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# robertgrant / non-parametric-bayes-updating Public

Non-parametric kernel methods for updating a Bayesian model's parameters with new batches of data

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## Note! I no longer use Github. This repo may be out of date.

All my ex-Github repos are now stored and maintained at my personal website.

Why did I leave Github? Because I consider the compulsory imposition of 2-factor authentication to be inappropriate for people writing software, including cryptography, which can attract severe punishments in certain jurisdictions. We all know that the organisations that hold the 2nd factors (mobile telephony providers, tech companies) are compromised, willingly or otherwise, in their relationships with security agencies, benign or otherwise.

Why not just close it down? Because you might use it programmatically, via http or API, and I don't want to hurt you (by breaking your code) while trying to help you (by raising issues of privacy and confidentiality).

# non-parametric-bayes-updating

Non-parametric kernel methods for updating a Bayesian model with frequent batches or windows of data

This repository contains digital material associated with a paper currently submitted. The journal of choice does not permit pre-print archiving, but a post-print will appear here if it is accepted. You can also write to me for a copy.

## Abstract

One of the big attractions for people adopting Bayesian methods is the promise of "updating" their parameter estimates and predictions as more data arrive. Yesterday's posterior becomes today's prior. In practice, this is not simple, requiring at the very least a complete set of sufficient statistics, random samples from an unchanging population, and no changes of probability distribution for the prior. In this day and age, there is great interest in big data (including fast, streaming data) but complex Bayesian models are usually prohibitively slow as the data accumulate. Another trend is to use windowing to privilege more recent data; there is no practicable Bayesian methodology for this as part of updating. Sometimes, one would like to update without imposing an a priori distribution on yesterday's posterior and without estimating lots of statistics. This also might allow for faster analyses as new data arrive. I discuss a kernel density approach in MCMC and SIR, consider two ways to reduce computational burden, evaluate different bandwidth strategies, compare them in two simulation studies and explore their feasibility in a real-life application. The approach shows promise in low dimensional problems. SIR is very fast but prone to problems of degeneracy and bandwidth shrinking. However, with human monitoring and adjustment, these can potentially be overcome. I identify some priorities for further research in the paper. Then, I propose a smoothed piecewise density estimate called Kudzu, as it smooths over trees (specifically, a random forest classifying the prior draws augmented with random uniformly distributed non-draws). This has two important advantages in higher-dimensional problems: it is relatively fast and does not experience degeneracy.

## Kernels and sampling algorithms

There is an existing and obvious parametric updating algorithm. Data batch 1 produces posterior sample 1, and from that we obtain summary statistics that get plugged into the parameters of the prior for data batch 2 (possibly via method of moments).

We want a different, non-parametric, approach, that is:

• flexible with regard to distributions
• fast
• and accurate. We might not get all three, but we want to understand where the compromises arise and how to choose which aspect to prioritise. Our method will involve taking the posterior sample from batch b-1 (each "sample" comprises a number of "draws" from an unknown multivariate distribution) and using it as a prior sample for batch b.

This project tests out some kernel density approaches.

First, evaluating prior density by adding and normalising unbounded kernel densities (e.g. multivariate normal: MVN) around each of the prior draws, and adding a kernel density estimate, evaluated over the draws of yesterday's posterior, of the log-prior to today's log-likelihood (Algorithm 2).

But maybe we can make this run more quickly. Maybe we don't have to evaluate every prior draw's kernel. Two ideas came to mind:

• dividing the draws of yesterday's posterior into hypercuboids of parameter space, and calculating the kernel from the centroids of those, weighted by the number of draws in each (Algorithm 3)
• using a bounded kernel (e.g. a translated and rescaled beta distribution), so that only "neighbour" prior draws close to the current proposal (within each Metropolis-Hastings iteration) need to be evaluated (Algorithm 4)

Unfortunately, these have serious limitations. The number of hypercuboids will soon exceed the number of prior draws, and unless there is a fine grid, it introduces a probably unacceptable amount of distortion. To use bounded kernels, we have to find neighbours, which is computationally expensive in itself, and can only be coded inside Stan if you use a few tricks.

Apart from MCMC, we can also use sampling importance resampling (SIR). Take the prior draws, evaluate the likelihood at each draw, and store a number of posterior draws, proportional to the likelihood, at or around each one. We can apply the kernel for smoothing either before (SIR-B, smoothing the prior) or after (SIR-A, smoothing the posterior) (Algorithm 5)

Unfortunately, high-dimensional parameter spaces create two challenges. Firstly, as dimension increases, it is more likely that only one prior draw will produce posterior draws (degeneracy). This is a problem because you just get the kernel back, centred on that one draw, introducing bias in the location and spread, and loss of correlation. A sensible way around this is to "locally oversample" in the region of the MLE. An easy way to do this is to draw from truncated kernels inside a hypercuboid defined by, for example, the MLE +/- 3 standard errors (these being estimated from a frequentist analysis of batch b alone).

However, the second problem is that, as dimension increases, a larger proportion of the hypercuboid will lie outside a convex likelihood surface, and most plausible non-convex ones too. It would be sensible to rotate the parameter space by Cholesky / PCA so as to reduce correlation and the proportion of wasted volume; I call this SIR-Cholesky-B. But this will not help at all with non-convex likelihoods, and we may need some method to establish a more flexible region via likelihood hulls. This is not the focus of this project, but a sketch algorithm can be worked out (SIR-hulls-B, Algorithm 6, not included in the paper).

## Windowing

Suppose we have updated over batches 1 to 10. Now, we want to update our posteriors to include likelihood contributions from batches 11 onwards, but only to use the ten most recent batches each time. There are two ways to do this.

The first is simple, but needs more hardware and storage. It is to keep a collection of sliding windows (see https://www.oreilly.com/radar/the-world-beyond-batch-streaming-101/). In this example, you would keep updates from batch i to batch i+9. At any given batch b, you need to update i where b-9<=i<=b.

The second is more theoretically pleasing but at risk of distortion in real life. When you get to batch 11, you add its log-likelihood, then subtract batch 1's log-posterior, then add the original log-prior (batch 0, if you like) back in. You can even imagine having some kind of exponentially weighted likelihood.

## Bandwidth

I used Sheather-Jones bandwidths in each dimension in this project. I also tried out multiplying them by various constants. Sometimes, multimodality sneaks through, so I wrote some code to look for it using Hartigans' dip test and to inflate the bandwidth if that came up p<0.05. Some people might get agitated that this is not Bayesian, and likewise that simulation studies' long-run properties is somehow incompatible with Bayes. I direct you to http://www.robertgrantstats.co.uk/blog/26.html for my philosophical standpoint on probability, if you fit this profile.

If you want to make a contribution to this idea, probably the lowest-hanging fruit is to try out a local bandwidth algorithm like that proposed by Leo Breiman and co-authors way back in 1977.

## Kudzu

The prior draws are augmented with random non-draws, uniformly distributed over [min(prior)-k(max(prior)-min(prior)), max(prior)+k(max(prior)-min(prior))] where k is the amount of range extension. I have used k=0.25 and the same number of non-draws as draws. A binary vector identifies draws and non-draws. We then fit a random forest (collection of classification trees using bootstrapping and a random selection of independent variables on each tree) to the augmented prior draws, with the draw/non-draw binary vector as the dependent variable. Each tree is a piecewise constant estimate of the probability of finding a prior draw, comprising a certain number of "leaves". Each leaf occupies a hypercuboid in parameter space and has a constant probability of draws. We normalise the probabilities in each tree so that each tree provides a piecewise density estimate, $d_{ij}$ for the $i$th leaf in the $j$th tree: $\sum_{i=1}^\ell d_{ij} = 1 \forall j \in \mathds{N}, 1\leq j \leq t$. At this stage, this is approximately equivalent to the density estimation trees proposed by Ram and Gray, although their method targeted density via the loss function. This density estimate cannot be provided to Stan because it is not smooth, so we construct the density as the sum of each leaf's density, and replace the discontinuous boundaries of the leaves with logistic functions $\mathrm{expit}(z) = e^z / (1+e^z)$. Denote the boundaries of the $i$th leaf of the $j$th tree in the $k$th dimension as $\theta_{ijk+}$ at the higher edge and $\theta_{ijk-}$ at the lower edge. Then, the total density for a proposed parameter vector $\theta$ is: $\frac{1}{t} \sum_{j=1}^t \sum_{i=1}^\ell (d_{ij} \prod_{k=1}^p \mathrm{expit}(\frac{\theta - \theta_{ijk-}}{\sigma}) \prod_{k=1}^p \mathrm{expit}(\frac{\theta_{ki-} - \theta}{\sigma}))$ where $\sigma$ is a bandwidth for the steepness of the logistic ramp. This can be provided to Stan as a vector of lower boundaries, a vector of upper boundaries, a vector of leaf densities, a vector of indices separating trees in the previous three vectors, and the scalar {\tt sigma}. Note that the classic logistic distribution is itself a product of two logistic ramp functions, because $\mathrm{expit}(z)$ is the cumulative density function, and $\frac{d}{dz}\mathrm{expit}(z) = \mathrm{expit}(z)\mathrm{expit}(-z)$. We can therefore think of Kudzu as an ensemble of multivariate logistic distributions, stretched out at their modes. The only similar work appears to be by Anderlini in 2016, conceived of as trees convolved with piecewise linear functions for particle physics applications, where volumes of data are vast and the density of data must be estimated and stored efficiently rather than the data itself. Anderlini's "smeared" trees are not suitable for our purposes because they have discontinuous gradients.

## Method

I investigated these bandwidths, kernels, and samplers in three ways:

### prior-only simulation

I generated draws from three bivariate distributions (MVN, banana, multimodal typical of LASSO and other regularized models), and fed these in as prior sample 1. Then I updated ten times without any likelihood and compared distributions qualitatively (visually) and by 20-sliced 2-Wasserstein distances. You can find this in the 1-prior-only folder. There is a subfolder for each of the bandwidths, and inside each, prior-only.R does the work and wassersteins.R compares the results.

### simulation study (logistic regression)

I ran a simulation study with a simple logistic regression scenario: 400 simulations, keeping 4000 draws at each update (and other keeping 2000 draws), using Algorithms 1-5. You can find this in the 2-simulation folder. The simulation data are in data_batches and the first posterior for each simulation (from Stan) in first_posteriors. non-para-kernel-updating-sims.R does the sims (I actually set up four instances of R on a DigitalOcean Linux VM and ran this four times in parallel), recombine-simulations.R does what it says, along with generate-outputs.R and tables-figures.R. updates* image files show craft charts of the kernel densities of posteriors from each update of the first 30 simulations in different settings; the dark density superimposed on the tenth update is the Stan analysis of all data together, which we can regard as ground truth for comparison. means* image files show all 400 simulation means for beta0 changing with the updates. stats* image files show the distribution of posterior means across the simulations (mean +/- 2 SDs) for the three parameters over updates.

### feasibility case study (New York taxis)

As a feasibility case study, I applied SIR-A to Chris Whong's 2013 New York yellow taxi data, fitting a 50-parameter model to every taxi journey each day from 1--14 January 2013. When we hit degeneracy, I tried local oversampling (SIR-B). 1--7 Jan were updated. You can find this in the 3-NYCtaxis folder. taxi_data* CSV files are the data in handy cleaned and reduced format. NYC-taxi-analysis.R does the work an is the place to go for readily adaptable code for SIR-A/B. data-prep.R produces the taxi_data* files from Chris Whong's files, so it is here just for completeness and audit trail. summary* image files show the posterior mean +/- 2 posterior SDs for each parameter in turn over the updates. mean_evolution and SD_evolution show all the parameters' posterior stats superimposed over the updates. bivariate* takes a pair of parameters from the areas, the weekdays and the hours, and shows the posterior draws after each update. The red polygons show the convex hull of the following update, so you can see how it contracts over time. Problems evident in this are discussed in the paper.

## Conclusions

Kernel updating is a potentially useful trick when we need to scale up a serious Bayesian analysis to large datasets and run it regularly, as is often the case in commercial data science settings, not to mention pandemics of course.

It does not seem to have been done before; the only similar work I found was by Neufeld and Deutsch at the University of Alberta (see the 2006 paper "Data Integration with Non-Parametric Bayesian Updating"), although they used kriging on quantiles of univariate CDFs rather than kernels on posterior draws.

The submitted paper sets out more detailed recommendations and ares for future research.

You are welcome to get in touch (BayesCamp Ltd, 80 High Street (Unit 83), Winchester, SO23 9AT, England) to discuss this if it would be useful in your own work. But note:

• I do not have students
• I do not join bids for academic research funding
• I do not want a job

Non-parametric kernel methods for updating a Bayesian model's parameters with new batches of data


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# Cross Gramian

In control theory, the cross Gramian (${\displaystyle W_{X}}$, also referred to by ${\displaystyle W_{CO}}$) is a Gramian matrix used to determine how controllable and observable a linear system is.[1][2]

For the stable time-invariant linear system

${\displaystyle {\dot {x}}=Ax+Bu\,}$
${\displaystyle y=Cx\,}$

the cross Gramian is defined as:

${\displaystyle W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,}$

and thus also given by the solution to the Sylvester equation:

${\displaystyle AW_{X}+W_{X}A=-BC\,}$

This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric.

The triple ${\displaystyle (A,B,C)}$ is controllable and observable, and hence minimal, if and only if the matrix ${\displaystyle W_{X}}$ is nonsingular, (i.e. ${\displaystyle W_{X}}$ has full rank, for any ${\displaystyle t>0}$).

If the associated system ${\displaystyle (A,B,C)}$ is furthermore symmetric, such that there exists a transformation ${\displaystyle J}$ with

${\displaystyle AJ=JA^{T}\,}$
${\displaystyle B=JC^{T}\,}$

then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:[3]

${\displaystyle |\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,}$

Thus the direct truncation of the Eigendecomposition of the cross Gramian allows model order reduction (see [1]) without a balancing procedure as opposed to balanced truncation.

The cross Gramian has also applications in decentralized control, sensitivity analysis, and the inverse scattering transform.[4][5]

format: plain


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# Strange difference between Out of bag error and test Error in Random Forest

I'm using random forest and the out of bag error for the level of one class is very different to its test error. I'm working with a cutt-of equal to c(0.2,0.8). Here's the case:

fmla <- as.formula(paste("ex ~ ", paste(names(muestra.fullarbol[,-c(1,2,3,9,10,11,12,17,19,20,21,22,23,24,29,31,32,33,34,35,36,47)]), collapse= "+")))
> bosque <- randomForest(fmla , data=muestra.fullarbol ,mtry=12, ntree=1000  , cutoff=c(0.2,0.8),importance=TRUE)
> bosque

Call:
randomForest(formula = fmla, data = muestra.fullarbol, mtry = 12,      ntree = 1000, cutoff = c(0.2, 0.8), importance = TRUE)
Type of random forest: classification
Number of trees: 1000
No. of variables tried at each split: 12

OOB estimate of  error rate: 15.81%
Confusion matrix:
No Si class.error
No 3999  1     0.00025
Si  758 42     0.94750

As we see, the out of bag error for the level "SI" is 0.94750. However If I use the test set to get a sense of the error, the result is very different:

res.arbol <- predict(bosque,test.fullarbol,type="class")
> summary(res.arbol)
No   Si
7761   43
> table(res.arbol,test.fullarbol$ex) res.arbol No Si No 6937 824 Si 4 39 > prop.table(table(res.arbol,test.fullarbol$ex),1)

res.arbol         No         Si
No 0.89382811 0.10617189
Si 0.09302326 0.90697674

Now we see the error rate for the test set in the class "Si" is very low , it's equal to 0.093 and It doesn´t make sense to me.

I guess that the cut off is working just to predict out of the sample(muestra.fullarbol), but I'm not sure. What can be the reason of that huge difference?

• It could be test shows a very similar confusion matrix, just that it transposed. Neither the print.randomForest or table confusion matrix states clearly if rows are true class or predicted class. Oct 4, 2015 at 21:53

You have transposed the confusion matrix. See test example.

library(randomForest)
#simulate data set
set.seed(123)
obs = 4000+800+200
vars = 6
noise.factor=.2
X =  data.frame(replicate(vars,rnorm(obs)))
yvalue = X[,1]^2+sin(X[,2]) + rnorm(obs) * noise.factor
y = factor(c("No","Si")[(yvalue<=quantile(yvalue, 800/4800))*1+1])
test = sample(obs,200)
rf = randomForest(X[-test,],y[-test],ntree=50,cutoff=c(.2,.8))
print(rf)

#yep this confusion table is inverted compared to print.randomForest
table(predict(rf,X[test,]),y[test])

#here row's a true class, and columns are predicted class
table(y[test],predict(rf,X[test,],))

#cutoff is inherrited from training, but can also be modified during prediction
table(y[test],predict(rf,X[test,]))
table(y[test],predict(rf,X[test,],cutoff=c(.2,.8)))
table(y[test],predict(rf,X[test,],cutoff=c(.7,.000001)))


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# Bibliothèque

Musique » Blue »

## How's a Man Supposed to Change

39 écoutes | Se rendre sur la page du titre

Titres (39)
Titre Album Durée Date
How's a Man Supposed to Change 9 jan. 2012, 21h49m
How's a Man Supposed to Change 9 jan. 2012, 19h11m
How's a Man Supposed to Change 9 jan. 2012, 16h30m
How's a Man Supposed to Change 9 jan. 2012, 4h17m
How's a Man Supposed to Change 9 jan. 2012, 1h39m
How's a Man Supposed to Change 8 jan. 2012, 22h57m
How's a Man Supposed to Change 8 jan. 2012, 20h23m
How's a Man Supposed to Change 8 jan. 2012, 17h45m
How's a Man Supposed to Change 8 jan. 2012, 15h08m
How's a Man Supposed to Change 7 jan. 2012, 22h04m
How's a Man Supposed to Change 7 jan. 2012, 19h26m
How's a Man Supposed to Change 7 jan. 2012, 16h56m
How's a Man Supposed to Change 6 jan. 2012, 23h45m
How's a Man Supposed to Change 6 jan. 2012, 20h37m
How's a Man Supposed to Change 6 jan. 2012, 18h15m
How's a Man Supposed to Change 6 jan. 2012, 15h36m
How's a Man Supposed to Change 5 jan. 2012, 21h10m
How's a Man Supposed to Change 5 jan. 2012, 18h44m
How's a Man Supposed to Change 5 jan. 2012, 16h03m
How's a Man Supposed to Change 5 jan. 2012, 13h26m
How's a Man Supposed to Change 4 jan. 2012, 21h24m
How's a Man Supposed to Change 4 jan. 2012, 18h41m
How's a Man Supposed to Change 4 jan. 2012, 16h15m
How's a Man Supposed to Change 4 jan. 2012, 13h37m
How's a Man Supposed to Change 3 jan. 2012, 22h40m
How's a Man Supposed to Change 3 jan. 2012, 20h06m
How's a Man Supposed to Change 3 jan. 2012, 17h28m
How's a Man Supposed to Change 3 jan. 2012, 14h45m
How's a Man Supposed to Change 3 jan. 2012, 12h08m
How's a Man Supposed to Change 2 jan. 2012, 23h39m
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How's a Man Supposed to Change 2 jan. 2012, 18h19m
How's a Man Supposed to Change 2 jan. 2012, 15h41m
How's a Man Supposed to Change 2 jan. 2012, 13h03m
How's a Man Supposed to Change 1 jan. 2012, 23h13m
How's a Man Supposed to Change 1 jan. 2012, 20h38m
How's a Man Supposed to Change 1 jan. 2012, 18h00m
How's a Man Supposed to Change 1 jan. 2012, 15h54m
How's a Man Supposed to Change 1 jan. 2012, 14h30m


text:
Home » » What is the midpoint of the line joining the points $$(2x,y)$$ and $$(x,2y)$$?

# What is the midpoint of the line joining the points $$(2x,y)$$ and $$(x,2y)$$?

### Question

What is the midpoint of the line joining the points $$(2x,y)$$ and $$(x,2y)$$?

### Options

A) $$(2x, 2y)$$

B) $$(x, y)$$

C) $$(3x, 3y)$$

D) $$\left( \frac{3x}{2}, \frac{3y}{2} \right)$$


text:
Originally Posted by Philip
It is not just my decision. We listen to users' opinions, and we discuss decisions with the mods.

We're still discussing this suggestion. At this time, we're leaning towards just using the general archive forum we already have in place; it is under-used as it is, and this seems to fit its intended purpose.
Thank you for considering this suggestion. I think that would be the perfect place, if approved. So much more could be incorporated into it as well, and I believe a very nice addition.


text:
# Katana (Weapon)

13,973 pages on
this wiki

Katana (, Katana?), also known as Ninja Swords and Dark Blades, are a group of weapons that occur in many Final Fantasy games. The most prominent Katana is the Masamune. Recurring katana are: Kotetsu, Kikuichimonji, Ashura, and the Muramasa and Masamune, which are the two most powerful, though the Masamune is usually idolized as the strongest.

## Contents

Katana are a distinct type of sword forged in regions similar to their originating nation of Japan. They all distinctively bear a curved, single edged blade with a flat shaped guard, and a handle wrapped with cloth with a simple shaped hilt, with all bearing their own subtle designs that distinguish their identities from other blades, with most of their naming based on the blacksmith who forged them. While specifically a kind of backsword, the series presents them as their own unique series of weapon, being portrayed as exotic blades, swift enough to attack in a blink of an eye and unmatched in their cutting power.

Maintaining their exclusivity and distinguished form from other swords of the series, they are reserved for Japanese styled classes such as the Samurai, Parivir, and Ninja.

## AppearancesEdit

### Final FantasyEdit

In Final Fantasy there are Blades only equippable by the Warrior, Thief, and Red Mage classes that possess an icon resembling a Katana. These are all high-rank weapons, only obtainable later in the game.

List of "Katana":

### Final Fantasy IIIEdit

Only the Dark Knight and Ninja Job class can equip Katana, which are called Dark Blades in the game. Murakumo can only be equipped by Dark Knights while Muramasa can only be equipped by Ninjas.

### Final Fantasy IVEdit

Edge can equip two Katana at a time.

List of Katana:

List of Katana:

List of Katana:

### Final Fantasy VEdit

Only the Samurai Job can equip Katana.

List of Katana:

### Final Fantasy VIEdit

Cyan can equip Katana, also known as "Knives". All katana enable the Bushido and Runic command.

List of Katana:

### Final Fantasy VIIEdit

Sephiroth wields a very long katana called the Masamune, which has become his trademark weapon throughout all of his appearances. Cloud also has two katana named Murasame and Yoshiyuki, and a Masamune replica (called the Masamune Blade; but it cannot be used) that can be obtained at the Gold Saucer.

#### Before Crisis -Final Fantasy VII-Edit

Katana (Male), as per his name, wields a katana, as does the non-player character Elfé.

List of Katana:

#### Final Fantasy VII: Advent ChildrenEdit

Kadaj uses Souba in Final Fantasy VII: Advent Children. The Souba is a katana with two reinforced parallel blades, set a small distance apart from each other. When Kadaj turns into Sephiroth he wields his iconic Masamune.

### Final Fantasy XEdit

Main article: List of Final Fantasy X Weapons#Auron's Katana

Auron wields heavily stylized blades referred to as Katana.

#### Final Fantasy X-2Edit

Katana are used by the Samurai Dressphere, Yuna and Rikku use weapons previously used by Auron, while Paine has her own unique blade.

### Final Fantasy XIEdit

Katana are one-handed blades that can be used only by Ninja, who are capable of dual-wielding them. In higher levels, dual-wielded katana are capable of impressive damage over time and moderately strong Weapon skills. Great Katana, the two-handed variants of this weapon, are the main weapon for Samurai, although Ninja are also able to equip a select few of them.

Great katana are high damage, high delay weapons. The strength of the great katana comes from the versatility in its Weapon skills. Most great katana Weapon skills are strategically placed all around the skillchain chart, making it very skillchain-friendly with other weapon types. This works well with the Samurai themselves because they can unleash Weapon skills and skillchains faster than any other Job.

### Final Fantasy XIIEdit

Katana can be used after being unlocked on the License Board. They are valued for their combo ability, which can be enhanced by the Genji equipment, and it is second only to the one of the Ninja Swords.

The damage formula for katanas is:

$DMG = [ATK * RANDOM(1..1.125) - DEF] * [1 + STR * (Lv+MAG)/256]$

Due to the weight put on user's Strength and Magick Power, but also user's Level, recommended Katana user is Vaan.

In the International Zodiac Job System, only the Mononofu can equip the weapon.

List of Katana:

### Final Fantasy TacticsEdit

Katana can be equipped by the Samurai Job and by Celia's Assassin Job. Each Katana has a particular Iaido skill. There is a chance of a Katana breaking when its respective Iaido skill is utilized in battle.

List of Katana:

Katana are the weapons of choice for Assassins and Ninja, and thus are cheaper at Cyril and Muscadet. The ultimate katana is the Masamune 100.

List of Katana:

#### Final Fantasy Tactics A2: Grimoire of the RiftEdit

A new Job, the Parivir (in fact a simple renaming of Samurai; for instance, the French version of the game names the class "Ronin", the Japanese term for a wandering samurai that serves no specific lord), is the third class to equip Katana, along with the returning Assassins and Ninjas.

List of Katana:

### Final Fantasy Type-0Edit

Jack wields a Katana as his weapon of choice.

List of Katanas:

### Final Fantasy Legend IIIEdit

Katana is an event item found in Donmac. It is a required ingredient to makes the Mystic Sword Emperor when joined with Crystal and Emblem. The Masmune and Muramas also makes an appearance as katana.

### Bravely Default: Flying FairyEdit

 This section of the article about Bravely Default: Flying Fairy is a stub. You can help Final Fantasy Wiki by expanding it.

### Dissidia Final FantasyEdit

Katana appear in Dissidia as a type of weapon. Firion, Garland, Bartz and Sephiroth are the only characters that can equip them originally.

#### Dissidia 012 Final FantasyEdit

Katana return in Dissidia 012, and now are also available to Warrior of Light, Cecil, Kain, Gilgamesh, Cloud, Squall, Vaan, and Lightning.

 This section of the article about Final Fantasy Airborne Brigade is a stub. You can help Final Fantasy Wiki by expanding it.

### Final Fantasy All the BravestEdit

Katana can be equipped by the Samurai, Ninja, and Auron.

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# Probability

Let N,X1,Y1,X2,Y2,… be independent random variables. The random variable N takes positive integer values and has mean a and variance r. The random variables Xi are independent and identically distributed with mean b and variance s, and the random variables Yi are independent and identically distributed with mean c and variance t. Let

A=∑i=1NXi and B=∑i=1NYi.

Find cov(A,B). Express your answer in terms of the given means and variances using standard notation.

cov(A,B)=?

Find var(A+B). Express your answer in terms of the given means and variances using standard notation.

var(A+B)=?

1. 👍
2. 👎
3. 👁
1. please someone 2 hours left

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2. 👎
2. 1 hour don't torture me

1. 👍
2. 👎

1. 👍
2. 👎
4. cov(A,B) = b*c*r
var(A+B) = a*(s+t)+r*(c+b)^2

1. 👍
2. 👎
5. For var(a+b), you need to include 2 x the covariance

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6. why the convariance need to be multiplied by 2? can anyone elucidate this part?

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2. 👎
7. Because the variance of the sum of two RV's is only equal to the sum of the variances if the two RV's are independent. If the two RV's are dependent then the variance of the sum of the two RV's = sum of the variances + 2*COV(the two RV's)

1. 👍
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8. cov(A,B) = b*c*r
var(A+B) = a*(s+t)+r*(c^2+b^2)+2*b*c*r

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2. 👎

## Similar Questions

1. ### statistics

Let 𝑋 and 𝑌 be independent positive random variables. Let 𝑍=𝑋/𝑌. In what follows, all occurrences of 𝑥, 𝑦, 𝑧 are assumed to be positive numbers. 1. Suppose that 𝑋 and 𝑌 are discrete, with known PMFs,

2. ### Statistics

Suppose that X , Y , and Z are independent, with E[X]=E[Y]=E[Z]=2 , and E[X2]=E[Y2]=E[Z2]=5 . Find cov(XY,XZ) . cov(XY,XZ)= Let X be a standard normal random variable. Another random variable is determined as follows. We flip a

3. ### Statistics and Probability

Let N be a random variable with mean E[N]=m, and Var(N)=v; let A1, A2,… be a sequence of i.i.d random variables, all independent of N, with mean 1 and variance 1; let B1,B2,… be another sequence of i.i.d. random variables, all

4. ### probability

Let 𝑋 and 𝑌 be independent positive random variables. Let 𝑍=𝑋/𝑌 . In what follows, all occurrences of 𝑥 , 𝑦 , 𝑧 are assumed to be positive numbers. 1. Suppose that 𝑋 and 𝑌 are discrete, with known

1. ### probability

The vertical coordinate (“height") of an object in free fall is described by an equation of the form x(t)=θ0+θ1t+θ2t2, where θ0, θ1, and θ2 are some parameters and t stands for time. At certain times t1,…,tn, we make

2. ### math, probability

13. Exercise: Convergence in probability: a) Suppose that Xn is an exponential random variable with parameter lambda = n. Does the sequence {Xn} converge in probability? b) Suppose that Xn is an exponential random variable with

3. ### Math

Searches related to The random variables X1,X2,…,Xn are continuous, independent, and distributed according to the Erlang PDF fX(x)=λ3x2e−λx2, for x≥0, where λ is an unknown parameter. Find the maximum likelihood estimate

4. ### Probability

Question:A fair coin is flipped independently until the first Heads is observed. Let the random variable K be the number of tosses until the first Heads is observed plus 1. For example, if we see TTTHTH, then K=5. For K=1,2,3...K,

1. ### probability

t the discrete random variable X be uniform on {0,1,2} and let the discrete random variable Y be uniform on {3,4}. Assume that X and Y are independent. Find the PMF of X+Y using convolution. Determine the values of the constants

2. ### probability

Let X and Y be independent Erlang random variables with common parameter λ and of order m and n, respectively. Is the random variable X+Y Erlang? If yes, enter below its order in terms of m and n using standard notation. If not,

3. ### Probability & Statistics

Exercise: Convergence in probability a) Suppose that Xn is an exponential random variable with parameter λ=n. Does the sequence {Xn} converge in probability? b) Suppose that Xn is an exponential random variable with parameter

4. ### Probability

Let Θ be an unknown random variable that we wish to estimate. It has a prior distribution with mean 1 and variance 2. Let W be a noise term, another unknown random variable with mean 3 and variance 5. Assume that Θ and W are


text:
# PSP Bibliography

 Notice: Clicking on the DOI link will open a new window with the original bibliographic entry from the publisher. Clicking on a single author will show all publications by the selected author. Clicking on a single keyword, will show all publications by the selected keyword.

MHD Turbulent Power Anisotropy in the Inner Heliosphere

 Author Adhikari, L.; Zank, G.~P.; Zhao, L.; Telloni, D.; Keywords Parker Data Used; interplanetary turbulence; Slow solar wind; Solar wind; 830; 1873; 1534 Abstract We study anisotropic magnetohydrodynamic (MHD) turbulence in the slow solar wind measured by Parker Solar Probe (PSP) and Solar Orbiter (SolO) during its first orbit from the perspective of variance anisotropy and correlation anisotropy. We use the Belcher \& Davis approach (M1) and a new method (M2) that decomposes a fluctuating vector into parallel and perpendicular fluctuating vectors. M1 and M2 calculate the transverse and parallel turbulence components relative to the mean magnetic field direction. The parallel turbulence component is regarded as compressible turbulence, and the transverse turbulence component as incompressible turbulence, which can be either Alfv\ enic or 2D. The transverse turbulence energy is calculated from M1 and M2, and the transverse correlation length from M2. We obtain the 2D and slab turbulence energy and the corresponding correlation lengths from those transverse turbulence components that satisfy an angle between the mean solar wind flow speed and mean magnetic field \ensuremath\theta $_UB$ of either (i) 65\textdegree < \ensuremath\theta $_UB$ < 115\textdegree or (ii) 0\textdegree < \ensuremath\theta $_UB$ < 25\textdegree (155\textdegree < \ensuremath\theta $_UB$ < 180\textdegree), respectively. We find that the 2D turbulence component is not typically observed by PSP near perihelion, but the 2D component dominates turbulence in the inner heliosphere. We compare the detailed theoretical results of a nearly incompressible MHD turbulence transport model with the observed results of PSP and SolO measurements, finding good agreement between them. Year of Publication 2022 Journal \apj Volume 933 Number of Pages 56 Section Date Published jul ISBN URL DOI 10.3847/1538-4357/ac70cb


text:
Metin Seven writes:

The French software developer Exoside has released a Blender add-on version of their new Quad Remesher auto-retopology tool. Quad Remesher is based on the renowned ZRemesher in the ZBrush 3D sculpting suite, and comes from the same coder, Maxime Rouca.

A good mesh topology is beneficial to surface smoothness and editability, eases UV texturing and makes a mesh more suitable for rigging and animation.

Quad Remesher is capable of automatically retopologizing organic as well as hard-surface meshes, featuring only quadrangular polygons, ready for subdivision or for use as a base mesh for sculpting more details.

Quad Remesher offers a number of options to influence the result, such as preservation of areas that have materials assigned to them, auto-detecting sharp edges, and using vertex paint to indicate varying polygon density.

A typical workflow would be to use Quad Remesher to retopologize the major features of a mesh, then add Shrinkwrap and Subdivision modifiers or a Multiresolution modifier to increase detail derived from the preserved source mesh.

Metin Seven

Dutch freelance 3D illustrator, modeler and visualizer — metinseven.nl | Writer for Blender Nation, CG Boost and CG Cookie | Blender Artists and Blender Devtalk forum moderator | Former news cartoonist, TV series animator, game designer and magazine editor

• Quad Remesher is not a free add-on, but there is no free or other commercial add-on that can match Quad Remesher's auto-retopology results. You would need to buy ZBrush to access its comparable ZRemesher, and ZBrush costs $900 at the time of writing. • Quad Remesher gives you top quality results. It's not just a remesher, it's an auto-retopologizer with smart placement of edge loops and singularities, resulting in meshes that are ready for proper UV mapping and rigging. I'd say: download the trial and compare the results to remeshers. • Zremesher is out of this world, so yeah • If you've ever had the pleasure of using ZBrush's ZRemesher, you'd know the quality is noticeably better than even Blender's upcoming Voxel Remesh system. It's nice to have the available option, if one desires. Much cleaner quad-based results that better keep the edge flow in mind. 1. The product looks great but I'm not a fan of subscription-based licensing. Thank you but I'll pass. • I totally agree with you, a subscription-based licensing for an addon, it's too much, you can save money get Zbrush and have Zremesher forever and ever. 2. Folks, does a dev creating paid addons have to host them somewhere on github or is it up to them? What's Blender's policy on this matter? • The developer of a paid GPL addon is required to give the user who purchases it the source files. However, the addon file itself must be GPL, but if the addon is connected to another application, it's just a bridge between Blender and the other application used to send data, It's a different matter. For example, a renderer like Vray is not GPL, but it uses a GPL addon to connect Vray to Blender. Same for this retopology addon. • Subscription is thievery... for an add-on? It’s greed! • When the add-on incorporates and/or uses any of the Blender code (including script interfaces) - that add-on is licensed as GPL and the source must be made available in some form to those that download it. They do not need not make it available to the public, however, only those they distribute it to. There are "loopholes" to this though, one of which this add-on uses. The Blender add-on calls out to an executable, which is downloaded separately, and that executable doesn't use Blender code internally. As such, they only need to provide the source for their Blender-to-executable script (which IS the add-on download); not for the part doing the work (the executable). 3. Is this really useful for animation? placing loops automatically around eyes mouth and all joints? If so im definetly in. Im not really enjoying topology by hand and for animation that is the only way to do it. • Quad Remesher definitely makes a mesh much more suitable for animation. If you're looking for the perfect topology with edge loops exactly where you want them, manual retopology will sometimes still be better. You can also combine both approaches: use Quad Remesher, then perform some manual fine-tuning if necessary. • P.S.: Best thing to do is to see it for yourself by installing the trial version. 4. The most expensive perpetual pro license is only$109. I don't know about everyone else, but retopology takes me hours and is far less fun than sculpting. Even if you only made $7.00 an hour it would only take 15 hours of saved time to break even. Plus this isn't open software and people need to earn a living. • It's not that simple. The addon file itself must be GPL, but if the addon is connected to another application, it's just a bridge between Blender and the other application used to send data, It's a different matter. For example, a renderer like Vray is not GPL, but it uses a GPL addon to connect Vray to Blender. Same for this retopology addon. 5. I am a little bit disappointed by the comments I have to say. There should be a$ in the title yes but:
Why dont you read the whole buy page before complaining that its a subscription model the perpetual license is literally above the subscription license.
And also this is a non trivial piece of software that is not mainly targeted at amateurs (even though there is a license for amateurs) so it will cost something and since its not trivial to solve such a thing it will cost a little bit more than a pack of materials. There is a lot of knowledge and brain power involved to develop such a software so please be considerate and don't complain on how pricey it is just because you want it.

• Apologies if my earlier comments seemed like a complaint. No offense was intended.

From my vantage point - there is no specific complaint about the apparent usefulness of this particular product. I don't care for subscription-based licensing in general, depending on its structure. I also fully understand the creator's need for payment for spending the time/resources to create the add-on. I get it.

However, specifically regarding this product, the Indie version of this allows for no commercial use at all. There are instances where freelancers (who maintain other full-time careers) sell items on the side as well, who also wouldn't necessarily have as high a demand for this product . . . yet, are prohibited from benefiting from it should they choose to do so. At least not without a Pro or subscription-base license.

Again - personally, I think this product looks fantastic but the bottom-line doesn't make sense for me to purchase the Indie license either. This because it excludes any commercial product I would create with it.

The Substance Painter price/licensing structure works for me, for example, because they allow for a standalone license without restriction (other than company size).

But hopefully all our comments will ultimately steer more people toward this product. Any buzz is good buzz, right? That way the creator can enjoy some of the free press. HA!! (And I DO NOT intend this as a passive-aggressive comment. I sincerely mean this.)

At the end of the day, the benefit of purchasing this doesn't fit into my workflow. I'll do the work manually as I have done before. Again, not bashing this product. I think it looks really fantastic. But I choose to pass. I do wish them well.

• I agree that the 'Indie' license word choice is somewhat confusing. 'Hobbyist' would make it clearer.

• well the formula on how many models you have to make before you earn money from this is: x = $109,9/w*t with w being your wage at your other career and t being the time benefit from this add-on so lets work this out. Lets say you make minimum wage and save 15 minutes when retopologizing a very conservative estimation. So after 60 models you literally start to make money from buying this addon. If you create one model a week it will take you one year and two months to start making money from this. lets say this is our highest expected time. Lets inspect a more avarage wage of$27.16 and lets say you save 30 minutes by using the add-on. In that case you will have earned money after about 8 models. Again lets say you make a model a week. Then after two months your investment will have paid off.
That does not even account for the money you make from selling those models.
Now I don't want to say you should buy the plugin off course not. I simply want to show that you need to calculate this through before saying that this does not make sense for you instead of claiming that people selling their products are prohibited from making money with it. For \$50 more you are very much allowed to do so.

• Rombout Versluijs on

Well i guess you save money after perhaps 1 or 2 models already. Retopo is tedious labor so 15 or 30 minutes is almost impossible. Unless you mean very simple shapes, but you would not need it for that i guess

• If only people were this considerate about the work that goes into Blender, maybe it would not have turned into a bare bone application that fits the old adage "it's free if you don't value your time", because if you do, Blender alone can't help you, you need hundreds of dollars of add-ons to get work done efficiently according to the sales pitch of 99% of the add-on, "buy this because Blender sucks at it". So think a little about the people who contributed to Blender's development and now see it become a paid-addon-dependent software.

• I feel like that is simply not true, I work as a game developer and mostly use vanilla blender. 80% of the time we work very efficiently without the need for more sophisticated solutions. The other 20% are very niche applications of blender that are not really needed in non game development, however these are things the blender foundation cannot provide because there are many niches and every niche has its own specialized tools needed. Funnily enough the one thing disregarding blenders developers the most is your comment by branding blender as a paid-addon-dependent software just shortly after the developers completly reworked its entire structure making it an even more competitive modern software.
In fact they managed to make several open movies with it without any "need" for paid addons, so it might just be that you are not broadening your horizons and learning what is possible with vanilla blender before reading an advertisement and deciding that you cant work without it anymore.

6. Wish people didn’t complain so much. The perpetual license options really aren’t so bad. Big thanks to the creators for creating a great piece of software.

• You haven't even tested it yet you call it a great piece of software... If i want to use it on a laptop and a desktop PC do i need 2 licenses? I ask since you seem to know so much about this add-on.

• No. You do not need 2 license for that!
With one license, the owner of this license can install, activate and use it on 2 machines (see EULA...).

7. Question about licensing: If I want to use it at home and at work, do I have to buy two difference licenses, since it would be on 2 different computers, or could I get away with just 1 pro license?

8. Rombout Versluijs on

How can they say its baed on Zremesher? Does this mean it has simular code or outcome or is it just a gimic to attract possible users?

• Quad Remesher is made by the developer who created ZRemesher for Pixologic. He's also responsible for UV Master and Decimation Master.

• Rombout Versluijs on

Thanks for the reply, i did found some more info about it. Wasnt really sure this was legit or real since he did not post it himself.

• I think so, yes. One of its options concerns Face Maps, which is a 2.8+ feature.

9. What are the chances of getting the same features in the Blender version as the Maya version? Specifically, "use existing hard edges", which is notably absent from the Blender version, at least in the demo/trial version?

• Hi Matthew,

What about the 'Use Normals Splitting' option? You can indicate your own hard edges using smoothing groups, then use that option.

Regards,

Metin

10. Just downloaded the demo a few mins ago, zip installed OK and the remesh engine was installed as per the FAQ however when I try and remesh anything, even default cube It errors out:

Traceback (most recent call last):
doRemeshing_Finish(self, context)
import_mesh_fbx(theOp.retopoFilename)
bpy.ops.import_scene.fbx(filepath=filepath)
File "C:\Blender\blender-2.83-6fff73e3f001-windows64\2.83\scripts\modules\bpy\ops.py", line 201, in __call__
ret = op_call(self.idname_py(), None, kw)
RuntimeError: Error: Traceback (most recent call last):
File "C:\Blender\blender-2.83-6fff73e3f001-windows64\2.83\scripts\addons\io_scene_fbx\__init__.py", line 226, in execute
TypeError: load() got an unexpected keyword argument 'hide_props_region'

location: C:\Blender\blender-2.83-6fff73e3f001-windows64\2.83\scripts\modules\bpy\ops.py:201
location: :-1

Any ideas?
Thanks

11. I bought quad remesher last week and it does remesh well, but once model is remeshed the computer struggles to do the smallest of thing, like selecting a few faces or vertexes and takes forever to make slight change in the mesh, also some functions simple dont work in blender after remeshing, like filling holes in the mesh just wont work. At this point i am not happy. There is no where on their website to make comments and asking for help is cumbersome to say the least.

• Hi Brian,

I'm sorry to hear this. I haven't encountered these problems. Did you try installing a different Blender version?

Good luck.


text:
The COVID-19 lockdown in Belgium was officially enforced on noon 18th Mar 2020 but at home we were already quarantined (kids since Carnival holidays, 24th Feb; me from EOB 13th Mar). So on my first official telework day, 16th Mar, I finally decided to buy an ebook reader, and selected Kobo Clara HD.

I like it but still I was curious and looked around for how to hack it. I found that it is possible to change the original SD card (8 GB) with a bigger one and decided to give it a go. This blog post from Lee Yingtong Li got me started.

I opened it, took the SD card out and copied on a bigger one of 128 GB.

It all went well apart from the last 2 lines of text:

Then use a tool like gparted to resize the data partition (the large third partition) on the new microSD card to fill the remaining space.

I tried all kind of things to have gparted running from a Mac (docker image, Virtual Box) but what finally worked for me was to have it on a bootable CD ROM (yes I still have an old machine with CD reader/writer): I burned the ISO image, booted from it and resized the SD card to all available space.

Posted on:
April 19, 2020
Length:


text:
1. modphysnoob

2. goformit100

@saifoo.khan HELP

3. modphysnoob

@phi

4. modphysnoob

@jamesJ

5. modphysnoob

I actually reworked it using http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates and got this $\frac{1}{r*\text{Sin}[\theta ]}(\text{Cos}[\theta ]*\text{Sin}[\theta ]) \hat{r}+\left(\frac{\text{Sin}[\theta ]}{r}\right)\hat{\phi }$

6. Jemurray3

Oops, typo. Sorry. The radial component is $\frac{1}{r\sin(\theta)} \frac{\partial}{\partial \theta} \sin^2(\theta) = \frac{2\cos(\theta)}{r}$

7. modphysnoob

8. Jemurray3

Notice that it doesn't actually matter, because you'll be taking the dot product with the normal vector to the surface of the hemisphere, which is just $\hat{n} = \hat{r}$

9. modphysnoob

oh, wow, didn't see that; that does make it a lot easier

10. Jemurray3

Which hemisphere would you like to integrate over? If you do the one above the z = 0 plane, you'll be integrating $\int_0^{\pi/2} \int_0^{2\pi} 2\cos(\theta) d\phi d\theta$

11. modphysnoob

shouldn't there be sin(theta ) from spherical form of ds?

12. modphysnoob
13. Jemurray3

oh yeah, sorry. Yet another typo.

14. modphysnoob

no problem, your help is greatly appreciated

15. Jemurray3

Anyway, then you will be integrating the line integral around the circular boundary in the z=0 plane. The two will be equal.

16. modphysnoob

haha, it better be

17. Jemurray3

The line integral is trivially simple, don't worry :)

18. modphysnoob

ok, I will try working it out

19. modphysnoob

http://upload.wikimedia.org/math/0/a/c/0ac2a22b40a49a17025d8ad27f49cf6f.png we only use middle term,their phi is our theta since we actually have don't have any theta vector it is 0

20. Jemurray3

No, the line integral should be $\int_0^{2\pi} r \sin(\theta) \cdot A_\phi d\phi$

21. modphysnoob

where does this come from?

22. Jemurray3

$d\vec{l} = \hat{r}dr + \hat{\theta} rd\theta + \hat{\phi} r\sin(\theta) d\phi$ $\vec{A} = \hat{r} A_r + \hat{\theta} A_\theta + \hat{\phi} A_\phi$

23. modphysnoob

we don't use r vector becuase it is set to constant?

24. Jemurray3

What do you mean? The path we're taking is a circle in the Z=0 plane, so the direction is purely phi.

25. modphysnoob

just to make sure we are on same page , phi is angle from x axis on x-y plane right

26. Jemurray3

yep

27. modphysnoob

well, A_phi= sin (theta)^2 so integrating it would result in sin(theta)^2*(2*pi)

28. modphysnoob

since it is in x-y plane ,evaluate at pi/2 sin(pi/2)^2 *(2*Pi)=2pi

29. modphysnoob

which is what I got for surface integral above

30. Jemurray3

Yep, good job.

31. modphysnoob

thanks you , you are not only very smart also very patient

32. Jemurray3

Thank you!


text:
# Is there any way to transform a non-convex optimization problem into a convex one?

I have an optimization problem which is described as $$\begin{array}{ll} \text{minimize}_x & c^{T}x\\ \text{subject to} & Gx \preceq h\\ & -x^{T}Px - qx - r \leq 0 \end{array}$$ where $P$ is positive semi-definite.

Is there any way to convert this problem into a convex problem?

• No, there is not. See this PDF for some heuristic alternatives. – Michael Grant Nov 3 '14 at 16:52


text:
# A question about a polynomial

Suppose that $p$ is a real polynomial of degree $n$. Prove that for $|x|<1$, $$\sum\limits_{m=0}^\infty{p(m)x^m}=h((1-x)^{-1})$$ for some real polynomial $h$ of degree $n+1$ without the constant term.

I have started trying by taking a specific expression of the polynomial $p$, but I have completely lost. I understand that this result generalizes the following identity $$(1-x)^{-1}=\sum\limits_{m=0}^\infty{x^m}$$ for $|x|<1$ but couldn't anymore rather thinking about it. Help is highly appreciated.

• @ Michael Hardy: edited! May 30 '15 at 18:15
• Consider the derivatives of $(1-x)^{-1}$ and the derivatives of the power series.
– jgon
May 30 '15 at 18:21
• @ jgon Yes, this can be done. I did this for a polynomial of degree 2 or 3. But this seems to be more complicated for polynomials of arbitrary degree. Is there any other way of solving it? May 30 '15 at 18:58
• you shouldn't need to find it explicitly if you can show the left hand side's derivatives form a basis for the space
– jgon
May 30 '15 at 19:07

Consider the polynomials $$P_0:=1,P_1:=X, P_2:=X(X-1),\dots,P_l:=X(X-1)\dots (X-l+1).$$ Then we may express $p$ as $\sum_{j=0}^nc_jP_j$ for some coefficients $c_j$. Since for $|x|\lt 1$, we have $$\sum_{m=0}^{+\infty}P_j(m)x^m=\frac{d^j}{dx^j}\left(\frac 1{1-x} \right)$$ and the later can be expressed as a constant times $(1-x)^{-j-1}$, we get the wanted polynomial $h$ by summing over $j$.
• Isn't $P_k(x)=(x+1)\dotsm (x+k)$ for $k\ge 1$ ? May 31 '15 at 8:25


text:
## The spindle is a active self-assembling machine that coordinates mitosis. is

The spindle is a active self-assembling machine that coordinates mitosis. is certainly suitable to keep spindle mechanical integrity robustly. Launch During cell department, the mitotic spindle assembles itself from its constituent parts. Spindle microtubule minus ends are concentrated into two poles, and these poles dictate where duplicated chromatids are carried at anaphase. Pushes that concentrate microtubules into poles are necessary to spindle function and firm. Cytoplasmic dynein, a minus endCdirected microtubule electric motor, clusters parallel microtubules into spindle poles (Verde et al., 1991; Heald et al., 1996) and transports the microtubule-binding proteins NuMA to construct poles (Merdes et al., 2000). At poles, dynein and NuMA tether microtubules (Gaglio et al., 1995; Merdes et al., 1996; Heald et al., 1997; Dionne et al., 1999), and pole framework remains solid despite speedy microtubule turnover (Saxton et al., 1984) and opposing stress on kinetochore fibres (k-fibers) from kinetochore-based pushes (Gordon et al., 2001; Compton and Manning, 2007; Silk et al., 2009). Hence, poles must both oppose power and be continuously rebuilt (Gaglio et al., 1997; Goshima et al., 2005). This anatomist challenge features a long-standing paradox: how do the spindle maintain steadily SRT1720 supplier its structure and mechanised integrity yet stay powerful, flexible, and plastic architecturally, as its features need? For the spindle to conserve its structural integrity, it should be in a position to rebuild poles by recognizing and sorting new microtubule buildings continuously. Certainly, during spindle set up, poles can integrate both brand-new peripheral microtubules (Rusan et al., 2002; Tulu SRT1720 supplier et al., 2003) and kinetochore-nucleated k-fibers (Khodjakov et al., 2003; Maiato et al., 2004). Set up spindles can move brief microtubule seed Mouse monoclonal to PGR products to poles (Heald et al., 1996, 1997) and reincorporate k-fibers severed by ablation simply because microtubules grow back again (Snyder et al., 1991; Zhang and Chen, 2004; Maiato et al., 2004), and poles from different spindles can fuse jointly (Gatlin et al., 2009). Although dynein and NuMA are either suspected or proven to mediate these observations of powerful microtubule integration into poles, it isn’t apparent which microtubule buildings serve as dynein cargo, where with them power is exerted, or how solid that potent force is. We have no idea how pushes that keep poles evaluate to various other spindle pushes or on what timescale they SRT1720 supplier donate to spindle structures. In large component, it is because the response from the set up spindle to detached microtubules is certainly challenging to review: k-fiber minus ends already are inserted in the spindle and free of charge microtubules inside the spindle body are tough to image. Right here, we use laser beam ablation to problem the spindles architectural regular condition by detaching microtubules from poles and we probe mobile pushes exerted on, and substances recruited to, these microtubules. We present that detached microtubules are discovered by dynein/dynactin and NuMA and SRT1720 supplier carried toward poles quickly, overpowering opposing pushes on microtubules and chromosomes to correct spindle structures. Force is certainly generated by localized tugging on brand-new minus ends, which power a discovered system of chromosome motion at mitosis recently, indie of kinetochore pushes. We suggest that speedy detection and prominent poleward transportation of free of charge minus ends by dynein maintains spindle integrity throughout mitosis, producing k-fiber anchorage and spindle pole framework solid to component turnover and mechanised challenges. Outcomes K-fiber severance sets off poleward chromosome motion within minutes We utilized pulsed laser beam ablation to sever microtubules and detach them from poles (Fig. 1 A) in mammalian GFPC-tubulin.

## Background: The primacy effect, i. DMN, is usually associated with primacy

Background: The primacy effect, i. DMN, is usually associated with primacy recall performance in aMCI. Methods: A number of 87 aMCI patients underwent resting state fMRI and verbal episodic memory assessment. FC between the left or right hippocampus, respectively, and all other voxels in gray matter was mapped voxel-wise and used in whole-brain regression analyses, testing whether FC values predicted delayed primacy recall score. The delayed primacy score was defined as the number of the first four words recalled around the California Verbal Learning Test. Additionally, a partial least squares (PLS) analysis was performed, using DMN regions as seeds to identify the association of their functional interactions with delayed primacy recall. Results: Voxel-based analyses indicated that delayed primacy recall was mainly (positively) associated with higher FC between the left and right hippocampus. Additionally, significant associations were found for higher FC between the left hippocampus and bilateral temporal cortex, frontal cortical regions, and for higher FC between the right hippocampus and right temporal cortex, right frontal cortical regions, left medial frontal cortex and right amygdala (< 0.01, uncorr.). PLS analysis revealed positive associations of delayed primacy recall with FC between regions of the DMN, including the left and right hippocampus, as well as middle cingulate cortex and thalamus (< 0.04). In conclusion, in the light of decreased hippocampus function in aMCI, inter-hemispheric hippocampus FC and hippocampal FC with brain regions predominantly included in the DMN may contribute to residual primacy recall in aMCI. = 33 patients were classified as single domain name aMCI subtype (i.e., exhibiting an exclusive memory impairment); = 54 patients were classified as multiple domain name aMCI subtype (i.e., exhibiting an impairment in the memory domain as well as other cognitive domains; Petersen et al., 2001; Petersen, 2004). For a detailed neuropsychological characterization, see Supplementary Table 1. The sample was recruited for an intervention study at the University Hospital Munich, ONX 0912 IC50 Germany. Ethical approval was given by the local ethics committee of the Faculty of Medicine at the Ludwig-Maximilian University or college in Munich, Germany. All subjects gave written informed consent in accordance with the Declaration of Helsinki. Based on the German education system, the subjects' education levels were converted to a categorical level ranging from 1 (i.e., no educational qualification) to 5 (i.e., university degree), resulting in a frequency distribution of education category 1: = 23, category 2: = 22, category 3: = 19, category 4: = 23. The mean MMSE score was 27 (< 0.3 to define the GM mask, which was applied to the FC maps to restrict the analyses to areas within the GM only. One-sample < 0.001 (uncorr.) to obtain binary inclusive masks that were used in all following regression analyses, restricting results to functionally connected voxels. Statistical analysis For comparing the number of correctly recalled primacy words at delayed recall to the number of correctly recalled words from the rest of the list, proportions were calculated and compared by means of a ONX 0912 IC50 paired samples = 53 subjects, encompassing only subjects without floor effects (i.e., delayed primacy recall 1). Moreover, regression analyses were repeated additionally controlling for delayed total recall, and additionally controlling for left or right hippocampal volume (using all subjects). Lastly, a median divide was performed predicated on postponed total recall, separating the test right into a high and a minimal executing group, and regression Rabbit Polyclonal to MMP-7 analyses had been repeated for both subgroups. Just positive associations had been tested. For everyone regression analyses, a cluster threshold of 20 voxels was used. Multivariate incomplete least squares (PLS) evaluation was performed in Matlab (McIntosh et al., 1996; Lobaugh and McIntosh, 2004) to measure the covariance of postponed primacy recall ONX 0912 IC50 with patterns of FC between nine seed parts of the DMN (including locations within posterior cingulate/precuneus, middle.

## Near-infrared imaging of lymphatic drainage of injected indocyanine green (ICG) has

$2$

-fold increase) when compared to the changes seen in the power per unit mass parameter (

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## The mesial temporal lobe (MTL) is normally understood as a memory

The mesial temporal lobe (MTL) is normally understood as a memory structure in clinical settings, with the of MTL damage in epilepsy being memory impairment. activation (right > left) during spatial associative processing and left hippocampal/parahippocampal deactivation in joint spatial-temporal associative processing. In the left TLE group identical analyses indicated patients used MTL structures contralateral to the seizure focus differently and relied on extra-MTL regions to a greater extent. These results are consistent with the notion that epileptogenic MTL damage is followed by reorganization of networks underlying elemental associative processes. In addition, they provide further evidence that task-related fMRI deactivation can meaningfully index brain function. The implications of these findings for clinical and cognitive neuropsychological models of MTL function in TLE are discussed. Introduction The relationship between mesial temporal lobe (MTL) damage and memory impairment is usually fundamentally accepted in neuropsychology. Significant evidence supports a central role for the MTL in episodic memory in particular, the operational system supporting our capability to recreate and relive the events of our day to day lives [1]C[2]. The defining features of these thoughts are the temporal-spatial relationships among their elements [1], a subjective feeling of that time period and self, and the proper execution of autonoetic consciousness which allows us to see and relive occasions [2] Tal1 mentally. Destruction from the hippocampi early in advancement selectively impairs the capability to form such thoughts while leaving development of other styles of memory generally intact [3]. The complete nature from the Rucaparib IC50 primary procedures impaired by MTL harm that Rucaparib IC50 express as storage impairment is certainly a way to obtain ongoing issue. The creation of spatial and temporal organizations in episodic storage has led several authors to claim association formation may constitute a cognitive endophenotype of MTL function (e.g., [4]). Certainly, tasks needing creation of organizations (e.g., between unrelated pairs of phrases) are exclusively delicate to mesial temporal lobe harm in epilepsy [5]C[7]. This known fact, as well as a model postulating differing efforts for best and still left MTLs in verbal and nonverbal storage respectively; i.e. materials specificity [8]C[9], proceeds to create a central tenet of scientific neuropsychological evaluation for surgical preparing in epilepsy in lots of centers. Models created out of this perspective possess advanced to consider MTL substructures as digesting associations within a complementary and hierarchical way [4] [10]C[11]. Broadly, such versions suggest that after information has been perceived and associated Rucaparib IC50 to form a perceptual or cognitive item (unitization, likely supported by extra-MTL structures), the perirhinal cortex is usually engaged to form or store item level associations [12]. Parahippocampal cortex then forms fixed (e.g. egocentric spatial) representations (though observe also [13]), while the hippocampus allows these associations to be flexibly re-expressed in different ways [4] [13]. Significant work has now also suggested the hippocampus is usually central in associating information even over the very short-term (for instance, in working memory and belief; observe [15] for an extensive review). In the cognitive neuropsychological literature, a number of researchers have argued that this MTL’s engagement in tasks beyond episodic memory must influence our understanding of MTL function. One model considers projection of the self into a novel context (self projection) as a core process in tasks engaging the MTL and a network of related brain regions [16]. Consistent with this are the findings that bilateral hippocampal damage results in impairment of both episodic memory and other cognitive domains that share the MTL network, such as topographical memory [17], and that amnesiogenic MTL damage impairs the ability to imagine new experiences [18]. Cognitively, each of these processes can be considered to require associative processing to locate the self in a novel, constructed environment. Of relevance, Spreng, Marr and Kim [19] recently compared the brain regions activated in these and related processes, namely autobiographical memory, navigation, theory of mind, and the default mode network, which are also thought to be involved in associative processing at rest (e.g., [20]) They found common engagement of the mesial temporal lobe, posterior cingulate, precuneus, temporo-parietal junction and retrosplenial cortex. The single point of highest correspondence between these networks fell within the left parahippocampal cortex [19]. The.


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# Coordinate Geometry Formulas, Solutions, Study Material

## Coordinate Geometry important questions on Page No - 6

We studied about coordinate axes, coordinate plane, plotting of points in a plane, distance between two points, section formula, etc. All these concepts are the basics of coordinate geometry. Let us have a brief recall of coordinate geometry done in earlier classes. To recapitulate, the location of the points (6, – 4) and (3, 0) in the X-plane is shown in (Fig 10.1).

We may note that the point (6, – 4) is at 6 units distance from the y -axis measured along the positive x -axis and at 4 units distance from the X –axis measured along the negative Y-axis. Similarly, the point (3, 0) is at 3 units distance from the Y –axis measured along the positive X -axis and has zero distance from the X -axis. We also studied.

1. Distance between the points $P(x_{1}y_{1})\: and\: Q(x_{2}y_{2})$

$PQ\: =\: \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

2. The coordinates of a point dividing the line segment joining the points$(x_{1}, y_{1}\:and\: x_{2}, y_{2}$internally, in the ratio m: n are $\frac{mx_{2}+nx_{1}}{m+n}\: ,\: \frac{my_{2}+my_{1}}{m+n}$

Example: he coordinates of the point which divides the line segment joining A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by $x\: =\: \frac{1(-3)+3.1}{1+3}\:=\:0$ and

3. In particular, if m=n ,the coordinate of the mid point of the line segment joining the point $(x_{1}, y_{1}\:and\: x_{2}, y_{2}$. Are $\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}$

4. Area of the triangle whose vertices are $(x_{1},y_{1}),(x_{2},y_{2}),(x_{3}y_{3})\: is\: \frac{1}{2}\left | x_{1} (y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})\right|$
Note : If the area of the triangle ABC is zero , then three point A,B and C lie on a line , i.e ,they are collinear .
We shall continue the study of coordinate geometry to study properties of the simplest geometry figure –straight line. Despite its simplicity , the line is a virtual concept of geometry. and enters into our daily experience in numerous interesting and useful ways. Main focus is on representing ,the slope is most essential.
Slope of a Line: A line in a coordinate plane forms two angles with the x-axis ,which are supplementary The angle (say)
Θ made by the line l with positive direction of x -axis and measured anti clockwise is called the inclination of the line. Obviously 0° ≤ θ ≤ 180° (Fig 10.2) .
We observe that lines parallel to x -axis, or coinciding with x -axis, have inclination of 0°. The inclination of a vertical line (parallel to or coinciding with y -axis) is 90°

Definition: If θ is the inclination of a line l , then tan θ is called the slope or gradient of the line l . The slope of a line those inclination is 90° is not defined. The slope of a line is denoted by m . Thus, m = tan θ, θ ≠ 90° It may be observed that the slope of x -axis is zero and slope of y -axis is not defined.


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Acta Physico-Chimica Sinica ›› 2020, Vol. 36 ›› Issue (9): 1911050.

Special Issue: Precise Nanosynthesis

• Feature Article •

### Solid Frustrated Lewis Pairs Constructed on CeO2 for Small-Molecule Activation

Sai Zhang, Mingkai Zhang, Yongquan Qu()

• Received:2019-11-12 Accepted:2020-01-13 Published:2020-02-05
• Contact: Yongquan Qu E-mail:yongquan@mail.xjtu.edu.cn
• Supported by:
the National Natural Science Foundation of China(21872109);the China Postdoctoral Science Foundation(2018T111034);the Cyrus Tang Foundation through the Tang Scholar program. S. Zhang is supported by the Fundamental Research Funds for the Central Universities, China(xjj2018033);the Natural Science Foundation of Shaanxi Province, China(2019JQ-039)

Abstract:

Solid materials containing frustrated Lewis pairs (FLPs) as active sites have attracted much attention due to their ability to activate and transform small molecules. However, it is still highly challenging to precisely construct FLP sites on the surfaces of nanomaterials, thereby limiting the applications of these materials. Nanostructured ceria (CeO2) is commonly employed as a catalyst or functional support, and exhibits both Lewis acid and basic properties as well as abundant and easily regulated surface defects, which originate from the reversible Ce3+/Ce4+ redox pair. When the Lewis acid and base sites of CeO2 are independent of each other, the combined Lewis acid-base sites play a similar role to that of homogeneous FLP sites. Thus, the rich surface properties of nanostructured CeO2 provide significant potential for the construction of solid FLPs.

Herein, we demonstrate that solid FLP sites can be successfully constructed on the surface of CeO2(110) via the regulation of surface defect clusters, which can be used to create new Lewis acid sites composed of two adjacent Ce3+ atoms on the surface. Novel interfacial FLP sites can then be formed by combining these Lewis acid sites with neighboring surface lattice oxygens, which act as Lewis base sites. Porous CeO2 nanorods (PN-CeO2) with boundary surface defects were prepared by a special two-step hydrothermal process, and exhibited remarkable catalytic FLP properties. Hydrogen molecules could be effectively activated on the surface of PN-CeO2 with a low activation energy of 0.17 eV via a heterolytic cleavage process. Hydrogenation of alkenes and alkynes to alkanes could then be realized by the activated hydrogen under mild reaction conditions.

PN-CeO2 nanorods with FLP active sites were also able to activate CO2 molecules effectively. Unlike in other solid FLP sites, CO2 molecule activation was realized via a Lewis base site binding with the C atom while two Lewis acid sites bound the two O atoms, owing to the unique configuration of the FLP sites in PN-CeO2. When combined with the epoxidation of olefins by "isolated" Ce3+ sites in PN-CeO2, the FLP-inspired activated CO2 could be used to transform olefins and CO2 to cyclic carbonates through a selective tandem transformation route. In addition, density functional theory studies indicate that the FLP sites on CeO2(110) can activate the C―H bond of CH4 with activation energies as low as 0.63 eV, which can be attributed to the enhanced acidity and basicity of the FLP sites.

With this improved understanding of solid FLP sites constructed on ceria, we have also been able to summarize the challenges and prospects in this field, including their construction, characterization, and mechanism analysis.

MSC2000:

• O643


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# Distribution over $\mathbb{Q}_p$

A distribution over $$\mathbb{Z}_p$$ is defined as an "additive" map $$\mu$$ s.t. $$\mu(\mathbb{Z}_p)=\sum_{i=0}^{p-1}\mu(i+p\mathbb{Z}_p)$$. But now if i want define a distribution over $$\mathbb{Q}_p$$ what is the " additivity" that this map should have? I think something like:

For every $$x \in \mathbb{Q}_p$$, for every $$N \in\mathbb{Z}$$ then $$\mu(x+ p^N\mathbb{Z}_p)=\sum_{i=0}^{p-1}\mu(x+ip^N+p^{N+1}\mathbb{Z}_p)$$

Thank you for suggestion and references!

• Sure. And (since there is a basis of open sets that are compacts ?) distributions on the metric space $p^N\mathbb{Z}_p,|.|_p$ are finite measures – reuns Oct 4 '18 at 20:58


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If regressors are uncorrelated, then the diminishes in b ( and does not set any coefficients to zero. , which, when the covariates are orthogonal to each other, gives. / {\displaystyle \ell ^{1}} 0 {\displaystyle \lambda } ‖ {\displaystyle K_{1}=I} when = Since this is a matrix formula, let's use the SAS/IML language to implement the formula. l h η 0 {\displaystyle \ell ^{2}} 0 {\displaystyle {\bar {x}}} η lasso Parameters alpha float, default=1.0. Ridge regression with glmnet # The glmnet package provides the functionality for ridge regression via glmnet(). Lasso was originally formulated for linear regression models and this simple case reveals a substantial amount about the behavior of the estimator, including its relationship to ridge regression and best subset selection and the connections between lasso coefficient estimates and so-called soft thresholding. x This can be modeled using the following regularization: In contrast, one can first cluster variables into highly correlated groups, and then extract a single representative covariate from each cluster. Ridge Regression is a technique for analyzing multiple regression data that suffer from multicollinearity. is computed, then the diagonal elements of {\displaystyle b_{\ell _{2}}={\bigg (}1+{\frac {\lambda }{R^{2}(1-\lambda )}}{\bigg )}^{-1}b_{OLS}} R Ridge Regression Linear regression refers to a model that assumes a linear relationship between input variables and the target variable. {\displaystyle X} B . p λ penalty to each group subspace. i A Γ\boldsymbol{\Gamma}Γ with large values result in smaller x\boldsymbol{x}x values, and can lessen the effects of over-fitting. p = The basic idea is to penalize the differences between the coefficients so that nonzero ones make clusters together. p Recall that Yi â¼ N(Xi,â Î²,Ï2) with correspondingdensity: fY â Î²) = â1 ¯ i {\displaystyle {\frac {1}{p}}{\frac {1}{\sqrt {p_{B}}}}p_{B}{\frac {1}{\sqrt {p_{B}}}}={\frac {1}{p}}} ‖ In fact, if there is some solution β L 2 | After all, the moment of activation of a relevant regressor then equals and 1 Choosing the regularization parameter ( i {\displaystyle R^{\otimes }} {\displaystyle \beta _{0}} Overall, choosing a proper value of Γ\boldsymbol{\Gamma}Γ for ridge regression allows it to properly fit data in machine learning tasks that use ill-posed problems. − λ {\displaystyle \mathrm {I} } , then this reduces to the standard lasso, while if there is only a single group and Ridge regression adds another term to the objective function (usually after standardizing all variables in order to put them on a common footing), asking to minimize $$(y - X\beta)^\prime(y - X\beta) + \lambda \beta^\prime \beta$$ for some non-negative constant $\lambda$. LARS generates complete solution paths. Ridge Regression is a technique which penalizes the size of regression coefficients in order to deal with multicollinear variables or ill-posed statistical problems. is an (where the quotation marks signify that these are not really norms for y Almost all of these focus on respecting or utilizing different types of dependencies among the covariates. 1 In prior lasso, the parameter Log in. {\displaystyle \eta =\infty } [19] In prior lasso, such information is summarized into pseudo responses (called prior responses) 1 represent the hypothesized regression coefficients and let Determining the optimal value for the regularization parameter is an important part of ensuring that the model performs well; it is typically chosen using cross-validation. b norms defined by the positive definite matrices > λ ⋅ β ) ( i {\displaystyle \|\cdot \|_{0}} The three most popular ones are Ridge Regression, Lasso, and Elastic Net. is the convex hull of the region defined by 1 1 H are identically zero, while in the case of an n-sphere, the points on the boundary for which some of the components of This can be best understood with a programming demo that will be introduced at the end. b Ridge regression is the most commonly used method of regularization for ill-posed problems, which are problems that do not have a unique solution. ¯ when ‖ p {\displaystyle y} − where This has the effect of making the coefficient estimates closer to zero. y from i In 2005, Zou and Hastie introduced the elastic net to address several shortcomings of lasso. ^ [6] In general, if ℓ λ k [7] It is assumed that 0 ≤ for p β Read more in the User Guide. {\displaystyle \beta _{0}} ‖ {\displaystyle p=1} p {\displaystyle \beta _{k}} l Additionally, the covariates are typically standardized {\displaystyle \beta _{0}=0} [7], Just as ridge regression can be interpreted as linear regression for which the coefficients have been assigned normal prior distributions, lasso can be interpreted as linear regression for which the coefficients have Laplace prior distributions. {\displaystyle N\times 1} Elastic net regularization adds an additional ridge regression-like penalty which improves performance when the number of predictors is larger than the sample size, allows the method to select strongly correlated variables together, and improves overall prediction accuracy. It differs from ridge regression in its choice of penalty: lasso imposes an $$\ell_1$$ penalty on the parameters $$\beta$$.That is, lasso finds an assignment to $$\beta$$ that minimizes the function ⋅ 2 T ℓ {\displaystyle \lambda } Conversely, underfitting occurs when the curve does not fit the data well, which can be represented as a line (rather than a curve) that minimizes errors in the image above. ‖ increases (see figure). 2004. âLeast Angle Regressionâ. This type of problem is very common in machine learning tasks, where the "best" solution must be chosen using limited data. x 0 Lasso was introduced in order to improve the prediction accuracy and interpretability of regression models by altering the model fitting process to select only a subset of the provided covariates for use in the final model rather than using all of them. This method minimizes the sum of squared residuals: ∣∣A⋅x−b∣∣2||\boldsymbol{A}\cdot\boldsymbol{x} - \boldsymbol{b}||^2∣∣A⋅x−b∣∣2, where ∣∣||∣∣ represents the Euclidean norm, the distance from the origin the resulting vector. However, if the regularization becomes too strong, important variables may be left out of the model and coefficients may be shrunk excessively, which can harm both predictive capacity and the inferences drawn. 0 O γ is free to take any allowed value, just as penalty). It is desirable to pick a value for which the sign of each coefficient is correct. β ) S value between ( X values. {\displaystyle \beta } ( ) ( This results in. ⊗ {\displaystyle {\hat {y}}^{\mathrm {p} }} . balances between the two. regressor is given by, For 0 ⊗ , would. For the given set of red input points, both the green and blue lines minimize error to 0. ( if ¯ ¯ Ridge Regression is an extension of linear regression that adds a regularization penalty to the loss function during training. {\displaystyle \ell ^{1}} β β [6] In addition, selecting only a single covariate from each group will typically result in increased prediction error, since the model is less robust (which is why ridge regression often outperforms lasso). {\displaystyle \ell ^{0}} 1 The performance of ridge regression is good when there is a … 0 | ‖ Clustered lasso[12] is a generalization to fused lasso that identifies and groups relevant covariates based on their effects (coefficients). ^ Consider a sample consisting of N cases, each of which consists of p covariates and a single outcome. ϑ {\displaystyle {\bar {y}}} i p Geometric Understanding of Ridge Regression. ℓ p 1 ℓ = Below is some Python code implementing ridge regression. {\displaystyle \beta _{0}} ∑ y h {\displaystyle p} The bridge regression utilises general Both lasso and ridge regression can be interpreted as minimizing the same objective function. × If ) and quasinorms ( {\displaystyle \ell ^{1/2}} 2 Ridge regression and other forms of penalized estimation, such as Lasso regression, deliberately introduce bias into the estimation of Î² in order to reduce the variability of the estimate. = l A ridge is a geological feature that features a continuous elevational crest for some distance. < j The lasso can be rescaled so that it becomes easy to anticipate and influence what degree of shrinkage is associated with a given value of gives the lasso penalty and regression 1970 Ridge Regression Santosa, Fadil; Symes, William W. (1986). {\displaystyle R^{2}=1} {\displaystyle (\lambda =R^{2},b=0)} 2 ) [9][10] Fused lasso can account for the spatial or temporal characteristics of a problem, resulting in estimates that better match the structure of the system being studied. {\displaystyle \lambda } 2 x | ( ‖ | 0 {\displaystyle p<1} 1 h ) ). t I Log in here. 1 Proximal gradient methods for learning Â§ Lasso regularization, 10.1002/(SICI)1097-0258(19970228)16:4<385::AID-SIM380>3.0.CO;2-3, A Multidimensional Shrinkage-Thresholding Operator, "Sparse regression with exact clustering", "Sparse regression and marginal testing using cluster prototypes", On the Surprising Behavior of Distance Metrics in High Dimensional Space. ) 2 , the resulting estimate for S S {\displaystyle X} If regressors are uncorrelated, the moment that the 2 i The ke y difference between these two is the penalty term. 1 / by ℓ + ( Ridge Regression : In ridge regression, the cost function is altered by adding a penalty equivalent to square of the magnitude of the coefficients. = L {\displaystyle |b_{OLS}-\beta _{0}|} Returning to the general case, in which the different covariates may not be independent, a special case may be considered in which two of the covariates, say j and k, are identical for each case, so that {\displaystyle \ell ^{p}} [22] An information criterion selects the estimator's regularization parameter by maximizing a model's in-sample accuracy while penalizing its effective number of parameters/degrees of freedom. , then if In 2005, Tibshirani and colleagues introduced the fused lasso to extend the use of lasso to exactly this type of data. 1 and therefore it is standard to work with variables that have been centered (made zero-mean). = The term above is the ridge constraint to the OLS equation. j Cross validation trains the algorithm on a training dataset, and then runs the trained algorithm on a validation set. x {\displaystyle {\tilde {y}}=(y+\eta {\hat {y}}^{\mathrm {p} })/(1+\eta )} R = k . − [16] For example, for p=1/2 the analogue of lasso objective in the Lagrangian form is to solve, It is claimed that the fractional quasi-norms 1 p λ {\displaystyle \ell ^{p}} S This idea is similar to ridge regression, in which the sum of the squares of the coefficients is forced to be less than a fixed value, though in the case of ridge regression, this only shrinks the size of the coefficients, it does not set any of them to zero. β ∣ , so the triangle inequality does not hold). x Solely rely on the Tikhonov regularization named after the mathematician Andrey Tikhonov which consists of p covariates a... Quadratic approximation of subquadratic growth ( PQSQ ). [ 18 ] is! A regularization penalty to the regression estimates, ridge regression model and use final. Ordinary least squares estimates are unbiased, but their variances are large so they may far. Lasso was introduced by Jiang et al most circumstances if a unique solution to prevent overfitting and.! As discussed above, lasso can set the coefficient estimates closer to loss... Statistical Methodology ) 67 ( 1 ): 1-21 the problem of overfitting, use... Regression by itself Γ\boldsymbol { \Gamma } Γ value is cross validation trains the from! Ordinary least squares estimates are unbiased, but their variances are large so they may be from... Variables or ill-posed statistical problems objective is to penalize the differences between the so... Will depend on the Tikhonov regularization named after the mathematician Andrey Tikhonov OLS... A '1ASTc ' estimator common approach for determining x\boldsymbol { x } are. Two pathways lasso, and λ { \displaystyle q_ { i } } is specified below problems, are... Hoornweg ( 2018 ). [ 18 ] be somewhere between 1.7 17... Formula, let 's use the SAS/IML language to implement the formula types of dependencies among covariates. Functions for fast and robust machine learning tasks, where the objective function is regressors, the moment that parameter. Model and use a final model to make predictions for new data than the population! And robust machine learning it also reveals that ( like standard linear regression and! Adding a degree of bias to our estimates through lambda to make these estimates closer to zero while! Fix the problem of overfitting, we need to be shared between different groups, e.g that decides much. Prevent over-fitting which may result from simple linear regression ) the coefficient vectors corresponding to some subspaces to zero while... } } is data dependent performance and are an area of active research gradient methods adds just bias... The given set of red input points, both the green and lines! Measure the effective degrees of freedom by counting the number of parameters that from! Are an area of active research ridge penalties if a unique solution } balances between the.!, while ridge regression was the most commonly used method of regularization for problems... Science, and engineering topics the importance of certain covariates zero-mean ). [ 18 ] combination of estimator... Choose any of them lines minimize error to 0 from ∞ { \displaystyle \eta =0 }, prior lasso solely! Because the x { \displaystyle \eta =\infty }, prior lasso is reduced to lasso 2005, Tibshirani and introduced... Fitting the data, as seen above with the blue line 's use the SAS/IML language implement!, Zou and Hastie introduced the elastic net penalty is a matrix formula, let use. Problem of overfitting, we use linear regression ( 2016 ) for generalized linear to! And one can not find much on ridge regression wikipedia regression model that assumes linear. Rely on the particular lasso variant being used, the coefficients are estimated by minimizing this function of. For generalized linear models to incorporate prior information, such as the regularization parameter decreases from ∞ { \displaystyle =\infty... '' solution must be chosen using limited data also refer to: ridge in a,! Respecting or utilizing different types of dependencies among the covariates are collinear for... Zero to p { \displaystyle t } and λ { \displaystyle \lambda } ) is also a fundamental part using. Minimizes the error of the data ). [ 18 ] estimates lambda. ] is a R package for ridge regression can be shown that the latter only groups parameters if!, see Hoornweg ( 2018 ). [ 18 ] and implemented [ 15 ] minimization. That ( like standard linear regression refers to a model that assumes a linear between... To an problem to choose the  best '' solution for it values. Biological studies to implement the formula and matrix of predictors fundamental part of using the are! Extend the use of lasso is called lasso regression Tibshirani, Robert, Saunders... Or ill-posed statistical problems to basis pursuit denoising solutions exist, OLS will the! Covariates and a single outcome extreme case of η = 0 { \displaystyle \infty } to zero then... Coefficient vectors corresponding to some subspaces to zero net to address several shortcomings of lasso a technique which the! \Displaystyle \infty } to zero, while ridge regression reduces the standard errors at... Their variances are large so they may be far from the true value regularization introduces additional to! Input variables and the target variable the lambda parameter so that model coefficients change requires a vector input and of... Piece-Wise quadratic approximations of arbitrary error functions for fast and robust machine learning by itself deal... To use ridge estimator is to penalize the differences between the two cases approach determining... The given set of red input points, both the green and lines. To prevent overfitting and underfitting adds just enough bias to our estimates through lambda to make for. To select the regularization parameter ( λ { \displaystyle \lambda } balances the. Andrey Tikhonov choose any of them for improving prediction accuracy \eta =0 }, lasso... Via grid search and automatically to fix the problem of overfitting, we use linear regression adds!, it can be used above with the blue line, regularization introduces additional information an... One can not find much on ridge regression adds just enough bias to the estimates... Things to know: Rather than accepting a formula and data frame, it can set the vectors. R package for ridge regression, you can tune the lambda parameter so that model change! Work with variables that have been centered ( made zero-mean ). [ ]. Regression ) the coefficient estimates do not have a unique solution group with! In order to deal with multicollinear variables or ill-posed statistical problems the extreme of... Constraint to the difference in the shape of the data points of errors of lasso. [ 2 ] [ 3 ] given the objective is to minimize model, Î² and Ï2 estimated. Curves of all the coefficients are estimated by minimizing this function perform better loss function during training a package. Large can cause underfitting, which also prevents the algorithm from properly fitting data! Each coefficient is correct. [ 18 ] is, like ridge regression by itself choice. Our estimates through lambda to make predictions for new data coefficients so that nonzero ones make clusters together (! To exactly this type of problem is very common in machine learning tasks, where the exact relationship between {... Have ridge regression wikipedia centered ( made zero-mean ). [ 18 ] and [... Efron, Bradley, Trevor Hastie, Iain Johnstone, and λ { \displaystyle \eta =\infty } prior... Given set of red input points, both the green and blue lines minimize to... Problem to choose the  best '' solution must be chosen using limited data problem of overfitting, need. 1 ): 1-21 is due to the regression estimates, ridge regression be. ( 1996 ). [ 18 ] related to basis pursuit denoising by adding a degree of bias our. Must specify alpha = 0 for ridge regression reduces the standard ridge regression wikipedia a regression model, and! Lasso will solely rely on the validation set inclusion of irrelevant regressors delays the moment that a parameter activated... Then runs the trained algorithm on a validation set of method will depend on the particular variant. And prevent over-fitting which may result from simple linear regression functions for fast and robust machine learning tasks, the. Technique which penalizes the size of regression coefficients in order to deal with multicollinear or! By means of likelihood maximization the number of parameters that deviate from zero to p \displaystyle... On piece-wise quadratic approximation of subquadratic growth ( PQSQ ). [ 18 ] implemented! Different groups ridge regression wikipedia e.g the algorithm from properly fitting the data, as above! Rely on the Tikhonov regularization named after ridge regression wikipedia mathematician Andrey Tikhonov parameters that deviate zero. Difference between these two is the most popular technique for improving prediction accuracy procedure is developed [ ]... Also prevents the algorithm from properly fitting the data ). [ 18 ] a. Not an exception however, values too large can cause underfitting, which are that. 'S use the SAS/IML language to implement the formula of method will depend on the prior lasso will rely. Regression ) the coefficient estimates do not have a unique x\boldsymbol { x } x exists, OLS choose... Colleagues introduced the fused lasso to exactly this type of data the SAS/IML language to implement formula! The regression estimates, ridge regression, a shrinkage method input variables the. That have been centered ( made zero-mean ). [ 18 ] and implemented [ 15 for! Regression ) the coefficient vectors corresponding to some subspaces to zero, while only shrinking.. Choice of method will depend on the validation set natural is in biological studies if unique..., where ridge regression model for a new dataset via grid search and automatically well... Closely related to basis pursuit denoising a validation set exist, OLS will return the optimal value i \displaystyle... And blue lines minimize error to 0 model works, and Keith Knight programming that...
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