Draft position for players in the NBA for the 2020-21 season

Imagen tomada de Reddit.

Statistical Odds & Ends

When the 2022 NBA draft happened almost a month ago, I thought to myself: do players picked earlier in the draft (i.e. higher-ranked) actually end up having better/longer careers?

If data wasn’t an issue, the way I would do it would be to look at players chosen in the draft lottery (top 60 picks) in the past 10/20 years. For each player, I would look at how many years he played in the NBA and see if there was a correlation between that and draft position. (Here, number of years in the NBA is a proxy for how successful an NBA career is. There are other possible ways to define success, e.g. minutes played, points scored.)

Unfortunately data is an issue, so I ended up looking at a related question: What are the draft positions of players currently in the NBA? If players picked earlier in the draft are more…

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Karl Marx’s philosophy detailed – how his work continues to influence

This excellent short summary of Karl Marx’s outlook and lasting influence was published by Newshub 5 July 2022. It was written by Christopher Pollard for The Conversation In 1845, Karl Marx declared: “philosophers have only interpreted the world in various ways; the point is to change it”. Change it he did. Political movements representing masses of […]

Karl Marx’s philosophy detailed – how his work continues to influence

A New Theory in Physics Claims to Solve the Mystery of Consciousness

Interesante artículo, sin embargo, debe tenerse cuidado de no caer en la tentación del reduccionismo. Tentación a la que parecen sucumbir los investigadores cuando señalan que “la conciencia debe investigarse con las mismas herramientas matemáticas que los físicos usan para otros fenómenos relativistas conocidos”. A pesar de ello, “Ella no usa órganos sensoriales, mide sus representaciones neuronales directamente por la interacción entre una parte de su cerebro con otras partes. Mide sus representaciones neuronales según sus relaciones con otras representaciones neuronales.” es un notable hallazgo en términos de la relación del todo y las partes y en términos de que lo fundamental en los fenómenos son sus relaciones internas. Esto refuerza diversos principios del Materialismo Dialéctico.

Source: Bar-Ilan University How do 1.4 kg of brain tissue create thoughts, feelings, mental images, and an inner world? The ability of the brain to create consciousness has baffled some for millennia. The mystery of consciousness lies in the fact that each of us has subjectivity, something that is like to sense, feel and think. […]

A New Theory in Physics Claims to Solve the Mystery of Consciousness

The IRA and the four horsemen of the climate apocalypse

Michael Roberts Blog

The announcement that US President’s Biden’s Inflation Reduction Act (IRA) has got the backing of pro-business, coal-mining owner Democratic Senator Manchin has been greeted with a wave of optimism that the US target of cutting carbon emissions in half before the end of this decade (or 40% compared with 2005 levels), can be met.  “This bill will really turbocharge that transition to clean energy, it will transform markets where already solar PV, wind and batteries are in many cases cheaper than incumbent fossil fuels,” said Anand Gopal, executive director of policy at Energy Innovation, an open source research body.  “Increasingly I’m more optimistic that keeping the temperature rise under 2C (3.6F) is more reachable. 1.5C is a stretch goal at this point.” 

The bill will cut US emissions by between 31% and 44% below 2005 levels by 2030, according to Rhodium Group, a non-partisan research firm. A separate…

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Written by William A. Harris


Review by Gunnel Minett

The average adult human brain weighs about 1.5 kg and accounts for about 2% of the adult body weight. It is very similar in shape and form for most human beings. Still it is unique to each individual. By the time a child is born the brain has billions of neurones that are wired together through trillions of interconnections into what can only be described as the most sophisticated supercomputer we can only begin to imagine.

This book describes the journey from the fertilised egg to the moment a new human being is born, and beyond. It is written by an experimental neurobiologist, describing the development of neurobiology which has lead to the increase in understanding of human evolution as well as the human brain.

Still, the story of how this development takes place is filled…

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Ukraine: the invasion of capital

Imagen tomada de Translator without Borders.

Michael Roberts Blog

Last week, Ukraine’s foreign private creditors agreed to the country’s request for a two-year freeze on payments on about $20bn of foreign debt.  This would enable Ukraine to avoid defaulting on its overseas borrowings.  Unlike other ‘emerging economies’ in debt distress, it seems that foreign bondholders are happy to help Ukraine out – if only for two years.  The move will save Ukraine $6bn over the period, helping to reduce pressure on central bank reserves, which slid by 28 per cent year-to-date, despite significant foreign aid.

Ukraine’s economy is, not surprisingly, in a desperate state. Real GDP is projected to decline by more than 30% in 2022 and the unemployment rate is at 35% (Constantinescu et al. 2022, Blinov and Djankov 2022, National Bank of Ukraine 2022). “We are grateful for the private sector support of our proposal in such terrible times for our country,” responded Yuriy Butsa, Ukraine’s deputy…

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Don’t Pay UK: Mass civil disobedience is risky – but it could work

Imagen del encabezado tomada de Left Voice.

Don’t Pay UK is still in its infancy, but the momentum is encouraging. Over the next month, we will see whether it can mount a serious campaign which will force the government’s hand, or simply wither away.

Don’t Pay UK: Mass civil disobedience is risky – but it could work

Russia under Putin

Imagen tomada de worldometers.

Michael Roberts Blog

In my last post, I described how Western capital is planning to take over and control Ukraine’s resources and exploit its labour force to the maximum in order to boost the profitability of both Ukraine’s domestic capitalists (oligarchs) and foreign multi-nationals.

However, there is a problem for Western capital and Ukraine’s oligarchs: it’s Russia. The war has already led to Russian forces gaining control of at least $12.4trn worth of Ukraine’s resources in energy (cola), metals and mineral deposits, apart from agricultural land. If Putin’s forces succeed in annexing Ukrainian land seized during Russia’s invasion, Kyiv would permanently lose almost two-thirds of its deposits. Moscow now controls 63% of Ukraine’s coal deposits, 11% of its oil, 20% of its natural gas, 42% of its metals, and 33% of its rare earths.

So any rebuilding effort funded by Western capital has a major obstacle.  “Not only will Ukraine have lost…

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Rank and trace are equal for a real symmetric idempotent matrix

Mathematical Odds & Ends

Proposition. Let $latex mathbf{X} in mathbb{R}^{n times n}$ be a matrix that is symmetric ($latex mathbf{X}^top = mathbf{X}$) and idempotent ($latex mathbf{X}^2 = mathbf{X}$). Then the rank of $latex mathbf{X}$ is equal to the trace of $latex mathbf{X}$. In fact, they are both equal to the sum of the eigenvalues of $latex mathbf{X}$.

The proof is relatively straightforward. Since $latex mathbf{X}$ is real and symmetric, it is orthogonally diagonalizable, i.e. there is an orthogonal matrix $latex mathbf{U}$ ($latex mathbf{U}^top mathbf{U} = mathbf{I}$) and a diagonal matrix $latex mathbf{D}$ such that $latex mathbf{D} = mathbf{UXU}^top$ (see here for proof).

Since $latex mathbf{X}$ is idempotent,

$latex begin{aligned} mathbf{X}^2 &= mathbf{X},
mathbf{U}^top mathbf{D}^2 mathbf{U} &= mathbf{U}^T mathbf{DU},
mathbf{D}^2 &= mathbf{D}. end{aligned}$

Since $latex mathbf{D}$ is a diagonal matrix, it implies that the entries on the diagonal must be zeros or ones. Thus, the number of ones on the diagonal (which is $latex text{rank}(mathbf{D})…

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Asymptotic distribution of the Pearson chi-square statistic

Imagen tomada de ResearchGate.

Statistical Odds & Ends

I recently learned of a fairly succinct proof for the asymptotic distribution of the Pearson chi-square statistic (from Chapter 9 of Reference 1), which I share below.

First, the set-up: Assume that we have $latex n$ independent trials, and each trial ends in one of $latex J$ possible outcomes, which we label (without loss of generality) as $latex 1, 2, dots, J$. Assume that for each trial, the probability of the outcome being $latex j$ is $latex p_j > 0$. Let $latex n_j$ denote that number of trials that result in outcome $latex j$, so that $latex sum_{j=1}^J n_j = n$. Pearson’s $latex chi^2$-statistic is defined as

$latex begin{aligned} chi^2 = sum_{text{cells}} dfrac{(text{obs} – text{exp})^2}{text{exp}} = sum_{j=1}^J dfrac{(n_j – np_j)^2}{np_j}. end{aligned}$

Theorem. As $latex n rightarrow infty$, $latex chi^2 stackrel{d}{rightarrow} chi_{J-1}^2$, where $latex stackrel{d}{rightarrow}$ denotes convergence in distribution.

Before proving the theorem, we prove a lemma that we will…

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