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# Engineering Jobs combining programming with hardware?

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1. Mar 19, 2017

Hi, I'm a high school student in a STEM program, and as of now my primary interests have been in Computer Science, Math, and Robotics. Currently, as I research colleges, my intent is to double major in either Math and Computer Science, or in Computer Science and Electrical Engineering. However, I'm worried that there may not be jobs combining these interests. I've been told that any job I might get would be in only one of these areas, not both, and that worries me because I don't want to be a purely electrical engineer without doing any programming, and I don't want to be a Computer Scientist without ever touching or dealing with hardware in any way. I definitely tend to prefer Computer Science, if I were to have to choose only one discipline, but I really would prefer to be able to combine the two in a kind of way similar to Arduino projects that I've done. Are there jobs like that out there where I would be involved in both, not just specializing in one or the other? And also, how heavily do those jobs use math? I also worry that a job that is too specialized would not scratch my math itch, as I've heard that many EE jobs boil down to formulas.

Any help is appreciated, and let me know if I'm being too idealistic or my information is wrong.

2. Mar 19, 2017

### TChris

Don't do it. Become a doctor or a lawyer instead. A lot of people are promoting STEM jobs right now. But it's not worth it. As a scientist or engineer, you may not be able to get a job and if you do, you'll earn a little above average and always have to worry about getting laid off. If scientists, engineers and programmers are in such demand, why do doctors and lawyers get paid 6-figure salaries and STEM workers don't? The reality is, the jobs that have always paid big salaries or the ones that still do for the most part. As a technical professional you're just another replaceable peon to the corporate world. I know, I'm an engineer, went to Cal Tech and now wondering if I'll ever have enough money to retire. Think about it kid. Most of the people urging you to be an engineer probably are NOT engineers themselves and they don't really know what it's like. Best thing you can do is go to indeed.com and look for engineering jobs and then look at the outrageously long list of requirements that most employers have now for engineering jobs. If I had as many skills as some of these jobs demand, I wouldn't be looking for a job, I'd start my own company.

3. Mar 19, 2017

### StatGuy2000

TChris, I'm not sure if you're trolling here (this is your first post), but frankly your post here isn't helpful to the question the OP is asking. But let's look at the claims you make.

First of all, do you think most lawyers earn 6 figures, or that lawyers don't have to worry about being laid off? Consider the following:

https://qz.com/206705/the-us-lawyer-bubble-has-conclusively-popped/

According to the article above, the unemployment rate nine months after law school graduation for the class of 2013 was 11.2% (not to mention that the salaries for lawyers are on the decline, in large part because of a large surplus of law school graduates). By contrast, the unemployment rate for electrical engineers in 2013 spiked to a high of 6.5%, and was closer to 3.5% for the years 2010 and 2011 (again, I don't have the latest numbers for either figure).

https://www.bls.gov/opub/mlr/2015/article/stem-crisis-or-stem-surplus-yes-and-yes.htm

So from the above 2 articles alone, it seems pretty clear to me that studying engineering is a much safer bet than studying law.

Second, even if I presume you're a Caltech-trained engineer who is worried whether you'll be able to retire -- without knowing more about your circumstances, to me, this sounds to me to be more of a problem with your ability to save money rather than anything to do with the marketability or employability of engineers.

4. Mar 19, 2017

### Päällikkö

Many of the big companies out there have slashed hardware engineers (e.g. Intel) and from what I've heard (but this is not first hand experience), that career path is on the decline. It makes sense as a general trend because general purpose hardware has become quite cheap. That said, do note that one of the most hyped fields right now is "Internet of Things". There's plenty of ideas and startups out there needing people who can both code and understand hardware at the level of Arduino stuff, i.e. be able to interface with it, but not necessarily design one's own DSPs.

As for programming or STEM being a bad career, Google starting salary is somewhere ~120k base + benefits (anecdotal evidence which Glassdoor.com seems to support), which is well into 6 figures.

5. Mar 20, 2017

### TChris

Now who's trolling? I have about 35 years of experience with this problem. I know many engineers who didn't get jobs or wound up doing something different. You pick lawyers to try to make your case, but I notice you say nothing about doctors. Are you going to deny that they make from $170,000 up$400,000? What's the source of statistics and why should I believe them? I can tell you this there are many unemployed engineers who aren't even counted as unemployed especially if they graduate and never land that first job.

6. Mar 20, 2017

### symbolipoint

As a way to respond to some of the discouragement there, one must actually find people and TALK to them. This can be done at conferences, hiring fairs, and also by traveling to a company site and bringing resume and telling someone what you are hoping to do.

7. Mar 20, 2017

### StatGuy2000

TChris, I have been working in the STEM field as a statistician for the past 14 years, and during that time I've known many engineers (primarily electrical, mechanical, or computer engineers, a few industrial engineers as well) and computer scientists, all of whom are gainfully employed and earning solid salaries (certainly well above the average salary in Canada and the US, close to 6 figures). I don't dispute that there are unemployed engineers, but oftentimes this is dependent on what sector you are working in, and what field. So the notion that there are no jobs for engineers is simply not true across the country.

As for being laid off, in the modern economy, anyone and everyone can be laid off at any time. The question is whether there are other jobs available for engineers or those with a training in that field. I have never met an engineer in my life who has been unemployed for an extended period of time (i.e. unemployed for longer than 6 months).

I mentioned lawyers because you mentioned lawyers in response to the OP's original post -- you were advising the OP to go to law school instead of pursuing engineering or computer science. What I had done was present sources which show that your advice is not good advice. As for sources, the link I gave you are the sources, and there are many others.

1. Not everyone is cut out to become a doctor. IMHO, only those who are truly dedicated to serving and helping patients should consider pursuing medicine, given that one typically needs to finish at minimum 3 years of pre-med studies before going to medical school (and usually complete a full 4 year degree beforehand), spend another 4 years in med school at considerable cost in tuition (most medical school students are not offered scholarships or grants to pursue medicine, so the have to take on debt to pay for med school, on top of whatever student debt these students may have), and then complete another 3-7 years of residency before they are eligible for medical licensing.

2. The salary range of doctors vary considerably depending on specialty and where they happen to practice. In general, specialists like neurosurgeons or cardiologists earn the top dollars you talk about, while primary care doctors earn much less. Also, doctors in the big cities tend to earn more than doctors in rural areas. But the key shortage of doctors are precisely in those areas where they happen to earn the least (i.e. rural areas, primary-care physicians -- a problem that also affects Canada)

3. Related to point #1, medical school graduates are often saddled with huge debts, which cuts in to the 6 figures that you talk about. Also keep in mind that doctors are typically in school or in training for much longer than engineers or computer scientists (you can go into engineering of computer science immediately after graduating with a 4-year BS, whereas it takes anywhere from 10-16 years to become a fully licensed doctor). So the earning potential of those additional years of schooling/training needs to be kept in mind.

8. Mar 20, 2017

### StatGuy2000

Back on topic.

To the OP:

To follow up with what Paallikko stated earlier, if your primary interest is in working directly with hardware (i.e. working in semiconductors or integrated chips), then opportunities, while still available, have been on the decline.

That being said, if as you state, you are interested in combining an interest in programming/software and hardware, there are increasing opportunities in areas like robotics, which at least by word-of-mouth has been a growth industry (one of my first jobs out of grad school was working for an engineering company specializing in robotics). I can also foresee opportunities in areas like the Internet of Things (where software is embedded into many products to ensure connectivity to the Internet), among others.

You had also stated earlier that you had an interest in combining computer science with math. If so, then there are also opportunities that are open to you in areas like data science/machine learning (a major hot area right now), or in areas like cybersecurity/data privacy/encryption (cybersecurity will become increasingly important, especially as more and more data will become available through the Internet and more businesses and other organizations rely on being connected).

So to summarize, I feel that you have a solid set of opportunities ahead of you, so long as you are flexible and willing to learn and work hard in college/university. You're still young, and still have time to explore your options.

Best of luck!

9. Mar 20, 2017

### Staff: Mentor

10. Mar 21, 2017

### Staff: Mentor

Thread re-opened after some cleanup. Let's please stay on-topic in helping the OP. Thanks.

11. Mar 21, 2017

Thank you very much for that information! I guess a better way to phrase one of my questions I had, though is like this: Lots of people that I've talked to have told me that even in fields like Robotics which work in both software and hardware, I would still be doing one or the other. As in software specialists and hardware specialists collaborating, not with people who actively work in both aspects. Is this true, and if so to what extent?

12. Mar 21, 2017

### StatGuy2000

I personally don't work in the robotics field (at least not directly for over a decade), so I may not be best placed to answer this question, but from my limited knowledge (among people I've known in the field), it is true that there tends to be software specialists and hardware specialists collaborating, which is understandable given the complex nature of the robotics field (robotics encompasses everything from control systems in hardware to incorporating intelligent decision making in robots).

That being said, from what people have told me and what I observed when I worked for a robotics firm a decade ago, the software specialists still need to have an understanding of what the hardware component and vice versa, and there is scope to work on both aspects.

13. Mar 27, 2017

### reasonableman

I have a point to add that does not seem to have been covered here. OP seems to have a fundamental misunderstanding of professional engineering. Generally people are hired to solve a problem. They say, "if you have a hammer, every problem is a nail", similarly, if you are hired by computer science people they will see a computer science problem and visa versa. If you have insights to both that can be very valuable but it will not be advertised that way. This is related to the second point about EE jobs being formula based. Most real world problems are hard and formulae that are used in industry are usually hard earned and used 'because they work'. Trying to derive, or re-derive, formulae is usually a waste of time and get in the way of actually doing the job. There are exceptions of course but for many problems in industry solving a problem is more important than understanding it.

14. Mar 28, 2017

I don't mean to sound like I think engineering is easy, by any means. I guess my problem lies more so in that I'm not really sure what specific field is for me, yet.

15. Mar 28, 2017

### Aufbauwerk 2045

First, I don't think a double major in math and CS is a good solution. Most of the math you would take as a math major has little to do with computer science. The CS dept will let you know what math you really need for CS.

Your don't need to worry about never working with electronics or hardware in general. In fact lots of programming involves interfacing with hardware. Look at embedded systems, which is a huge field. But you don't need an EE degree. What a CS type needs to know really is basic digital electronics along with some knowledge of how computer systems work, how do CPUs and GPUs work, how do networks work, etc. But a CS major does not need to get bogged down in hardware details.

As far as jobs, I think there is no question that someone with a B.S. in CS is in a much better situation than with a B.S. in EE or math.

Last edited: Mar 28, 2017
16. Mar 28, 2017

### Aufbauwerk 2045

P.S. if you do want to get into hardware design as a CS major, then you could learn about FPGAs or related technology, and learn how to design a custom chip.

You mentioned the Arduino. Eventually you might want to combine your interests in software and hardware by designing your own system. You could always team up with a hardware specialist if you need to. No one can know it all.

My "spare time" project, if I ever have spare time, will be to design my own LISP computer from the chip level up, running my own version of a LISP operating system.

Here is a fascinating talk by the great Prof. Wirth on his experience designing an Oberon-only system, including a custom chip.

17. Mar 30, 2017


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# How do we write the Lewis structures of cyanide, and isocyanide ions?

Apr 5, 2017

Typically, we represent the negative charge on CARBON.........

#### Explanation:

The typical Lewis structure for cyanide anion is ""^(-):C-=N:; i.e. the carbon centre is associated with 7 electrons, and is therefore a formal anion. Certainly in the parent hydrogen cyanide, the (acidic) hydrogen is bound to carbon, i.e. $H - C \equiv N$

When cyanides are used (in my opinion, VERY DESTRUCTIVELY) to extract metal ores, for instance those of gold according to the later equation, certainly the carbon centre is the atom that coordinates to gold:

4 Au + 8 NaC-=N + O_2 + 2H_2O rarr 4{Na^(+)""^(-)[Au(C-=N)_2]} + 4NaOH

For $\text{isocyanide}$, $R - {N}^{+} \equiv C {:}^{-}$, the nitrogen centre is formally cationic with no lone electron pairs available for donation.

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# non-commutating graph

Is there any research work about automorphism group of non-commutating graph?

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do you mean the non-commuting graph of a group? If not, could you define what you mean? –  Nick Gill Nov 9 '12 at 14:14

Given that you're based in Esfahan, and given that there are a number of Iranian mathematicians working on the non-commuting graphs of groups, I'll assume this is what you're asking about.

So the non-commuting graph of a group is the graph with vertices equal to elements of $G$, and two elements connected if they do not commute. Let's throw away the central elements of $G$ as these are just isolated vertices that will only confuse things.

Clearly the automorphism group of the non-commuting graph of a group $G$ will be the same as the automorphism group of the commuting graph of $G$ since the latter is the complement of the former.

To my knowledge the only completely general statement that one can make about this automorphism group is that it contains $G/Z(G)$ as a subgroup, since $G$ acts naturally on the vertices of the graph by conjugation.

Generally research on commuting graphs has not focused on the automorphism group. Instead people have been interested in recognition questions (when does the commuting graph of a group characterise the group), or to graph-theoretic questions (e.g. computing the diameter, chromatic number, clique number etc). I asked a MO-question on this recently that may be helpful.

The only specific reference I have been able to find to automorphism groups of commuting graphs is from this conference in which Mirzargar gave a talk showing that the automorphism group "is a non-abelian group such that its order is not prime power and square-free number." (I don't quite know how to interpret this statement, as the use of English is a little confusing.) So far as I'm aware this result has not appeared in the literature.

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Thanks for making sense of the question! –  Alain Valette Nov 9 '12 at 15:03
thank you so much –  sebastian Nov 10 '12 at 17:53
soroosh, my pleasure. if you like the answer click the tick' to the side of it. this will accept' the answer and take if off the unanswered stack. –  Nick Gill Nov 14 '12 at 12:12
@ Nick! OK! so thanks for your attetion! –  sebastian Jan 13 '13 at 21:11


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# Plus or Minus Sign \pm \pm: solving quadratic equations

In this activity I have a question about ([Copy of] 5.26 Solving Equations with Squares • Activity Builder by Desmos) I took code from this activity which I don’t have a question about ([Copy of] Solve Equations with Step Checker (Updated 5/4/21) • Activity Builder by Desmos)

When solving 2(x-5)^4=512 you get to a point where x-5= plus or minus 4, and when I type this equation into the table, I do not get a checkmark for it, using x-5= \pm 4. Is there a way to code the question so that Desmos recognizes the plus or minus sign?

I can get checkmarks for x-5=4 and x-5 = -4 using the current code that I have (which only currently has been applied to row 1 column 1 of the table).


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# Shannon's Entropy for Images

Image Entropy turned out to be really interesting. I wanted to find the maximum entropy for a given image size and resolution and started with the assumption that it would occur with all pixel values equally probable. For most images we use 8 bits per RGB channel so its easy to figure out the maximum but what about real numbers that have infinite resolution? What about small images when we dont have enough pixels to represent the 256 values for 8 bit?

### Image Entropy

To calculate the Shannon’s entropy for data $$x$$ we use,

\begin{align} H = -\sum_i {\bf{p}}(x_i)\log_2 {\bf{p}}(x_i), \end{align}

for probabilities $${\bf{p}}(x_i)$$ and entropy calculated in bits.

For an image we need to determine the pixel depth $$p$$, often $$p=8$$ for 8 bits or 256 levels per pixel or $$p=\infty$$ for real numbers, and then bin the pixel values to determine their probability.

### Maximum Entropy value for Images

To find the maximum possible entropy for a specific image size $$m\times n$$ and pixel depth $$p$$ we observe that entropy is maximized when all of the probabilities are equal. So for a $$16\times 16$$ image with $$p=8$$ we can define 4 pixels per pixel value ($$n\times m / 2^p$$).

For small image sizes there may be insufficient pixels to support all pixel values. For example a $$7\times 7$$ image can only support a pixel depth of $$\log_2 7^2$$ or 5.614.

Our maximum entropy, for an $$n\times m$$ image with pixel depth $$p$$ is,

\begin{align} H_{\mathrm{max}}^p(m,n) &= -\frac{mn}{2^{2\hat p}}\log_2 \left(\frac{mn}{2^{2\hat p}}\right)\times 2^{\hat p}, \\ \label{eqn:maxent} &= -\frac{1}{2^{\hat p}}nm\left(log_2\left( nm\right) - 2\hat p\right), \end{align}

where $$\hat p= \min(p, \log_2( nm))$$.

We note that for the common case for arrays of float values, where the pixel depth is infinite, the effective’’ pixel depth is $$\log_2( nm))$$ and the maximum entropy Equation above can be simplified to,

\begin{align} H_{\mathrm{max}}^{\infty}(m,n) &= \log_2 \left(mn\right). \end{align}

If you want to know what a maximum entropy image looks like here is such a thing,

Thanks!


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# Abcd is a Trapezium in Which Ab || Dc. If Ar (δAbd) = 24 Cm2 and Ab = 8 Cm, Then Height of δAbc is - Mathematics

MCQ

ABCD is a trapezium in which AB || DC. If ar (ΔABD) = 24 cm2 and AB = 8 cm, then height of ΔABC is

•  3 cm

• 4 cm

•  6 cm

•  8 cm

#### Solution

Given: (1) ABCD is a trapezium, with parallel sides AB and DC

(2) Area of ΔADB = 24 cm2

(3) AB = 8 cm

To find: Height of ΔABC.

Calculation: We know that ,” two triangles with the same base and between the same parallels are equal in area.”

Here we can see that, ΔADB and ΔACB are on the same base AB.

Hence,

Area of ΔACB = Area of ΔADB

Area of ΔACB = 24

1/2 (8 xx h) =24

h = (24 xx 2)/8

h = 6 cm

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 14 Areas of Parallelograms and Triangles
Q 16 | Page 61


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# A Look Inside InDesign CS5

InDesign CS5 has arrived with lot of new tools, panels and features. Let's have a closer look at these changes and discover how they can improve our workflow, productivity and, best of all, our creative process. Time to also think outside of the box, since InDesign is starting to explore new grounds of interactivity and digital publications with growing enthusiasm. So, what's new?

## Live Corners and Corner Options

Let's start with the Live Corners. Remember in InDesign CS4, where you were forced to use scripts if you wanted to make a rounded corner in only one part of a frame? Well, that's gone. Now all rectangular frames, both for graphic and text, will have a small yellow square at the upper side of its right border. Click on it and the frame will display four yellow squares at all the corners. Click on one yellow square and drag it to create four rounded corners around the frame. You can also press the Shift button while dragging one square to create a rounded corner only where you clicked, ignoring the other three corners. Quick and handy.

However, if you don't think this feature is useful or you don't like the sight of that yellow square, just go to View > Extras > Hide Live Corners.

You may prefer to use more precise measures instead of adjusting your corners manually. You may also prefer to use other types of corners instead of the rounded corner. Go to Object > Corner Options, and you will find an improved menu where you can apply different corner sizes and shapes to each individual corner. Good-bye scripts!

## Content Grabber

Another easy to spot feature. Now each graphic frame lets you manipulate its content directly by simply clicking on the Content Grabber, the translucent black circle surrounded by a translucent white ring in the middle of the frame (or in the middle of the visible frame, in case you zoomed in so much that part of the frame is outside of your view). The Grabber may not be visible if the frame is too small relative to your actual workspace, in that case you will have to zoom in or make the frame bigger.

When the Selection Tool is over the Content Grabber, it automatically turns into the Hand icon, letting you move the frame's content with a single click. No more need to click twice or go for the Direct Selection Tool. It takes half a second to press the A button and use that white arrow but why use two tools for selection if you can optimize production with only one? This is really a time saver.

It also doesn't matter if objects are grouped, because you can directly select the Content Grabber of each frame individually. Another handy feature is that, when you rotate the content of a frame, the black circle will display an inclined line, indicating that the image has been rotated at the same inclination of the line. So if the line is horizontal, that means the content is upside down, due to a 180° rotation.

By the way, you're not forced to use the Content Grabber, so if you think it's more annoying than useful just go to View > Extras > Hide Content Grabber.

## The Tool Hints panel

A really useful panel to explore all the possibilities of your tools. Go to Window > Utilities > Tool Hints and the panel will open. Now all you need to do is click on a tool from the Tools Panel, any tool, from Direct Selection to Zoom, and the Tool Hints panel will display information about the tool you selected. It shows its main functions, all it does when you press modifiers keys and its shortcuts. Good for beginners and for experts too, since you have new tools such as the Gap tool and the Page tool that require some explaining.

## Multiple Page Sizes and the Page Tool

Now you can change all page sizes inside a single document. For example, if you have a spread, you can choose that the left page will have a bigger width. Why would you need pages of different sizes? Well, brochures, flyers and other commercial publications can have an asymmetrical layout, where the spine of the folded pages is not in the middle of the spread. Also, magazines and books can have flaps on the cover, so the cover page needs to be larger.

To change a single page size you will need to use the new Page tool (Shift + P). With the tool you can click on any page and it will become selected. You can press Shift and Click on more pages to extend the range of pages selected. You can also press Command to add or remove a page from the selection. With the pages selected, you can change their size at the Control panel, just like you would change objects sizes: the "W:" box changes the width and the "H:" changes the height.

You can choose the page orientation (portrait or landscape) and a page preset like A4, A5 etc, from a list. You can also create your own custom preset by clicking on the Custom page size (last on the list).

You can also change the page size on the Pages panel. There's an "Edit page size" icon at the bottom of the panel. Select the pages you want and click on the icon, then choose a page preset or create your own custom preset by clicking on the "Custom page size" (last on the list).

You can move your page by dragging it with the Page tool or changing the Y value at the control panel. Only vertical change is allowed. Obviously we can't change the X-axis, since we can't move a page away from a spine or invade the space of a facing page. You can only move one page of a spread since moving the two pages together would be redundant and have no result at all. The content of the page will not move with the page itself, unless you check the "Objects Move with Page" on the Control panel.

"Enable Layout Adjustment" is an option that makes the page content adjust automatically when the page changes size (I think its safer to make all your page design before even adding content but everyone has their personal workflow).

Finally, the "Show Master Page Overlay" is a Control panel option that just seems to put a translucent blue box with a thick border over your page and indicates the Master name at the bottom of the box. It's actually much more useful if the Master page has a different page size than your current document page. Can the document page have a different size than the Master? Yes, of course. Just override the document page size with the Page tool. Now when you select the page again you can not only see the Master Page Overlay but also move all the Master Page items by dragging the thick blue border of the Overlay box with the Page tool.

## The Gap Tool

The Gap tool (U) is useful for quick alignment changes. What it does is change the gap between page items, which means the space between two items like graphic frames, text frames, lines and vectors. You can select a gap by simply placing the Gap tool over it, a double arrow icon will appear behind your cursor to indicate if you're going to adjust an horizontal or a vertical gap and the grey area that appears on the background indicate the extension of the gap you selected.

Dragging the Gap tool will resize all items aligned with the selected gap. Remember that adjusting the gap will not adjust the content of a graphic frame, only the frame itself. The Gap tool also does not keep vector images proportions while resizing them, so they will always become deformed, be aware!

If you want to resize only two page items instead of resizing everything that is aligned to the gap, just move your cursor exactly between the two items and use the tool with the Shift button pressed.

With the Command button you can resize the gap instead of moving it.

Finally, with the Option button you can move all aligned page items on the gap instead of resizing them.

And if your cursor is between two items to your left and right and other two items above and below, just press the arrow keys and you will select a different gap to move (from vertical to horizontal gap or the other way around).

## The Mini Bridge Panel

You don't need to minimize your InDesign application and open Bridge to search for files. Now you have a Mini Bridge panel inside InDesign! You can navigate your files and preview then. Dragging and dropping files from Mini Bridge will also place then in your document. You can also filter your files by rating it with stars and selecting only your high rated files. There's also the useful Slide show and Review modes inside the Preview options that let you see the first two pages of a InDesign file in thumbnail form.

You can view more than 2 pages of a INDD file if you go to InDesign > Preferences > File Handling and use the Saving InDesign Files box. There you can choose to save from one to all of your files as thumbnails or even in larger sizes. Remember that this will make your files size bigger! View your INDD pages in Review mode instead of Slide show, it's better since you navigate the pages at your own pace. This is still a mini version of Bridge, not the entire application, so you can't add metadata.


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## Tuesday, October 20, 2009

### Amicus Brief: AAIDD (Dufour v Florida, 2009)

I just posted a new Amicus Brief (see Amicus Brief section of blog; click here to go to brief)  filed by AAIDD in Dufour v Florida (2009).  The brief focuses primarily on the need to incorporate the Standard Error of Measurement (SEm) in establishing a range for intellectual disability determination and the need for objective measurement of adaptive behavior (and not to place undue emphasis on specific adaptive behavior strengths).  The brief aruges that Florida law is not consistent with the Atkins v Virginia SCOTUS decision.

I'm hoping to locate copies of decisions related to the Dufour case and post them when I can.

Technorati Tags: , , , , , , , , , , , , , , , , , , , , , , , , ,


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# Group of Proper Symmetries of Painted Cube

What is the proper symmetry group of a cube in which three faces, coming together at one vertex, are painted green and the other faces are red?

I know that the axis of rotation for which the rotations of $$0$$, $$2\pi/3$$, and $$4\pi/3$$ produce a proper rotational group is $$x=y=z$$ with the intersection points being the intersection of the similarly painted faces. But I'm not sure how to formulate this into a proper symmetry group. Any help would be great, thank you in advance!

• "proper symmetry group" means reflections are excluded? – Parcly Taxel Mar 10 at 5:46
• I think you want the dihedral group of the triangle. – Angela Richardson Mar 10 at 6:03
• Yes, @ParclyTaxel. – James Done Mar 10 at 7:19
• "Proper" has already many meanings, not to be used for another one. "Orientation-preserving", "direct", "group of motions" etc already mean what you want. – YCor Mar 10 at 13:51


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# Constructing a graphic sequence from a subsequence

Is it possible to determine, given an $N$ and a non-increasing sequence $(a_i)_{i=1}^k$ where $k \leq N$, if a graphic sequence $(d_i)_{i=1}^N$ exists containing $(a_i)$ as a subsequence? Is it further possible to construct one such graphic sequence if any should exist?

For example, for $N = 5$, given the sequence $(4,2,1)$, we can "complete" this as the graphic sequences $(4,2,2,1,1)$ or $(4,3,2,2,1)$, but obviously no such completion exists for the sequence $(4,0)$.

I suppose a brute-force method would be to enumerate all possible degree sequences containing the subsequence, and then checking each if they satisfy the Erdős–Gallai theorem, but for the particular problem I am working on $N$ is quite large and $k$ relatively small, so this is not feasible.

Thanks everyone!

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1 Introduction

## Abstract

The Binary Space Partitioning (BSP)-Tree process is proposed to produce flexible 2-D partition structures which are originally used as a Bayesian nonparametric prior for relational modelling. It can hardly be applied to other learning tasks such as regression trees because extending the BSP-Tree process to a higher dimensional space is nontrivial. This paper is the first attempt to extend the BSP-Tree process to a -dimensional () space. We propose to generate a cutting hyperplane, which is assumed to be parallel to dimensions, to cut each node in the -dimensional BSP-tree. By designing a subtle strategy to sample two free dimensions from dimensions, the extended BSP-Tree process can inherit the essential self-consistency property from the original version. Based on the extended BSP-Tree process, an ensemble model, which is named the BSP-Forest, is further developed for regression tasks. Thanks to the retained self-consistency property, we can thus significantly reduce the geometric calculations in the inference stage. Compared to its counterpart, the Mondrian Forest, the BSP-Forest can achieve similar performance with fewer cuts due to its flexibility. The BSP-Forest also outperforms other (Bayesian) regression forests on a number of real-world data sets.

## 1 Introduction

Several machine learning methods, such as decision trees for regression and relational modelling for identifying interaction patterns, are concerned with space partitioning strategies to identify meaningful “blocks” in a product space. Models may be fitted to the data in each block, within which the data will exhibit certain types of homogeneity. These techniques have found application in relational modeling [14, 1], community detection [22, 13], collaborative filtering [24, 17], and random forests [16] – This space-partitioning strategy has shown its promising prospect in real-world applications. As a result, a number of structured space-partitioning priors have been developed, including the Mondrian process [28, 27, 26], the Rectangular Tiling process [21], the Ostomachion process [9], and the Binary Space Partitioning (BSP)-Tree process [8]. Other strategies are also developed to complete the task, e.g., the Rectangular Bounding Process [7] is recently proposed to use a bounding strategy to partition the space and it claims to obtain a parsimonious result.

Among the aforementioned approaches, the BSP-Tree process [8] is an efficient way to partition the two-dimensional space. Instead of axis-aligned cuts adopted in most conventional approaches [14, 28, 21], the BSP-Tree process uses angled cuts to better describe the potential dependency between each dimension. The BSP-tree process is attractive because it is self-consistent, which ensures distributional invariance while restricting the process from a larger domain to a smaller one. However, as the BSP-Tree process is originally proposed for relational modelling, which only requires partitions on a two-dimensional space, it can hardly be extended to a -dimensional () space. This restriction prohibits its applications to decision tree-style models, which usually consist of more than two dimensions of features. Unfortunately, we cannot straightforwardly extend the BSP-Tree process to a higher dimensional space because it would violate the self-consistency. It is possible to consider a proper extension that is able to retain the self-consistency property, however, for a -dimensional space, the hyperplane usually lies in the -dimensional space and the measure of all potential cutting hyperplanes involves complicated integrals which would incur unacceptable calculations for a machine learning task.

In this work, we make the first endeavor to extend the domain of the BSP-Tree process to -dimensional () space while still keeping its self-consistency. To simplify the process, we propose to generate a cutting hyperplane, which is assumed to be parallel to dimensions, to cut each node in the -dimensional BSP-tree. That is to say, each cutting hyperplane is allowed to have two degrees (dimensions) of freedom during the generative process. The current node, which is a convex polyhedron, is first projected onto a pair of dimensions – The pair is sampled in proportion to the perimeter of its projection onto the sampled pair of dimensions among all possible configurations. The subsequent cut on this node is then generated on the projected convex polygon through the same way of the BSP-Tree process and the rest dimensions of the cutting hyperplane are parallel to the other dimensions. This geometrically simple construction permits a flexible multi-dimensional extension to the BSP-tree process that provably retains the self-consistency property.

Our second contribution is to construct an ensemble of BSP-trees – the BSP-forest – to enable complex and flexible regression modelling. In contrast to Bayesian additive regression trees (BART) [5] and the Mondrian Forest  [16], which only implement node cuts in a single dimension to generate the tree structure, the BSP-forest uses two dimensions jointly to form a hyperplane cut in the feature space. As a result, the BSP-forest can achieve higher performance with a lower-depth hierarchical tree structure. In addition, because it consists of multiple BST-trees who select different pairs of dimensions to describe the data in a local region, the BSP-Forest is able to capture all-round pairwise dimensional dependence.

In the inference stage, an efficient Particle Gibbs sampler is developed to infer the BSP-Forest. Particularly, instead of cutting the entire node, we can further simplify the partitioning process by only conducting the hyperplane cut in the convex hull of the training data within the node to circumvent the complicated polyhedron related evaluation. Thanks to the self-consistency of the proposed extended BSP-Tree process, these two strategies lead to the same equilibrium distribution; we can thus largely simplify the sampling procedure and improve the efficiency.

The effectiveness of the proposed BSP-Forest is validated through an extensive study on the famous Friedman’s function and then five real-world data sets. Compared to its counterpart, the Mondrian Forest, the BSP-Forest can achieve similar performance with fewer cuts due to its flexibility. The BSP-Forest also outperforms other (Bayesian) regression forests on a number of real-world data sets.

## 2 Preliminary: The BSP-Tree Process

The BSP-tree process [8] generates partitions, , on an arbitrary two-dimensional complex polygon . The BSP-tree process is an almost surely right-continuous Markov jump process on , where is a pre-fixed budget. Let denote the BSP-tree partition at time . is a hierarchical partition of , which can be represented as a triplet . is a finite binary space partitioning tree, and and respectively specify, for each intermediate node of the tree, the orthogonal slope () for the cutting line and the cut position () on the projected segment of the node.

The probability density function of is proportional to the length of the projected segment , and the cut position is uniformly distributed on . represents a hierarchical partition of into the root node (; the base of the domain), intermediate nodes () and terminal nodes. Intermediate nodes are generated from their parental node through a node split and are themselves split into two child nodes through the cutting line

 {x∈□j:(x−uj)(1;tanθj)⊤<0}, {x∈□j:(x−uj)(1;tanθj)⊤>0}.

Terminal nodes are the final generated blocks, which do not contain cutting lines.

Defining the time for the -th cut as , the incremental cut time depends only on the partitions at time . Given an existing partition , the time to the next cut follows an Exponential distribution:

 (τl−τl−1)|⊞τl−1∼Exp(l∑k=1PE(□(k)τl−1)) (1)

where denotes the perimeter of the -th block in partition . Each cut divides one block (chosen with probability proportional to its perimeter) into two new blocks and forms a new partition. If the time index of the new cut exceeds the budget , the BSP-tree process terminates and returns the partition as the final realisation.

#### Self-consistency

Self-consistency of the BSP-Tree process refers to that, when restricting the BSP-tree process on a convex polygon to any sub-region , the resulting partitioning on the sub-region is distributed as if it is directly generated on through the BSP-tree’s generative process. This property can be usefully exploited in settings such as online learning and domain expansion. See [8] for more detailed description of the BSP-tree process.

## 3 Extending the BSP-tree process to d-dimensional space

One motivation for extension of the BSP-tree process to multi-dimensional space is regression. Here, we might have labelled datapoints , where we aim to predict the unknown labels from the -dimensional predictors . Other applications, such as classification, simply require straightforward changes to the likelihood constructions for the labels.

The proposed multidimensional BSP-Tree process works similarly as its original version. It is still a continuous-time Markov jump process where, for , the value taken at , which is denoted as , is a BSP-Tree partition in a convex polyhedron and also a further refinement of the value taken at time  (see Figure 1). For the domain (root node) , the partition result is composed of a set of convex polyhedrons and is recursively generated through a series of cutting hyperplanes.

In order to extend the domain of the BSP-Tree process to -dimensional () space while still keeping its self-consistency, we consider a reduced generative process where each cutting hyperplane is only allowed to have two degrees (dimensions) of freedom. In this way, each potential cutting hyperplanes on is assumed to be parallel to the rest dimensions, except the selected two.

Given the current partition and , the next cutting hyperplane is generated in the following steps (see the illustrations for Steps 1–4 in Figure 2):

(1)

Sample a candidate polyhedron (leaf node) from all the existing leaf nodes in proportion to , where denotes an arbitrary pair of dimensions from the dimensions, denotes the projection of onto the dimensions of , and denotes the perimeter of the projection (i.e., a 2-dimensional polygon);

(2)

Sample a pair of free dimensions from all possible pairs in proportion to ;

(3)

On the projection , sample a direction from , where the probability density function is in proportion to the length of the line segment , onto which is projected in the direction of ; and sample the cutting position uniformly on the line segment . The proposed cutting hyperplane is formed as the straight line passing through and crossing through the projection , orthogonal to in the dimensions of and parallel to the rest dimensions1;

(4)

Sample the incremental time for the new cut as . If , reject the proposed cutting hyperplane and return as the final partition structure; otherwise accept the proposed cutting hyperplane, increase to and go back to Step (1).

Sampling a two-dimensional pair (Step (2)) is the novel key step that helps extend the BSP-tree process to -dimensional spaces; all other steps are the natural and logical extensions of the generative process of the two-dimensional BSP-tree process.

Through the above generative process, the cutting hyperplane can be parameterised as , where denotes the index of the selected polygon (leaf node), denotes the -th element of vector , and is a two-dimensional vector denoting the position on the dimensions of . The cutting hyperplane is parallel to all dimensions except and such that it is fully characterised on .

Moreover, the cutting hyperplane is only meaningful (i.e. it intersects with ) if and only if it intersects with the projection on dimensions . As is a convex polygon, from Proposition 2 of [8], the measure of the hyperplane cuts on dimensions is uniform over the perimeter of . Thus, the measure of the hyperplane cuts for is uniform over the sum of the perimeters of on all unique pairs of dimensions i.e. .

When , the above extended BSP-tree process reduces to the original two-dimensional version of [8]. According to Propositions 1 and 2 in [8], the likelihood of the cutting hyperplane on the two dimensions for is . Extending this hyperplane to -dimensions, the likelihood is . That is, all potential partitions are equally favoured and the hyperplane cut is uniformly distributed, without involving additional (prior) knowledge about the cutting hyperplane.

#### Self-consistency of the extended BSP-tree process:

As a result of the particular technique of generating -dimensional cutting hyperplanes, the BSP-tree process retains the self-consistency property in -dimensions.

###### Proposition 1

The extended BSP-tree process is self-consistent in -dimensional () space, and maintains distributional invariance when restricting its domain from a convex polyhedron to a sub-domain.

That is, if the extended BSP-Tree process on a finite convex polyhedron is then restricted to a sub-region , then the resulting partitions restricted to are distributed as if they were directly generated on through the BSP-tree generative process. Subsequent application of the Kolmogorov Extension Theorem [15] states that the self-consistency property of the BSP-tree process then enables it to be defined on infinite multidimensional space.

This useful property can strongly simplify the sampling procedure and improve algorithmic efficiency in -dimensional () implementations, which would otherwise be extremely complicated to implement (see inference sections, below).

## 4 Binary space partitioning (BSP)-forests

We consider the label-predicting regression task outlined earlier, so that for observed data we aim to predict the unknown labels from the -dimensional predictors .

### 4.1 Model

We apply the extended BSP-tree process in a random forest style model. For the regression we attach mean intensity parameters, , to the leaf nodes of each tree structure, and use the sum of the intensities from BSP-trees to predicting the unknown label. That is,

 y=m∑j=1g(x;⊞(j),μ(j))+ϵ,ϵ∼N(0,σ2), (2)

where denotes the set of mean parameters associated with the leaf nodes of the -th BSP-tree, is the number of leaf nodes in the -th BSP-tree, if belongs to the -th leaf node of the -th BSP-tree, and denotes the observation variance.

Determination of node assignments for follows the typical procedure used for a decision tree. To begin, belongs to the root node of the BSP-tree . On any non-leaf node, the cutting hyperplane would then be used to determine the child node that belongs to: or . Through a sequence of such binary node assignments from the top to the bottom of the tree, would finally fall into a single leaf node of , with then allocating the intensity to the regression.

In common with other regression forests, each individual tree only contributes proportionally of the model for label , which prevents any one of the trees dominating the prediction. Further, because the ensemble comprises multiple BST-trees, each potentially selecting different dimensional pairs to describe the data in local regions, the BSP-forest is able to capture all pair-wise dependence in -dimensions. Similarly, the potentially different depths of each tree allows the BSP-forest to reflect different granularities of interactions between dimensions.

Each regression tree in the BSP-forest presents a hierarchical partition structure on -dimensional space. From this perspective, the sum-of-trees forest mechanism provides a compositional “smoothing” of the partition structure for local regions of the space, thereby greatly increasing the flexibility of the regression model. Given , a BSP-forest is determined by , which includes the BSP-tree structures and their associated leaf node mean parameters, and the observational variance parameter .

#### Prior specification for σ2,μ,m:

We specify the prior distributions for , the number of trees in the BSP-forest , and the mean parameters for all trees following [5]. Namely: (1) where, given an estimate of the sample variance , is set to satisfy the condition , where is the Inverse-Gamma c.d.f.; (2) As the labels are typically standardised, the prior for each mean parameter is specified as ; (3) While a prior for can be specified, and inference performed by a reversible-jump type algorithm [29], for simplicity we fix which we find performs well in practice.

#### Comparison with BART and MF:

The BSP-forest is directly related to two popular Bayesian regression-tree models: Bayesian additive regression trees (BART) and the Mondrian forest (MF). The key differences and advantage here is that the BSP-forest uses both angled cuts, and hyperplane cuts constructed from two dimensions, rather than the one-dimensional, axis-aligned cuts in both BART and MF. This clearly provides a more flexible way to model the observed data, and as a result, lower tree depth is required to obtain the same or even improved modelling capability.

### 4.2 Inference

Posterior inference for the BSP-forest is implemented using Markov chain Monte Carlo (MCMC). The joint distribution of all parameters, including the partition structures of extended BSP-trees , the leaf-node mean parameters of each tree , and the observational variance can be written as:

 p({⊞(j),μ(j)}j,σ2,Y|X,λ)=p(Y|{⊞(j),μ(j)}j,X,σ2) ⋅m∏j=1[p(⊞(j)|λ)p(μ(j))]p(σ2),

where and . Algorithm 1 outlines the MCMC algorithm, which iteratively samples the variance  (Line 3) and each BSP-tree structure with mean parameters ) (Line 5).

#### Sampling σ2:

Through conjugacy, the full posterior conditional distribution of is

 σ2∼IGamma(3+N2,^λ+Ess2) (3)

where .

#### Sampling {⊞(j),μ(j)}mj=1

We use the Conditional-Sequential Monte Carlo (C-SMC) sampler [2] to infer the partition structures of all the extended BSP-trees and the mean parameters . There is a key difference between our sampler and that used in [16, 8]: These previous works take node splitting events on the time line as the dimensions of the sampled variable. The particles at the same step are under different budget constraints and the target distribution of the C-SMC samplers may not be well mixed. In our implementation, we adopt fixed intervals () on the time line as the dimensions of the variable sampled by the C-SMC sampler. step , all the particles are under the same budget constraints and each dimension might involve one, more than one, or even no splitting events. This helps to fix the number of “dimensions” in C-SMC and improve the efficiency of the sampler. Algorithm 2 shows the detail strategy to infer the tree structure in the -th iteration for the -th BSP-tree.

#### Implementing cutting hyperplane on convex hull:

Cutting the full -dimensional polyhedron generated from a series of cutting hyperplanes on the original domain, is both a challenging and unnecessary task. This is because: (1) complete indexing of this polyhedron requires extensive geometric calculations, including calculating the intersection of multiple hyperplanes, deciding if two hyperplanes intersect in the existence of other hyperplanes, and listing all the vertices of the polyhedron; (2) in some cases, the cutting hyperplane occurs outside the convex hull (i.e. the minimum convex polyhedron that contains all the points in this block), it does not partition the available data and thereby influence the regression. In practice, while data usually lies in a small portion of the feature space, implementing cutting hyperplanes directly on this polyhedron is particularly inefficient.

To address these issues, we implement the hyperplane cut on the convex hull only. The projection of the convex hull on any two dimensions can be obtained by simply slicing the coordinates on these two dimensions and then using a conventional convex hull detection algorithm (such as the Graham Scan algorithm [12]. Algorithm 3 describes our approach to generate the hyperplane cut in the convex hull and Fig. 3 visualises the convex hull projection in -dimensional space.

Due to the self-consistency property of the (extended) BSP-tree process, we conveniently have that:

###### Proposition 2

The hyperplane restricted on the convex hull is distributed the same as if we first partition on the “full” polyhedron and then restrict attention to the convex hull. Both of these two methods lead to the idential equilibrium distribution in the MCMC algorithms.

#### Computational cost

The computational cost of the BSP-Forest is almost the same as the batched Mondrian Forest, except that the number of candidate features (used to generate a hyperplane) increases from to .

## 5 Related Work

Random forests have been proposed for decades and the volume of literature may be too huge to be reviewed here. The interested readers can refer to [6] for a recent review. For the structure of a tree in the forest, the node splitting in a binary tree usually consists of two steps: (1) choose the most suitable dimension to cut; (2) find the best cutting position on the chosen dimension. A widely adopted strategy is to optimize some information gain related criterion in a greedy way. In terms of the decision tree, a random forest uses the sum-of-trees strategy to further improve the model capability. Some additional strategies to decorrelate the trees include bagging and subsampling the data set.

Here we use the Breimain-Random Forest [3] and Gradient Boosting regressor [10] as two representative works. The Breimain-RF adopts the bagging strategy and chooses a subset of the dimensions for each tree. The Gradient Boosting regressor uses the boosting strategy, which is to propose regression trees to cater for the negative gradient of the loss function at the current step.

For the Bayesian decision trees, Bayesian Additive Regression Trees (BART) [5] and the Mondrian Forest [16] are two representative counterparts of the proposed BSP-Forest. BART assigns probability distributions to the structure variables of all the trees, while MF uses the Mondrian process as the prior of the tree structure for the forest. In contrast to these two methods which merely select one dimension to conduct node cut to generate the tree structure, the BSP-Forest adopts two dimensions together to form a hyperplane cut in the feature space.

Multivariate decision trees [4] (single-tree model) may be the closest one to the proposed BSP-tree in term of the inter-dimensional dependence. Its less popularity might be due to its high computational cost in finding an optimized cutting hyperplane. Compared to our BSP-tree, the lack of prior information in the multivariate decision tree requires more regularization techniques.

## 6 Experiments

### 6.1 Toydata: Friedman’s Function

We first evaluate the performance of the BSP-Forest on the Friedman’s function [10, 5, 20]. In this test setting, each data point is generated from the uniform distribution, while its label takes the following form:

 y=10sin(πx1x2)+20(x3−12)2+10x4+5x5+ϵ

where denotes the white noise effect and denotes the -th dimension of . The Friedman’s function consists of two nonlinear terms, two linear terms and an interaction term between the dimensions.

#### Posterior Mean Estimation

We begin with a simple application on posterior mean estimates of the BSP-Forest. The number of dimensions is set to , such that considers only the first dimensions of the data points and the rest five dimensions are irrelevant; the number of data points is set to . The results in Figure 4 show the estimated against the groundtruth on the training and test label sets. There is a vertical line on each point to indicate the posterior confidence interval. We can see that, on the training data set, the predicted values correlate very well with the true values , and the confidence intervals tend to cover the true values as well. On the test data set, one can observe a slight degradation of the correlation and wider intervals, which indicate a larger variance estimation of at new values.

#### Partial Dependence

Partial dependence is commonly used in Statistics to show the effect of adding features to a model. Figure 5 shows the plots of points and interval estimates of the partial dependence functions for from the MCMC samplers. The nonzero marginal effects of and the zero marginal effects of seem to be completely consistent with the form of .

#### Dimension Usage

The BSP-Forest can also be used to identify the dependent dimensions that are most correlated to the function . This can be completed by recording the pair of dimensions used for each regression tree and count the times of involvements of these dimensions. Figure 6 plots the proportions of for the number of trees . As gets smaller, the fitted sum-of-trees models increasingly incorporate only those dimensions that are critical to explain the label . Without making use of any assumptions or information about the actual function, the BSP-Forest has exactly identified the underlying dimensions that determine .

#### Performance Evaluation w.r.t. Budget

For the Mondrian Forest and the BSP-Forest, the parameter determines the expected depth of the BSP-tree, the budget , has more impact on their performance. Figures 7 and 8 show the Root Mean Absolute Error (RMAE) and the average number of cuts for the BSP-Forest and the Mondrian Forest, while budget is taking different values from to . We can see that the BSP-Forest can obtain better RMAE performance on all budget values. Also, the number of cuts in the BSP-Forest is always smaller than that in the Mondrian Forest. The efficiency in the number of cuts is consistent with our expectation on the BSP-Forest. Although the curve seems quite flat across values of considered, the reason may be that the average number of cuts does not change much: it only changes from 4 to 5 in Figure 8, as the budget varies from 0.4 to 1.4.

### 6.2 Real-world Datasets

The performance of the BSP-Forest is compared with other tree-based models on a number of real-world data sets. The comparison methods we include Bremain’s Random Forest [3], Gradient Boosting regressor [11], Bayesian Additive Regression Trees (BART) [5], batch version of the Mondrian Forest (MF) [19]. The implementations of Bremain’s Random Forest and Gradient Boosting regressor are imported from the Python module of Scikit-Learn [23]. BART is implemented through the R package of BART [25]. We implement the Mondrian Forest through the same strategy as the BSP-Forest here.

For a fair comparison, we set the number of trees in all the comparison methods as . Except for this, the other parameters of Bremain’s RF, Gradient Boosting regressor and BART are set as default in the modules. The budget of MF and the BSP-Forest is set to . In addition, we add scaling parameters to the distribution of the cost variable such that the expectation of the cost at stage is for both MF and BSP-Forest.

Five real-world datasets [18] are used for the performance evaluation (see Table 2). We normalize the data into . Each data set is randomly divided into five equal parts where out of are used for training while the rest for testing. We report the Root Mean Absolute Error (RMAE) as a robust measure over random runs on the datasets in Table 1. It is easily to see that our BSP-Forest performs the best among all the compared methods.

## 7 Conclusion

This paper makes the first endeavor to extend the BSP-Tree process to a -dimensional () space. By designing a subtle strategy to only sample two free dimensions each time from the space, the extended BSP-Tree process can retain the essential self-consistency property. We further take the extended BSP-trees to construct the BSP-Forest for regression. Compared to other (Bayesian) regression forests, the BSP-Forest can perform best due to its flexibility.

## 8 Acknowledgements

Xuhui Fan and Scott A. Sisson are supported by the Australian Research Council through the Australian Centre of Excellence in Mathematical and Statistical Frontiers (ACEMS, CE140100049), and Scott A. Sisson through the Discovery Project Scheme (DP160102544). Bin Li is supported by Fudan University Startup Research Grant and Shanghai Municipal Science & Technology Commission (16JC1420401).

### Footnotes

1. Generating a cutting line parameterized by and on the projection (two-dimensional polygon) follows the same sampling method used in the BSP-Tree process [8].

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Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01gm80hx773
 Title: New Sampling Strategies for Bayesian Optimization of Functions with Location-Dependent Sample Noise Authors: McGrory, Conor Advisors: Pillow, Jonathan Department: Computer Science Class Year: 2016 Abstract: The problem of finding the global extremum of noisy, black-box functions has applications in many different fields, and it has been studied extensively in the literature. Bayesian optimization is a technique for solving this problem where a set of samples of the objective function, collected by the optimizer, is used to infer a probability distribution over possible values of the objective function, which is used to estimate its global maximum or minimum. One essential component of the Bayesian optimization approach is the strategy for collecting samples, which sequentially picks new sample points based on the current distribution over the function’s possible values. Although there are many cases where the noise associated with a sample value depends greatly on the location of the sample point, no sampling strategies have been developed to take this information into account if it is known to the optimizer. In this paper, we present two sampling strategies for Bayesian optimization in the case of location-dependent sample noise, and show that they outperform two of the most commonly used sampling strategies on randomly-generated objective functions. Extent: 39 pages URI: http://arks.princeton.edu/ark:/88435/dsp01gm80hx773 Type of Material: Princeton University Senior Theses Language: en_US Appears in Collections: Computer Science, 1988-2017

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# Definition:Bounded Linear Transformation

## Definition

Let $H, K$ be Hilbert spaces.

Let $A: H \to K$ be a linear transformation.

Then $A$ is a bounded linear transformation if and only if

$\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_K \le c \left\Vert{h}\right\Vert_H$

## Bounded Linear Operator

Let $H$ be a Hilbert space.

Let $A: H \to H$ be a linear operator.

$A$ is a bounded linear operator if and only if

$\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_H \le c \left\Vert{h}\right\Vert_H$

That is, a bounded linear operator is a bounded linear transformation from a Hilbert space to itself.


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The third law of thermodynamics states that: “The entropy of all the perfect crystalline solids is zeros at absolute zero temperature”. A more fundamental statement was later labelled the 'zeroth law'. The first law asserts that if heat is recognized as a form of energy, then the total energy of a system plus its surroundings is conserved; in other words, the total energy of the universe remains constant.. It has had great influence on thermodynamics. 4.3 The third law of thermodynamics. This is the lowest point on the Kelvin scale. Traditionally, thermodynamics has stated three fundamental laws: the first law, the second law, and the third law. The "theorem" is given as a restatement of the consequences of the zeroth, first, second, and third laws of thermodynamics, with regard to the usable energy of a closed system: 0. Called thermal equilibrium, this state of the universe is unchanging, but at a temperature higher than absolute zero. This allows us to define a zero point for the thermal energy of a body. Amy Dusto is a high school science teacher and a freelance writer. To become perfectly still, molecules must also be in their most stable, ordered crystalline arrangement, which is why absolute zero is also associated with perfect crystals. This makes sense because the third law suggests a limit to the entropy value for different systems, which they approach as the temperature drops. (consequence of first law of thermodynamics) 2. We can extrapolate from experimental data that the entropy of a perfect crystal reaches zero at absolute zero, but we can never demonstrate this empirically. Keywords: Nernst postulate, thermodynamics, entropy, quantum laws This scale is built on a particular physical basis: Absolute zero Kelvin is the temperature at which all molecular motion ceases. The system and surroundings are separated by a boundary. Evaluation of absolute entropy is not the only consequence of the third law. Most importantly, the third law describes an important truth of nature: Any substance at a temperature greater than absolute zero (thus, any known substance) must have a positive amount of entropy. The Third Law of Thermodynamics. The Third Law of Thermodynamics is the lesser known of the three major thermodynamic laws. This allows an absolute scale for entropy to be established that, from a statistical point of view, determines the … The consequence of the Third Law of thermodynamics is that “it would require an infinite number of steps to reach absolute zero, which is not possible but if there is some possibility to reach absolute zero, it would violate the Second Law of thermodynamics, because if we had a heat sink at absolute zero, then we could build a machine that would be 100 percent efficient.”. These consequences are summed up in the Third Law of Thermodynamics. 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The third law of thermodynamics also refers to a state known as “absolute zero”. In other words, in any isolated system (including the universe), entropy change is always zero or positive. The entropy of a perfectly ordered crystal at 0 K is zero. To get to absolute zero it requires an infinite number of steps. Of all the laws of thermodynamics or all the thermodynamic laws, we tend to be more familiar with the First Law Of Thermodynamics and the second law of thermodynamics more than the third law of thermodynamics. This also suggests that absolute molar entropies can be calculated by $S = \int_o^{T} \dfrac{C}{T} dT$ where $$C$$ is the heat capacity. P.J. This is the Nernst–Planck–Simon statement of the Third Law of Thermodynamics. as we know G = H - TS & F = U - TS so as S tends to zero G = H & F = U as T tends to zero Physically , it means that there is perfect order and all energy is available for work. behavior of Thermodynamic Potentials follows the consequences . This was true in the last example, where the system was the entire universe. Substances with similar molecular structures have similar entropies. In its simplest form, the third law of thermodynamics reads like this, the entropy or randomness of the matter is related to its absolute temperature. The third law of thermodynamics. At that point, the universe will have reached thermal equilibrium, with all energy in the form of thermal energy at the same nonzero temperature. Those values make sense only relative to other values. This principle states that the entropy of a system at the temperature of absolute zero is a well-defined constant. This is often referred to as the heat death of the universe. These laws defined are of paramount importance in the field of science and research; everything in this universe abides by them. Scientists everywhere, however, use Kelvins as their fundamental unit of absolute temperature measurement. Putting together the second and third laws of thermodynamics leads to the conclusion that eventually, as all energy in the universe changes into heat, it will reach a constant temperature. The laws of thermodynamics are absolute physical laws - everything in the observable universe is subject to them. Consequences from the third law of thermodynamics are analyzed from the standpoint of low-temperature phase equilibria. Because a temperature of absolute zero is physically unattainable, the Third Law may be restated to apply to the real world as: the entropy of a perfect crystal approaches zero as its temperature approaches absolute zero. Most entropy calculations deal with entropy differences between systems or states of systems. So 0 Kelvin becomes the lowest temperature in … The laws of thermodynamics are deceptively simple to state, but they are far-reaching in their consequences. 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# Consider the following data set below. Find the UPPER INTERVAL of the population mean at 80\% confidence interval. Given: 12,10,9,11

Consider the following data set below. Find the UPPER INTERVAL of the population mean at $80\mathrm{%}$ confidence interval. (write your answer in two decimal places) Given: 12,10,9,11
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It is given that,
12,10,9,11
Thus,
Sample mean, $\stackrel{―}{x}=\frac{\sum _{t-1}^{n}{X}_{t}}{n}=10.5$
Sample standard deviation, $s=\sqrt{\frac{\sum _{t-1}^{n}{\left({X}_{t}-\stackrel{―}{X}\right)}^{2}}{n-1}}=1.29099$
Since, Population standard deviation is unknown thus t distribution is used.
Sample size, $n=4$
Degree of freedom $=n-1=4-1=3$
The value of ${t}_{\frac{\alpha }{2},n-1}$ at $80\mathrm{%}$ confidence interval is $=1.638$ [From t table]
Step 2
Now, the $80\mathrm{%}$ confidence interval for the population mean can be calculated as:
$CI=\stackrel{―}{x}±{t}_{\frac{\alpha }{2},n-1}×\frac{s}{\sqrt{n}}$
$=10.5±1.638×\frac{1.29099}{\sqrt{4}}$
$=\left(9.44,11.56\right)$
Upper limit $=11.56$