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Home » Van Halen - Deluxe [6LP Box] (180 Gram, includes Tokyo Dome Live In Concert, Van Halen, and 1984)
Van Halen - Deluxe [6LP Box] (180 Gram, includes Tokyo Dome Live In Concert, Van Halen, and 1984)
VAN HALEN DELUXE VAN HALEN // 1984 // TOKYO DOME LIVE IN CONCERT Limited Vinyl 6LP Box Set
TOKYO DOME LIVE IN CONCERT is set to become Van Halen's first-ever live album to feature original singer David Lee Roth. Recorded on June 21, 2013 at the famed Tokyo Dome in Tokyo, Japan, the album includes 23 songs, spanning all seven of the band's albums with Roth.
Van Halen reinvented the sound of hard rock in 1978 with its influential and wildly successful eponymous debut album and then achieved even greater success six years later with 1984. Both albums would define a generation while selling more than 10 million copies each in the U.S. alone.
TOKYO DOME LIVE IN CONCERT features an electrifying performance by singer David Lee Roth, guitarist Eddie Van Halen, drummer Alex Van Halen, and bassist Wolfgang Van Halen. Mixed by award-winning engineer and mixer Bob Clearmountain (Rolling Stones, Bruce Springsteen, Roxy Music, Nine Inch Nails, and many more), and mastered by Chris Bellman, the album features such classic anthems as "Dance The Night Away," "Beautiful Girls," "And The Cradle Will Rock…" "Unchained," and "Everybody Wants Some!!" as well as blistering versions of "Tattoo," "She's The Woman," and "China Town" from the group's most recent studio album, A Different Kind of Truth.
This deluxe collection features Tokyo Dome Live In Concert and the newly remastered Van Halen and 1984 albums into a four-CD set and a six-LP vinyl set in special packaging.
1. Runnin' With The Devil (2015 Remastered Version)
2. Eruption (2015 Remastered Version)
3. You Really Got Me (2015 Remastered Version)
4. Ain't Talkin' 'Bout Love (2015 Remastered Version)
5. I'm The One (2015 Remastered Version)
1. Jamie's Cryin' (2015 Remastered Version)
2. Atomic Punk (2015 Remastered Version)
3. Feel Your Love Tonight (2015 Remastered Version)
4. Little Dreamer (2015 Remastered Version)
5. Ice Cream Man (2015 Remastered Version)
6. On Fire (2015 Remastered Version)
1. 1984 (2015 Remastered Version)
2. Jump (2015 Remastered Version)
3. Panama (2015 Remastered Version)
4. Top Jimmy (2015 Remastered Version)
5. Drop Dead Legs (2015 Remastered Version)
1. Hot For Teacher (2015 Remastered Version)
2. I'll Wait (2015 Remastered Version)
3. Girl Gone Bad (2015 Remastered Version)
4. House Of Pain (2015 Remastered Version)
1. Unchained (Live At The Tokyo Dome June 21, 2013)
2. Runnin' With The Devil (Live At The Tokyo Dome June 21, 2013)
3. She's The Woman (Live At The Tokyo Dome June 21, 2013)
4. I'm The One (Live At The Tokyo Dome June 21, 2013)
1. Tattoo (Live At The Tokyo Dome June 21, 2013)
2. Everybody Wants Some!! (Live At The Tokyo Dome June 21, 2013)
1. Somebody Get Me A Doctor (Live At The Tokyo Dome June 21, 2013)
2. China Town (Live At The Tokyo Dome June 21, 2013)
3. Hear About It Later (Live At The Tokyo Dome June 21, 2013)
1. (Oh) Pretty Woman (Live At The Tokyo Dome June 21, 2013)
2. Drum Struck (Live At The Tokyo Dome June 21, 2013)
3. You Really Got Me (Live At The Tokyo Dome June 21, 2013)
1. Dance The Night Away (Live At The Tokyo Dome June 21, 2013)
2. I'll Wait (Live At The Tokyo Dome June 21, 2013)
3. And The Cradle Will Rock... (Live At The Tokyo Dome June 21, 2013)
Disc: 10
1. Hot For Teacher (Live At The Tokyo Dome June 21, 2013)
2. Women In Love... (Live At The Tokyo Dome June 21, 2013)
3. Romeo Delight (Live At The Tokyo Dome June 21, 2013)
1. Mean Street (Live At The Tokyo Dome June 21, 2013)
2. Beautiful Girls (Live At The Tokyo Dome June 21, 2013)
3. Ice Cream Man (Live At The Tokyo Dome June 21, 2013)
4. Panama (Live At The Tokyo Dome June 21, 2013)
1. Eruption (Live At The Tokyo Dome June 21, 2013)
2. Ain't Talkin' 'Bout Love (Live At The Tokyo Dome June 21, 2013)
3. Jump (Live At The Tokyo Dome June 21, 2013)
2015 Rhino Records
The Big Chill Original Soundtrack [LP] Marvin Gaye, Temptations, Aretha, Smokey $ 21.98
The Velvet Underground - New York Rehearsal 1966: The Factory Broadcasts [2LP] Gatefold, import $ 36.98
Megadeth - Live In Los Angeles 1995 [LP] $ 29.98
Doctor Who - Evil Of The Daleks (Original Soundtrack) [4LP] Limited Red Colored Vinyl $ 89.98


The Classic Dog hot dog steamer is the economical version of the Dog Hut with many of the same advanced features. Also built with heavy-duty 20 gauge stainless steel construction makes, it compact size still holds 192 hot dogs or 96 dogs and 30 buns.
Actual Dimensions WxDxH: 13" x 16" x 14.5"
Shipping Dimensions WxDxH: 19" x 22" x 19"


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Events and trends that matter
Social Media Meets the Corporation
One of the best social media panels I've heard in a while came in an unexpected context: A seminar sponsored by software vendor Visible Path, on Monday at the Olympic Club in San Francisco.
The developing corporate uses of social media are fascinating: Once top-down companies are now embracing a range of applications to increase collaboration and encourage innovation. The session simply reinforced my perception that we're rapidly moving beyond blogs and wikis to a variety of new tools that will both flatten the organization and blur former corporate boundaries.
Socialtext's always-insightful Ross Mayfield said that social media offers corporations the chance to rethink the way they approach decision-making. "What would happen," he asked rhetorically, "if we decoupled information rights and decision rights?" That is, what if everybody had access to the same information, so it wouldn't just be top decision-makers who hoarded all the data? And what would happen if a particular decision was made more participatory – again, in a process enhanced by social media?
Motorola's director of Internet and Collaborative Technology, Matt Beveridge, talked extensively about his company's uses of blogs and wikis, but pointed out that the biggest challenge is integration: data is spread across too many disparate tools. Forrester's Jeremiah Owyang pointed out a second problem: Information context, making sure the right data is asked in the right setting. He gave the example of Facebook, which asked for intimate information he wouldn't normally want to give in a business environment.
That's a problem that Anthony Lye, Oracle's Senior Vice President of CRM On Demand, said his company's new offering would address. Traditional Customer Relationship Management software, he said, was simply an enforcement function. That "sales 1.0" process could be summed up as, "Report more, sell less," he stated flatly. But Lye said that the "sales 2.0" process is the reverse: Sell more, report less. "Reporting should be a by-product of productivity," he claimed. However, he wouldn't give more specifics in advance of the company's announcement – though he was scheduled to talk at the Sales 2.0 conference the next day.
Visible Path's offering is interesting: A "social map" of contacts, illustrating the strength of ties by watching a person's Microsoft Outlook activity such as email traffic and calendar entries. These kinds of tools are increasingly being integrated into salesforce automation tools like Salesforce.com, to increase the value of individual and corporate social networks.
Gary - Sounds like it was definitely a "better than usual" panel, very interesting thoughts. Need to catch up on the re-birth of Visible Path. Seeing them mentioned in the Suite Two offering at the Enterprise 2.0 Conference, I would've fallen off my chair, except I was standing at the time! I'd spent a good 18 months pretty deeply intertwined in the personal and corporate social networking phenomenon in 2003/2004, including the suspects at the time, Visible Path, Contact Networks, SRD (since acquired by IBM), offerings coming from the KM world (and typically labeled as "expertise location" services). Great that the story is continuing to unfold.
Don't know the context of your quote from Ross, but this decoupling, or more likely, just sharing of the information wealth is what open source, agile development, prediction markets, and the raft of topics covered in Wikinomics are largely about, right? But like many things, the message hasn't quite penetrated into the mass market, and as much as I'm a fan of emergent behavior and collective intelligence, I don't think it works in all situations.
Great description on Sales 1.0 vs Sales 2.0, although perhaps we're on Sales 3.0, because I would argue that much Sales activity is not reported on, simply floundering, random luck; 2.0 added reporting and early adoption of CRM; and 3.0 is simply embedding "smarts" inteh system and the output is accomplished just by doing the work.
If you follow TRIZ (the Theory of Inventive Problem Solving), that's called a move towards "ideality" - where the system disappears, and the ultimate result happens as fluidly as possible. Thank god usability is getting baked into systems as we move into "2.0" mode - it's about time we left the cruft of overcomplicated interfaces behind!
Director of Market Intelligence at AIIM.org
Posted by: Dan Keldsen | October 31, 2007 at 06:38 PM
Pop!Tech launches "Accelerator", names HIV/AIDS project Pop!Tech's major domo Andrew Zolli launched "The Pop!Tech Accelerator" Friday, an initiative he said is geared to put world changing ideas into action by bringing new tools and approaches to global issues with the assistance of the network of attendees...
TED08@Aspen: Tastes Great, Less Local Let's say you're Chris Andersen, steward of the TED conferences. Your high-class problem is that way more than 1000 people want to attend your annual Monterey event, paying $6000 more than a year in advance. (You do the math.) In...
Gary Bolles
Dan Keldsen: Gary - Sounds like it was definitely a better t... | more »
On Social Media Meets the Corporation
Graeme Thickins: Great recap, Gary. Thanks -- youre the best!... | more »
On D5: Jobs & Gates
James: Hi Gary, I invite you to the beta if you want t... | more »
On Demo07: Opening Morning Session

redpajama_set_name: RedPajamaCommonCrawl


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Глубоковский — русская фамилия, и топоним.

Известные носители
 Глубоковский, Борис Александрович (1894 — после 1932) — актёр московского Камерного театра, журналист, прозаик, драматург, режиссёр.
 Глубоковский, Матвей Никанорович (1857—1903) — русский журналист, издатель, изобретатель.
 Глубоковский, Михаил Константинович (1948) — доктор биологических наук, депутат Государственной Думы.

Населённые пункты
 Глубоковский — хутор в Серафимовичском районе Волгоградской области.
 Глубоковский — посёлок в Суворовском районе Тульской области.

Другое
 Глубоковский район — район на севере Восточно-Казахстанской области в Казахстане.
 Глубоковский сельский совет — входит в состав Харьковского района Харьковской области Украины.
 Глубоковский сельсовет — административно-территориальное и муниципальное образование в Завьяловском районе Алтайского края России.
 Глубоковский сельсовет - административно-территориальная единица в Коношском районе Архангельской области.

redpajama_set_name: RedPajamaWikipedia


A Cat's Meow by Cotton Ginny's 7"x 11"Cat Price is for pattern only.
Blue Jean Bunnies by Cotton Ginny's 16" Rabbit Price is for pattern only.
Bonnie by Patches & Stitches Price is for pattern only.
Dala Horse by Rosalie Quinlan 18" Tall Price is for pattern only.
Gwen & George by Hatched & Patched 53cm Tall Price is for pattern only.
Henrietta 21" Country Chook by Raggedy Stitches Price is for pattern only.
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The August 28 monsoon has turned out to be one of the worst in recent history, perhaps extended history. A condo building in downtown Phoenix lost its roof, several trees were uprooted, and a friend of mine caught a pic of a palm tree flattening a car near ASU. Add to this that the 100 mile/hour winds in parts of the city rivaled a Category 1 hurricane and well, you've got a city in shambles.
The following photos came from here.


The title of this blog is perhaps misleading; the subtitle, hopefully less so. Regarding the latter, I've been a fan of rock music for decades, with my roots in that much-maligned subcategory known as "classic rock" but venturing from there into grunge and, principally, heavy metal. But in particular I'm fascinated by the history of rock music, the twists and turns of its evolution over its more than six decades of existence, and the interplay between rock and the world it inhabits.
This blog is an opportunity to voice my thoughts on these topics, while hopefully avoiding sounding too pretentious. I doubt there will be much here that hasn't been more artfully expressed by others, so I'll do my best to credit the guilty party where I'm sure what I've typed isn't a product of my weird, meandering mind.
As for the title, it came from a line in Cameron Crowe's 2000 movie "Almost Famous", in which the young protagonist scribe is meeting with rock journalist Lester Bangs for the first time. Bangs, taken by the kid's passion and enthusiasm, tells him to "give me a thousand words on Black Sabbath." I've heard a story since then about how "a thousand words on Black Sabbath" has from time to time been an actual assignment given to aspiring music scribblers. The story may be apocryphal, but hey…. it's good nonetheless.


Fresh off wowing Wall Street with its its fourth quarter financial results, Apple is rolling out updates to its consumer-oriented desktop and notebook lines just in time for the holiday buying season…plus introduced a new multi-touch "Magic Mouse" that handles multi-finger gestures for things like scrolling and panning—without the need for gizmos like scroll wheels.
First up, Apple's new iMac all-in-one desktop computers feature built-in 21.5-inch and 27-inch LED-backlit displays, with a new edge-to-edge glass design and aluminum cases. The 21.5-inch model offers a 1920 by 1,080 pixel resolution, while the 27-inch display scoots all the way out to 2,560 by 1,440 pixels, making for some serious work area—and both displays offer a 178° viewing angle. In addition to big screens, the new iMacs pack some power under the hood, featuring Intel Core 2 Duo processors at speeds up to 3 GHz, 4 GB of RAM (expandable to 16 GB), Nvidia GeForce 9400M or ATI Radeon graphics, 500 GB to 1 TB hard drives, a slot-loading 8× dual-layer Super Multi DVD burner, and (in a first for an Apple desktop) an integrated SD card slot. The iMacs also sport Apple's Mini DisplayPort video output, gigabit Ethernet, 4 USB 2.0 ports, a FireWire 800 port, an integrated iSight webcam, and 802.11n and Bluetooth 2.1 wireless networking.
Pricing for the new iMacs starts at $1,199 for the low-end 21.5-inch version—although that does pack the 3 GHz processor and the 4 GB of RAM. The 27-inch version starts at $1,699 and sports 1 TB of storage and ATI Radeon HD 4670 graphics. If you really want a top-flight iMac, Apple is also offering a $1,999 version with the 27-inch display, ATI Radeon HD 4850 graphics, and a 2.66 GHz Intel Core i5 quad-core processor…and there's a built-to-order option for a 2.8 GHz Intel Core i7.
Next, Apple has revved up its diminutive Mac mini, offering three models: a $599 unit with an Intel Core 2 Duo CPU, a base 2 GB of RAM, Nvidia GeForce 9400M graphics, and a 160 GB hard drive, and a $799 Mac mini with a 2.53 GHz Core 2 Duo, 4 GB of RAM, and a 320 GH hard drive. All the minis feature five USB 2.0 ports, along with FireWire 800, gigabit Ethernet, 802.11n networking, and a SuperDrive DVD burner. Folks looking for mini servers will not be disappointed either: for $999, Apple is offering a mini pre-configured with Mac OS X Snow Leopard Server and two 500 GB hard drives…all in the same tiny case.
Apple has also refreshed its MacBook consumer notebook, adding a unibody polycarbonate case and a glass multi-touch trackpad to the unit…along with some of the same battery technology Apple packed into the 17-inch MacBook Pros to give the MacBook a claimed 7-hour battery life. The new MacBook features a 1.3-inch 1,280 by 800-pixel LED-backlit display, a 2.26 GHz Intel Core 2 Duo CPU, Nvidia GeForce 9400M graphics, 2 GB of RAM (expandable to 4 GB), a 250 GB hard drive, a slot-loading 8× dual later superdrive, a Mini DisplayPort video output, gigabit Ethernet, 802.11n Wi-Fi and Bluetooth wireless networking, an integrated iSight webcam, and two USB 2.0 ports—built-to-order options include more RAM and storage, and pricing starts at $999.
Finally, Apple also debuted the Magic Mouse, which the company claims is the world's first multi-touch mouse. Instead of using scroll wheels and a flurry of buttons, the Magic Mouse uses multitouch technology to enable users to pan, zoom, scroll, and navigate their documents and computer using gestures rather than gizmos. The Magic Mouse features a seamless no-button surface and a laser tracker for operating smoothly on a wide variety of surfaces. And, of course, it's wireless, relying on Bluetooth to talk to a computer. The Magic Mouse is shipping now with Apple's new iMacs, and will be available for $69 at Apple Stores; it can be configured via System Preferences, and requires Mac OS X 10.5.8 or newer.


Bodhisattva Statues represent the path to awalkening. Bodhisattva comprise the great assemblage of Bodhisattvas are equally powerful and equally beneficial to countless beings. Bodhisattvas are considered to be various degrees of attainment or rank relating to their level on towards buddhahood.
Our Bodhisattva statues are casted by our master artisans in Nepal according to exact tradation specification using the ancient lost wax process. Bodhisattvas statue we offer are of high qulaity and are made of copper, bronze, silver & gold plated.


Can Mrs Doyle's green tea can help you lose weight? yes it can ! this seems to be due to the presence of lots of antioxidants and nutrients that can provide lots of health benefits.
Many studies have shown that green tea can increase fat burning. Green tea has a high concentration of vitamins and minerals, as well as it's low calorie content, which can increase your metabolism so improving your body's ability to burn energy.
The large range of antioxidants , called catechins can help burn fat and boost metabolism .As Green tea is full of antioxidants which help combat cell damaging free-radicals so boosting your immunity system. Green tea also contains an ingredient known as theanine, which is a type of amino acid , which can help reducing stress and again this can help in losing weight.


16 Document de l'UIT-T L.1300 portant Recommandations des Meilleures Pratiques des Centres Ecologiques d'Informations Recommendation ITU-T L.1300: states that reducing energy consumption and GHG emissions should be considered in the design and construction of data centers, and that constant monitoring will be required to consistently manage and improve energy consumption while the data center is in operation. (Approved in January 2012) Best practices are outlined for the use, management and planning of data centers, for cooling and power equipment, for the optimum design of data centre buildings, and for the monitoring of data centers after construction. For example, applying best practice to cooling could reduce the energy consumption of a typical data center by more than 50 per cent.
17 Quel est le problème? The electronics industry has revolutionized the world: electrical and electronic products have become globally ubiquitous technologies. The widespread use of electrical and electronic equipment (EEE) has raised public awareness of its positive effects such as the reduction of the digital divide, but also of the negative environmental and health effects associated with the inefficient waste management of end-of-life electrical and electronic equipment (WEEE or e-waste). There have been alarming reports of e-waste mismanagement in many countries, particularly in less developed nations and countries with economies in transition. E-waste is a significant contributor to the ICT industry's impact on the environment, and urgent global action to address this issue is essential if the industry is to fulfill its commitment to a sustainable future.
21 Cloud Computing Recommendation ITU-T L.1300: states that reducing energy consumption and GHG emissions should be considered in the design and construction of data centers, and that constant monitoring will be required to consistently manage and improve energy consumption while the data center is in operation. (Approved in January 2012) Best practices are outlined for the use, management and planning of data centers, for cooling and power equipment, for the optimum design of data centre buildings, and for the monitoring of data centers after construction. For example, applying best practice to cooling could reduce the energy consumption of a typical data center by more than 50 per cent.
SG15 Accès Bande Large Internet de l'UIT &Normes Transport Accès: ADSL: Document UIT-T G.992 PLT: Document UIT-T G.9960 (G.hn) FTTX: GPON Document UIT-T G.984 Fibres Accommodantes: Document UIT-T G.657 Transport Optique: SDH: Document UIT-T G.707 Ethernet Transporteur : ITU-T Y.1731 Synchronisation: Liaison Terrestre Mobile Fiber cable can be very flexible, but traditional fiber's loss increases greatly if the fiber is bent with a radius smaller than around 30 mm. This creates a problem when the cable is bent around corners or wound around a spool, making FTTX installations more complicated. "Bendable fibers", targeted towards easier installation in home environments, have been standardized as ITU-T G.657.
27 Groupes Focus UIT-T NOUVELLE Télédistribution Numérique!
Etablir le Pont: de l'Innovation aux Normes Systèmes Secours, Réseau de Résilience et Relance Service Layer M2M Accessibilité Audiovisuelle Média Accessibility Driver Distraction Communication Auto - Non-members can get a taste of ITU's work by participating in our Focus Groups which may create deliverables that are eventually input into the formal standards process. - Focus Groups are an instrument created by ITU-T that augment the Study Group work programme by providing an alternative working environment for the quick development of specifications in their chosen areas. - Focus Groups are now widely used to address industry needs as they emerge, and when they are not covered within an existing Study Group. - The key difference between Study Groups and Focus Groups is the freedom that they have to organize and finance themselves. Focus Groups can be created very quickly, are usually short-lived and can choose their own working methods, leadership, financing, and types of deliverables.
Prochaine Kaléidoscope Construire des Communautés Durables Université Kyoto, Avril 2013 Dépôt des Communications Date butoire: 10 Septembre 2012 3ème dièdre de Jules Verne Université de l'Exposition Locale 25 Avril – atelier sur l'Education des Normes Kaleidoscope 2013 will be held at Kyoto University from 22 to 24 April. This academic conference is calling for original academic papers offering innovative and bold approaches in research and development to build smart, ethical, and sustainable communities. In addition to an exhibition of local universities' curricula, the presence of high-level keynote speakers, and the presentation of invited papers, Kaleidoscope 2013 will host the third Jules Verne's corner. On 25 April a Workshop on Standards Education will be held at the same venue.
17-21 Septembre 2012 – Paris, France Organisée conjointement avec Microsoft et TechAmerica Objectif Principal: Sensibiliser sur l'importance et les opportunités à utiliser les normes TIC pour construite une économie verte et améliorer la gestion de l'énergie durable.
Télécharger ppt "Vijay Mauree, Coordonnateur de Programme, TSB/UIT"


A gymnast growing up, Huckaby is a favorite for gold in two snowboarding events in South Korea.
With a background in gymnastics, Brenna Huckaby's first reaction when she tried snowboarding was to compare the board to the balance beam she'd spent so much time mastering.
Her first time learning to ride wasn't easy, and her mother Kristie Huckaby watched her struggle through it on a rehabilitation trip to the National Ability Center in Utah. Brenna was part of a group from MD Anderson in Houston, where she received treatment in 2010 for osteocarcoma, a type of cancer that develops in bone. Her right leg was amputated after complications with medication and chemotherapy.
Instructors at the National Ability Center initially wanted Brenna to try skiing, but she insisted on snowboarding, according to her mom.
"I knew when she put her mind to it, that she was going to do it. That's what she does," Kristie Huckaby said.
By the end of the trip, Kristie Huckaby saw the light in Brenna's eyes as she gained more confidence riding.
"That's what I wanted to see every time I saw her," Kristie said. "She jumped off that snowboard, (and said) 'Mama, it's like a beam.' When she said those words, I knew that was it.
Now 22, Brenna Huckaby is one of the top riders on the U.S. team that will compete in the Paralympics when snowboarding begins Monday. She's a favorite for gold in two events, snowboardcross and banked slalom in Pyeongchang.
The path to her first Paralympics began in Baton Rouge, La., where she took up gymnastics as a child.
Snowboarding provided Brenna somewhat of a fresh start in life after months of undergoing cancer treatment, her mother says.
After Brenna lost her right leg, Kristie Huckaby helped put things into perspective for her daughter. "You've got to redefine yourself, find a new life. And I knew that I would help her in that journey," Kristie Huckaby said.
The journey brought mother and daughter back to Utah, where they settled in so Brenna could snowboard. She began competing in 2013, while training with the National Ability Center. At her first national championships in 2014, Huckaby finished third.
In the months leading up to the Paralympics, Huckaby has further established herself as the one to beat. She won a crystal globe as the overall World Cup winner in her division, SB-LL1, after winning snowboardcross and banked slalom races in February at Big White in British Columbia.
The world outside snowboarding has also taken notice of Huckaby's story and rise in the sport. In February she was featured on the cover of Parents – she and fiancé Tristan have a daughter Lilah, who will be 2 in May. Huckaby was also a model in the recent Sports Illustrated swimsuit edition.


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Burglars cause misery for residents as crime wave continues in Crosby
THIEVES are still on the prowl in Crosby after a further eight burglaries were reported in the past week.
Cars, jewellery and small electrical goods were stolen in a wave of crime.
There have also been number plates stolen from three cars in the area.
Burglars got into a home on Vogan Avenue, Crosby, through an unlocked kitchen door on Monday, January 10 between 5.30pm and 11pm.
They took a handbag along with car keys before stealing a silver Golf car.
Jewellery and two televisions were taken from a house on Eshe Road North, Blundellsands, between 1.40pm and 3pm last Wednesday, after offenders forced a side door to get in.
A Vauxhall Astra was stolen from outside a house on Esplen Avenue, Crosby, later that day between 3pm and 6pm.
The thief got in through an unlocked front door and stole the car keys.
Thieves got in through an unlocked rear door on the same day, between midday and 3.45pm, on Thorndale Road, Waterloo. They stole an iPod, the iPod dock and a PlayStation 3.
A handbag and mobile phone were stolen that day from a house on Seafield Avenue, Crosby, at around 5.30pm.
Another home on Edgemoor Drive, Thornton, was targeted between January 7 and January 14 but nothing was taken. Burglars smashed rear patio doors to enter a home on Dowhills Road, Blundellsands and stole a Mercedes. It happened between 1.50pm and 7.10pm on January 14.
A Ford Focus was also taken from Mersey Road, Blundellsands, between midnight and 8.30pm on Sunday.
Sergeant Keith Francis, said: "We would like to reassure people that we have dedicated extra resources in the area to tray and tackle the problem. We would encourage people to be vigilant and to do everything they can to make it difficult for burglars, such as making sure doors and windows are locked at all times."
Number plates were ripped off cars on Brooklands Avenue and The Serpentine.
One was taken from a Renault Clio on Brooklands Avenue between January 8 and January 9.
A few days later plates were stolen from a Renault Kangoo and a Citroen C5, which were parked on The Serpentine, on January 11.
Crosby Police can install security screws on number plates to prevent incidents.
Sgt Francis said: "We would ask people to be aware and if they see anyone suspicious hanging around their vehicles to contact police."
l TO report a crime call Crosby Police on 0151 777 3634 or Crimestoppers on 0800 555 111.
Money Saving ExpertITV Good Morning Britain's Martin Lewis warning all parents should knowEven families with younger children should start doing this now
AnglingAngling: Bigger the betterJoe Farrell brings you all the latest angling news


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↑ Eron, Carol (1979). "Women in Medicine and Health Care". In O'Neill, Lois Decker. The Women's Book of World Records and Achievements. Anchor Press. p. 204. ISBN 0-385-12733-2. "First Hindu Woman Doctor"
↑ "Historical Photos Depict Women Medical Pioneers", Public Radio International, 2013-07-12. Retrieved on 2013-10-29.
↑ "IRDS Awards 2011". Irdsindia.com. Retrieved 29 October 2013. "आनन्दाबाई कष्टपरम्परायां सत्यां महता परिश्रमेण अमेरिकादेशस्य पेनिल्वेनियाविश्वविद्यालयतः १९ शतके एव आधुनिके वैद्यकीयशिक्षणे स्नातकोत्तरपदवीं प्राप्नोत् ।"
Mrs. Caroline Healey Dall (1888). The Life of Dr. Anandabai Joshee. Roberts Brothers, Boston. http://archive.org/stream/lifeofdrananabai00dalliala#page/n7/mode/2up.


My husband will be in California for the summer for work, and the whole family is tagging along. Bum luck, huh? Although summer in Maine is a shame to miss, too! We've been to San Diego twice before; here are some of our favorite spots. I highly recommend that you check some of them out!
Probably one of the best known tourist destinations in San Diego, Balboa Park is a collection of museums and attractions housed in beautiful buildings interspersed with native plants. The famous San Diego Zoo is located here and is definitely worth a visit. Let's be honest, you can relieve your wallet of quite a bit of cash at Balboa Park, but you can also spend a very pleasant day with a picnic just strolling around the grounds. The Timken Art Museum is always free, and we were able to get into a couple of museums for free with our ATSC reciprocal membership card from back home. A bus will drop you right out front if you are using public transportation.
Old Town San Diego is one of my favorite destinations, featuring historic buildings and really capturing an "old west" flavor. It is a great learning opportunity and many of the buildings are free to visit; kids eat free in some of the restaurants as well. Again, if you are using public transportation, this is a great destination with a transit center right outside.
It's a quick stop, but if you're any kind of train or architecture fan, it's worth a look. You can easily work it in on your way to or from the airport.
You know you're from Maine when you go to a place like California, known for perpetually sunny weather, and immediately start looking for indoor play areas because, among other things, it's "too hot" to be outside, ha! Actually, since we were there long term for work, I needed a place where I could work, as well. Kidventures has a parent's cafe in the middle with video monitors to every room, so I can write and watch them as well. It's not so much a tourist attraction, but it is a great place to play! I highly recommend it.
If you want to get out of the city, try a visit to the Anza-Borrego Desert. The drive to the desert was itself a brand new experience, passing farms (including a camel farm!), vineyards, and boulder strewn hillsides unlike anything I've ever seen. Check out the visitor's center, really a mini museum. When we were there, the valley where the visitor's center is located featured many large dinosaur sculptures, another fun feature to look at with the kids.
Julian is an old gold-mining town that makes a fun day trip. Wander the shops and make sure to get a slice of pie at the Julian Pie Company!
These suggestions just scratch the surface of fun places to visit in and around San Diego, California! If you're familiar with the area or are planning a trip, I'd love to hear from you in the comments!
The post California, Here We Come: Best Places to Visit in San Diego appeared first on Thinking Outside The Sandbox Family DIY, Recipes, Autism, Kids.


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text:
Jul 7 VIDEO (C-SPAN): L.A. Times Supreme Court correspondent: 'Scalia/Ginsburg' libretto is 'wonderful'
In a July 2, 2015 panel discussion on C-SPAN, David Savage (Supreme Court Correspondent for the Los Angeles Times) calls the Scalia/Ginsburg libretto "wonderful" — and follows up on his June 2015 article about the Scalia/Ginsburg friendship and opera:
VIDEO: David Savage of the Los Angeles Times describes Derrick Wang's opera 'Scalia/Ginsburg' in a panel discussion with fellow Supreme Court correspondents, moderated by Arthur Spitzer (Legal Director, ACLU of the National Capital Area). (July 2, 2015) (Duration: 3:07)
TRANSCRIPT (EXCERPT):
Arthur Spitzer (Moderator; Legal Director, ACLU of the National Capital Area):
We now have an opera that's going to have its world premiere [on Saturday, July 11, 2015] called Scalia/Ginsburg. David, you wrote a piece about them and the opera [on June 22: "BFFs Ruth Bader Ginsburg and Antonin Scalia Agree to Disagree"]…
David Savage (Supreme Court Correspondent, Los Angeles Times):
The [editor] who was running this [story in the LA Times] called and said, "Do you normally cover opera?" He wanted me to come out and write about some other operas. And I said, "No, I don't…"
As far as the opera…[the libretto of Scalia/Ginsburg] is just wonderful to read! And the fellow Derrick Wang who [created] this has footnoted all of it with actual citations of [the Justices'] statements. I mean, he's used their words. Scalia is singing about how "you justices are blind, you find things in the Constitution," and then he gets imprisoned for "excessive dissenting" — and then Ruth Ginsburg breaks through a glass ceiling to…rescue him.
Anyway, I have not seen it yet. It's July 11, [17,] and 19, but it sounds like — maybe I will switch to covering operas, it sounds delightful.
A.S.:
I think a line I got from reading your story was that the composer of the opera said that a lot of what he heard Scalia saying dissenting from the bench reminded him of these rage arias in Italian opera and it sounded like a very good character.
D.S.:
[Derrick Wang, the composer and librettist of Scalia/Ginsburg,] was a music major at Harvard and at Yale and then went to law school, and he actually says he was reading these dissents and says "Now here's an operatic character: Scalia." So that was the origin of the opera.…
VIDEO: Supreme Court Correspondents discuss Derrick Wang's opera 'Scalia/Ginsburg', C-SPAN [excerpt from 'Reporters on 2014 Supreme Court Term,' July 2, 2015]
David Savage, "BFFs Ruth Bader Ginsburg and Antonin Scalia Agree to Disagree, Los Angeles Times (June 22, 2015)
David Savage, Los Angeles Times, Scalia/Ginsburg, Scalia, Ginsburg, Antonin Scalia, Ruth Bader Ginsburg, RBG, opera, Derrick Wang, libretto, C-SPAN
Jul 8 Justice Ginsburg: 'Scalia/Ginsburg' 'quite funny,' Derrick Wang 'very talented' (Washington Post)
Jul 6 Dallas Opera CEO: 'Scalia/Ginsburg' is 'one of the most intriguing concepts' of 2015 opera season


text:
Holder Visits Tower of London, Talks About 21st-Century Torture
Nick Gillespie | 4.27.2009 8:32 AM
United States Attorney General Eric Holder visited the Tower of London during a European swing through three cities during which he's "seeking allies' help to close Guantanamo Bay, fight terrorism, and catch cyber-criminals."
The attorney general did not say how much longer he thought it would take to relocate the Guantanamo [Bay] detainees. Before officials can meet President Barack Obama's January deadline, the U.S. must first decide which detainees to put on trial and which to release to the U.S. or other countries.
Holder said the first step is to decide how many total detainees will be freed.
"We're doing these all on a rolling basis," he said. "I think we're probably relatively close to making some calls."
Whole story here.
Below, Glenn Greenwald waterboards Obama's iffy stance on civil liberties. And here, Reason.tv's Michael C. Moynihnan asking whether the president's statements about reversing Bush admin detention policies are just a "head fake":
NEXT: Vernor Vinge on the Singularity and the End of Humanity & Christmas Cards & More
Nick Gillespie is an editor at large at Reason and host of The Reason Interview With Nick Gillespie.
PoliticsCivil LibertiesCultureWorldTerrorismReason.tvBarack Obama


text:
V. A. Shepelev
Institute of Radio Astronomy, Kharkiv
V. N. Melnik
R. V. Vashchishin
Gravimetrical Observatory of IGP
Домашня сторінка > Том 31 (2018) >	Shepelev
PRELIMINARY RESULTS OF INTERFEROMETRIC OBSERVATIONS OF THE QUIET SUN AT THE FREQUENCIES 8 – 32 MHZ
V. A. Shepelev, V. N. Melnik, R. V. Vashchishin
We studied a radio emission of the quiet Sun at decameter wavelengths to determine its angular dimensions. Observations were conducted using interferometers composed of antenna sections of the UTR-2 radio telescope. A number of interferometers with baselines of 225 to 1400 m, oriented along the meridian, operated at two frequencies of 20 and 25 MHz with a bandwidth 250 kHz. We used them to estimate a shape of a source brightness distribution. To measure the angular size of the Sun in a wide frequency band of 8-32 MHz another set of interferometers was used. It consisted of two pairs of the interferometers with the baselines of 225 and 450 m elongated both along the parallel and the meridian. They used digital wideband receivers operating at the frequencies of 8 to 32 MHz to record received signals. There was some sporadic solar activity during observational campaign in March-April 2015 that complicated determination of parameters of the quiet Sun radiation. A large number of radio frequency interference also noted. To mitigate their influence we applied а few data cleaning algorithms. The use of the algorithms in spectral strips with a low level of interferences in periods free from powerful sporadic solar radiation allowed us to make estimation of the angular size of the quiet Sun in two orthogonal directions at some separate frequencies in the range of 20 to 32 MHz. Calculation of the dimensions was performed in an assumption of Gaussian radio brightness distribution of the source. Our estimates of the quiet Sun sizes agree well with the data of other authors. The simplicity and convenience of determining the angular size using the data obtained with two interferometers with different baselines are noted. Continuation such study will allow to extend the frequency range down to the lowest frequencies where observations from the Earth's surface are possible and give us a possibility to study how the angular size of the quiet Sun depend on a phase of solar activity. This study will allow us to construct a model of solar corona at distances of 1.5-3 solar radii directly from the radio observations and to find out how the effective temperature of the Sun changes with the frequency.
quiet Sun; angular diameter; interferometer; decameter range
Abranin E.P. and Bazelian L.L.: 1986, Preprint #304, (in Russian), Kharkov, IRE NASU
Aubier M., Leblanc Y.,Boischot A.: 1971, Astron. & Astrophys., 12, 435
Braude S.Ya., Megn A.V. Sodin L.G.: 1978, An-tenny iss 26, (in Russian), Moskow, Sviaz
Megn A.V., Braude S.Ya., Rashkovsky S.L. et al.: 1997, Radiophys. and Radioastron.( in Russian), 2, 4, 385
Melnik V.N., Shepelev V.A., Poedts S. et al.: 2018, Solar Phys., 293, 97
Rashkovsky S.L., Belov A.S., Ivanov et al.: 2012, Radiophys. and Radioastron.( in Russian), 17, 3, 207
Stanislavsky A.A., Koval A.A., Konovalenko A.A: 2013, Astron. Nachr, 334, 10, 1086
Subramanian K.R.: 2004, Astron. & Astroph, 426, 329
Shepelev V.A.: 2015, Radiophys. and Radioastron.( in Russian), 20, 1, 20
Zakharenko V., Konovalenko A.A., Zarka P. et al.: 2016, Astron. Instrum, 5, 4


text:
\subsection{RNN}

Now from \eqref{eq:rnn1}, we can write
\begin{align}
(\mathbf{X}_t|\mathbf{X}_{t-1}{=}\mathbf{x}_{t-1},...,\mathbf{X}_{1}{=}\mathbf{x}_{1};\mathbf{X}_{0}{=}\mathbf{x}_{0}) &= f(\mathbf{x}_{t-1}) - \Delta t  (\mathbf{r}_t|\mathbf{X}_{t-1}{=}\mathbf{x}_{t-1},...,\mathbf{X}_{1}{=}\mathbf{x}_{1};\mathbf{X}_{0}{=}\mathbf{x}_{0})
\end{align}

and given this relationship between $\mathbf{X}_t$ and $\mathbf{r}_t$, a change of variables for the likelihood gives
\begin{equation}
    \log\mathrm{Pr}(\mathbf{x}_t|\mathbf{x}_{t-1},...,\mathbf{x}_{1};\mathbf{x}_{0}) = \log\mathrm{Pr}(\mathbf{r}_t|\mathbf{r}_{t-1},...,\mathbf{r}_{1}) -K\log(\Delta t) \label{change_of_variables}
\end{equation}
\eqref{general_loglik_decomp} is therefore
\begin{align}
\begin{split}
     &= \log\mathrm{Pr}(\mathbf{r}_1) +  \log\mathrm{Pr}(\mathbf{r}_2|\mathbf{r}_1)  +  \log\mathrm{Pr}(\mathbf{r}_3|\mathbf{r}_2,\mathbf{r}_1)  \\{}& \qquad \qquad  + ... + \log\mathrm{Pr}(\mathbf{r}_{n}|\mathbf{r}_{n-1:1}) - Kn\log(\Delta t)
\end{split} \label{eq:rnn_deriv_0} \\
&=\log\mathrm{Pr}(\mathbf{r}_1) + \sum_{t=2}^{n}\log\mathrm{Pr}(\mathbf{r}_{t}|\mathbf{l}_{t})  - Kn\log(\Delta t) \label{eq:rnn_deriv_1}\\
&=  \sum_{k=1}^K\Big(\log\mathrm{Pr}(r_{k,1}) + \sum_{t=2}^{n}\log\mathrm{Pr}(r_{k,t}|l_{k,t})
  \Big)
 - Kn\log(\Delta t) \label{eq:rnn_deriv_2}
\end{align}
where \eqref{eq:rnn_deriv_1}--\eqref{eq:rnn_deriv_2} follow from the independencies in the graphical model (Figure \ref{rnn_architecture}).

\subsection{Polynomial Model}

For the polynomial model, the log-likelihood of the $\mathbf{X}_t$ sequence is
\begin{linenomath*}
\begin{align}
    \log\mathrm{Pr}(\mathbf{x}_1,...,\mathbf{x}_n;\mathbf{x}_0)
 &= \sum_{k=1}^{K}\Big(\log\mathrm{Pr}(h_{k,1})+  \sum_{t=1}^{n-1}\log\mathrm{Pr}(h_{k,t+1}|h_{k,t})\Big) - Kn\log( \Delta t)
\end{align}
\end{linenomath*}
and the derivation follows below.

The graphical model is shown in Figure \ref{fig:other_graphical_models}a. From \eqref{eq:poly}, we can write
\begin{linenomath*}
\begin{align}
(\mathbf{X}_t|\mathbf{X}_{t-1}{=}\mathbf{x}_{t-1},...,\mathbf{X}_{1}{=}\mathbf{x}_{1};\mathbf{X}_{0}{=}\mathbf{x}_{0}) &= f(\mathbf{x}_{t-1}) - \Delta t  (\mathbf{h}_t|\mathbf{X}_{t-1}{=}\mathbf{x}_{t-1},...,\mathbf{X}_{1}{=}\mathbf{x}_{1};\mathbf{X}_{0}{=}\mathbf{x}_{0})
\end{align}
\end{linenomath*}
and given this relationship between $\mathbf{X}_t$ and $\mathbf{h}_t$, a change of variables for the likelihood gives
\begin{linenomath*}
\begin{equation}
    \log\mathrm{Pr}(\mathbf{x}_t|\mathbf{x}_{t-1},...,\mathbf{x}_{1};\mathbf{x}_{0}) = \log\mathrm{Pr}(\mathbf{h}_t|\mathbf{h}_{t-1},...,\mathbf{h}_{1}) -K\log(\Delta t) \label{change_of_variables_poly}
\end{equation}
\end{linenomath*}
\eqref{general_loglik_decomp} is therefore \begin{linenomath*}
\begin{align}
\begin{split}
     &= \log\mathrm{Pr}(\mathbf{h}_1) +  \log\mathrm{Pr}(\mathbf{h}_2|\mathbf{h}_1)  +  \log\mathrm{Pr}(\mathbf{h}_3|\mathbf{h}_2,\mathbf{h}_1)  \\{}& \qquad \qquad  + ... + \log\mathrm{Pr}(\mathbf{h}_{n}|\mathbf{h}_{n-1:1}) - Kn\log(\Delta t)
\end{split} \\
    &= \log\mathrm{Pr}(\mathbf{h}_1) + \sum_{t=1}^{n-1}\log\mathrm{Pr}(\mathbf{h}_{t+1}|\mathbf{h}_{t}) - Kn\log(\Delta t) \label{poly_temp} \\
 &= \sum_{k=1}^{K}\Big(\log\mathrm{Pr}(h_{k,1})+  \sum_{t=1}^{n-1}\log\mathrm{Pr}(h_{k,t+1}|h_{k,t})\Big) - Kn\log(\Delta t) \label{poly_spatial_depend}
\end{align}
\end{linenomath*}
where \eqref{poly_temp}--\eqref{poly_spatial_depend} follow from the independencies described in \eqref{eq:red}.

\begin{figure*}
    \includegraphics[width=300pt, trim = 1cm 7cm 1.2cm 1cm]{figs/chapter_app/james_paper_other_models.pdf}
    \caption{Graphical model for (a) Polynomial and (b) GAN with added white noise.}
    \label{fig:other_graphical_models}
\end{figure*}

\subsection{GAN}

We approximate the GAN's likelihood by calculating the likelihood of a model which functions and gives results almost identical to the GAN (one with a small amount of  white noise added) using importance sampling and the reparameterization method as in the Variational Autoencoder \cite{vae}. Here, \eqref{gan1} is altered to
\begin{linenomath*}
\begin{align}
X_{k,t+1} &= X_{k,t} + \omega_k(\mathbf{X}_t) - \Delta t\ u(X_{k,t},h_{k,t+1}) \label{eq:gan3}
\end{align}
\end{linenomath*}
where $u(X_{k,t},h_{k,t+1}) = U(X_{k,t},h_{k,t+1}) + N_{k,t+1}$ and $N_{k,t} \sim \mathcal{N}(0,0.001^2)$. The log-likelihood is
\begin{linenomath*}
\begin{align}
      \log\mathrm{Pr}(\mathbf{x}_1,...,\mathbf{x}_n;\mathbf{x}_0)   &= \log\mathbb{E}_{\mathbf{h} \sim \mathrm{Pr}_\mathbf{\tilde{H}}.} \frac{\mathrm{Pr}(\mathbf{u}.|\mathbf{h}.,\mathbf{x}_0) \mathrm{Pr}_\mathbf{H.}(\mathbf{h}.)}
    {\mathrm{Pr}_\mathbf{\tilde{H}.}(\mathbf{h}.)}
    -Kn\log(\Delta t)
    \notag
\end{align}
\end{linenomath*}
where the expectation can now be approximated by a large sum if there is a suitable importance sampler $\mathrm{Pr}_\mathbf{\tilde{H}.}(\mathbf{h}.)$. 50 samples were used for the importance sampling, and this was repeated 50 times to give 95\% confidence intervals. We note that given the finite number of samples taken it is of course possible that the GAN likelihood is larger, but a well-trained importance sampler should minimise the chance of this. The derivation and form of the importance sampler is detailed below.

The graphical model of the GAN with a small amount of white noise added is shown in Figure \ref{fig:other_graphical_models}b. $\mathbf{X}_t$ is related to $\mathbf{u}_{t}$ as in \eqref{eq:gan3} so
\begin{linenomath*}
\begin{align}
(\mathbf{X}_t|\mathbf{X}_{t-1}{=}\mathbf{x}_{t-1},...,\mathbf{X}_{1}{=}\mathbf{x}_{1};\mathbf{X}_{0}{=}\mathbf{x}_{0}) &= f(\mathbf{x}_{t-1}) - \Delta t  (\mathbf{u}_t|\mathbf{X}_{t-1}{=}\mathbf{x}_{t-1},...,\mathbf{X}_{1}{=}\mathbf{x}_{1};\mathbf{X}_{0}{=}\mathbf{x}_{0})
\end{align}
\end{linenomath*}
and a change of variables for likelihood gives
\begin{linenomath*}
\begin{equation}
    \log\mathrm{Pr}(\mathbf{x}_t|\mathbf{x}_{t-1},...,\mathbf{x}_{1};\mathbf{x}_{0}) = \log\mathrm{Pr}(\mathbf{u}_t|\mathbf{u}_{t-1},...,\mathbf{u}_{1};\mathbf{x}_0) -K\log(\Delta t)
\end{equation}
\end{linenomath*}
\eqref{general_loglik_decomp} is therefore
\begin{linenomath*}
\begin{align}
\begin{split}
     &=  \log\mathrm{Pr}(\mathbf{u}_1|\mathbf{x}_0) +   \log\mathrm{Pr}(\mathbf{u}_2|\mathbf{u}_1,\mathbf{x}_0)  +  \log\mathrm{Pr}(\mathbf{u}_3|\mathbf{u}_2,\mathbf{u}_1,\mathbf{x}_0)  \\{}& \qquad \qquad  + ... + \log\mathrm{Pr}(\mathbf{u}_{n}|\mathbf{u}_{n-1:0},\mathbf{x}_0) -Kn\log(\Delta t)\notag
\end{split} \\
    &= \log\mathrm{Pr}(\mathbf{u}.|\mathbf{x}_0) -Kn\log(\Delta t) \notag
\end{align}
\end{linenomath*}
and just decomposing the first term below:
\begin{linenomath*}
\begin{align}
    \log\mathrm{Pr}(\mathbf{u}.|\mathbf{x}_0) &= \log\mathbb{E}_{\mathbf{h} \sim \mathrm{Pr}_\mathbf{H}.}\mathrm{Pr}(\mathbf{u}.|\mathbf{h}.,\mathbf{x}_0) \notag\\
    &= \log\mathbb{E}_{\mathbf{h} \sim \mathrm{Pr}_\mathbf{\tilde{H}}.} \frac{\mathrm{Pr}(\mathbf{u}.|\mathbf{h}.,\mathbf{x}_0) \mathrm{Pr}_\mathbf{H.}(\mathbf{h}.)}
    {\mathrm{Pr}_\mathbf{\tilde{H}.}(\mathbf{h}.)} \label{approx_lik_gan}
\end{align}
\end{linenomath*}
where \eqref{approx_lik_gan} can be approximated with a sufficiently large sum given a good enough encoder $\mathrm{Pr}_\mathbf{\tilde{H}.}(\mathbf{h}.)$.

For training purposes, the lower-bound is used:
\begin{linenomath*}
\begin{align}
    &\geq \mathbb{E}_{\mathbf{h} \sim \mathrm{Pr}_\mathbf{\tilde{H}}.} \log\frac{\mathrm{Pr}(\mathbf{u}.|\mathbf{h}.,\mathbf{x}_0) \mathrm{Pr}_\mathbf{H.}(\mathbf{h}.)}
    {\mathrm{Pr}_\mathbf{\tilde{H}.}(\mathbf{h}.)} \label{elbo_gan}
\end{align}
\end{linenomath*}

The terms in the numerator decompose as follows, using the independencies from the graphical model and associated equations:
\begin{linenomath*}
\begin{align*}
   \log\mathrm{Pr}(\mathbf{u}.|\mathbf{h}.,\mathbf{x}_0) &=  \sum_{t=1}^{n}\log\mathrm{Pr}(\mathbf{u}_t|\mathbf{h}_t,\mathbf{x}_{t-1}) \\
   &=  \sum_{t=1}^{n}\sum_{k=1}^{K}\log\mathrm{Pr}(u_{k,t}|h_{k,t},x_{k,t-1})
\end{align*}
\end{linenomath*}
and
\begin{linenomath*}
\begin{align*}
   \log\mathrm{Pr}_\mathbf{H.}(\mathbf{h}.)&= \log \mathrm{Pr}_\mathbf{H}(\mathbf{h}_1) +  \sum_{t=2}^{n}\log\mathrm{Pr}_\mathbf{H}(\mathbf{h}_t|\mathbf{h}_{t-1})  \\
   &= \sum_{k=1}^{K} \Big(\log \mathrm{Pr}_{H}(h_{k,1}) +  \sum_{t=2}^{n}\log\mathrm{Pr}_H(h_{k,t}|h_{k,t-1}) \Big)
\end{align*}
\end{linenomath*}

The perfect importance sampling distribution would be proportional to $\mathrm{Pr}(\mathbf{h}.|\mathbf{u}.,\mathbf{x}_0)$. Therefore we want $\mathrm{Pr}_\mathbf{\tilde{H}.}(\mathbf{h}.)$ to allow for the same dependencies. We choose it by carrying out the following steps:
\begin{linenomath*}
\begin{align}
   \log\mathrm{Pr}_\mathbf{\tilde{H}.}(\mathbf{h}.)&\propto \log\mathrm{Pr}(\mathbf{h}.|\mathbf{u}.,\mathbf{x}_0) \notag \\
   &= \log\mathrm{Pr}_{\mathbf{H}_1}(\mathbf{h}_1|\mathbf{u}.,\mathbf{x}_0) + \sum_{t=2}^{n}\log\mathrm{Pr}_{\mathbf{H}_t}(\mathbf{h}_t|\mathbf{h}_{t-1},\mathbf{u}.,\mathbf{x}_0) \label{is_1}
\end{align}
\end{linenomath*}
where \eqref{is_1} follows from using the laws of probability to decompose the term above, followed by applications of the independencies from the graphical model. Note that although the process is Markov it is not necessarily time-homogeneous. To deal with this, an RNN is used to model $\mathrm{Pr}_{\mathbf{H}_t}(\mathbf{h}_t|\mathbf{h}_{t-1},\mathbf{u}.,\mathbf{x}_0)$ as $\mathrm{Pr}_\mathbf{H}(\mathbf{h}_t|\mathbf{h}_{t-1},\mathbf{u}.,\mathbf{x}_0,\mathbf{s}_t)$ where the state $\mathbf{s}_t$ can in principle account for the non-homogeneous updating procedure of $\mathbf{h}_t$. The log-likelihood is therefore
\begin{linenomath*}
\begin{align}
  &= \sum_{k=1}^K \Big( \log\mathrm{Pr}_{H_1}(h_{k,1}|u_{k.},x_{k,0}) + \sum_{t=2}^{n}\log\mathrm{Pr}_{H_t}(h_{k,t}|h_{k,t-1},u_{k.},x_{k.},s_{k,t}) \Big) \label{is_2}
\end{align}
\end{linenomath*}
where the spatial independencies introduced in \eqref{is_2} follow from the graphical model and $u_{k,.} = u_{k,1:n}$ and $x_{k,.} = x_{k,0:n-1}$. The only simplifying assumption used here is that the same update model for $h_{k,t}$ is used for each component $k$. Finally, the sequence of $(u_{k.},x_{k.})$ is summarised using a bidirectional RNN to give $w_{k,t}$. Therefore, the final form of the log-likelihood of our importance sampler is
\begin{linenomath*}
\begin{align}
  &= \sum_{k=1}^K \Big( \log\mathrm{Pr}_{H_1}(h_{k,1}|w_{k,1},x_{k,0}) + \sum_{t=2}^{n}\log\mathrm{Pr}_{H_t}(h_{k,t}|h_{k,t-1},w_{k,t}, x_{k,t-1},s_{k,t}) \Big)
\end{align}
\end{linenomath*}

The training of the importance sampler is done by maximising \eqref{elbo_gan}, with the learnable parameters being those of the model used to learn $h_{k,1}$, the RNN used to model the update of $h_{k,t}$, and the bidirectional RNN.

\section{Introduction}
\label{chapter_intro}
A major source of inaccuracies in climate models is due to `unresolved' processes. These occur at scales smaller than the resolution of the climate model (`sub-grid' scales) but still have key effects on the overall climate. In fact, most of the inter-model spread in how much global surface temperatures increase after $\textrm{CO}{_2}$ concentrations double is due to the representation of clouds \citep{schneider_clouds,zelinka2020causes}. The typical approach to deal with the problem of unresolved processes has been to model the \textit{effects} of these unresolved processes as a function of the resolved ones. This is known as `parameterization'.

Red noise is a key feature in many parameterization schemes \citep{christensen_regimes,johnson2019seas5,molteni2011new,palmer2009stochastic,skamarock2019description,walters2019met}. \citet{hasselmann1976stochastic} developed the theory underpinning this, showing that the coupling of processes with different time-scales leads to red noise signals in the longer time-scale process, analogous to what takes place in Brownian motion. In parameterization, the resolved processes typically have far longer time-scales than the unresolved ones, motivating the inclusion of red noise in these schemes. The usefulness of tracking the history of the system for parameterization follows from the importance of red noise.

We use probabilistic machine learning to propose a data-driven successor to red noise in parameterization. The theory explaining the prevalence of red noise in the climate was developed in an idealized model, within the framework of physics-based differential equations. Such approaches may be intractable outside of the idealized case. Recently, there has been much work looking at uncovering relationships from \textit{data} instead \citep{arcomano2022hybrid,beucler2020towards,beucler2021enforcing,bolton2019applications,brenowitz2018prognostic,brenowitz2019spatially,chattopadhyay2020data,gan_hannah,gentine,krasnopolsky,ogorman_2018,rasp2018deep,vlachas2022multiscale,yuval_rf,yuval2021}. We develop a Recurrent Neural Network in a probabilistic framework, combining the benefits of stochasticity with the ability to learn temporal patterns in more flexible ways than permitted by red noise. We use the Lorenz 96 model \citep{lorenz1996predictability}, henceforth the L96, as a proof-of-concept. Our model is competitive with, and often outperforms, a bespoke baseline. It also generalises to unseen scenarios.

\subsection{Numerical Models}

Numerical models represent the state of the Earth system at time $t$ by a state vector $\mathbf{X}_t$. The components of $\mathbf{X}_t$ may include, for example, the mean temperature and humidity at time $t$ in every cell of a grid that covers the Earth. The goal is to parameterize the effects of unresolved processes in such a way that the model can reproduce the evolution of this finite-dimensional representation of the state of the real Earth system. A simple way to model the evolution would be
\begin{linenomath*}
\postdisplaypenalty=0
\begin{align}
    X_{k,t+1} &= X_{k,t} + f_k(\mathbf{X}_t) \label{eq:model0}
\end{align}
\end{linenomath*}
where $X_{k,t}$ is the value that the state variable $X$ takes at the spatial coordinate $k$ and time point $t$ and $X_{k,t} \in \mathbb{R}^d$, $\mathbf{X}_t \in \mathbb{R}^{dK}$, and $f$ is a function for the updating process.

\subsection{Stochastic Schemes}

Stochasticity can be included using the following form
\begin{linenomath*}
\postdisplaypenalty=0
\begin{align}
    X_{k,t+1} &= X_{k,t} + f_k(\mathbf{X}_t,h_{k,t+1}) \label{eq:model1} \\
    h_{k,t+1} &= \beta(h_{k,t},z_{k,t+1}) \label{eq:hidden}
    \\
    z_{k,t} &\sim \mathcal{N}(0,I)     \label{eq:noise}
\end{align}
\end{linenomath*}
where invented hidden variables (discussed below) are denoted $h_{k,t} \in \mathbb{R}^H$ and are tracked through time, $\beta$ is a function for updating these, and $z_{k,t} \in \mathbb{R}^Z$ is a source of stochasticity.

Introducing stochasticity through \eqref{eq:model1} improved on models of the form \eqref{eq:model0}. Results included better ensemble forecasts \citep{buizza1999stochastic,leutbecher2017stochastic,palmer2012stochastic}, and improvements to model mean state \citep{berner2012systematic} and climate variability \citep{christensen2017stochastic}. The motivation for using stochasticity comes from the understanding that the effects of the unresolved (sub-grid) processes cannot be effectively predicted as a deterministic function of the resolved ones due to a lack of scale separation between them. Stochasticity allows us to capture our uncertainty about those aspects of the unresolved processes which may affect the resolved outcomes.

\subsection{Hidden Variables and Red Noise}

Hidden variables --- $h_{k,t}$ in~\eqref{eq:model1} and \eqref{eq:hidden} --- are defined here as being variables separate to the observed state, $X_{k,t}$, which if tracked help better model $X_{k,t}$. Using them is key to numerical weather and climate models using stochastic parameterizations, allowing temporal correlations to be better modelled. Red noise results when the hidden variables evolve with a first-order autoregressive (AR1) process such as
\begin{linenomath*}
\begin{equation}
    h_{k,t+1} =\phi h_{k,t} +  \sigma (1-\phi^2)^{1/2}z_{k,t+1} \label{eq:red}
\end{equation}
\end{linenomath*}
One example is the stochastically perturbed parameterization tendencies (SPPT) scheme \citep{buizza1999stochastic,palmer2009stochastic} which is widely used in forecasting models \citep{leutbecher2017stochastic,molteni2011new,palmer2009stochastic,sanchez2016improved,skamarock2019description,stockdale2011ecmwf}. Here, the AR1 process results in far better weather and climate forecasting skill than a simple white noise model, with good modelling of regime behaviour requiring correlated noise \citep{christensen_regimes,dawson2015simulating}.

There is no intrinsic reason why an AR1 process is the best way to deal with these correlations. It is simply a modelling choice.

\subsection{Machine Learning for Parameterization}

Learning the parameters of a climate model, either the simple form~\eqref{eq:model0} or the general form~\eqref{eq:model1}--\eqref{eq:noise}, requires deciding on a parametric form for the functions $f$ and $\beta$. For full-scale climate models, it is difficult to find appropriate functions. The machine learning (ML) approach is to \emph{learn} these from data. Various researchers have proposed ML methods for learning the deterministic model~\eqref{eq:model0} \citep{beucler2020towards,beucler2021enforcing,bolton2019applications,brenowitz2018prognostic,brenowitz2019spatially,gentine,krasnopolsky,ogorman_2018,rasp2018deep,yuval_rf,yuval2021}.

Amongst ML-trained stochastic models \citep{gan_hannah,guillaumin2021stochastic}, various ones with red noise were proposed by \citet{gan_hannah}, using Generative Adversarial Networks (GANs) \citep{gan_goodfellow}. Full details of the architecture are in \citet{gan_hannah} and we will refer to one of their best-performing models (which they call X-sml-r) as the \emph{GAN}. A wider range of generative models can be trained using such an advesarial approach as opposed to maximum likelihood (the standard way to train models in ML, discussed in Section \ref{chapter_methodology}) but such methods are notoriously unstable due to the nature of the minimax loss in training. Although they used ML to learn $f$ in \eqref{eq:model1}, it was not used to learn $\beta$, which they modelled with an AR1 process.

Recurrent Neural Networks (RNNs) are a popular ML tool for modelling temporally correlated data, eliminating the need for update functions to be manually specified. RNNs have had great success in the ML literature in a variety of sequence modelling tasks, including text generation \citep{graves2013generating,sutskever2011generating}, machine translation \citep{sutskever2014sequence} and music generation \citep{eck2002first,mogren2016c}. The state-of-the-art RNNs are gated ones, principally the long short-term memory (LSTM) networks \citep{lstm} and gated recurrent unit (GRU) networks \citep{gru}, which provide major improvements to the standard issue of unstable gradients.

Current parameterization work using ML to model temporal correlations has used deterministic approaches, including deterministic RNNs and echo state networks \citep{arcomano2022hybrid,chattopadhyay2020data,chattopadhyay2020sp,vlachas2018data,vlachas2020backpropagation,vlachas2022multiscale} and found notable success. This suggests such ML approaches are effective ways to model temporal dynamics in physical systems. However, these studies do not incorporate stochasticity.

Other off-the-shelf models are not obviously suited for the parameterization task. Transformers and attention-based models \citep{vaswani2017attention} perform well for sequences but require all the previous data to be tracked for each simulation step, providing a computational burden which increases simulation cost. Random Forests (RFs) have been used for parameterization \citep{ogorman_2018,yuval_rf} and shown to be stable at run-time (due to predicting averages form the training set) but it is not obvious how they would learn and track hidden variables.

\subsection{Overview}

The L96 set-up and baselines are presented in Section \ref{chapter_background}. Our model is detailed in Section \ref{chapter_methodology}, with the experiments and results in Section \ref{chapter_results}. This is followed by a discussion on the use of `likelihood' in evaluating probabilistic climate models (Section \ref{chapter_likelihood}), with our conclusions in Section \ref{chapter_conclusion}.

\subsection{Lorenz 96 Model}

We use the two-tier L96 model, a toy model for atmospheric circulation that is extensively used for stochastic parameterization studies \citep{hannah_bespoke,crommelin2008subgrid,gan_hannah,kwasniok2012data,rasp2020}. We use the configuration described in \citet{gan_hannah}. It consists of two scales of variables: a large, low-frequency variable, $X_k$, coupled to small, high-frequency variables, $Y_{j,k}$. These are dimensionless quantities, evolving as follows:
\begin{linenomath*}
\begin{equation*}
    \frac{dX_k}{dt} =\underbrace{ -X_{k-1}(X_{k-2} - X_{k+1})}_{\textrm{advection}} \underbrace{- X_{k}}_{\textrm{diffusion}} \ \underbrace{+ F}_{\textrm{forcing}} - \underbrace{\frac{hc}{b} \sum_{j=J(k-1)+1}^{kJ} Y_j}_{\textrm{coupling}}  \mkern70mu k = 1, ..., K
\end{equation*}
\begin{equation*}
    \frac{dY_{j,k}}{dt} = \underbrace{-cbY_{j+1,k}(Y_{j+2,k} - Y_{j-1,k})}_{\textrm{advection}} \: \underbrace{- \, cY_{j,k}}_{\textrm{diffusion}} \: \underbrace{ - \frac{hc}{b} X_k}_{\textrm{coupling}}   \mkern150mu j = 1, ..., J
\end{equation*}
\end{linenomath*}
where in our experiments, the number of $X_k$ variables is $K=8$, and the number of $Y_{j,k}$ variables per $X_{k}$ is $J=32$. The value of the constants are set to $h=1,\:b=10$ and $\:c=10$. These indicate that the fast variable evolves ten times faster than the slow variable and has one-tenth the amplitude.

The L96 model is good for parameterization work as we can consider the $X_k$ to be coarse processes resolved in both low-resolution and high-resolution simulators, whilst the $Y_{j,k}$ can be considered as those that can only be resolved in high-resolution, computationally expensive simulators. In this study the coupled set of L96 equations are treated as the `truth', and the aim is to learn a good model for the evolution of $X$ alone using this `truth' data. The effects of $Y_{j,k}$ must therefore be parameterized. Success would be if the modelled evolution of $X$  matched that from the truth.

The L96 is also useful as it contains separate persistent dynamical regimes, which change with different values of $F$ \citep{christensen_regimes,lorenz2006regimes}. The dominant regime exhibits a wave-two pattern around the ring of $X$ variables, whilst the rarer regime exhibits a wave-one type pattern. In the atmosphere, regimes include persistent circulation patterns like the Pacific/North American (PNA) pattern and the North Atlantic Oscillation (NAO).

It is easy to use models with no memory which do well on standard loss metrics but fail to capture this interesting regime behaviour. This is seen in the L96, where including temporally correlated (red) noise improves the statistics describing regime persistence and frequency of occurrence \citep{christensen_regimes} as well as improving weather forecasting skill \citep{hannah_bespoke,palmer2012stochastic}. Temporally correlated noise also improves regime behaviour (frequency of occurrence and persistence) in operational models \citep{dawson2015simulating}, as well as forecasting skill \citep{palmer2009stochastic}.

\subsection{Parameterization Models in the Literature}

The below models are stochastic. They use AR1 processes to model temporal correlations, but this need not be the case, as we show in Section \ref{chapter_methodology}.

\subsubsection{Stochastic Non-ML Model with Non-ML Hidden Variables}

\citet{hannah_bespoke} propose a model which we refer to as the \emph{polynomial} model (in light of the polynomial in~\eqref{eq:poly}) which is
\begin{linenomath*}
\postdisplaypenalty=0
\begin{align}
   X_{k,t+1} &= X_{k,t} + \omega_k(\mathbf{X}_t) - \Delta t\,\Bigl(a X_{k,t}^3 + b X_{k,t}^2 + c X_{k,t} + d + h_{k,t+1} \Bigr) \label{eq:poly}
\end{align}
\end{linenomath*}
where $h_{k,t}$ evolves as in \eqref{eq:red} with $h_{k,1} = \sigma z_{k,1}$, and where
\begin{linenomath*}
    \begin{equation}
    \omega_k(\mathbf{X}_t) = \lambda_k\Big(\mathbf{X}_t + \lambda(\mathbf{X}_t/2)\Big)
    \label{eq:rk} \end{equation}
\end{linenomath*}
and
\begin{linenomath*}
    \begin{equation}
    \lambda_k(\mathbf{X}_t) = \Delta t\,\Big(-X_{k-1,t}(X_{k-2,t} - X_{k+1,t}) - X_{k,t} + F\Big)
    \label{eq:lambda}
    \end{equation}
\end{linenomath*}
where \eqref{eq:rk} is the implementation of a second order Runge-Kutta method and $[\lambda(\mathbf{a})]_k=\lambda_k(\mathbf{a})$. $h_{k,t}$ only depends on $h_{k,t-1}$.

\subsubsection{Stochastic ML Model with Non-ML Hidden Variables}

The GAN from \citet{gan_hannah} replaces \eqref{eq:poly} by
\begin{linenomath*}
\postdisplaypenalty=0
\begin{align}
    X_{k,t+1} &= X_{k,t} + \omega_k(\mathbf{X}_t) - \Delta t\ U(X_{k,t},h_{k,t+1} ) \label{gan1}

\end{align}
\end{linenomath*}
where the function $U$ is implemented by a neural network (NN), $\omega$ is defined in \eqref{eq:rk}--\eqref{eq:lambda}, $h_{k,t}$ evolves as in \eqref{eq:red} with $\sigma = 1$, and $h_{k,1} = z_{k,1}$.
\subsection{Stochastic ML Model with ML Hidden Variables}

Our model, henceforth denoted the \textit{RNN}, follows the general form in \eqref{eq:model1}--\eqref{eq:noise} but splits the hidden variable into two parts: $h_{k,t} = (r_{k,t},l_{k,t})$. The model is
\begin{linenomath*}
\postdisplaypenalty=0
\begin{align}
    X_{k,t+1} &= X_{k,t} + \omega_k(\mathbf{X}_t) - \Delta t \Big(g_\theta(X_{k,t}) + r_{k,t+1} \Big) \label{eq:rnn1}\\
    r_{k,t+1} &= b_\theta(l_{k,t+1}) + \sigma z_{k,t+1} \label{eq:rnn2} \\
    l_{k,t+1} &= s_\theta(l_{k,t},r_{k,t}) \label{eq:rnn3}
\end{align}
\end{linenomath*}
where $g$, $b$ and $s$ are NNs with weights $\theta$, $z_{k,t}$ is exogenous noise as defined in \eqref{eq:noise}, $\omega$ is specified in equation \eqref{eq:rk} above (implements a second order Runge-Kutta method and expressing physical quantities like advection), and $\theta$ and $\sigma$ are parameters to be learnt. The dimensions of the NNs are: $g_\theta:\mathbb{R}^1 \rightarrow \mathbb{R}^1$, $b_\theta:\mathbb{R}^8 \rightarrow \mathbb{R}^1$ and $s_\theta:\mathbb{R}^9 \rightarrow \mathbb{R}^8$. It is structurally based on a Recurrent Neural Network (RNN) as it has hidden variables, the same update procedure is used each step, and $g$, $b$ and $s$ are NNs.

The key insight is that our model allows more flexibility than the standard AR1 processes for expressing temporal correlations. To clarify this, equation \eqref{eq:rnn1} is identical in form to \eqref{eq:poly}, but the hidden state is evolved in a more flexible manner than the AR1 process in \eqref{eq:red}.

Figure \ref{rnn_architecture} shows the mechanism of generation and the NN architectures used to learn the functions. The dotted lines denote the action of deterministic functions, with the solid lines showing the sources of stochasticity. The main architectural details are that $s$ in \eqref{eq:rnn3} is implemented using two GRU layers, each composed of four units, and $b$ in \eqref{eq:rnn2} is represented with a dense layer of size one. $g$ in \eqref{eq:rnn1} is implemented using three fully-connected layers.

Here, as well as in the baselines, the parameterization models are `spatially local' meaning that the full $\mathbf{X}_t$ vector is not taken in as input when modelling $X_{k,t+1}$ anywhere apart from in $\omega_k(\mathbf{X}_t)$. This is done to mimic parameterizations in operational weather and climate models. For all forecast models, $\Delta t = 0.005$ model time units (MTU). One MTU is approximately five atmospheric days when considering the time it takes for errors to double.

\begin{figure*}[t]
\includegraphics[width=12cm]{figs/chapter3/james_paper_rnn.pdf}
\caption{RNN graphical model showing how each $\mathbf{X}_{t}$ is generated. $\mathbf{X}_{t+1}$ is a function of $\mathbf{X}_{t}$ and $\mathbf{r}_{t+1}$, and $\mathbf{r}_t$ is a function of $\mathbf{l}_t$ and $\mathbf{z}_t$. $\mathbf{z}_t$ is the source of stochasticity. $b_{\theta}$, $s_{\theta}$ and $g_{\theta}$ are NNs. $\mathbf{l}_t$ consists of a stack of $l_{t}^1$ and $l_{t}^2$, each $\in \mathbb{R}^4$.}
\label{rnn_architecture}
\end{figure*}

\subsection{Training Probability Models Using Likelihood}
\label{chapter_submethod_likelihood}

The RNN is probabilistic and trained using likelihood. The `likelihood of a sequence' is denoted $\mathrm{Pr}(x_1,x_2,...,x_n )$ and can be interpreted as the probability assigned to a given sequence of variables $(x_1,x_2,...,x_n)$ by a given model.

For our RNN, the log-likelihood (the natural logarithm of the likelihood) of the sequence of $\mathbf{X}_t$ is
\begin{linenomath*}
\begin{equation*}
    \log\mathrm{Pr}(\mathbf{x}_1,...,\mathbf{x}_n;\mathbf{x}_0)
 =  \sum_{k=1}^K\Big(\log\mathrm{Pr}(r_{k,1}) + \sum_{t=2}^{n}\log\mathrm{Pr}(r_{k,t}|l_{k,t})
  \Big)
 - Kn\log(\Delta t)
\end{equation*}
\end{linenomath*}
where $r$ and $l$ are deterministic functions of $\mathbf{x}_0,...,\mathbf{x}_n$ derived from equations \eqref{eq:rnn1}--\eqref{eq:rnn3}. The full derivation is in \ref{chapter_appendix}.

A common way to train probabilistic models is to maximise the likelihood of the training data. This involves finding a set of parameters ($\theta$ and  $\sigma$ here) which maximise the likelihood. Other enhancements to training involve including regularisation penalties such as dropout. The explicit form of the RNN's likelihood makes training relatively easy.

\subsection{RNN Training}

Truth data for training was created by running the full L96 model five separate times with the following values of $F$: $(19,20,20.5,21,21.5)$. We gave our model data from different forcing scenarios so it could learn how perturbations in the forcing may affect the evolution of $\mathbf{X}_t$. The GAN and polynomial were trained only on $F = 20$ as was done by their authors. The truth data was created by solving the two-level L96 equations using a fourth-order Runge-Kutta timestepping scheme and a fixed time step $\Delta t = 0.001$ MTU, with the output saved at every 0.005 MTU. A training set of length 2,500 MTU was assembled from the truth data, consisting of three equal 500 MTU components with $F = (19,20.5,21)$ and one 1,000 MTU component with $F = 20$, with a $F=21.5$ set of length 500 MTU kept as a validation hold-out set.

The RNN was trained using truncated back propagation through time on sequences of length 700 time steps for 100 epochs with a batch size of 32 using Adam \citep{kingma2014adam}, with $\theta$ and $\sigma$ being the learnable parameters. A variable learning rate was used, starting at 0.0001 for the first 70 epochs and decayed to 0.00003 for the remaining 30. The model parameters which gave the lowest loss on the validation set were saved.
\subsection{Weather Evaluation}

The models were evaluated in a weather forecast framework. 745 initial conditions were randomly selected from the truth attractor and an ensemble of 40 forecasts each lasting 3.5 MTU were produced from each initial condition. Figure \ref{weather_figure} shows the spread and error terms for these experiments over time. The error is defined as
\begin{linenomath*}
\begin{equation*}
    \text{error}(t) = \sqrt{\frac{1}{M}\sum_{m=1}^{M}\big(X_{m}^{O} (t)-\overline{X_{m}^{\text{sample}}(t)}\big)^2}
\end{equation*}
\end{linenomath*}
where there are $M$ different initial conditions, $X_{m}^{O} (t)$ is the observed state at time $t$ for the $m^{\text{th}}$ initial condition, and $\overline{X_{m}^{\text{sample}}(t)}$ is the ensemble mean forecast at time $t$, initialized at $t_{init}$, such that $t = t_{init} + \tau$.

The spread is defined as
\begin{linenomath*}
\begin{equation*}
    \text{spread}(t) = \sqrt{\frac{1}{M}\sum_{m=1}^{M}
    \frac{1}{N}\sum_{n=1}^{N}
    \big(X_{m,n}(t)-
    \overline{X_{m}^{\text{sample}}(t)}\big)^2}
\end{equation*}
\end{linenomath*}
where $N$ is the number of ensemble members and $X_{m,n}(t)$ is the state of the $n^{\text{th}}$ member at time $t$ for the $m^{\text{th}}$ initial condition.

As noted by \citet{leutbecher2008ensemble}, for a perfectly reliable forecast (defined as one where $X_{m,n}(t)$ and $X_{m}^{O} (t)$ are independent samples from the same distribution), for large $N$ and $M$ the error should be equal to the spread. A reduction in error indicates an ensemble forecast that better tracks observations, and a spread/error ratio close to one indicates a reliable forecast. Figure \ref{weather_figure} shows that the polynomial and RNN have the smallest errors, but the RNN has the better spread/error ratio after 0.75 MTU. The GAN is underdispersive, with the greatest error but smallest spread/error ratio, suggesting it is overconfident in its predictions.

\begin{figure*}[t]
\includegraphics[width=250pt]{figs/chapter4/weather_plot.pdf}
\caption{Error and spread for weather experiments. We expect a smaller error for a more accurate forecast model. The spread/error ratio would be close to one in a `perfectly reliable' forecast.}
\label{weather_figure}
\end{figure*}

\subsection{Climate Evaluation}

The ability to simulate the climate of the L96 was evaluated using the 50,000 MTU simulations. The histograms in Figure \ref{fig:climate_hist}a are approximations of each model's marginal distribution of $X_{k,t}$. Successful reproduction of the L96 climate would result in a model's histogram matching the truth model's. Qualitatively, all the models are similar to the truth.

\begin{figure*}[t]
    \includegraphics[width=400pt]{figs/chapter4/climate_eval_histograms.pdf}

    \caption{Histograms of $X_{k,t}$. (a) Comparisons for the full simulation data. (b) Depiction which shows sampling error. The truth histogram is shown in green and \textbf{bold}. The closer a model's histogram to the truth, the better it models the probability density of $X_{k,t}$.}
    \label{fig:climate_hist}
\end{figure*}

We can quantitatively compare how well histograms match the truth using KL-divergence. We compute this as follows: we discretize our modelled continuous distribution of a variable by binning into a histogram, and do the same for the truth model. We then evaluate the KL-divergence between $q(a)$, the distribution described by the histogram of a variable $A$ in the true L96 model, and $p(a)$, the distribution described by the histogram of that variable in our model
\begin{linenomath*}
 \begin{equation*}
 \text{KL}(q(a)||p(a)) = \sum_{a} q(a) \log \frac{q(a)}{p(a)}
 \end{equation*}
\end{linenomath*}
where the sum is over all the discrete values which $A$ takes. The smaller this measure, the better the goodness-of-fit.  The results (Table \ref{tab:kl} column one) confirm that the RNN best matches the truth model here.

\begin{table*}
\caption{\textit{KL-divergence (goodness-of-fit) between 1) the truth and 2) the distributions shown in the noted Figures. The smaller the KL-divergence, the better the match between the true and modelled distributions. The best model in each case is shown in \textbf{bold}}.}
\label{tab:kl}
\centering
\begin{tabular}{llllllll}
\hline
Model      & Figure \ref{fig:climate_hist}a        & Figure \ref{fig:regimes}     & Figure \ref{fig:regimes_decomposed}a      & Figure \ref{fig:regimes_decomposed}b               & Figure \ref{fig:regimes_clim_change}a       & Figure \ref{fig:regimes_clim_change}b       \\ \hline
Polynomial & 0.004          & 3.9          & 0.03 & 0.040          & - & -          \\
RNN        & \textbf{0.001} & \textbf{1.8} & \textbf{0.02}          & \textbf{0.004}          & \textbf{0.008}          & 0.006          \\
GAN        & 0.010          & 11.6         & 0.14          & 0.050                & 0.009          & \textbf{0.004}
\end{tabular}
\end{table*}

It it is also important to check whether the histograms are composed of enough data such that they are good representations of each model's actual distribution of $X_{k,t}$. Figure \ref{fig:climate_hist}b seeks to answer this by accounting for sampling error. By splitting a particular model's simulation run into five equally sized sets and plotting these histograms on top of each other, variability can be made apparent. Having done this, there is no noticeable difference between Figure \ref{fig:climate_hist}a and b, suggesting that the error from the sampling process is small relative to the differences between each model.

The L96 model used here displays two distinct regimes with separate dynamics. We use the approach from \citet{christensen_regimes} to examine the regimes. First, the time series are temporally smoothed with a running average over 0.4 MTU to help identify regimes \citep{stephenson2004existence,straus2007circulation}.  The dimensionality of the system is then reduced using Principal Component Analysis, as often done when studying atmospheric data \citep{selten2004preferred,straus2007circulation}. For the truth time series, the components $PC1$ and $PC2$ are degenerate and are in phase quadrature, representing wavenumber two oscillations. $PC3$ and $PC4$ are also degenerate and in phase quadrature, representing wavenumber one oscillations. All model runs are projected onto the truth model's components. Given the degeneracies, the magnitude of the principal component vectors, $||[PC1,PC2]||$ and $||[PC3,PC4]||$ are computed and histograms of the system are plotted in this space.

\begin{figure*}[t]
    \includegraphics[width=350pt]{figs/chapter4/regime_plots.pdf}
    \caption{Regime characteristics of the L96. 2D histograms show the magnitudes of the projections of $X_{k,t}$ on to the principle components from the truth series.}
    \label{fig:regimes}
\end{figure*}

The presence of two regimes is apparent in Figure \ref{fig:regimes}a where there is a large maximum corresponding to the major regime, A, but also a smaller peak corresponding to the minor regime, B. The polynomial fails to capture the two regimes, whereas the ML models succeed. The RNN puts an appropriate amount of density in Regime B, whilst the GAN puts too much. Comparison is made easier by decomposing the 2D Figure \ref{fig:regimes} into two, 1D density plots (Figure \ref{fig:regimes_decomposed}). We can use the KL-divergence to measure how well our models match the truth across all three histograms. Table \ref{tab:kl} shows the RNN best matches the truth and so captures the regime characteristics best. There are still deficiencies though as it does not put enough density in the right-hand side tail of Figure \ref{fig:regimes_decomposed}a.

\begin{figure*}[t]
    \includegraphics[width=350pt]{figs/chapter4/non_clim_change_regime_hists.pdf}
    \caption{Density plots for each regime for $F=20$. The truth is shown in green and \textbf{bold}. (a) Magnitude of [$PC1$,$PC2$]. (b) Magnitude of [$PC3$,$PC4$]. }
    \label{fig:regimes_decomposed}
\end{figure*}

\subsection{Generalisation Experiments}

We set the L96 forcing to $F =28, 32, 35$ and $40$, and examine how the models can capture changes due to varying external forcings. These $F$ values are notably different to those in the training data so allow us to test generalisation. For example, the change from $F = 21.5$ (validation set) to $F=28$ results in the following changes to the regime structure: the centroid location of the rarer regime shifts to higher values of $||[PC1,PC2]||$ and $||[PC3,PC4]||$ (Figure \ref{fig:regimes_diff_21_5}), and the proportion of time spent in the rarer regime increases from $38\%$ to $50\%$. The range of $X_{k,t}$ also increases from $[-12,19]$ to $[-24,29]$. The differences in wave behaviour between regimes means the above changes result in different system dynamics.

\begin{figure}[t]
    \includegraphics[width=0.5\textwidth]{figs/chapter5/clim_change_diff.pdf}
    \caption{Difference between the $F=21.5$ and $ F = 28$ densities in principal component space. The principal components are those from the $F=20$ truth series.}
    \label{fig:regimes_diff_21_5}
\end{figure}

In all the below experiments the polynomial model exploded so is omitted. This is merely due to the specification of a third-order polynomial. It significantly deviates from its target values when $X_{k,t}$ values notably different from those in training (such as $X_{k,t} \geq 19$) are taken as inputs.

The to simulate the $F = 28$ climate was explored in a similar manner to the $F=20$ one. First, for the histograms of $X_{k,t}$ (analogous to Figure \ref{fig:climate_hist}) the RNN and GAN both had small KL divergences (0.0015 vs 0.0025). Next, the principal component projections were examined (analogous to Figure \ref{fig:regimes}). The same components as determined from the $F=20$ truth data set are used. In this space, the RNN has a smaller KL-divergence (0.6) than the GAN (0.7). Figures for the above two plots are omitted due to it being hard to visually distinguish the models. As before, further comparison can be made by examining the two, 1D density plots in Figure \ref{fig:regimes_clim_change} (analogous to Figure \ref{fig:regimes_decomposed}). Both models perform well (KL divergence in Table \ref{tab:kl}), but neither put enough density in the right-hand side tails.

\begin{figure*}[t]
    \includegraphics[width=350pt]{figs/chapter5/clim_change_regime_hists.pdf}
    \caption{Density plots for each regime for $F=28$. The truth is shown in green and \textbf{bold}. (a) Magnitude of [$PC1$,$PC2$]. (b) Magnitude of [$PC3$,$PC4$]. The right-hand side tails are not giving sufficient density by the RNN nor the GAN.}
    \label{fig:regimes_clim_change}
\end{figure*}

The models were also evaluated in the weather forecasting framework for $F=28, 32, 35$ and $40$. 750 initial conditions were randomly selected from the truth attractors and an ensemble of 40 forecasts each lasting 2.5 MTU were produced. Figure \ref{fig:weather_generalise} shows the RNN and GAN have similar errors, but the RNN's spread is best matched with its error, whereas the GAN continues to be underdispersive.

\begin{figure*}[t]
    \includegraphics[width=350pt]{figs/chapter5/weather_plots_generalise.pdf}
    \caption{Error and spread for weather experiments for varying forcing values. The RNN's spread is best matched to its error unlike the GAN.}
    \label{fig:weather_generalise}
\end{figure*}

\subsection{Computational Cost}

We wish our models to have a lower cost than the full L96. The computational cost is calculated by considering the number of floating-point operations per simulated time step ($\Delta t = 0.005$). The RNN  (8,682) is notably cheaper than the truth model (88,000) and the GAN (14,074), so meets the computational cost objective of parameterization work. The polynomial (334) is the cheapest.

\subsection{Hold-out Likelihood}

We also evaluate using the likelihood (explained in Section \ref{chapter_submethod_likelihood}) of hold-out data. This is a standard approach in the ML literature. Evaluation is done on hold-out sets to ensure that overly complex models which just overfit the training data are not selected.

Here, hold-out sets for $F=20$ and $F=28$ were created by taking a 10,000 MTU subset from the 50,000 MTU truth sets created in Section \ref{chapter_results}. The hold-out log-likelihoods are shown in Table \ref{tab:loglik_comparisons}. The likelihood for the polynomial model and its derivation is in \ref{chapter_appendix}. We also present an approach to approximate the GAN likelihood. This is despite its full form being intractable (involving integrals which cannot be efficiently approximated using Monte Carlo sampling) and so it not being typically used to evaluate GANs. This is also detailed in \ref{chapter_appendix}. The RNN has a lower likelihood on $F=28$ than the polynomial, despite the polynomial exploding. This is discussed below.

\subsection{What is the Use of Likelihood}

\subsubsection{For Evaluation}

Other metrics only capture snapshots of model performance. Likelihood is a composite measure which assesses a model's full joint distribution. For example, in the univariate case, the mean-squared-error and variance of a model tell two separate things about its performance, with implicit assumptions being made about variables being normally distributed (whenever mean-squared-error is used). The likelihood captures both of these, and without such assumptions.

Likelihood also captures information about more complex, joint distributions, saving the need for custom metrics to be invented to assess specific features. To illustrate this, consider we wish to assess a model's temporal correlation. The likelihood already contains this information. Although custom metrics could be invented to assess this, the likelihood is an off-the-shelf metric which is already available. We suggest it is wasteful not to use it.

Being a composite metric brings challenges though. Poor performance in certain aspects may be overshadowed by good performance elsewhere. This can result in cases where increased hold-out likelihoods do not correspond to better sample quality, as noted in the ML literature  \citep{gan_goodfellow,flow_gan,eval_generative_models,zhao2020bridging} and seen in our results: the RNN has a worse $F=28$ hold-out likelihood than the polynomial, yet the polynomial model explodes unlike the RNN. In this case, the phenomenon of `explosion' is not significantly penalising the likelihood. Likelihood is therefore a complement, not replacement, of other metrics which are important to the end-user.

\subsubsection{For Diagnostics}

Likelihood's composite nature makes it a helpful diagnostic tool. Just like KL-divergence, the further away a model is --- in any manner --- from the data in the hold-out set, the worse the likelihood will be. If a model has a poor likelihood despite performing well on a range of standard metrics, this suggests there are still deficiencies in the model which need investigating. For example, with the RNN at $F=28$, the poor average likelihood is caused by a tiny number of segments of the $X_{k,t}$ sequences being extremely poorly modelled (and therefore having extremely poor likelihoods assigned). On inspection, the issue is due to the choice of a fixed $\sigma$ in equation \eqref{eq:rnn2} --- this is appropriate for most of the time series, but for parts which are difficult to model it is too small, preventing the model from expressing sufficient uncertainty. This could be rectified by allowing $\sigma$ to vary, and we suggest future work explore this.

\begin{table}
\caption{\textit{Log-likelihood on hold-out data. The RNN has the best likelihood in both cases.}}
\label{tab:loglik_comparisons}
\centering
\begin{tabular}{lll}
\hline
Model        & $F = 20$       & $F = 28$     \\ \hline
Polynomial   & $4.98$        & $\textbf{4.05}$          \\
RNN & \textbf{6.53} & $2.69$ \\
GAN   & $\approx -37.69 \pm 0.01$            & $\approx -147.02 \pm 0.08$
\end{tabular}
\end{table}
\subsection{Conclusion}

We have shown how to build on the benefits of red noise in parameterizations by using a probabilistic ML approach based on an RNN. This now needs testing on more complex systems --- both to specifically improve on AR1 processes, and to learn new, more flexible models of $\mathbf{X}_t$. An interesting area for further work is to examine what information is tracked in the hidden variables of the RNN, and how this relates to the timescales of the required memory in the modelled system. The field is ripe for other probabilistic ML tools to be used and we suspect that further customisation of these will lead to many improved parameterization models.

Likelihood can often be fairly easily calculated, and where this be the case, we propose that the community also evaluate the hold-out likelihood for any devised probabilistic model (ML or otherwise). It is a useful debugging tool, assessing the full joint distribution of a model. It would also provide a consistent evaluation metric across the literature. Given the challenges relating to sample quality, likelihood should complement not replace existing metrics.

Finally, we have used demanding tests to show that ML models can generalise to unseen scenarios. We cannot hope for ML models to generalise to all settings. We do not expect this from our physics-based models either. But ML models \emph{can} generalise outside of their training realm. And it is by using challenging hold-out tests that we can assess their ability to do so, and if they fail, begin the diagnosis of why.

\section{Parameterization in the Lorenz 96}
\input{chapter2}

\section{Our Proposed RNN Model}
\input{chapter3}

\section{Results}
\input{chapter4}

\section{Evaluation and Diagnostics Using Likelihood}
\input{chapter6}

\conclusions[Discussion and Conclusion]
\input{chapter7}

\codedataavailability{All code used in this study (including the code required to create the data) is publicly available at \url{https://github.com/raghul-parthipan/l96_rnn}.}

redpajama_set_name: RedPajamaArXiv


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text: CGGBP1 affects CTCF occupancy and histone modifications at known CTCF-binding sites. a, b CTCF read distribution was plotted for CT, KD and OE at the LAD boundaries in the 50 kb flanks with bins of 1 kb. The average read density with standard error of the mean was plotted for LAD start sites (a) and LAD end sites (b). CTCF binding is reported to occur in 10 kb flanks of the LAD start and end sites. c Correlation heatmap was plotted for H3K4me3, H3K9me3 and H3K27me3 ChIP-seq peaks in CT and KD. The correlation values (r) corresponding to CT and KD are mentioned for each histone modification in the heatmap. Unlike H3K4me3 and H3K27me3, the CT and KD samples for H3K9me3 were significantly different from each other. d PCA analysis using input as control shows the distance between H3K9me3 CT and KD as the most affected in KD. CGGBP1 depletion minimizes the distinction of H3K9me3 from H3K4me3 and H3K27me3 along PC2 only while along PC1, the two repressing marks remain closer to each other than the activating mark. e Disturbances in the H3K9me3 is observed in KD compared to CT. Genome browser views (a 40 kb region on chr11 starting from approximately 1983 kb) showing reads of CTCF (data range on Y-axis 0–25), H3K9me3 (data range on Y-axis 0–25), H3K27me3 (data range on Y-axis 0–8) and H3K4me3 (data range on Y-axis 0–35) for CT and KD along with the CTCF-binding sites and repeats at the H19-ICR locus. The gain of CTCF binding upon CGGBP1 knockdown is highlighted by green arrow and changes in the histone modification profile in 10 kb flanks are highlighted by gray boxes. f Genome browser view (36 kb region on chr11 starting from approximately 5294 kb) of the beta-globin locus control region undergoing change in chromatin barrier function upon CGGBP1 depletion. The plot contains tracks of CTCF reads (data range on Y-axis 0–25), H3K9me3 (data range on Y-axis 0–25), H3K27me3 (data range on Y-axis 0–20) and H3K4me3 (data range on Y-axis 0–35) reads, hg38 genes, DNase hypersensitivity clusters and repeats. The gain and loss of CTCF binding upon CGGBP1 loss of function are highlighted with an arrowhead and an arrow, respectively