The Bed Of Procrustes PDF Free Download

10/3/2021by admin

The Bed of Procrustes: Philosophical and Practical Aphorisms – by Nassim Nicholas Taleb
Date read: 8/18/17. Recommendation: 9/10.

Dec 5, 2019 - DOWNLOAD PDF Incerto Deluxe Edition Fooled by Randomness The Black Swan The Bed of Procrustes Antifragile Skin in the Game Free Epub/MOBI/EBooks. Nassim Nicholas Taleb’s Incerto series consists of 4 books: Fooled by Randomness, The Black Swan, The Bed of Procrustes and Antifragile. This is the third in the series, and the only book I hadn. 926 downloads 4213 Views 2MB Size Report. The Bed of Procrustes: Philosophical and Practical Aphorisms.

Great introduction to Taleb's take on uncertainty, which he discusses in detail in his other books that make up the Incerto series: Antifragile, Fooled by Randomness, and The Black Swan. This book offers a succinct look into how we deal with what we don't know. Taleb examines our tendency to package and reduce ideas into neat narratives that fit within the constraints of our limited knowledge.

See my notes below or Amazon for details and reviews.

My Notes:

Every aphorism here is a about a Procrustean bed of sorts–we humans, facing limits of knowledge, and things we do not observe, the unseen and the unknown, resolve the tension by squeezing life and the world into crisp commoditized ideas, reductive categories, specific vocabularies, and prepackaged narratives.

These aphorisms are standalone compressed thoughts revolving around my main idea of how we deal, and should deal, with what we don't know.

The person you are most afraid to contradict is yourself.

An idea starts to be interesting when you get scared of taking it to its logical conclusion.

To bankrupt a fool, give him information.

An erudite is someone who displays less than he knows; a journalist or consultant, the opposite.

Your brain is most intelligent when you don't instruct it on what to do–something people who take showers discover on occasion.

In nature we never repeat the same motion; in captivity (office, gym, commute, sports), life is just repetitive-stress injury. No randomness.

If you know, in the morning, what your day looks like with any precision, you are a little bit dead–the more precision, the more dead you are.

When we want to do something while unconsciously certain to fail, we seek advice so we can blame someone else for the failure.

The most painful moments are not those we spend with uninteresting people; rather, they are those spent with uninteresting people trying hard to be interesting.

The characteristic feature of the loser is to bemoan, in general terms, mankind's flaws, biases, contradictions, and irrationality–without exploiting them for fun and profit.

You exist if and only if you are free to do things without a visible objective, with no justification and, above all, outside the dictatorship of someone else's narrative.

To be completely cured of newspapers, spend a year reading the previous week's newspapers.

The opposite of success isn't failure; it is name-dropping.

Modernity needs to understand that being rich and becoming rich are not mathematically, personally, socially, and ethically the same thing.

The fastest way to become rich is to socialize with the poor; the fastest way to become poor is to socialize with the rich.

You will be civilized on the day you can spend a long period doing nothing, learning nothing, and improving nothing, without feeling the slightest amount of guilt.

Someone who says 'I am busy' is either declaring incompetence (and lack of control of his life) or trying to get rid of you.

To see if you like where you are, without the chains of dependence, check if you are as happy returning as you were leaving.

Modernity: we created youth without heroism, age without wisdom, and life without grandeur.

People focus on role models; it is more effective to find antimodels–people you don't want to resemble when you grow up.

People usually apologize so they can do it again.

It is those who use others who are the most upset when someone uses them.

The three most harmful addictions are heroin, carbohydrates, and a monthly salary.

My only measure of success is how much time you have to kill.

With terminal disease, nature lets you die with abbreviated suffering; medicine lets you suffer with prolonged dying.

Only in recent history has 'working hard' signaled pride rather than shame for lack of talent, finesse, and, mostly, sprezzatura.

We are hunters; we are only truly alive in those moments when we improvise; no schedule, just small surprises and stimuli from the environment.

You are alive in inverse proportion to the density of cliches in your writing.

We unwittingly amplify commonalities with friends, dissimilarities with strangers, and contrasts with enemies.

True humility is when you can surprise yourself more than others; the rest is either shyness or good marketing.

For the robust, an error is information; for the fragile an error is an error.

Games were created to give nonheroes the illusion of winning. In real life, you don't know who really won or lost (except too late), but you can tell who is heroic and who is not.

Knowledge is subtractive, not additive–what we subtract (reduction by what does not work, what not to do), not what we add (what to do).

The best way to spot a charlatan: someone (like a consultant or a stock broker) who tells you what to do instead of what not to do.

They think that intelligence is about noticing things that are relevant (detecting patterns); in a complex world, intelligence consists in ignoring things that are irrelevant (avoiding false patterns).

The curious mind embraces science; the gifted and sensitive, the arts; the practical, business; the leftover becomes an economist.

The only definition of an alpha male: if you try to be an alpha male, you will never be one.

At any stage, humans can thirst for money, knowledge, or love; sometimes for two, never for three.

Love without sacrifice is like theft.

Our overreactive brains are more likely to impose the wrong, simplistic narrative than no narrative at all.

The mind can be a wonderful tool for self-delusion–it was not designed to deal with complexity and nonlinear uncertainties.

Counter to common discourse, more information means more delusions: our detection of false patterns is growing faster and faster as a side effect of modernity and the information age.

Thus my classical values make me advocate the triplet of erudition, elegance, and courage; against modernity's phoniness, nerdiness, and philistinism.

Procrustes superimposition. The figure shows the three transformation steps of an ordinary Procrustes fit for two configurations of landmarks. (a) Scaling of both configurations to the same size; (b) Transposition to the same position of the center of gravity; (c) Rotation to the orientation that provides the minimum sum of squared distances between corresponding landmarks.

In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name Procrustes (Greek: Προκρούστης) refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off.

In mathematics:

  • an orthogonal Procrustes problem is a method which can be used to find out the optimal rotation and/or reflection (i.e., the optimal orthogonal linear transformation) for the Procrustes Superimposition (PS) of an object with respect to another.
  • a constrained orthogonal Procrustes problem, subject to det(R) = 1 (where R is a rotation matrix), is a method which can be used to determine the optimal rotation for the PS of an object with respect to another (reflection is not allowed). In some contexts, this method is called the Kabsch algorithm.

When a shape is compared to another, or a set of shapes is compared to an arbitrarily selected reference shape, Procrustes analysis is sometimes further qualified as classical or ordinary, as opposed to Generalized Procrustes analysis (GPA), which compares three or more shapes to an optimally determined 'mean shape'.

Introduction[edit]

To compare the shapes of two or more objects, the objects must be first optimally 'superimposed'. Procrustes superimposition (PS) is performed by optimally translating, rotating and uniformly scaling the objects. In other words, both the placement in space and the size of the objects are freely adjusted. The aim is to obtain a similar placement and size, by minimizing a measure of shape difference called the Procrustes distance between the objects. This is sometimes called full, as opposed to partial PS, in which scaling is not performed (i.e. the size of the objects is preserved). Notice that, after full PS, the objects will exactly coincide if their shape is identical. For instance, with full PS two spheres with different radii will always coincide, because they have exactly the same shape. Conversely, with partial PS they will never coincide. This implies that, by the strict definition of the term shape in geometry, shape analysis should be performed using full PS. A statistical analysis based on partial PS is not a pure shape analysis as it is not only sensitive to shape differences, but also to size differences. Both full and partial PS will never manage to perfectly match two objects with different shape, such as a cube and a sphere, or a right hand and a left hand.

In some cases, both full and partial PS may also include reflection. Reflection allows, for instance, a successful (possibly perfect) superimposition of a right hand to a left hand. Thus, partial PS with reflection enabled preserves size but allows translation, rotation and reflection, while full PS with reflection enabled allows translation, rotation, scaling and reflection.

Optimal translation and scaling are determined with much simpler operations (see below).

Ordinary Procrustes analysis[edit]

Here we just consider objects made up from a finite number k of points in n dimensions. Often, these points are selected on the continuous surface of complex objects, such as a human bone, and in this case they are called landmark points.

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The shape of an object can be considered as a member of an equivalence class formed by removing the translational, rotational and uniform scaling components.

Translation[edit]

For example, translational components can be removed from an object by translating the object so that the mean of all the object's points (i.e. its centroid) lies at the origin.

Mathematically: take k{displaystyle k} points in two dimensions, say

((x1,y1),(x2,y2),,(xk,yk)){displaystyle ((x_{1},y_{1}),(x_{2},y_{2}),dots ,(x_{k},y_{k})),}.

The mean of these points is (x¯,y¯){displaystyle ({bar {x}},{bar {y}})} where

x¯=x1+x2++xkk,y¯=y1+y2++ykk.{displaystyle {bar {x}}={frac {x_{1}+x_{2}+cdots +x_{k}}{k}},quad {bar {y}}={frac {y_{1}+y_{2}+cdots +y_{k}}{k}}.}

Now translate these points so that their mean is translated to the origin (x,y)(xx¯,yy¯){displaystyle (x,y)to (x-{bar {x}},y-{bar {y}})}, giving the point (x1x¯,y1y¯),{displaystyle (x_{1}-{bar {x}},y_{1}-{bar {y}}),dots }.

Uniform scaling[edit]

Likewise, the scale component can be removed by scaling the object so that the root mean square distance (RMSD) from the points to the translated origin is 1. This RMSD is a statistical measure of the object's scale or size:

s=(x1x¯)2+(y1y¯)2+k{displaystyle s={sqrt {{(x_{1}-{bar {x}})^{2}+(y_{1}-{bar {y}})^{2}+cdots } over k}}}

The scale becomes 1 when the point coordinates are divided by the object's initial scale:

((x1x¯)/s,(y1y¯)/s){displaystyle ((x_{1}-{bar {x}})/s,(y_{1}-{bar {y}})/s)}.

Notice that other methods for defining and removing the scale are sometimes used in the literature.

Rotation[edit]

Removing the rotational component is more complex, as a standard reference orientation is not always available. Consider two objects composed of the same number of points with scale and translation removed. Let the points of these be ((x1,y1),){displaystyle ((x_{1},y_{1}),ldots )}, ((w1,z1),){displaystyle ((w_{1},z_{1}),ldots )}. One of these objects can be used to provide a reference orientation. Fix the reference object and rotate the other around the origin, until you find an optimum angle of rotation θ{displaystyle theta ,!} such that the sum of the squared distances (SSD) between the corresponding points is minimised (an example of least squares technique).

A rotation by angle θ{displaystyle theta ,!} gives

(u1,v1)=(cosθw1sinθz1,sinθw1+cosθz1){displaystyle (u_{1},v_{1})=(cos theta ,w_{1}-sin theta ,z_{1},sin theta ,w_{1}+cos theta ,z_{1}),!}.

Procrustean Bed

where (u,v) are the coordinates of a rotated point. Taking the derivative of (u1x1)2+(v1y1)2+{displaystyle (u_{1}-x_{1})^{2}+(v_{1}-y_{1})^{2}+cdots } with respect to θ{displaystyle theta } and solving for θ{displaystyle theta } when the derivative is zero gives

θ=tan1(i=1k(wiyizixi)i=1k(wixi+ziyi)).{displaystyle theta =tan ^{-1}left({frac {sum _{i=1}^{k}(w_{i}y_{i}-z_{i}x_{i})}{sum _{i=1}^{k}(w_{i}x_{i}+z_{i}y_{i})}}right).}

When the object is three-dimensional, the optimum rotation is represented by a 3-by-3 rotation matrixR, rather than a simple angle, and in this case singular value decomposition can be used to find the optimum value for R (see the solution for the constrained orthogonal Procrustes problem, subject to det(R) = 1).

Shape comparison[edit]

The difference between the shape of two objects can be evaluated only after 'superimposing' the two objects by translating, scaling and optimally rotating them as explained above. The square root of the above mentioned SSD between corresponding points can be used as a statistical measure of this difference in shape:

d=(u1x1)2+(v1y1)2+.{displaystyle d={sqrt {(u_{1}-x_{1})^{2}+(v_{1}-y_{1})^{2}+cdots }}.}

This measure is often called Procrustes distance. Notice that other more complex definitions of Procrustes distance, and other measures of 'shape difference' are sometimes used in the literature.

Superimposing a set of shapes[edit]

We showed how to superimpose two shapes. The same method can be applied to superimpose a set of three or more shapes, as far as the above mentioned reference orientation is used for all of them. However, Generalized Procrustes analysis provides a better method to achieve this goal.

Generalized Procrustes analysis (GPA)[edit]

GPA applies the Procrustes analysis method to optimally superimpose a set of objects, instead of superimposing them to an arbitrarily selected shape.

Generalized and ordinary Procrustes analysis differ only in their determination of a reference orientation for the objects, which in the former technique is optimally determined, and in the latter one is arbitrarily selected. Scaling and translation are performed the same way by both techniques. When only two shapes are compared, GPA is equivalent to ordinary Procrustes analysis.

The algorithm outline is the following:

  1. arbitrarily choose a reference shape (typically by selecting it among the available instances)
  2. superimpose all instances to current reference shape
  3. compute the mean shape of the current set of superimposed shapes
  4. if the Procrustes distance between mean and reference shape is above a threshold, set reference to mean shape and continue to step 2.

Variations[edit]

There are many ways of representing the shape of an object.The shape of an object can be considered as a member of an equivalence class formed by taking the set of all sets of k points in n dimensions, that is Rkn and factoring out the set of all translations, rotations and scalings. A particular representation of shape is found by choosing a particular representation of the equivalence class. This will give a manifold of dimension kn-4. Procrustes is one method of doing this with particular statistical justification.

Bookstein obtains a representation of shape by fixing the position of two pointscalled the bases line. One point will be fixed at the origin and the other at (1,0)the remaining points form the Bookstein coordinates.

The bed of procrustes pdf free download 64 bit

It is also common to consider shape and scale that is with translational and rotational components removed.

Examples[edit]

Shape analysis is used in biological data to identify the variations of anatomical features characterised by landmark data, for example in considering the shape of jaw bones.[1]

One study by David George Kendall examined the triangles formed by standing stones to deduce if these were often arranged in straight lines. The shape of a triangle can be represented as a point on the sphere, and the distribution of all shapes can be thought of a distribution over the sphere.The sample distribution from the standing stones was compared with the theoretical distribution to show that the occurrence of straight lines was no more than average.[2]

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See also[edit]

References[edit]

  1. ^'Exploring Space Shape'Archived 2006-09-01 at the Wayback Machine by Nancy Marie Brown, Research/Penn State, Vol. 15, no. 1, March 1994
  2. ^'A Survey of the Statistical Theory of Shape', by David G. Kendall, Statistical Science, Vol. 4, No. 2 (May, 1989), pp. 87–99
  • F.L. Bookstein, Morphometric tools for landmark data, Cambridge University Press, (1991).
  • J.C. Gower, G.B. Dijksterhuis, Procrustes Problems, Oxford University Press (2004).
  • I.L.Dryden, K.V. Mardia, Statistical Shape Analysis, Wiley, Chichester, (1998).

The Bed Of Procrustes Pdf Free Download 64 Bit

External links[edit]

Wikimedia Commons has media related to Procrustes analysis.
  • Extensions to continuum of points and distributions Procrustes Methods, Shape Recognition, Similarity and Docking, by Michel Petitjean.
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