prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How are smartphones and tablets different from a technical perspective?

Read the following text and answer briefly.

Context 1: The future of mobile CPUs, part 1: Today’s fork in the road | Ars Technica
2013 may be a big year for the evolution of smartphones and tablets 
Mobile computing's rise from niche market to the mainstream is among the most significant technological trends in our lifetimes  And to a large extent, it's been driven by the bounty of Moore’s Law—the rule that transistor density doubles every 24 months  Initially, most mobile devices relied on highly specialized hardware to meet stringent power and size budgets  But with so many transistors available, devices inevitably grew general-purpose capabilities  Most likely, that wasn't even the real motivation  The initial desire was probably to reduce costs by creating a more flexible software ecosystem with better re-use and faster time to market  As such, the first smartphones were very much a novelty, and it took many years before the world realized the potential of such devices  Apple played a major role by creating innovative smartphones that consumers craved and quickly adopted 
To some extent, this is where we still stand today  Smartphones are still (relatively) expensive and primarily interesting to the developed world  But over the next 10 years, this too will change  As Moore’s Law rolls on, the cost of a low-end smartphone will decline  At some point, the incremental cost will be quite minimal and many feature phones of today will be supplanted by smartphones  A $650 unsubsidized phone is well beyond the reach of most of the world compared to a $20 feature phone, but a $30 to $40 smartphone would naturally be very popular 
In this grand progression, 2013 will certainly be a significant milestone for mobile devices, smartphones and beyond  It's likely to be the first year in which tablets out-ship notebooks in the US  And in the coming years, this will lead to a confluence of high-end tablets and ultra-mobile notebooks as the world figures out how these devices co-exist, blend, hybridize, and/or merge 
Against this backdrop, in this two-part series, we'll explore the major trends and evolution for mobile SoCs  More importantly, we'll look to where the major vendors are likely going in the next several years 
Tablet and phone divergence
While phones and tablets are mobile devices that often share a great deal of software, it's becoming increasingly clear the two are very different products  These two markets have started to diverge and will continue doing so over time 
From a technical perspective, smartphones are far more compact and power constrained  Smartphone SoCs are limited to around 1W, both by batteries and by thermal dissipation  The raison d’etre of a smartphone is connectivity, so a cellular modem is an absolute necessity  For the cost sensitive-models that make up the vast majority of the market, the modem is integrated into the SoC itself  High-end designs favor discrete modems with a greater power budget instead  The main smartphone OSes today are iOS and Android, though Windows is beginning to make an appearance (perhaps with Linux or BlackBerry on the horizon)  Just as importantly, phone vendors like HTC must pass government certification and win the approval of carriers  There is very much a walled-garden aspect, where carriers control which devices can be attached to their networks, and in some cases devices can only be sold through a certain carrier  The business model places consumers quite far removed from the actual hardware 
In contrast, tablets are far more akin to the PC both technically and economically  The power budget for tablet SoCs is much greater, up to 4W for a passively cooled device and as high as 7-8W for systems with fans  This alone means there is a much wider range of tablet designs than smartphones  Moreover, the default connectivity for tablets is Wi-Fi rather than a cellular modem  The vast majority of tablets do not have cellular modems, and even fewer customers actually purchase a wireless data plan  As a result, cellular modems are almost always optional discrete components of the platform  The software ecosystem is relatively similar, with Microsoft, Apple, and Google OSes available  Because tablets eschew cellular modems, the time to market is faster, and they are much more commonly sold directly to consumers rather than through carriers  In terms of usage models, tablets are much more PC-like, with reasonable-sized screens that make games and media more attractive 
Looking forward, these distinctions will likely become more pronounced  Many tablets today use high-end smartphone SoCs, but the difference in power targets and expected performance is quite large  As the markets grow in volume, SoCs will inevitably bifurcate to focus on one market or the other  Even today, Apple is doing so, with the A6 for phones and the larger A6X for tablets  Other vendors may need to wait a few years to have the requisite volume, but eventually the two markets will be clearly separate 
Horizontal business model evolution
Another aspect of the mobile device market that is currently in flux and likely to change in the coming years is the business model for the chip and system vendors  Currently, Apple is the only company truly pursuing a vertically integrated model, where all phones and tablets are based on Apple’s own SoC designs and iOS  The tight integration between hardware and software has been a huge boon for Apple, and it has yielded superb products 
Samsung is one of the few others companies that takes a vertically integrated approach to phones and tablets, although in truth its strategy seems to be ambivalent on that point  Unlike Apple, Samsung’s SoCs are readily available to third parties, and some Samsung devices, such as the S7562 Galaxy S Duos, use SoCs from competitors  More recently though, there has been a trend of Samsung devices using Samsung SoCs, at least for the premier products  For the moment, Samsung’s approach is best characterized as a hybrid, particularly as the company lacks a bespoke OS 
The rest of the major SoC vendors (e g , Intel, Qualcomm, Nvidia, TI, Mediatek, etc ) have stayed pretty far away from actual mobile devices  These companies tend to focus on horizontal business models that avoid competing with customers or suppliers 
In the long term, mobile devices are likely to evolve similarly to the PC and favor a horizontal business model  The real advantage is one of flexibility; as costs drop and the market expands, it will be increasingly necessary for vendors like HTC to offer a wide range of phones based on radically different SoCs  While a vertically integrated company like Apple can focus and maintain leadership in a specific (and highly lucrative) niche, it would be very difficult to expand in many growing areas of the market  The differences between an iPhone 6 and a $20 feature phone are tremendous and would be very difficult for a single company to bridge 
However, SoC vendors will attempt to reap the benefits of vertical integration by providing complete reference platforms to OEMs  Conceptually, this is a form of "optional" system integration, where the phone vendor or carrier can get the entire platform from the SoC supplier  This has the principal advantages of reducing time to market while also providing a baseline quality and experience for consumers  Currently, this approach has mostly been tested in emerging markets, but it's likely to become more common over time  There is a crucial distinction between reference platforms and vertical integration  Namely, OEMs can always choose to customize a platform to differentiate, and the SoC vendor avoids dealing with consumers directly  Typically, most of the customization is in terms of software on top of a base operating system 
Quote:Moreover, that will make the transition to a 10nm node even more difficult, as the foundries will have to move from 20nm interconnects to 10nm interconnects and skip a generation The advances in technology lately allowing components on such a small scale to even be envisioned, much less planned for, are truly amazing 
Off topic: show
I present the first generation 'non-technical' rock:
I don't think your horizontal market development theory is supported by facts  Samsung and Apple are more vertically oriented than their competition, for starters  I know this article is narrowly focused on the hardware, but MS and Intel getting into hardware, Amazon getting into hardware, Google buying Moto, this is all vertical integration  How can you support the idea that this trend will be reversed with no real justification? I'm sure mobile chips will continue to specialize, but I don't think this means what you think it means  Automobile companies started making their own engines and with rare exceptions, never went back to being more horizontal  Same with retail and their store brands  Same with cloud companies and their servers  Same with mobile companies and their OSs  The horizontal market of PCs created by long-lasting standards and loose hegemony is the exception, not the norm 
Why wouldn't the foundries be able close the process gap with Intel? Is it a matter of money? Scale?
I'm not so sure about several things:1- Moore's law's relevance  Moore's Law is about ICs  ICs are not as big a part of mobile computers as they are of desktops, even of laptops: screens, batteries, radios are a huge part of tablets' and phones' costs, as opposed to the bare SoC + RAM 2- The tablet vs phone dichotomy  For some reason (probably price insensitivity due to subsidies), Phones have a tendency to be more powerful than Tablets, ie phone SoCs are more than good enough for tablets  Since the OS and peripherals are the same, it makes more sense to design and build just one type of SoC, and just disable the phone-modem part of it (even the other radios are still required: BT, Wifi, GPS   ), same as Intel disable cache and cores for their entry-level CPUs  Once you're fabbing a SoC, it makes more sense to make more of the same than to setup a separate run of a cut-down SoC on an older process, unless volumes are huge  We might still be getting previous-generation, well amortized SoCs in cheaper tablets, though 3- On the contrary, I see a tablet and phone convergence (the ugly phablet)  I'm patiently waiting for the new 6"+ phones to replace my Nook Color and Galaxy Note 1 with a single device 4- The advantage of diversity ? Software is becoming ever more important than hardware  Multiplying SoCs means multiplying product development costs, making support and updates more difficult    Again, unless volumes are huge, OEMs are probaly better off going the way of the car industry and using modular "platforms" housed in different chassis with various screen sizes, keyboards, radios, digitizers   I'm wondering why the "single device" trend does not figure in your analysis  Is it stillborn ? Does it have no impact nor dependency on/with SoCs ?
Samsung has its own bespoke OS: Bada and it is used on an extensive line of devices  I think there are numbers somewhere that it outsold Windows Phone 7 for a time 
gypsumfantastic wrote:Why wouldn't the foundries be able close the process gap with Intel? Is it a matter of money? Scale?First mover advantage 
SoC? System on a Chip I guess?
You're way off on the Moore's Law/cost of smartphones point  The processors used in today's high-end smartphones are already cheap, around $25  And there are less expensive options if you want a lower end product  In fact, the hardware in the whole smartphone is relatively cheap  Analyst's estimate the Z10's materials cost around $160, the iPhone 5 around $140  They're using expensive glass and metals, then there's the battery, memory, etc  which means the processor is a small factor of the cost And then there's the jump from $140 in materials to the unsubsidized costs  The reason these phones cost $650 is because of the high margins these companies are able to get and the high cost of hardware design and/or software development  But the point is that making the processors 4 times better/cheaper isn't going to change the economics of the smartphone  What will change the economics is commoditized designs and software and cheaper materials all around  Then you'll have a $50 smartphone that's decent 
Last edited by ggeezz on Wed Feb 13, 2013 9:17 am
bigterp wrote:SoC? System on a Chip I guess?Yup 
gypsumfantastic wrote:Why wouldn't the foundries be able close the process gap with Intel? Is it a matter of money? Scale?Money and momentum, the x86 market is a huge money maker for Intel so it is able to recoup its huge investments for advanced foundries 
Quote:Currently, the only products using 3D integration are FPGAs from Xilinx,Doesn't Sony use it in the PS Vita? I thought I read somewhere that they had the CPU, main memory (2 dies) and video memory, so 4 dies in total, sitting on top of each other all on the same chip 
renoX wrote:gypsumfantastic wrote:Why wouldn't the foundries be able close the process gap with Intel? Is it a matter of money? Scale?Money and momentum, the x86 market is a huge money maker for Intel so it is able to recoup its huge investments for advanced foundries Exactly and I would clarify that it's all about margins, the difference between what it costs to make a chip and what it sells for  The margins for desktop and server processors is huge because a) the whole product is expensive so $200 to $1000 for the chip is acceptable, and b) Intel has huge advantages in that space and little competition So Intel can afford to do the R&D to stay ahead of the curve and keep their position  When your smartphone chip sells for $25 you can't do the R&D to leapfrog a company that sells Xeons for $1000 and Core i5's for $200 
I am happy to see Kanter here at Ars, I like his writing and he maintains Real World Tech, where Linus Torvalds often shows up to comment on CPU arch and other interesting topics 
ggeezz wrote:renoX wrote:gypsumfantastic wrote:Why wouldn't the foundries be able close the process gap with Intel? Is it a matter of money? Scale?Money and momentum, the x86 market is a huge money maker for Intel so it is able to recoup its huge investments for advanced foundries Exactly and I would clarify that it's all about margins, the difference between what it costs to make a chip and what it sells for  The margins for desktop and server processors is huge because a) the whole product is expensive so $200 to $1000 for the chip is acceptable, and b) Intel has huge advantages in that space and little competition So Intel can afford to do the R&D to stay ahead of the curve and keep their position  When your smartphone chip sells for $25 you can't do the R&D to leapfrog a company that sells Xeons for $1000 and Core i5's for $200 Spot on Intel are able to piggyback other development efforts off the highly lucrative mainstream x86 market which generates the huge sums of money to fund their amazing fab technology The question for the future is how the economics will stack up when overall device costs fall significantly and there is a big downward pressure on SoC prices  In that situation, can Intel still justify bringing their A-game to a market where products are essentially commoditised and you have processors selling for a only a few dollars each?The lesson from their previous involvement in the DRAM market is that they probably won't want to be involved because there isn't enough money to be made to justify manufacturing phone SoCs on a cutting edge, or near cutting edge process  In that scenario, Intel may not totally abandon the market but they might just stick to manufacturing SoCs on nodes that are a step or two behind the state of the art 
solomonrex wrote:I don't think your horizontal market development theory is supported by facts  Samsung and Apple are more vertically oriented than their competition, for starters  I know this article is narrowly focused on the hardware, but MS and Intel getting into hardware, Amazon getting into hardware, Google buying Moto, this is all vertical integration  How can you support the idea that this trend will be reversed with no real justification? I'm sure mobile chips will continue to specialize, but I don't think this means what you think it means  Automobile companies started making their own engines and with rare exceptions, never went back to being more horizontal  Same with retail and their store brands  Same with cloud companies and their servers  Same with mobile companies and their OSs  The horizontal market of PCs created by long-lasting standards and loose hegemony is the exception, not the norm Yea, each year Amazon, MS, Apple and Google look more and more the same 
Intel cannot abandon the phone/tablet market  Desktop/laptop sales are stagnating/decreasing and phones/tablets are on the rise  This trend is only going to increase going forward But you're right, they're going to have use their fabs that are a step or two behind the cutting the edge  But they're going to have to up their game in the tablet space to even be able to do that 
gypsumfantastic wrote:Why wouldn't the foundries be able close the process gap with Intel? Is it a matter of money? Scale?Intel's called Chipzilla for a reason up
Lagrange wrote:The question for the future is how the economics will stack up when overall device costs fall significantly and there is a big downward pressure on SoC prices  In that situation, can Intel still justify bringing their A-game to a market where products are essentially commoditised and you have processors selling for a only a few dollars each?The lesson from their previous involvement in the DRAM market is that they probably won't want to be involved because there isn't enough money to be made to justify manufacturing phone SoCs on a cutting edge, or near cutting edge process  In that scenario, Intel may not totally abandon the market but they might just stick to manufacturing SoCs on nodes that are a step or two behind the state of the art I think the processing is a bigger advantage than many realize  If Intel can stay ahead in process design - which this article seems to indicate - they should have a major advantage  All else being equal a 14nm chip should be significantly faster and more efficient than the same chip at 22nm  Add in the fact that yields increase geometrically - you can fit a lot more 14nm chips on a given wafer size vs 22nm (or 32nm for the other manufacturers ) and you have a very appealing proposition  And then add in the fact that Intel actually has a pretty good graphics stack and IP  It's not a sure thing by any means, but I suspect ARM may have just prodded a sleeping giant edit: Also worth noting, Intel, TSMC, and Samsung are the only manufacturers who are building out 450nm wafers  This will increase yields dramatically  Of course Samsung and TSMC will build ARM out, but it definitely puts quite a bit of pressure on all other manufacturers  As the article mentions Intel and Samsung are the only ones who control production top to bottom, and Samsung must share some of the benefits with ARM 
As someone who has worked in the semiconductor industry for longer than contemporary fanboys have existed, I'm getting a bit fed-up seeing many of these articles which rewrite history with analyses distorted by the lens of the contemporary Apple or Samsung fanboy The mobile industry moved to horizontal integration a long time ago  Better indicators than these odd contemporary obsessions with the relatively non-innovative Samsung and Apple where when Motorola spun out its semiconductor division as Freescale, Nokia stopped making it's own custom designs with TI and ST, and Ericsson spun out it's Ericsson Mobile Platforms division and formed ST-Ericsson with ST The true innovation in the mobile market was done a decade or more ago mainly by Moto/Nokia/Ericsson/TI/Qualcomm, and Samsung and Apple had little to do with it  Under the hood most stuff has been on a simple linear evolutionary path by the SoC providers since then  The phone manufacturers have then mainly been simply sticking these off-the-shelf SoCs (and their included low-level software stacks) in a box, a job made all the easier with the SoC manufacturer collaboration providing the bulk of the work for the AOSP 
Last edited by paul5ra on Wed Feb 13, 2013 11:06 am
introiboad wrote:I am happy to see Kanter here at Ars, I like his writing and he maintains Real World Tech, where Linus Torvalds often shows up to comment on CPU arch and other interesting topics Indeed  Most tech writing in this area is atrocious  This piece is one of the few well informed articles I've read in a long time 
ggeezz wrote:Intel cannot abandon the phone/tablet market  Desktop/laptop sales are stagnating/decreasing and phones/tablets are on the rise  This trend is only going to increase going forward But you're right, they're going to have use their fabs that are a step or two behind the cutting the edge  But they're going to have to up their game in the tablet space to even be able to do that The word you're looking for is Haswell, as far as I know 
Mabsark
ggeezz wrote:Intel cannot abandon the phone/tablet market  Desktop/laptop sales are stagnating/decreasing and phones/tablets are on the rise  This trend is only going to increase going forward But you're right, they're going to have use their fabs that are a step or two behind the cutting the edge  But they're going to have to up their game in the tablet space to even be able to do that Actually, that trend will not simply keep increasing going forward  The reason desktop/laptop sales are stagnating/decreasing is due to the fact that most people already have one and therefore don't need to buy another one  The exact same thing will happen with tablets as well  Sales are increasing now because people without tablets are buying them  When most people already own a tablet, they won't be buying a new one every year and therefore sales will stagnate/decrease The PC market is saturated and in a couple of years, the tablet market will be saturated too  Basically, in order to increase sales in a saturated market, you need to increase the population growth or decrease the longevity of the product 
gypsumfantastic wrote:Why wouldn't the foundries be able close the process gap with Intel? Is it a matter of money? Scale?Probably a mix of a lot of things  One big thing was during this recession, Intel was the ONLY fab company that didn't scale back their R&D  That alone gave Intel a large advantage Intel has almost always been ahead  One of the reasons could be that Intel works with much higher margins than many of the commodity companies like Samsung and TSMC Outside of the P4 flop and some of the monopolistic abuses, Intel has typically been selling to high end customers that are willing to pay a premium for "the best" Intel has a large benefit of having a relatively "good name" when it comes to CPUs, so they can effectively charge a brand-name premium I'm sure there are other reasons, and probably better reasons, but these are the main ones that I think of 
Mabsark wrote:Actually, that trend will not simply keep increasing going forward  The reason desktop/laptop sales are stagnating/decreasing is due to the fact that most people already have one and therefore don't need to buy another one  The exact same thing will happen with tablets as well  Sales are increasing now because people without tablets are buying them  When most people already own a tablet, they won't be buying a new one every year and therefore sales will stagnate/decrease The PC market is saturated and in a couple of years, the tablet market will be saturated too  Basically, in order to increase sales in a saturated market, you need to increase the population growth or decrease the longevity of the product That's true as long as most people are still buying both a tablet and a laptop when each needs to be replaced  I think the assumption is that, as you say, the tablet market will saturate, with people just replacing existing ones, but the desktop/laptop market could decrease much farther than that, if most people stop replacing them at all  I'm not sure of the likelihood of that, but I think that's where this idea comes from 
ggeezz wrote:Intel cannot abandon the phone/tablet market  Desktop/laptop sales are stagnating/decreasing and phones/tablets are on the rise  This trend is only going to increase going forward But you're right, they're going to have use their fabs that are a step or two behind the cutting the edge  But they're going to have to up their game in the tablet space to even be able to do that The upcoming Haswell chip is showing to consume 1/3 the power of IvyBridge at peak, consumes 1/20th the power at idle, all the while maintaining Identical or better performance This chip should actually compete with ARM CPUs on both power/performance and idle I am expecting a large war 
Apple once again is dictating the performance in the mobile industry  Nice to see others being able to keep the pace, as well 
paul5ra wrote:As someone who has worked in the semiconductor industry for longer than contemporary fanboys have existed, I'm getting a bit fed-up seeing many of these articles which rewrite history with analyses distorted by the lens of the contemporary Apple or Samsung fanboy The mobile industry moved to horizontal integration a long time ago  Better indicators than these odd contemporary obsessions with the relatively non-innovative Samsung and Apple where when Motorola spun out its semiconductor division as Freescale, Nokia stopped making it's own custom designs with TI and ST, and Ericsson spun out it's Ericsson Mobile Platforms division and formed ST-Ericsson with ST The true innovation in the mobile market was done a decade or more ago mainly by Moto/Nokia/Ericsson/TI/Qualcomm, and Samsung and Apple had little to do with it  Under the hood most stuff has been on a simple evolutionary path by the SoC providers since then Yeah, and most of the innovation in the automobile industry came about before Henry Ford came into the business  Doesn't change the fact that cars would probably have been an asterisk in the history books under "toys for rich people" if it weren't for him The same applies to to mobile computing for Apple, Samsung, et al 
SheldonRoss wrote:Lagrange wrote:The question for the future is how the economics will stack up when overall device costs fall significantly and there is a big downward pressure on SoC prices  In that situation, can Intel still justify bringing their A-game to a market where products are essentially commoditised and you have processors selling for a only a few dollars each?The lesson from their previous involvement in the DRAM market is that they probably won't want to be involved because there isn't enough money to be made to justify manufacturing phone SoCs on a cutting edge, or near cutting edge process  In that scenario, Intel may not totally abandon the market but they might just stick to manufacturing SoCs on nodes that are a step or two behind the state of the art I think the processing is a bigger advantage than many realize  If Intel can stay ahead in process design - which this article seems to indicate - they should have a major advantage  All else being equal a 14nm chip should be significantly faster and more efficient than the same chip at 22nm  Add in the fact that yields increase geometrically - you can fit a lot more 14nm chips on a given wafer size vs 22nm (or 32nm for the other manufacturers ) and you have a very appealing proposition  And then add in the fact that Intel actually has a pretty good graphics stack and IP  My point was that Intel might have a one or two process advantage over the rest of the industry at the cutting edge but that doesn't mean that they can afford to manufacture on those processes for very low margin parts  If the SoC market becomes increasingly commoditised, there isn't going to be the money to justify making them in a state of the art fab Remember that one of the big selling points of Itanium was that it would make use of process advantages that were effectively paid for by the mainstream x86 market  That didn't quite work out in practice and Itanium processors were often well behind Xeons in process technology 
paul5ra wrote:As someone who has worked in the semiconductor industry for longer than contemporary fanboys have existed, I'm getting a bit fed-up seeing many of these articles which rewrite history with analyses distorted by the lens of the contemporary Apple or Samsung fanboy The mobile industry moved to horizontal integration a long time ago  Better indicators than these odd contemporary obsessions with the relatively non-innovative Samsung and Apple where when Motorola spun out its semiconductor division as Freescale, Nokia stopped making it's own custom designs with TI and ST, and Ericsson spun out it's Ericsson Mobile Platforms division and formed ST-Ericsson with ST The true innovation in the mobile market was done a decade or more ago mainly by Moto/Nokia/Ericsson/TI/Qualcomm, and Samsung and Apple had little to do with it  Under the hood most stuff has been on a simple linear evolutionary path by the SoC providers since then  The phone manufacturers have then mainly been simply sticking these off-the-shelf SoCs (and their included low-level software stacks) in a box, a job made all the easier with the SoC manufacturer collaboration providing the bulk of the work for the AOSP Just goes to show that people who have worked in an industry for a long time don't always understand what that industry is doing You haven't been working in it long enough to seem to know that it was Acorn and Apple that invented the mobile ARM CPU in the first place  All those companies you've mentioned have just been working off Acorn and Apple's pioneering work  Now, Apple is back at it again, very successfully, and all the companies you mentioned that produce chips with ARM IP in them are licensing them from the company Acorn and Apple formed—ARM 
Last edited by melgross on Wed Feb 13, 2013 11:13 am
Mark Havel wrote:ggeezz wrote:Intel cannot abandon the phone/tablet market  Desktop/laptop sales are stagnating/decreasing and phones/tablets are on the rise  This trend is only going to increase going forward But you're right, they're going to have use their fabs that are a step or two behind the cutting the edge  But they're going to have to up their game in the tablet space to even be able to do that The word you're looking for is Haswell, as far as I know If tablets move into the $100-200 range, is there going to be room for Haswell?So long as there is a higher-end tablet market, then Haswell will be able to shine, but it's going to be a much more powerful and costly part than the sort of ARM based hardware that often runs tablets  If we see a race to the bottom where price is the dominant motivator behind purchases, then a high performance SoC will struggle to make its mark 
melgross wrote:paul5ra wrote:As someone who has worked in the semiconductor industry for longer than contemporary fanboys have existed, I'm getting a bit fed-up seeing many of these articles which rewrite history with analyses distorted by the lens of the contemporary Apple or Samsung fanboy The mobile industry moved to horizontal integration a long time ago  Better indicators than these odd contemporary obsessions with the relatively non-innovative Samsung and Apple where when Motorola spun out its semiconductor division as Freescale, Nokia stopped making it's own custom designs with TI and ST, and Ericsson spun out it's Ericsson Mobile Platforms division and formed ST-Ericsson with ST The true innovation in the mobile market was done a decade or more ago mainly by Moto/Nokia/Ericsson/TI/Qualcomm, and Samsung and Apple had little to do with it  Under the hood most stuff has been on a simple linear evolutionary path by the SoC providers since then  The phone manufacturers have then mainly been simply sticking these off-the-shelf SoCs (and their included low-level software stacks) in a box, a job made all the easier with the SoC manufacturer collaboration providing the bulk of the work for the AOSP Just goes to show that people who have worked in an industry for a long time don't always understand what that industry is doing You haven't been working in it long enough to seem to know that it was Acorn and Apple that invented the mobile ARM CPU in the first place  All those companies you've mentioned have just been working off Acorn and Apple's pioneering work  Now, Apple is back at it again, very successfully, and all the companies you mentioned that produce chips with ARM IP in them are licensing them from the company Acorn and Apple formed—ARM Of course I realise ARM IP has indeed been a major driving factor too (though only one if several architectures before ARM became dominant), though I see ARM's influence on the mobile industry as having nothing to do with modern day Apple and only one small piece of the puzzle  My point is that the hard electrical engineering, mathematics, DSP, semiconductor physics/chemistry, RF engineering, analogue design, CAD etc  that make modern telecommunications possible has very little to do with the fashion companies who consumers (and unfortunately much of the tech media) associate with it and give the credit (though in this respect Samsung does deserve a bit more credit for their work on NAND flash and displays)  The industry simply would not exist TODAY without the overwhelming horizontal integration that already dominates 
Quote:In the long term, mobile devices are likely to evolve similarly to the PC and favor a horizontal business model  The real advantage is one of flexibility; as costs drop and the market expands, it will be increasingly necessary for vendors like HTC to offer a wide range of phones based on radically different SoCs  You don't mention in the article that each SoC necessarily requires a bit of parallel dev work unlike the PC  In the PC world there is a standard BIOS and HW architecture that allows for pluggable designs  On a highly integrated SoC this is untrue  HTC suffers because it has to support radically different SoCs, their drivers and boot loaders, etc  Quote:While a vertically integrated company like Apple can focus and maintain leadership in a specific (and highly lucrative) niche, it would be very difficult to expand in many growing areas of the market  The differences between an iPhone 6 and a $20 feature phone are tremendous and would be very difficult for a single company to bridge It's only difficult because Apple chooses to ignore that market, not because they can't  If they can release a $99 Apple TV, they can surely cobble together a $20 feature phone if they chose to eschew 8GB of NAND, BT, WiFi, a specialized dock connector, LTE, and their specialized processors  In other words, build the equivalent of an iPod shuffle with a horrible screen and no OS to speak of 
paul5ra wrote:melgross wrote:paul5ra wrote:As someone who has worked in the semiconductor industry for longer than contemporary fanboys have existed, I'm getting a bit fed-up seeing many of these articles which rewrite history with analyses distorted by the lens of the contemporary Apple or Samsung fanboy The mobile industry moved to horizontal integration a long time ago  Better indicators than these odd contemporary obsessions with the relatively non-innovative Samsung and Apple where when Motorola spun out its semiconductor division as Freescale, Nokia stopped making it's own custom designs with TI and ST, and Ericsson spun out it's Ericsson Mobile Platforms division and formed ST-Ericsson with ST The true innovation in the mobile market was done a decade or more ago mainly by Moto/Nokia/Ericsson/TI/Qualcomm, and Samsung and Apple had little to do with it  Under the hood most stuff has been on a simple linear evolutionary path by the SoC providers since then  The phone manufacturers have then mainly been simply sticking these off-the-shelf SoCs (and their included low-level software stacks) in a box, a job made all the easier with the SoC manufacturer collaboration providing the bulk of the work for the AOSP Just goes to show that people who have worked in an industry for a long time don't always understand what that industry is doing You haven't been working in it long enough to seem to know that it was Acorn and Apple that invented the mobile ARM CPU in the first place  All those companies you've mentioned have just been working off Acorn and Apple's pioneering work  Now, Apple is back at it again, very successfully, and all the companies you mentioned that produce chips with ARM IP in them are licensing them from the company Acorn and Apple formed—ARM Of course I realise ARM IP has indeed been a major driving factor too (though only one if several architectures before ARM became dominant), though I see ARM's influence on the mobile industry as having nothing to do with modern day Apple and only one piece of the puzzle  My point is that the hard electrical engineering, mathematics, DSP, semiconductor physics/chemistry, RF engineering, analogue design,etc  that make modern telecommunications possible has very little to do with the fashion companies who consumers (and unfortunately much of the tech media) associate with it and give the credit (though in this respect Samsung does deserve a bit more credit for their work on NAND flash and displays)  The industry simply would not exist TODAY without the overwhelming horizontal integration that already dominates Yes the efforts of these companies getting cellular communications standardized were immense  And the technology matured  And then they didn't do much with it  It took some youngin's to look at the problem fresh and add the UI that make today's smartphones work  As we have all seen, the moment your technology has matured is the moment you are screwed because someone else now has the opportunity to look at it as a black box and make something new  Each of those manufacturers knew that smartphones would eventually be awesome, but none of them had the UI and software design to make a truly breakout product  Imagine if Motorola would have been smart enough to buy the Android guys instead of Google  Instead, Google bought a bunch of patents on that cellular black box to try to defend it's platform And when you think about it, which consumes more man years of engineering effort per year at this point     iterating that cellular black box or developing the OS, services and apps that power today's smartphones?
Intel had better decide that they are competing in this space "for real", or they are screwed  They've already let the Atom languish for five years, during which ARM has completely caught up in performance Just like Tim Cook said, if you don't cannibalize your own markets someone else will do it for you Whether Intel will embrace that concept in time remains to be seen  Personally, I hope they don't; if Intel transforms into a chipless Fab company (like TSMC) everyone benefits 
I still think Samsung has the advantage long term because they have both the SOC and the memory products  As mentioned in the article, TSV's (Through Silicon Via's) are going to be quite a disruption  Today, people normally stack an LPDDR2 package on top of their SOC package (POP or Package On Package)  Within the LPDDR2 package, you could have a stack of DRAM die typically with wire bonding connecting the die within the package Once you more to TSV's, you can have a LOT more connections between the SOC and its DRAM's  While this is being standardized through JEDEC (http://www jedec org/category/technolog     a/3d-ics-0), Samsung has all the pieces in house to do whatever they want  You could see a 512 bit or higher bus from the SOC to the memory  The trick is that the memory and the SOC need to line up with each other when you stack them  This gives Samsung an inherent advantage This isn't just going to impact mobile either  Take a look at that JEDEC link  It also lists High Bandwidth Memory (HBM)  This is a 1024 bit bus that provides 128GBytes/s to 256GBytes/s of bandwidth to a stack of up to 8 DRAM's  Here is your processor that includes 8-16 cores and 4GBytes of really, really, fast DRAM    No DIMMs required  How many of them do you want in your server rack?If I was Intel or Apple, I would be thinking seriously about making some investments in Micron to guarantee they make some compelling DRAM's to integrate with their SOC's and processors    otherwise Samsung is going to blow them out of the water on bandwidth 
Great_Scott wrote:Intel had better decide that they are competing in this space "for real", or they are screwed  They've already let the Atom languish for five years, during which ARM has completely caught up in performance Just like Tim Cook said, if you don't cannibalize your own markets someone else will do it for you Whether Intel will embrace that concept in time remains to be seen  Personally, I hope they don't; if Intel transforms into a chipless Fab company (like TSMC) everyone benefits It's true that Atom has stood still for too long, but honestly it's pretty amazing how Atom is still competitive with current ARM chips  The Z2760 is even 32nm vs 28nm of the latest Krait and A15 chips But that's all changing with Atom moving to the tick tock schedule this year  It wouldn't even surprise me to see Apple move to Intel chips for IOS And I don't see how Intel moving to a chipless Fab company would help everyone  It certainly wouldn't help Intel 
Mabsark wrote:ggeezz wrote:Intel cannot abandon the phone/tablet market  Desktop/laptop sales are stagnating/decreasing and phones/tablets are on the rise  This trend is only going to increase going forward But you're right, they're going to have use their fabs that are a step or two behind the cutting the edge  But they're going to have to up their game in the tablet space to even be able to do that Actually, that trend will not simply keep increasing going forward  The reason desktop/laptop sales are stagnating/decreasing is due to the fact that most people already have one and therefore don't need to buy another one  The exact same thing will happen with tablets as well  Sales are increasing now because people without tablets are buying them  When most people already own a tablet, they won't be buying a new one every year and therefore sales will stagnate/decrease The PC market is saturated and in a couple of years, the tablet market will be saturated too  Basically, in order to increase sales in a saturated market, you need to increase the population growth or decrease the longevity of the product Yes and no  I'm not sure the tablet market will saturate in a "couple of years " It may be more like 5 years  But that's a quibble Here's the real issue  Right now Apple wants you to own an iPhone AND iPad AND Macbook AND iWatch AND Apple TV  Microsoft, OTOH, is making the Surface so that you could ditch your laptop and just use a Surface  Not everyone, but some people If 5 years from now, we're in a world where a significant number of people use a Surface-type device instead of a laptop, then the PC market is going to contract significantly  Maybe some of the tablet-like devices will use moderately expensive Intel chips, but some of them are going to use cheaper chips 
GravyGraphics wrote:I still think Samsung has the advantage long term because they have both the SOC and the memory products  As mentioned in the article, TSV's (Through Silicon Via's) are going to be quite a disruption  Today, people normally stack an LPDDR2 package on top of their SOC package (POP or Package On Package)  Within the LPDDR2 package, you could have a stack of DRAM die typically with wire bonding connecting the die within the package Once you more to TSV's, you can have a LOT more connections between the SOC and its DRAM's  While this is being standardized through JEDEC (http://www jedec org/category/technolog     a/3d-ics-0), Samsung has all the pieces in house to do whatever they want  You could see a 512 bit or higher bus from the SOC to the memory  The trick is that the memory and the SOC need to line up with each other when you stack them  This gives Samsung an inherent advantage This isn't just going to impact mobile either  Take a look at that JEDEC link  It also lists High Bandwidth Memory (HBM)  This is a 1024 bit bus that provides 128GBytes/s to 256GBytes/s of bandwidth to a stack of up to 8 DRAM's  Here is your processor that includes 8-16 cores and 4GBytes of really, really, fast DRAM    No DIMMs required  How many of them do you want in your server rack?If I was Intel or Apple, I would be thinking seriously about making some investments in Micron to guarantee they make some compelling DRAM's to integrate with their SOC's and processors    otherwise Samsung is going to blow them out of the water on bandwidth Why not AMD? Last I checked they still made memory   and processors/GPUs 
Context 2: Paper Info

Title: Efficient nonparametric estimation of Toeplitz covariance matrices
Publish Date: March 20, 2023
Author List: Karolina Klockmann (from Department of Statistics and Operations Research, Universität Wien), Tatyana Krivobokova (from Department of Statistics and Operations Research, Universität Wien)

Figure

Figure 1: Spectral density functions (first row) and autocovariance functions (second row) for examples 1, 2, 3 
Figure 2: Distance between the first atom and the first center of mass of aquaporin (left) and the opening diameter y t over time t (right) 
black line in the left plot) confirms that the covariance matrix estimated with our VST-DCT method almost completely decorrelates the channel diameter Y on the training data set Next, we estimated the regression coefficients β with the usual PLS algorithm, ignoring the dependence in the data Finally, we estimated β with PLS that takes into account dependence using our covariance estimator Σ Based on these regression coefficient estimators, the prediction on the test set was calculated The plot on the right side of Figure 2 shows the Pearson correlation between the true channel diameter on the test set and the prediction on the same test set based on raw (grey) and decorrelated data (black) 
Figure 3: On the left, the auto-correlation function of Y (grey) and of Σ−1/2 Y (black), where Σ is estimated with the VST-DCT method; On the right, correlation between the true values on the test data set and prediction based on partial least squares (in grey) and corrected partial least squares (black) 
Uniform distributionThe observations follow a uniform distribution with covariance matrices Σ 1 , Σ 2 , Σ 3 of examples 1, 2, 3, i e , Y i = Σ 1/2 j X i , j = 1, 3, with X 1 ,    the parameter innov of the R function arima sim is used to pass the innovations X 1 ,    , X n i i d Table4, 5 and 6 show respectively the results for (A) p = 5000, n = 1, (B) p = 1000, n = 50 and (C) p = 5000, n = 10 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(B) p = 1000, n = 50: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 

abstract

A new nonparametric estimator for Toeplitz covariance matrices is proposed  This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression  The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast 
Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices  These results are readily extended to estimation of inverses of Toeplitz covariance matrices  Also, an alternative version of the Whittle likelihood for the spectral density based on the Discrete Cosine Transform (DCT) is proposed 
The method is implemented in the R package vstdct that accompanies the paper 

Introduction

Estimation of covariance and precision matrices is a fundamental problem in statistical data analysis with countless applications in the natural and social sciences  Covariance matrices with a Toeplitz structure arise in the study of stationary stochastic n = 1, to the best of our knowledge, there is no fully data-driven approach for selecting the banding/tapering/thresholding parameter 
suggested first to split the time series into non-overlapping subseries and then apply the cross-validation criterion of   However, it turns out that the right choice of the subseries length is crucial for this approach, but there is no data-based method available for this  In this work, an alternative way to estimate a Toeplitz covariance matrix and its inverse is chosen 
Our approach exploits the one-to-one correspondence between Toeplitz covariance matrices and their spectral densities  First, the given data are transformed into approximate Gaussian random variables whose mean equals to the logarithm of the spectral density  Then, the log-spectral density is estimated by a periodic smoothing spline with a data-driven smoothing parameter 
Finally, the resulting spectral density estimator is transformed into an estimator for Σ or its inverse  It is shown that this procedure leads to an estimator that is fully data-driven, automatically positive definite and achieves the minimax optimal convergence rate under the spectral norm over a large class of Toeplitz covariance matrices 
In particular, this class includes Toeplitz covariance matrices that correspond to long-memory processes with bounded spectral densities  Moreover, the computation is very efficient, does not require iterative or resampling schemes and allows to apply any inference and adaptive estimation procedures developed in the context of nonparametric Gaussian regression 
Estimation of the spectral density from a stationary time series is a research topic with a long history  Earlier nonparametric methods are based on smoothing of the (log-)periodogram, which itself is not a consistent estimator   Another line of nonparametric methods for estimating the spectral density is based on the Whittle likelihood, which is an ap-proximation to the exact likelihood of the time series in the frequency domain 
For example, estimated the spectral density from a penalized Whittle likelihood, while used polynomial splines to estimate the log-spectral density function maximizing the Whittle likelihood  Recently, Bayesian methods for spectral density estimation have been proposed (see , but these may become very computationally intensive in large samples due to posterior sampling 
The minimax optimal convergence rate for nonparametric estimators of Hölder continuous spectral densities from Gaussian stationary time series was obtained by under the L p , 1 ≤ p ≤ ∞, norm  Only few works on spectral density estimation show the optimality of the corresponding estimators  In particular, and derived convergence rates of their estimators for the log-spectral density under the L 2 norm, while neglecting the Whittle likelihood approximation error 
In general, most works on spectral density estimation do not exploit further the close connection to the corresponding Toeplitz covariance matrix estimation  In particular, an upper bound for the L ∞ risk of a spectral density estimator automatically provides an upper bound for the risk of the corresponding Toeplitz covariance matrix estimator under the spectral norm 
This fact is used to establish the minimax optimality of our nonparametric estimator for the Toeplitz covariance matrices  The main contribution of this work is to show that our proposed spectral density estimator is not only numerically very efficient, but also achieves the minimax optimal rate in the L ∞ norm, which in turn ensures the minimax optimality of the corresponding Toeplitz covariance matrix estimator 
The paper is structured as follows  In Section 2, the model is introduced and ap-proximate diagonalization of Toeplitz covariance matrices with the discrete cosine transform is discussed  Moreover, an alternative version of the Whittle's likelihood is proposed  In Section 3, new estimators for the Toeplitz covariance matrix and the precision matrix are derived, while in Section 4 their theoretical properties are presented 
Section 5 contains simulation results, Section 6 presents a real data example, and Section 7 closes the paper with a discussion  The proofs are given in the appendix to the paper 

Set up and diagonalization of Toeplitz matrices

Let Y 1 ,       , Y n i i d  ∼ N p (0 p , Σ), where Σ is a (p × p)-dimensional positive definite covariance matrix with a Toeplitz structure, that is, Σ = {σ |i−j| } p i,j=1 0  The sample size n may tend to infinity or to be a constant  The case n = 1 corresponds to a single observation of a stationary time series, and in this case the data are simply denoted by Y ∼N p (0 p , Σ) 
The dimension p is assumed to grow  The spectral density function f , corresponding to a Toeplitz covariance matrix Σ, is given by so that for f ∈ L 2 (−π, π) the inverse Fourier transform implies Hence, Σ is completely characterized by f , and the non-negativity of the spectral density function implies the positive definiteness of the covariance matrix 
Moreover, the decay of the autocovariance σ k is directly connected to the smoothness of f   Finally, the convergence rate of a Toeplitz covariance estimator and that of the corresponding spectral density estimator are directly related via Σ ≤ f ∞ := sup x∈ |f (x)|, where • denotes the spectral norm (see  
As in , we introduce a class of positive definite Toeplitz covariance matrices with Hölder continuous spectral densities  For β = γ + α > 0, where The optimal convergence rate for estimating Toeplitz covariance matrices over P β (M 0 , M 1 ) depends crucially on β  It is well known that the k-th Fourier coefficient of a function whose γ-th derivative is α-Hölder continuous decays at least with order O(k −β ) (see  
Hence, β determines the decay rate of the autocovariances σ k , which are the Fourier coefficients of the spectral density f , as k → ∞  In particular, this implies that for β ∈ (0, 1], the class P β (M 0 , M 1 ) includes Toeplitz covariance matrices corresponding to long-memory processes with bounded spectral densities, since the sequence of corresponding autocovariances is not summable 
A connection between Toeplitz covariance matrices and their spectral densities is further exploited in the following lemma  Lemma 1  Let Σ ∈ P β (M 0 , M 1 ) and let x j = (j − 1)/(p − 1), j = 1,    , p, then where δ i,j is the Kroneker delta, O(•) terms are uniform over i, j = 1,       , p and divided by √ 2 when i, j ∈ {1, p} is the Discrete Cosine Transform I (DCT-I) matrix 
The proof can be found in Appendix A 1  This result shows that the DCT-I matrix approximately diagonalizes Toeplitz covariance matrices and that the diagonalization error depends to some extent on the smoothness of the corresponding spectral density  In the spectral density literature the discrete Fourier transform (DFT) matrix
, where i is the imaginary unit, is typically employed to approximately diagonalize Toeplitz covariance matrices  Using the fact that introduced an approximation for the likelihood of a single Gaussian stationary time series (case n = 1), the so-called Whittle likelihood (1) The quantity , where F j denotes the j-th column of F , is known as the periodogram at the j-th Fourier frequency 
Note that due to periodogram symmetry, only p/2 data points I 1 ,    , I p/2 are available for estimating the mean f (2πj/p), j = 1,       , p/2 , where x denotes the largest integer strictly smaller than x  The Whittle likelihood has become a popular tool for parameter estimation of stationary time series, e g , for nonparametric and parametric spectral density estimation or for estimation of the Hurst exponent, see e g , ;  
Lemma 1 yields the following alternative version of the Whittle likelihood where W j = (D t j Y ) 2   Note that this likelihood approximation is based on twice as many data points W j as the standard Whittle likelihood  Thus, it allows for a more efficient use of the data Y to estimate the parameter of interest, such as the spectral density or the Hurst parameter 
Equations ( ) or (2) invite for the estimation of f by maximizing the (penalized) likelihood over certain linear spaces (e g , spline spaces), as suggested e g , in or   However, such an approach requires well-designed numerical methods to solve the corresponding optimization problem, since the spectral density in the second term of (1) or ( ) is in the denominator, which does not allow to obtain a closed-form expression for the estimator and often leads to numerical instabilities 
Also, the choice of the smoothing parameter becomes challenging  Therefore, we suggest an alternative approach that allows the spectral density to be estimated as a mean in an approximate Gaussian regression  Such estimators have a closed-form expression, do not require an iterative optimization algorithm and a smoothing parameter can be easily obtained with any conventional criterion 
First Hence, for W j = (D t j Y ) 2 , j = 1,       , p it follows with Lemma 1 that where Γ(a, b) denotes a gamma distribution with a shape parameter a and a scale parameter b  Note that the random variables W 1 ,       , W p are only asymptotically independent  Obviously, E(W j ) = f (πx j ) + O(1), j = 1,      
, p  To estimate f from W 1 ,       , W p , one could use a generalized nonparametric regression framework with a gamma distributed response, see e g , the classical monograph by   However, this approach requires an iterative procedure for estimation, e g , a Newton-Raphson algorithm, with a suitable choice for the smoothing parameter at each iteration step 
Deriving the L ∞ rate for the resulting estimator is also not a trivial task  Instead, we suggest to employ a variance stabilizing transform of that converts the gamma regression into an approximate Gaussian regression  In the next section we present the methodology in more detail for a general setting with n ≥ 1 

Methodology

For Y i ∼ N p (0 p , Σ), i = 1,       , n, it was shown in the previous section that with Lemma 1 the data can be transformed into gamma distributed random variables     , n, j = 1,       , p, where for each fixed i the random variable W i,j has the same distribution as W j given in (3)  Now the approach of Cai et al  ( ) is adapted to the setting n ≥ 1 
First, the transformed data points W i,j are binned, that is, fewer new variables     , T   Note that the number of observations in a bin is m = np/T   In Theorem 1 in Section 4, we show that setting T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) leads to the minimax optimal rate for the spectral density estimator 
To simplify the notation, m is handled as an integer (otherwise, one can discard several observations in the last bin)  Next, applying the variance stabilizing transform (VST) ∼ where H(y) = {φ(m/2) + log (2y/m)} / √ 2 and φ is the digamma function (see   Now, the scaled and shifted log-spectral density H(f ) can be estimated with a periodic smoothing spline
where h > 0 denotes a smoothing parameter, q ∈ N is the penalty order and S per (2q − 1) a space of periodic splines of degree 2q − 1  The smoothing parameter h can be chosen either with generalized cross-validation (GCV) as derived in or with the restricted maximum likelihood, see   Once an estimator H(f ) is obtained, application of the inverse transform function H −1 (y) = m exp √ 2y − φ (m/2) /2 yields the spectral density estimator f = H −1 H(f )  
Finally, using the inverse Fourier transform leads to the fol- The precision matrix Ω is estimated by the inverse Fourier transform of the reciprocal of the spectral density estimator, i e , Ω = (ω |i−j| ) p i,j=1 with ωk = The estimation procedure for Σ and Ω can be summarised as follows  1  Data Transformation:
where D is the (p × p)-dimensional DCT-I matrix as given in Lemma 1 and D j is its j-th column  2  Binning: Set T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and calculate W i,j , k = 1,       , T 

VST:

where k are asymptotically i i d  Gaussian variables  Inverse VST: Estimate the spectral density f with f = H −1 H(f ) , where Note that Σ and Ω are positive definite matrices by construction, since their spectral density functions f and f −1 are non-negative, respectively  Unlike the banding and tapering estimators, the autocovariance estimators σk are controlled by a single smoothing parameter h, which can be estimated fully data-driven with several available automatic methods, which are numerically efficient and well-studied 
In addition, one can also use methods for adaptive mean estimation, see e g , , which in turn leads to adaptive Toeplitz covariance matrix estimation  All inferential procedures developed in the Gaussian regression context can also be adopted accordingly 

Theoretical Properties

In this section, we study the asymptotic properties of the estimators f , Σ and Ω  The results are established under the asymptotic scenario where p → ∞ and p/n → c ∈ (0, ∞], that is, the dimension p grows, while the sample size n either remains fixed or also grows but not faster than p  This corresponds to the asymptotic scenario when the sample covariance matrix is inconsistent 
Let f be the spectral density estimator defined in Section 3, i e , f = m exp{ √ 2 H(f ) − φ(m/2)}/2, where H(f ) is given in (4), m = np/T and φ is the digamma function  Furthermore, let Σ be the Toeplitz covariance matrix estimator and Ω the corresponding precision matrix defined in equations ( ) and (6), respectively 
The following theorem shows that both Σ and Ω attain the minimax optimal rate of convergence over the class and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and q = max{1, γ}, the spectral density estimator f , the corresponding covariance matrix estimator Σ and the precision matrix estimator Ω satisfy sup
For h {log(np)/(np)}   The proof of Theorem 1 can be found in the Appendix A 3 and is the main result of our work  The most important part of this proof is the derivation of the convergence rate for the spectral density estimator f under the L ∞ norm  In the original work, established an L 2 rate for a wavelet nonparametric mean estimator in a gamma regression where the data are assumed to be independent 
In our work, the spectral density estimator f is based on the gamma distributed data W i,1 ,       , W i,p , which are only asymptotically independent  Moreover, the mean of these data is not exactly f (πx 1 ),       , f (πx p ), but is corrupted by the diagonalization error given in Lemma 1  This error adds to the error that arises via binning and VST and that describes the deviation from a Gaussian distribution, as derived in  
Finally, we need to obtain an L ∞ rather than an L 2 rate for our spectral density estimator  Overall, the proof requires different tools than those used in   To get the L ∞ rate for f , we first derive that for the periodic smoothing spline estimator H(f ) of the log-spectral density  To do so, we use a closed-form expression of its effective kernel obtained in , thereby carefully treating various (dependent) errors that describe deviations from a Gaussian nonparametric regression with independent errors and mean f (πx i ) 
Note also that although the periodic smoothing spline estimator is obtained on T binned points, the rate is given in terms of the vector dimension p  Then, using the Cauchy-Schwarz inequality and a mean value argument, this rate is translated into the L ∞ rate for the spectral density estimator f   To obtain the rate for the Toeplitz covariance matrix estimator is enough to note that

Simulation Study

In this section, we compare the performance of the proposed Toeplitz covariance estimator, denoted as VST-DCT, with the tapering estimator of and with the sample covariance matrix  We consider Gaussian vectors Y 1 ,    , (3) such that the corresponding spectral density is Lipschitz continuous but not differentiable: f (x) = 1 44{| sin(x + 0 5π)| 1 7 + 0 45} 
In particular, var(Y i ) = 1 44 in all three examples  Figure shows the spectral densities and the corresponding autocorrelation functions for the three examples  A Monte Carlo simulation with 100 iterations is performed using R (version 4 1 2, seed 42)  For our VST-DCT estimator, we use a cubic periodic spline, i e , q = 2 is set in (4) 
The binning parameters are set to T = 500 bins with m = 10 points for (A) and T = 500 bins with m = 100 points for both (B) and (C)  To select the regularisation parameter for our estimator, we implemented the restricted maximum likelihood (ML) method, generalized cross validation (GCV) and the corresponding oracle versions, i e , as if Σ were known The tapering parameter
where T ap k (Σ ν 2 ) is the tapering estimator of with parameter k  If n = 1, that is, under scenario (A), suggest to split the time series Y into l non-overlapping subseries of length p/l and then proceed as before to select the tuning parameter k  To the best of our knowledge, there is no data-driven method for selecting this parameter l 
Using the true covariance matrix Σ, we selected l = 30 subseries for the example 1 and l = 15 subseries for the exam-ples 2 and 3  The parameter k can then be chosen by cross-validation as above  We employ this approach under scenario (A) instead of an unavailable fully data-driven criterion and name it semi-oracle 
Finally, for all three scenarios (A), (B) and (C), the oracle tapering parameter is computed using grid search for each Monte Carlo sample as kor = arg min k=2,3,   ,p/2 T ap k ( Σ) − Σ , where Σ is the sample covariance matrix  To speed up the computation, one can replace the spectral norm by the 1 norm, as suggested by  
In Tables , the errors of the Toeplitz covariance estimators with respect to the spectral norm and the computation time for one Monte Carlo iteration are given for scenarios (A), (B) and (C), respectively  To illustrate the goodness-of-fit of the spectral density, the L 2 norm f − f 2 is also computed 
The results show that the tapering and VST-DCT estimator perform overall similar in terms of the spectral norm risk  This is not surprising as both estimators are proved to be rate-optimal  Moreover, both the tapering and VST-DCT estimators are clearly superior to the inconsistent sample Toeplitz covariance matrix 
A closer look at the numbers shows that the VST-DCT method has better constants, i e , VST-DCT estimators have somewhat smaller errors in the spectral norm than the tapering estimators across all examples, but especially under scenario (C)  The oracle estimators show similar behaviour, but are slightly less variable compared to the data-driven estimators 
In general, both the tapering and VST-DCT estimators perform best for example 1, second best for example 3 and worst for example 2, which traces back to functions complexity  In terms of computational time, both methods are similarly fast for scenarios (A) and (B)  For scenario (C), the tapering method is much slower due to the multiple high-dimensional matrix multiplications in the cross-validation method 
It is expected that for larger p the tapering estimator is much more computationally intensive than the corresponding VST-DCT estimator  (1) polynomial σ ( (  To test how robust our approach is to deviations from the Gaussian assumption, we simulated the data from gamma and uniform distributions and conducted a simulation study for the same scenarios and examples 
The results are very similar to those of the Gaussian distribution, see supplementary materials for the details 

Application to Protein Dynamics

We revisit the data analysis of protein dynamics performed in Krivobokova et al  (2012) and 
Context 3: Brooksley Elizabeth Born (born August 27, 1940) is an American attorney and former public official who, from August 26, 1996, to June 1, 1999, was chair of the Commodity Futures Trading Commission (CFTC), the federal agency which oversees the U S  futures and commodity options markets  During her tenure on the CFTC, Born lobbied Congress and the President to give the CFTC oversight of off-exchange markets for derivatives, in addition to its role with respect to exchange-traded derivatives, but her warnings were ignored or dismissed, and her calls for reform resisted by other regulators <ref name="nytimes">Goodman, Peter S  The Reckoning - Taking Hard New Look at a Greenspan Legacy, The New York Times, October 9, 2008 </ref> Born resigned as chairperson on June 1, 1999, shortly after Congress passed legislation prohibiting her agency from regulating derivatives 

Early life and education
Born graduated from Abraham Lincoln High School (San Francisco, California) at the age of 16  She then attended Stanford University, where she majored in English and was graduated with the class of 1961   She initially wanted to become a doctor, but a guidance counsellor at Stanford advised her against medicine, so she majored in English literature instead 

She then attended Stanford Law School, one of only seven women in her class   She was the first female student ever to be named president of the Stanford Law Review  She received the "Outstanding Senior" award and graduated as valedictorian of the class of 1964 

Legal career
Immediately after law school Born was selected as a law clerk to judge Henry Edgerton of the U S  Court of Appeals for the District of Columbia Circuit  It was during this time that she met her first husband, Jacob C  Landau, who was a journalist covering the Federal courts at the time  Following her clerkship, she became an associate at the Washington, D C -based international law firm of Arnold & Porter  Born was attracted to Arnold & Porter because it was one of the few major law firms to have a woman partner at that time, Carolyn Agger, who was the head of the tax practice  Born took a two-year leave of absence from Arnold & Porter to accompany her first husband to Boston, where he had received a fellowship  During that time she worked as a research assistant to law professor Alan Dershowitz 

Born's early career at Arnold & Porter focused on international trade law, in which she represented a number of Swiss industries and the government of Switzerland  She developed a practice representing clients in numerous complex litigation and arbitration cases involving financial market transactions  Among her high-profile cases was the matter of the Hunt Brothers attempt to corner the market in silver in the 1970s  She made partner at Arnold & Porter, after moving to a three-day schedule to help raise her second child, and eventually rose to be the head of the firm's derivatives practice 

Born was among the first female attorneys to systematically address inequities regarding how the laws treated women  Born and another female lawyer, Marna Tucker, taught what is considered to have been the first "Women and the Law" course at Catholic University’s Columbus School of Law  The class exclusively concerned prejudicial treatment of women under the laws of the United States, past and present   Born and Tucker were surprised to discover that there was no textbook on the issue at the time   Born is also one of the co-founders of the National Women's Law Center  Born also helped rewrite the American Bar Association rules to make it possible for more women and minorities to sit on federal bench 

During her long legal career, and into her retirement, Born did much pro bono and other types of volunteer work



ground_truth: Smartphones are more compact and power constrained.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the advantage of decorrelating the data before running the PLS algorithm?

Read the following text and answer briefly.

Context 2:  We consider data generated by the molecular dynamics (MD) simulations for the yeast aquaporin (Aqy1) -the gated water channel of the yeast Pichi pastoris  MD simulations are an established tool for studying biological systems at the atomic level on timescales of nano-to microseconds 
The data are given as Euclidean coordinates of all 783 atoms of Aqy1 observed in a 100 nanosecond time frame, split into 20 000 equidistant observations  Additionally, the diameter of the channel y t at time t is given, measured by the distance between two centers of mass of certain residues of the protein 
The aim of the analysis is to identify the collective motions of the atoms responsible for the channel opening  In order to model the response variable y t , which is a distance, based on the motions of the protein atoms, we chose to represent the protein structure by distances between atoms and certain fixed base points instead of Euclidean coordinates 
That is, we calculated where A t,i ∈ R 3 , i = 1,       , 783 denotes the i-th atom of the protein at time t, B j ∈ R 3 , j = 1, 2, 3, 4, is the j-th base point and d(•, •) is the Euclidean distance  Figure shows the diameter y t and the distance between the first atom and the first center of mass  It can therefore be concluded that a linear model Y = Xβ + holds, where
  This linear model has two specific features which are intrinsic to the problem: first, the observations are not independent over time and second, X t is high-dimensional at each t and only few columns of X are relevant for Y   have shown that the partial least squares (PLS) algorithm performs exceptionally well on this type of data, leading to a small-dimensional and robust representation of proteins, which is able to identify the atomic dynamics relevant for Y  
Singer et al  ( ) studied the convergence rates of the PLS algorithm for dependent observations and showed that decorrelating the data before running the PLS algorithm improves its performance  Since Y is a linear combination of columns of X, it can be assumed that Y and all columns of X have the same correlation structure 
Hence, it is sufficient to estimate Σ = cov(Y ) to decorrelate the data for the PLS algorithm, i e , Σ −1/2 Y = Σ −1/2 Xβ + Σ −1/2 results in a standard linear regression with independent errors  Our goal now is to estimate Σ and compare the performance of the PLS algorithm on original and decorrelated data 
For this purpose, we divided the data set into a training and a test set (each with p = 10 000 observations)  First, we tested whether the data are stationary  The augmented Dickey-Fuller test confirmed stationarity for Y with a p-value< 0 01  The Hurst exponent of Y is 0 85, indicating moderate long-range dependence supported by a rather slow decay of the sample autocovariances (see grey line in the left plot of Figure ) 
Therefore, we set q = 1 for the VST-DCT estimator to match the low smoothness of the corresponding spectral density  Moreover, the smoothing parameter is selected with the restricted maximum likelihood method and T = 550 bins are used  Obviously, the performance of the PLS algorithm on the decorrelated data is significantly better for the small number of components 
In particular, with just one PLS component, the correlation between the true opening diameter on the test set and its prediction that takes into account the dependence in the data is already 0 54, while it is close to zero for PLS that ignores the dependence in the data  showed that the estimator of β based on one PLS component is exactly the ensemble-weighted maximally correlated mode (ewMCM), which is defined as the collective mode of atoms that has the highest probability to achieve a specific alteration of the response Y  
Therefore, an accurate estimator of this quantity is crucial for the interpretation of the results and can only be achieved if the dependence in the data is taken into account  Estimating Σ with a tapered covariance estimator has two practical problems  First, since we only have a single realization of a time series Y (n = 1), there is no datadriven method for selecting the tapering parameter 
Second, the tapering estimator turned out not to be positive definite for the data at hand  To solve the second problem, we truncated the corresponding spectral density estimator ftap to a small positive value, i e , f + tap = max{ ftap , 1/ log(p)} (see   To select the tapering parameter with cross-validation, we experimented with different subseries lengths and found that the tapering estimator is very sensitive to this choice 
For example, estimating the tapered covariance matrix based on subseries of length 8/15/30 yields a correlation of 0 42/0 53/0 34 between the true diameter and the first PLS component, respectively  Altogether, our proposed estimator is fully data-driven, fast even for large sample sizes, automatically positive definite and can handle certain long-memory processes 
In contrast, the tapering estimator is not data-driven and must be manipulated to become positive definite  Our method is implemented in the R package vstdct 

Discussion

In this paper, we proposed a simple, fast, fully data-driven, automatically positive definite and minimax optimal estimator of Toeplitz covariance matrices from a large class that also includes covariance matrices of certain long-memory processes  Our estimator is derived under the assumption that the data are Gaussian 
However, simulations show that the suggested approach yields robust estimators even when the data are not normally distributed  In the context of spectral density estima- , for mixing processes (see Theorem 5 3 of Rosenblatt, 2012), as well as for non-linear processes (see   Since DFT and DCT matrices are closely related, we expect that equation (3) also holds asymptotically for these non-Gaussian time series, but consider a rigorous analysis to be beyond the scope of this paper 
In fact, our numerical experiments have even shown that if the spectral density is estimated from W j = f (πx j ) + j , that is, as if W j were Gaussian instead of gamma distributed, then the resulting spectral density estimator has almost the same L ∞ risk (and hence the corresponding covariance matrix has almost the same spectral norm) 
Of course, such an estimator would lead to a wrong inference about f (πx j ), since the growing variance of W j would be ignored  Since our approach translates Toeplitz covariance matrix estimation into a mean estimation in an approximate Gaussian nonparametric regression, all approaches developed in the context of Gaussian nonparametric regression, such as (locally)
adaptive estimation, as well as the corresponding (simultaneous) inference, can be directly applied  Bayesian tools for adaptive estimation and inference in Gaussian nonparametric regression as proposed in can also be employed 

Appendix

Throughout the appendix, we denote by c, c 1 , C, C 1 ,       etc  generic constants, that are independent of n and p  To simplify the notation, the constants are sometimes skipped and we write for less than or equal to up to constants  We embed the p-dimensional Toeplitz matrix Σ = toep(σ 0 ,       , σ p−1 ) in a (2p − 2)dimensional circulant matrix Σ = toep(σ 0 ,      
, σ p−1 , σ p−2 ,       , σ 1 )  Then, Σ = with the conjugate transpose U * , and Λ is a diagonal matrix with the k-th diagonal value for k = 1,    , p given by Furthermore, Σ = V * ΛV , where V ∈ C (2p−2)×p contains the first p columns of U   In particular, b(j, r) = b(j, 2p−r) and c(j, r) = −c(j, 2p−r) for r = p+1,      
, 2p−2  Together, we have (A 1) Some calculations show that for r = 1,       p Using the Taylor expansion of cot(x) for 0 < |x| < π one obtains for r = 1,       p where the O term does not depend on j and the hidden constant does not depend on r, p  If i = j, equations (A 1) -(A 3) imply where the O terms do not depend on j 
Since the complex exponential function is Lipschitz continuous with constant L = 1, it holds λ r = λ j + L r,j |r − j|p −1 where −1 ≤ L r,j ≤ 1 is a constant depending on r, j  Then, , it is sufficient to consider j = 1,    , p − 1  We begin with first sum  For a shorter notation, we use k := r − 1 and l := j − 1 in the following 
Then, summing the squares of the first term in (A 4) for l = 0,    , p−2 on sums of reciprocal powers  If p is even, then the residual terms are given by where φ and φ (1) denote the digamma function and its derivative  If p is odd, similar remainder terms can be derived  To see that R i (l, p) = O(p −1 ) for i = 1, 2, 3 and uniformly in l we use that asymptotically φ(x)∼ log(x)−1/(2x) and
The mixed term are both of the order p −1   Furthermore, since the harmonic sum diverges at a rate of log(p)  Finally, λ j = f (x j )+O{log(p)p −β } by the uniform approximation properties of the discrete Fourier series for Hölder continuous functions (see   All together, we have shown that (DΣD) j,j = where the O terms are uniform over j = 1,    , p 
Case i = j and |i − j| is even In this case, (DΣD) i,j = a i a j uniformly in i, j  To show that a i a j 2p−2 r=1 λ r c(i, r)c(j, r) = O(p −1 ), we proceed similarly as before  Setting k=r−1, l=j−1, m=i−1 and using that l =m and |l−m| is even, one obtains where for even p the residual terms are given by If p is odd, analogous residual terms can be derived 
Using similar techniques as before, one can show that the two residual terms and the remaining mixed and square terms vanish at a rate of the order O(p −1 ) and uniformly in i, j  Case i = j and |i − j| is odd |r − i| and |r − j| are either odd and even, or even and odd  Without loss of generality, assume that |r − i| is even 
Then, (DΣD) i,j = a i a j 2p−2 r=1 λ r b(i, r)c(j, r)  Since b(i, •) is an even function, c(j, •) is an odd function and λ r = λ 2p−r , it follows (DΣD) i,j = 0  The structure of the proof is as follows  First, we derive the L ∞ rate of the periodic smoothing spline estimator H(f )  Then, using the Cauchy-Schwarz inequality and a mean value argument, the convergence rate of the spectral density estimator f is
∞ the first claim of the theorem follows  Finally, we prove the second statement on the precision matrices  For the sake of clarity, some technical lemmas used in the proof are listed separately in A 4  hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator H(f ) described in Section 3 with q = max{1, γ} satisfies
Proof : Application of the triangle inequality yields a bias-variance decomposition Set T = 2T − 2 and x k = (k − 1)/ T for k = 1,    , T   Using Lemma 4, we can write where Mirroring and renumerating ζ k , η k , k is similar as for Y * k , k = 1,    , T   Using the above representation, one can write First we reduce the supremum to a maximum over a finite number of points 
If q > 1, then W (•, x k ) is Lipschitz continuous with constant L > 0  In this case, it holds almost surely that sup ) is a piecewise linear function with knots at x j = j/ T   The factor (ζ k + ξ k ) can be considered as stochastic weights that do not affect the piecewise linear property  Thus, the supremum is attained at one of the knots x j = j/ T , j = 1,    , T , and (A 7) is also valid for q = 1 
Again with (a + b) 2 ≤ 2a 2 + 2b 2 we obtain We start with bounding   This requires a bound on 1 • ψ 2 denotes the sub-Gaussian norm  In case of a Gaussian random variable the norm equals to the variance  Thus with Lemma 2 and Lemma 4, we obtain Lemma 1 6 of ) then yields Recall that T = p υ for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
Using the inequality log(x) ≤ x a /a one can find constants x υ , C υ > 0 depending on υ but not on n, p such that log(2 T ) log(p) Next, we derive a bound for the second term The exponential decay property of the kernel K stated in Lemma 2 yields The first term in (A 9) can be bounded again with Lemma 1 6 of  
We use the fact that for not necessarily independent random variables X 1 ,    , X N R and R > 0 are constants  This is a consequence of Lemma 1 of which yields , it follows that N i=1 a i X i has a subGaussain distribution and the subGaussian norm is bounded by 2R( N i=1 a 2 i ) 1/2   See for further details on the subgaussian distribution 
T h   For the second inequality Lemma 2(ii) is used  Applying Lemma 1 6 of then yields To bound the second term in (A 9), we use the moment bounds for ξ k derived in Lemma 4  Then, for all integers > 1 Combining the error bounds (A 10) and (A 11) and choosing R=m −1/2 gives By assumption T = p υ and m = np (1−υ) for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
If is an integer such that ≥ 1/(1 − υ), then where we used log(x) ≤ x a /a with a = 1/(4 )  Consider 1/2 < β ≤ 1 and let 0 < χ < 1 be a constant  Applying log(x) ≤ x a /a twice with a = χ/(2 ) yields For any fixed υ∈((4 − 2 min{1, β})/3, 1) one can find an integer which is independent of n, p such that the right side of (A 12) holds 
Since p/n → c ∈ (0, ∞] and thus n/p = O(1) and p −1 = O(n −1 ), it follows for satisfying (A 12) that In total, choosing an integer Using the representation in Lemma 4 once more gives for each x ∈ [0, 1] The bounds on k in Lemma 4 imply Consider the case that β ≥ 1  In particular, q = γ and f (q) is α-Hölder continuous 
Since f is a periodic function with f (x) ∈ [δ, M 0 ] and H(y) ∝ φ(m/2)+ log (2y/m), it follows that {H(f )} (q) is also α-Hölder continuous  Extending g := H(f ) to the entire real line, we get Expanding g(t) in a Taylor series around x and using that h −1 K h is a kernel of order 2q, see Lemma 2(iii), it follows that for any x ∈ [0, 1]
where ξ x,t is a point between x and t  Using the fact that the kernel K h decays exponentially and that g (q) is α-Hölder continuous on [δ, M 0 ] with some constant L, the logarithm is Lipschitz continuous on a compact interval, it follows g = H(f ) is β-Hölder continuous  Expanding g to the entire line and using Lemma 2(iii) with
In a similar way as before, one obtains Note that T −β =o(h β ) as β > 1/2, T h → ∞ and h → 0 by assumption  Since the derived bounds are uniform for x ∈ [0, 1] it holds Putting the bounds A 13 and A 14 together gives If h > 0 such that h → 0 and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator f described in Section 3 with q = max{1, γ} satisfies
Proof : By the mean value theorem, it holds for some function g between H(f ) and To show that the second term on the right hand side of (A 15) is negligible we use the moment generating function of H(f ) ∞   In the next paragraph, we derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞, where λ > 0 may depend on n, p or not 
By the exponential decay property of the kernel K stated in Lemma 2 holds First, H(f ) ∞ is bounded with the maximum over a finite number of points  Calculating the derivative of s : Since δ δx s(x) > 0 almost surely for x = x k , the extrema occur at x k , k = 1,    , T   Thus, for λ > 0 the moment generating function of H(f ) ∞ is bounded by
Let M j = ( T h) −1 T k=1 γ h (x j , x k ), which by Lemma 2 is bounded uniformly in j by some global constant M > 0  By the convexity of the exponential function we obtain √ 2 and by assumption 0 ≤ δ ≤ f ≤ M 0   Using Lemma 3, Q k can be written as a sum of m = np/T independent gamma random variables, i e 
The moment generating function of | log(X)| when X follows a Γ(a, b)-distribution is given by where Γ(a) is the gamma function and γ(a, b) is the lower incomplete gamma function  In particular, To derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞ we first establish the asymptotic order of the ratio Γ(a + t)/Γ(a) for a → ∞ 
We distinguish the two cases where t is independent of a and where t linearly depends on a  Thus, for 0 < t < a and t independent of a, equation (A 17) implies for a → ∞ that Γ(a + t)/Γ(a) = O(a t )  Similarly, it can be seen that Γ(a − t)/Γ(a) = O(a −t )  If 0 < t < a and t linearly depends on a, i e  t = ca for some constant c ∈ (0, 1), then we get Γ(a ± t)/Γ(a) = O(a ±t exp{a}) for a → ∞ 
Hence, for a fixed λ not depending on n, p and such that 0 < λ < m/( √ 2M j ) we get for sufficiently large n, p If λ = cm such that 0 < λ < m/( √ 2M j ), then for sufficiently large n, p b∈{cδ/m,cM 0 /m} (bm/2) for some constant L > 1  Set K = min j=1,   , T 1/( √ 2M j ) which is a constant independent of n, p  Altogether, we showed that for 0 < λ < Km and n, p → ∞
Bounding the right hand side of (A 15) for some constants c 0 , c 1 > 0 and n, p → ∞ Since g lies between H(f ) and H(f ), and f almost surely pointwise  Thus, for C > f ∞ = M 0 it holds where c 1 := H(C − M 0 )  Applying Markov inequality for t = cm with c ∈ (0, K) and C = 2L 4/c + M 0 where c, K, L are the constants in gives
Together with Proposition 1 follows Using the fact that the spectral norm of a Toeplitz matrix is upper bounded by the sup norm of its spectral density we get sup According to the mean value theorem, for a function g between H(f ) and H(f ), it holds that some constant c 1 > 0 not depending on n, p  Chosing the same constant C as in section A 3 2 it follows
Noting that 1/f ∞ ≤ 1/δ and 2/m exp {φ(m/2)} ∈ [0 25, 1] for m ≥ 1, (A 18) implies for some constants c 2 , c 3 > 0 and n, p → ∞ Since the derived bounds hold for each Σ(f ) ∈ F β , we get all together sup This section states some technical lemmata needed for the proof of Theorem 1  The proofs can be found in the supplementary material 
The first lemma lists some properties of the kernel K h and its extension K h on the real line  The proof is based on   Lemma 2  Let h > 0 be the bandwith parameter depending on N   (i) There are constants 0 < C < ∞ and 0 < γ < 1 such that for all for x, t ∈ [0, 1] Lemma 3 states that the sum of the correlated gamma random variables in each bin can be rewritten as a sum of independent gamma random variables 
for i = 1,    , n and j = (k − 1)m + 1,    , km, and x j = (j − 1)/(2p − 2)  Finally, Lemma 4 gives explicit bounds for the stochastic and deterministic errors of the variance stabilizing transform  Thus, it quantifies the difference to an exact Gaussian regression setting  This result is a generalization of Theorem 1 of Cai et al 
(2010) adapted to our setting with n ≥ 1 observations and correlated observations  √ 2 can be written as where for the proof of the first statement  Furthermore, for x, t ∈ [0, 1] holds In particular, for some constants C 1 , C 2 > 0 depending on γ ∈ (0, 1) but not on h and x, it holds h (iii) See Lemma 15 of with p = 2q − 1 
It is sufficient to show the statement for n = 1 by independence of the Y i   Then, the number of points per bin is m = p/T   For simplicity, the index i is skipped in the following  First, we write Q k as a matrix-vector product and refactor it so that it corresponds to a sum of independent scaled χ 2 random variables 
In the second step, we calculate the scaling factors  Let E (km) be a diagonal matrix with ones on the (k − 1)m + 1,    , km-th entries and otherwise zero diagonal elements  Then, By Theorem 1 of for the gamma distribution it follows where Wi,j iid  ∼ Γ(1/2, 2 f (x * k )) and such that Cov( Wi,j , Wi,h ) = Cov(W i,j , W i,h ) for j = (k − 1)p/T + 1,    , kp/T and h ∈ {1,    , p} \ {(k − 1)p/T + 1,    , kp/T } 
Let θ be the maximum difference of the observations' means in each bin  Then, θ = max are defined via quantile coupling, it holds Z k = Φ −1 {F Q( Qk )} (see   Furthermore, define the uniform random variables Let ρ = Cov(Z k , Z l )  Then, the identity implies F Z,Z (x, y) − Φ(x)Φ(y) ≥ 0 for all x, y ∈ R ⇐⇒ ρ ≥ 0, (see  
Since Cov( Qk , Ql ) ≥ 0 and the ratio of two densities is non-negative, x = − 2/m, it follows that f Q(x) is monotone decreasing for x ≥ − 2/m  Furthermore, F Q(− m/2) ≤ 0 5 for all m ∈ N as f Q(x) is right-skewed  In particular, − m/2 ≤ F −1 Q (1/2) for all m ∈ N  Finally, since f Q(− 2/m) → φ(0) for m → ∞ there is a constant c > 0 not depending on m such that
The simulation study in Section 5 is performed in the same way, but with the uniform and the gamma distribution instead of the Gaussian distribution 
Context 3:  She was active in the American Bar Association, the largest professional organization of lawyers in the United States   Initially Born was named a member of the governing council of the ABA's Individual Rights Section, eventually becoming chairperson   Born and Tucker founded the ABA Women's Caucus, the first organization of female lawyers in the ABA   She held several other senior positions in the ABA, including being named the first woman member of the ABA's Standing Committee on the Federal Judiciary   As a member of the Judiciary Committee, Born provided testimony and opinion on persons nominated for federal judgeships  In 1980 she was named chair of the committee   As chair of the committee, Born was invited to address the U S  Congress regarding the nomination of Judge Sandra Day O'Connor to the U S  Supreme Court 

In 1993, Born's name was floated as a possible candidate for Attorney General of the United States, but Janet Reno was nominated 

In July 2009, Nancy Pelosi appointed Brooksley Born as a commissioner to the Financial Crisis Inquiry Commission (FCIC) 

Born and the OTC derivatives market
Born was appointed to the CFTC on April 15, 1994, by President Bill Clinton  Due to litigation against Bankers Trust Company by Procter and Gamble and other corporate clients, Born and her team at the CFTC sought comments on the regulation of over-the-counter derivatives, a first step in the process of writing CFTC regulations to supplement the existing regulations of the Federal Reserve System,  the Options Clearing Corporation, and the National Association of Insurance Commissioners  Born was particularly concerned about swaps, financial instruments that are traded over the counter between banks, insurance companies or other funds or companies, and thus have no transparency except to the two counterparties and the counterparties' regulators, if any   CFTC regulation was strenuously opposed by Federal Reserve chairman Alan Greenspan, and by Treasury Secretaries Robert Rubin and Lawrence Summers  On May 7, 1998, former SEC Chairman Arthur Levitt joined Rubin and Greenspan in objecting to the issuance of the CFTC's concept release  Their response dismissed Born's analysis and focused on the hypothetical possibility that CFTC regulation of swaps and other OTC derivative instruments could create a "legal uncertainty" regarding such financial instruments,  hypothetically reducing the value of the instruments  They argued that the imposition of regulatory costs would "stifle financial innovation" and encourage financial capital to transfer its  transactions offshore  The disagreement between Born and the Executive Office's top economic policy advisors has been described not only as a classic Washington turf war, but also a war of ideologies,  insofar as it is possible to argue that Born's actions were consistent with Keynesian and neoclassical economics while Greenspan, Rubin, Levitt, and Summers consistently espoused neoliberal, and neoconservative policies 

In 1998, a trillion-dollar hedge fund called Long Term Capital Management (LTCM) was near collapse   Using mathematical models to calculate debt risk, LTCM used derivatives to leverage $5 billion into more than $1 trillion, doing business with fifteen of Wall Street's largest financial institutions   The derivative transactions were not regulated, nor were investors able to evaluate LTCM's exposures   Born stated, "I thought that LTCM was exactly what I had been worried about"   In the last weekend of September 1998, the President's working group was told that the entire American economy hung in the balance   After intervention by the Federal Reserve, the crisis was averted   In congressional hearings into the crisis, Greenspan acknowledged that language had been introduced into an agriculture bill that would prevent CFTC from regulating the derivatives which were at the center of the crisis that threatened the US economy   U S  Representative Maurice Hinchey (D-NY) asked "How many more failures do you think we'd have to have before some regulation in this area might be appropriate?"  In response, Greenspan brushed aside the substance of Born's warnings with the simple assertion that "the degree of supervision of regulation of the over-the-counter derivatives market is quite adequate to maintain a degree of stability in the system"  Born's warning was that there wasn't any regulation of them   Born's chief of staff, Michael Greenberger summed up Greenspan's position this way: "Greenspan didn't believe that fraud was something that needed to be enforced, and he assumed she probably did  And of course, she did "  Under heavy pressure from the financial lobby, legislation prohibiting regulation of derivatives by Born's agency was passed by the Congress



ground_truth: Decorrelating the data before running the PLS algorithm improves the performance of the algorithm.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What award did Brooksley Born receive in 2009?

Read the following text and answer briefly.

Context 3:   Born resigned on June 1, 1999 

The derivatives market continued to grow yearly throughout both terms of George W  Bush's administration  On September 15, 2008, the bankruptcy of Lehman Brothers forced a broad recognition of a financial crisis in both the US and world capital markets   As Lehman Brothers' failure temporarily reduced financial capital's confidence, a number of newspaper articles and television programs suggested that the failure's possible causes included the conflict between the CFTC and the other regulators Faiola, Anthony, Nakashima, Ellen and Drew, Jill  The Crash: Risk and Regulation - What Went Wrong, The Washington Post, October 15, 2008 

Born declined to publicly comment on the unfolding 2008 crisis until March 2009, when she said: "The market grew so enormously, with so little oversight and regulation, that it made the financial crisis much deeper and more pervasive than it otherwise would have been " She also lamented the influence of Wall Street lobbyists on the process and the refusal of regulators to discuss even modest reforms 

An October 2009 Frontline documentary titled "The Warning"  described Born's thwarted efforts to regulate and bring transparency to the derivatives market, and the continuing opposition thereto  The program concluded with an excerpted interview with Born sounding another warning: "I think we will have continuing danger from these markets and that we will have repeats of the financial crisis -- may differ in details but there will be significant financial downturns and disasters attributed to this regulatory gap, over and over, until we learn from experience "

In 2009 Born, along with Sheila Bair of the FDIC, was awarded the John F  Kennedy Profiles in Courage Award in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis   According to Caroline Kennedy, "Brooksley Born recognized that the financial security of all Americans was being put at risk by the greed, negligence and opposition of  powerful and well connected interests     The catastrophic financial events of recent months have  proved them [Born and Sheila Bair] right "  One member of the President's working group had a change of heart about Brooksley Born   SEC Chairman Arthur Levitt stated "I've come to know her as one of the most capable, dedicated, intelligent and committed public servants that I have ever come to know", adding that "I could have done much better  I could have made a difference" in response to her warnings 

In 2010, a documentary film Inside Job further alleged that derivatives regulation was ineffective from the Clinton administration on  Along with fellow whistleblower, former IMF Chief Economist Raghuram Rajan, who was also scorned by the economic establishment, Brooksley Born was cited as one of the authorities arguing that financial derivatives increase economic risk 

 Personal life 
Born is married to Alexander E  Bennett (also retired from Arnold & Porter)   She has five adult children - two from a previous marriage to Jacob Landau and three stepchildren  Notably, Born was named a partner at Arnold & Porter while working part-time so she could raise her two young children   When both of her children were school-age, Born returned to practice full-time 

References

External links
Attorney profile at Arnold & Porter
Brooksley Born (2009 Winner) of the Profiles in Courage Award, with acceptance speech transcript and NECN video

Profile at MarketsWiki
Speeches and statements
"Testimony Of Brooksley Born Chairperson of the CFTC Concerning The Over-The-Counter Derivatives Market", before the House Committee On Banking And Financial Services, July 24, 1998 
"The Lessons of Long Term Capital Management L P ", Remarks of Brooksley Born, Chairperson of the CFTC, Chicago-Kent-IIT Commodities Law Institute, Chicago, Illinois, October 15, 1998 
 Interview: Brooksley Born for "PBS Frontline: The Warning", PBS, (streaming VIDEO 1 hour), October 20, 2009 
Articles
Manuel Roig-Franzia  "Credit Crisis Cassandra:Brooksley Born's Unheeded Warning Is a Rueful Echo 10 Years On", The Washington Post, May 26, 2009
 Taibbi, Matt  "The Great American Bubble Machine", Rolling Stone'', July 9–23, 2009

1940 births
American women lawyers
Arnold & Porter people
Clinton administration personnel
Columbus School of Law faculty
Commodity Futures Trading Commission personnel
Heads of United States federal agencies
Lawyers from San Francisco
Living people
Stanford Law School alumni
21st-century American women
Stanford University alumni



ground_truth: In 2009, Brooksley Born received the John F. Kennedy Profiles in Courage Award.