This post is motivated by the wonderful numberphile video here by @mathemaniac.
The left panel is shows a ‘chain’ of circles between parallel lines which intersect the green circle. The lines and small circles in the inverted (reflected) in the green circle to produce the right panel. The blue circle is the inversion of the top line and the big red outer circle the inversion of the bottom line. The outer red circle and blue circle touch (osculate) at the centre of the green cricle (not shown).
The graphic is not perfect but pretty enough for me…
I read the first edition of this book. It is an excellent concise ‘snapshot’ of hemodynamics. I particularly enjoyed the chapter on allometric scaling. The graphics are very instructive and the references will be a useful resource. The second edition looks better in presentation and style (as well as being up to date). However, it is $130.48 (USD).
This is a further post to Purulent Seas.
If you look at the full data set. It is clear that the number of infections relate to the number of patient days. The following present naive linear, quadratic and exponential models with 95% confidence prediction intervals. The adjusted R^2 are: 0.82, 0.83 and 0.71 respectively. There can be no dispute with the positive relationship. Ranking is a more difficult story.
If the process is modeled as an underlying Poisson distribution (correcting for peer group size) and hospital peer group as an independent explanatory variable then the following emerges:
If we model each peer group as Poisson distributed with estimate of rate per 10000 based on aggregate in peer group.
Simulations (1000) of peer group provides insight likelihood of observed rates for peer group:
The probability of the observed data given the model can be used to rank. In the graphic the three lowest ranked are displayed.
This “ranking” hides a number of issues. The Poisson distribution is discrete and the metric used here to rank is the same for many values.
If tied values are labeled as one:
Further, the peer groups are relevant. The following unpacks the peer groups and tied metrics are given the same rank. The outlier points still ‘pop out’.
I have made these posts to motivate deeper consideration of data, particularly to argue against facile conclusions based on point estimates. There are some simple (expected) findings: larger volumes are associated with high numbers of infections; hospitals with more vulnerable patients have high rates of infection. Ranking is a more complex exercise. I am merely advocating a deep and robust thoughtful assessment of data to generate testable hypotheses rather than reflexive responses. I am not making any conclusions. I do not have all the data, full understanding of the methodologies and particularly the limitations, esp in data collection (and any potential selection biases). I merely present a number of ways (not that I am agreeing with any of them) to look at what was publicly available (to discourage naive pejorative but stimulating headlines that may influence those in power).
A recent government press release regarding blood stream Staph aureus infections made headline news. There were many pejorative and inflammatory statements (excuse the pun). This post is motivated to look a little more closely at the data than the point estimates and the “ranking”. This information is important. However, it must interpreted with a deeper understanding of the methodology (and its limitations) and the observed differences should be hypothesis generating and not for simplistic inferences based on rankings.
The following was extracted (with some difficulty) from the downloadable interactive (*.mht) file. It is presented to amplify/provide other ways of visualizing and representing the data (not to promulgate a particular position). It behoves us all to think carefully about information provided to us.
The overall data is presented above (hovering over data points provides tooltips). The hospitals were broken down by type:
The following are histograms of the overall data and by hospital category:
The following table presents probability of cases > x per 10000 bed days for the different hospital categories:
This is a delightful book. The books them is asymmetry in the natural world from the quantum level, to molecules, our macroscopic world and the heavens. The book is a well written and enjoyable journey from the late 19 th century to the present. It provides the accretive account (but exciting and not predictable) development of the Standard Model of particle physics viewed from the lens of asymmetry (symmetry). The journey has colorful personalities, accidents of discovery and historical attribution (fortune favoring the prepared mind). The book is filled with useful analogies and allegories and I found layers of my own confusion and misconceptions falling away courtesy of these helpful descriptions. The author uses words and a few selected figures and no equations.
The book culminates in the meaning of the Higg’s field, the discovery of the Higg’s boson and the anticipation of a unifying theoretical framework of the four fundamental forces with supersymmetry (SUSY). The book does not aim to look at the mathematical or physical limitations ( testability, falsifiability issues). It hints at the future but provides us (provided me) a clearer and firmer understanding and integration of the concepts and discoveries that have got us to this point.
I continue to learn a lot from Mathematica Stackexchange. It is a great resource. I have had some long and stressful days of late (and there are more to come). I get a lot out of contemplating the questions on the site. I learn from the expert users, I have fun trying to come up with an answer and I improve my communication skills in trying to provide a coherent, hopefully helpful, answer. The endeavour helps me process the baggage of the day.
I have not been an active review and only very recently looked into metamathematica.stackexchange. I found a wonderful, living active community. This blog post arises out of an interesting question regarding analytics on the Mathematica Stackexhange site. (it is here:
1:eJxTTMoPKmZmYGAoLkoGABNTAwM=…if a MSE user happens to want to look).
I have rerun some of the code (NOT MINE). I found solace, that despite being a casual and not acitve enough user (in editing, reviewing, closing) that I am a member of this community. There is a larger set of users but this focused on 100 users ranked by reputation. This seriously overcalls me but, even if a statisitical anomaly (my penchant for answering low level questions), I am glad to be part of this community that the analytics has peered into.
Here is a The community graph:
I am in community group 3 (the blue square)…
My other communities are in major flux. The following graphic, a comic expression of extreme frustration has has very deep roots in a sad reality of the present.
I hope all the communities we inhabit grow, develop and are resilient to the storms that come.
I read the first edition of this book. It is excellent. It provides a systematic approach/ It covers fluid mechanics: steady flow, unsteady flow, viscoelastic properties of tissues, Windkessel and more sophisticated models of the circulation.
It provides clear examples and figures from the in-vitro and in-vivo data.
The section of valve physiology and pathophysiology was excellent. The discussion on prosthetic valves, grafts and other vascular scaffolds was very instructive, clearly written with illuminating figures.
The book ends with discussion of measurement techniques: pressure, flow, velocity including discussions of doppler echocardiography, magnetic resonance imaging, computer fluid dynamics.
I hope the subsequent editions continue in this tradition and I look forward to seeing them.
This is a delightful book. I ‘inhaled’ it. I had some ‘laugh out loud’ moments. Simon Singh weaves the Mathematical secrets with historical vignettes and insights into the wonderfully creative writers of the Simpsons and Futurama. There is also a five part exam of sorts.
I admit to being one of those who recognized the Homer’s epiphany after wearing Kissinger’s spectacles (retrieved from the toilet) as homage to the Wizard of Oz…a confident but incorrect assertion by the Scarecrow after having been bestowed a brain by the Wizard…
I have enjoyed Simon Singh’s books, esp Fermat’s Last Theorem and The Code Book. This was an incredibly enjoyable read (from the view point of a fan of Mathematics and The Simpsons).
This post is motivated by the New York Times NumberPlay puzzle: The Princess Problem.
I was very late to this problem. The solution is very pleasing. It reminds me of the intermediate value theorem and the puzzle about meeting the monk on his journey up and down the meeting.
The puzzle can be visualized as a discrete Markov process:
I have plotted the strategy and 100 simulations of the journey through the rooms by the Princess. The horizontal axis is time. The vertical axis is the rooms. The purple “mountain” is the strategy and the red points are when the Prince successfully knocks (the first point of relevance). This is not a proof but a visual motivation towards one perhaps.
Note all all the even rooms are solved on the “ascent” and all the odd number rooms on the “descent”.