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I bought this solid body scrub from LUSH at the weekend, it may not be new to the LUSH range but I had never seen this massive green thing before so asked the lady what it was.....
LUSH Sugar Scrub
'A skin cocktail of sugar, grated ginger and fresh fennel to liven up tired skin and sort out any lumpy bumpy bits.' To me it smells like cut grass and after reading reviews on the LUSH website people agree.
It is huge.....
And the price.....?
I thought this was a huge bargain! The lovely LUSH lady gave me a tip too - how to make it last longer. Most people use it as a sold bar but because it is sugar it disolves after a few uses and can make a mess, but, if you bash it up like so (I used some of my boyfriends weights as I wouldn't have a clue where a hammer or rolling pin was!).....
It lasts a lot longer, keep it in a little tub next to the shower and just use what you need, I have been using this every other day since Saturday *so twice so far) and my skin feels really moisturised, when you use it you can tell it has lots of essential oils in. I guess this will last me at least ten scrubs!
Have you tried any LUSH scrubs?
I also bought a 'Glorious Mud' shower scrub which I am yet to use so I will report back once I have.
4 Comments:
I bought the Australian Igloo back in January (it was part of the Christmas collection), which looks just like this except is it white and has a more minty smell... it was amazing! It felt so exfoliating, I definitely want to pick this one up now!ReplyDelete
Frances xx
Oh I haven't tried any! I totally want to now though. I've tried a lot of the bath bombs and glitter bars lol I get distracted by the sparkles!ReplyDelete
Good idea on how to smash it and store it! haha i always wonder how i should store it once it's wet :)ReplyDelete
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The LG CES 2015 keynote has just concluded and LG has announced as many as seven new OLED TVs. These new OLED TVs will span from screen sizes of 55-inches to 77-inches and will sport 4K UHD resolution.
LG has explored all variations for the large screen OLED TVs. There will be flexible, curved as well as flat models. The seven OLED TVs are: 77-inch Flexible Curved EG9900; 65-inch Art Slim Unibody EF9800; 77-inch Curved EG9700; 55-inch and 65-inch Floating curved EG9600 and 55-inch and 65-inch Floating Flat EF9500.
The top-of-the-line 77-inch LG 77EG9900 has some bleeding edge design chops. This is the first TV which can go from flat to curved and back again to flat with the press of a single button on the remote control. This gives consumers the option to play around the level of immersion with their TVs. It will sport 4K resolution and also offer passive 3D.
All of LG's 2015 4K OLED TVs utilise the company's proprietary WRGB technology. This pixel structure adds a white sub pixel, which allows for expanded colours. LG's Optimised True Color technology regulates colour stability regardless of variations in brightness and delivers a superb viewing experience with more natural, comfortable colours. All the LG OLED TVs will come with 4.2-channel Harman/Kardon audio solutions.
The 65-inch Art Slim TVs concentrate on the slimness factor. It makes use of a transparent stand which gives the TV a floating appearance.
On the user interface front, you will get the webOS 2.0 interface. It delivers a simpler, easier and more intuitive experience as compared to webOS according to LG.
"LG is driving innovation by making this superb TV technology available to more consumers," said In-kyu Lee, senior vice president and head of the TV and monitor division at the LG Electronics Home Entertainment Company. "LG believes this is where the future of television is headed and we're going to continue developing and expanding our OLED offerings until it becomes a viable option for all consumers."
CES 2015: LG announces OLED TVs spanning 55-inch to 77-inch size with 4K UHD resolution - Tech2

subset: commoncrawl


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The vision of St. Jude is to welcome everyone to become ONE Catholic family, with fullness
of life, living the Paschal Mystery, in the Holy Spirit, with Mary as our companion.
Linda Klekner and her family have been parishioners at St. Jude Church for 5 years. She has been the Assistant to the Director of Faith Formation since 2016. Linda has over 11 years experience as a Catechist from 1st grade to 8th grade and in the Rite of Christian Initiation for Children (RCIC). She has also served as a Eucharistic Minister for 8 years. She is a graduate of Montclair State University with a Bachelor's degree in Business with a concentration in Buying and Distribution.
Linda and Jim have been happily married for 20 years and have 2 beautiful children, Bella & Jimmy. After Linda and Jim started their family, she gave up her career to be an at home mom. For 8 years, Linda was extremely active with Girl Scouts as the Byram Service Unit Manager, Cookie Manager & Troop Leader. She was also an active volunteer at Byram and Mt. Olive Schools.


text: CHROMATICS share new Dear Tommy single "Blue Girl" by Chris June 12, 2018 CHROMATICS share track 13 (of 21!) from Dear Tommy. Video by CHROMATICS: Read More Filed Under: blue girl, chromatics, dear tommy, johnny jewel, ruth radelet, video Category: mp3, news, video Send to a friend! Share on Facebook Share on Twitter Pin it! Reddit This! Share on Tumblr!


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I'm learning division right now. It's hard - even harder than multiplication! I'm also getting extra science. Some in the morning with my mom, some with my class. I read about Hawk Mountain and woodpeckers today.
I get to read a lot too. I really like fantasy fiction. Last time I was at the library, the librarian showed me how to find the ones with stickers on the spine that have a spaceship or unicorn on them. My mom says that was her favorite kind of book when she was younger, and she loves to read my books to make sure I'll like them.
Thursday, October 30, 2014
Monday, October 20, 2014
Gifts that Help Others
I'm making things again this year to sell to raise money. Last year I donated to World Vision to help girls around the world go to school. I raised over $150. This year the money is going to special needs children and adults through Lutheran Special Education Ministries and the PACER Center. Here are some of the things I currently have. I can do requests too, but I don't have clay for charms anymore. You may donate as much or as little as you like, but please remember that I need to pay to mail them too. If you live far away, you can mail a check, or use PayPal. Email my mom at email@example.com for details.
|Charms: cat, octopus, dog, pig, purple donut (Kjirsten made that), hamburger, s'more, sprinkle donut, hot dog, fries, cinnamon roll, carrot cake muffin
|More bracelets
|Decorative potholders on the left (Kjirsten helped with the top one), purses and pencil holders on the right
Posted by Bekah at 7:49 PM No comments:
Saturday, October 18, 2014
Happy birthday Kjirsten!
Kjirsten's birthday was last week. It was fun to see my grandparents on Sunday. Then I got to see them again for Grandparents' Day on Wednesday. I did bowling with them, and showed them my classroom where I do school when it's not with my mom. Then I saw my mom's aunt and uncle Thursday and Friday and Oma and Opa again on Friday. Tomorrow I get to see them again! Lots of family time!
Posted by Bekah at 7:30 PM No comments:
Labels: news
Wednesday, October 8, 2014
Baptism Anniversary!
We celebrated my Baptism anniversary on Tuesday. My mom made a cake, and tried to make marshmallow fondant. It was yummy, but she didn't take a picture because she thought it turned out funny. She said the dove looked like a squirrel. I decorated the living room with my favorite flower blanket and we had an indoor picnic for dessert!
Posted by Bekah at 7:18 PM No comments:
Labels: news
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Here it is—the moment everyone's been waiting for! The music is playing and the guests have been seated. The coordinator is lining up the bridesmaids while giving the musicians a five-minute warning. You're about to walk down the aisle and see the love of your life. You've probably dreamed about this moment for a long time! Vows will be shared and promises will be made—this is what it's all about.
No matter where you decide to get married, here are a few tips to make sure you get the best wedding photographs possible.
The way your ceremony is lit is very important, especially if it's going to be outdoors.
The lower the sun is in the sky, the more romantic the lighting. If you decide to get married outside at 12PM, not only will there be harsh lines of light on your faces, you will also be squinting. Try to plan your ceremony later in the day if possible.
For outdoor ceremonies, the best light will be 2 hours before sunset if you plan on seeing each other beforehand. If you don't plan on sharing a first look, I recommend having your ceremony 3 hours before sunset. This will make sure you will have plenty of time for all of your portraits afterwards.
If you are having an indoor ceremony, make sure to have even lighting at the altar. If, for example, there are two spotlights on the bride and groom, they will be very bright and your bridal party will be very dark. It is much better to have soft, even light across everybody.
An unplugged wedding is when you kindly ask your guests to put away their phones, iPads, cameras and other devices for the duration of the wedding ceremony. I always recommend unplugged ceremonies, for a few reasons:
For one, your guests can be present and enjoy the ceremony, rather than focusing on taking pictures.
As you're walking down the aisle, what would you rather see—your guests beaming with happiness as they watch, or a bunch of phones and iPads blocking their faces?
Your ceremony photographs will look better without the clutter of electronic devices in the audience. When you eventually look back at the photographs of you walking down the aisle, it can be distracting if there are phones and iPads all around you.
As a photographer, one of the biggest ceremony mistakes I see is when a couple never faces their guests!
Talk to your officiant and ask them if you can face the audience instead of the alter. You want to make sure your guests can see your lovely faces, and not just the back of your heads! This also makes for better ceremony pictures, so it's a win win.
The ceremony is the whole reason you've planned your wedding, so don't forget to enjoy it! I recommend hiring an officiant you connect with who also has a sense of humor. A few jokes here and there can help you relax and really be present.
An added benefit of this are pictures where you and your guests look happy! Although everyone may feel over the moon, without a few smiles and laughs here and there it can be hard to convey those emotions in pictures.
Hiring either a full event planner or just a day-of coordinator is so important. You'll realize very quickly that if you do not have a coordinator, that responsibility will automatically fall on either the mother of the bride, the photographer or you!
When a photographer is in charge of the coordination, the images will suffer. All of the most stressful wedding situations I've seen were caused by a lack of organization and coordination. It's so important to make this a priority.
In a world full of wedding blogs, Pinterest and bridal magazines, it's easy to be overwhelmed and consumed by the "prettiness" of the event. While details do matter and planning is necessary, it's important to never lose sight of the real purpose of this amazing day—the connection and love you two share.


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Fred Prinz
The Story
Fred Prinz made his name during a short stint working as the cellar master for the Rheingau estate, Kloster Eberbach, which is the jewel of the Staatsweingut, the largest winery in and owned by Germany. Along with his service for the state he developed his own estate with a production of only 1,500 cases on the side. Fred is one of the few modern era "garagistes," in Germany.
Today, Fred's organically made wines (one of the few in Germany) are celebrated at home and abroad, resulting in a quick turn around with the release of each vintage. His wines go through natural processes and you can feel it in the depth of their range. There is always a little extra stuffing in each cuvée from the basic Estate Trocken to the highest pedigree labels. Interestingly, many of Fred's Rieslings carry subtle, gorgeous wild red berry notes tucked in the fold of these very complex wines. The wines are as serious as their maker and do most of the talking for this shy and humble artisan.
Lay of the Land
The Rheingau was once the king of wine in Germany, but has since fallen behind other regions as they have rested on their laurels for decades while others have made the move up the ladder. Today, it's the small producers—like Fred Prinz—that lead the way in quality standards in both the vineyards and the cellar.
A drive from the charming and very well kept city of Wiesbaden, on the right bank (north side) of the Rhine River, rests an extensive field of grapevines facing south (and slightly east). It runs about 30 kilometers before making a sharp turn toward the northwest, where it ends about six kilometers in a small town named, Lorch. The vines are planted on moderate and sometimes steep slopes sandwiched between the Tanus mountains and the river the entire way, which gives some much-needed cool air during the summertime from the forest and a good regulation from the river against temperatures both too hot and too cold. It's a near perfect winegrowing region; only global warming could stand in the way of it recapturing its former glory.
Fred's vineyard holdings are a little less than eight hectares (20 acres) between the sites of Hendelberg, Frühernberg, Jungfer and Schönhell, and all located in the town of Hallgarten high above the river with good ventilation, save for Frühernberg. His vineyards are made of mostly loess, loam, gravel, red slate and quartzite, most of which aid in good drainage for the soils. Jungfer has marl in addition to large variations of soils, and the Schönhell is heavier on the deeply loamy side. Jungfer provides the grapes for the fruity wines, whereas the Schönhell is used primarily to make dry styles.
Fred Prinz - 2015 Riesling, Hallgarten Jungfer, Kabinett
Out of stock


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If you're looking for a cool new ring to add to your collection, here's my recommendation: Bing Bang's Double Baguette Ring. I got two over the holidays and I can't stop wearing them; they're amazing!
If you know anything about the NY-based jewelry brand, then you know pretty much everything they make is Insta-worthy. This ring is especially standout, though, because it's simple and versatile. It's adjustable so you can wear it as a regular ring or a midi ring. Even better, it looks good all on its own, but it looks killer with a bunch of other rings, too.
Here's the catch: this ring is one of Bing Bang's bestsellers so it constantly sells out. So consider this your warning. It's currently available in gold, rose gold and silver, so snatch one up in every color stat.
Be sure to check out the best Valentine's Day nail polish and 25% off sale items at Vince.
[Photo: Bing Bang Instagram]


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February is a wonderful month to begin your travel. It is the time when cruel winter welcomes the refreshing and lively spring season. Many cultural programs and festivals are celebrated across India during February. With Valentine's Day in it, it makes a perfect month for couples to travel and spend some time together. Here's a list of places to visit in February in India with family and friends.
20 Places To Visit In February In India In 2020
Looking for places to visit in February in India? Here is the list of places you can explore in India on your trip. Take a look!
- Goa: The Party Capital Of India
- Mumbai: The City That Never Sleeps
- Agra: Historic City Crowned By The Taj Mahal
- Puri, Orissa: For A Spiritual Sojourn
- Thekkady: Bestowed With A Rich Wildlife
- Kaziranga National Park: Home Of One-Horned Rhinos
- Udaipur: The City Of Lakes
- Varanasi: One of India's Oldest Cities
- Kodaikanal: The Pride Of Tamil Nadu
- Gulmarg: A Meadow Like None Other
- Kutch, Gujurat: A Grand Festival In The Salt Dessert
- Ludhiana: The Traditional City In Punjab
- Kolkata: The City Of Joy
- Jaisalmer: The Golden City!
- Shekhawati, Rajasthan: The Vibrant Festival
- Delhi: The Capital City
- Jodhpur: The Blue City!
- Trivandrum, Kerala
- Bengaluru, Karnataka: The Tree Festival
- Ranthambore, Rajasthan: Spot The Tiger!
1. Goa – The Party Capital Of India
Goa among the topmost places to visit in February in India. This magical place never fails to amaze you. November to January are the peak months for business in Goa, but February is when the crowd goes away. But, Goa is as charming as month of the year during this time. Goa Carnival takes place in February, making the entire state look like a colourful stage.
Suggested Read: 12 Hacks For A Kick-Ass New Year In Goa On A Budget In 2020
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2. Mumbai – The City That Never Sleeps
Mumbai is quite hot during other months of the year, but February brings pleasant climate to the city. It is the best time to take long walks around this vibrant city. From the lively suburbs of Mumbai to the British era buildings that tell stories of the past, there's so much this city has to offer to you. Enjoy the busy markets, the happening dics, the swanky pubs, and the sea of Mumbai in February.
3. Agra – Historic City Crowned By The Taj Mahal
Image Source
Agra is perhaps one of the best places to visit in February in India. It is in February, the city hosts the interesting Taj Mahotsav. Also, it is extremely pleasant to explore the historical monuments of the town in the month. Anyone who admires a perfect blend of cultures, heritage, and people would love to explore Agra in Feb.
Suggested Read: 15 Charming Places To Visit In January In India
4. Puri, Orissa – For A Spiritual Sojourn
Puri is comparatively an offbeat destination, but it certainly is among the best places to visit in February in India. Also known as the 'Spiritual Capital of Odisha', it is known for its sacred places. Apart from the popular temples, Puri has serene and calm beaches where you can just lie down and forget about the world out there.
5. Thekkady – Bestowed With A Rich Wildlife
A never ending chain of lagoons that are interconnected by lakes, rivers, and canals run along the Arabian Sea in Kerala. It is probably one of the greatest award bestowed upon Kerala by the almighty God. Lush wildlife reserves, rolling Western Ghats, spice plantations, and the spectacular backwaters, all of this awaits you in Thekkady of Kerala, and there's no better time than February to visit the gorgeous place for wildlife and nature enthusiasts.
6. Kaziranga National Park – Home Of One-Horned Rhinos
Do your love the wildness of the jungles, the soothing music of the birds, and the gentle air that blows inside national parks? If so, you should not skip Kaziranga National Park. It is probably among the finest places to visit in India during February. This home to the great Indian one-horned rhinoceroses is best explored in the pleasant weather of February on a safari.
Suggested Read: Embrace The Winter Chill At These 15 Best Places To Visit In Rajasthan In November!
7. Udaipur – The City Of Lakes
Udaipur has lakes where you can sit and enjoy a warm sun and palaces that'll make you love history. This place of rich heritage, fine sand dunes, and warmest hospitality is a gem of Rajasthan. February is the best time to admire the beauty of Lake Pichola, Fateh Sagar, and take a romantic ride on the emerald green waters. This is absolutely a land of royals and you will see it in their culture and behavior.
8. Varanasi – One Of India's Oldest Cities
Varanasi or Banaras is a small town that hold an important place in Hindu religion. Situated on the banks of river Ganga, Banaras is a pilgrimage destination for Hindus. If your wish to experience the true essence and charm of this ancient city, then Maha Shivaratri is the best time to be a part of the community and celebrate life.
9. Kodaikanal – The Pride Of Tamil Nadu
Those who wish to get lost in the verdant meadows and smell the fresh air of South India must visit Kodaikanal in February. It is one of the best places to visit in February in south India for its trekking trails, a wide variety of flowers and other plants that look exotic, and stargazing. February being an off season gives you opportunity to explore the touristy places without dealing with so much of crowd, and that too, at slashed rates.
10. Gulmarg – A Meadow Like None Other
Missed the snowfall of this winter? Worry not, it snows in Gulmarg in February too. One of the most popular and prime skiing destinations in the entire world, Gulmarg is a heaven for those who want to go skiing in February. You can book your stay at Khyber resort, which offers stunning views of the snow-covered Himalayas and alpine forests.
11. Kutch, Gujurat – A Grand Festival In The Salt Dessert
If you are planning to visit India in February then you can't miss out on the Rann Utsav which is organized every year in Kutch, Gujarat. Kutch is one of the biggest salt desserts in the world and the grand Rann Utsav is an event which is quite famous all around the world. One can enjoy traditional dance, relish local food and indulge in other sightseeing activities when they attend this event.
12. Ludhiana – The Traditional City In Punjab
Founded in 1480 by the Lodhi dynasty, Ludhiana is a city which is famous for its woolen clothes and cotton t-shirts. In the month of February, MachAuto EXPO takes place at Glada Ground, Ludhiana and this event is attended by most of the businessmen from India as well as abroad. This event is organized to display latest Engineering Machinery, Auto Parts, Machine Tools, Hand Tools, Automation Technology and more. Explore the traditional city of Punjab while you attend MachAuto EXPO and don't forget to head out for a shopping spree.
13. Kolkata – The City Of Joy
Famed as the city of joy, Kolkata is a vibrant city that will be perfect for your vacation in India in February. Travelers from all around the world should explore the magnificent city where famous literary artists like Rabindranath Tagore were born. Once you are in Kolkata, do try the little bundle of joy called 'Sandesh' and other delicious sweets.
14. Jaisalmer – The Golden City!
Located in Rajasthan also known as the hottest state of India, Jaisalmer is also called the Golden City because of its stunning Yellowstone architecture. Jaisalmer in winters will be the perfect spot for all those who can't bear the scorching summer heat. But why should you visit Jaisalmer in February? The great Jaisalmer Dessert Festival allures many tourists in the month of February. You can enjoy camel races, turban tying games, polo matches and more interesting activities in Jaisalmer on your trip.
Dates: 7 Feb 2020 – 9 Feb 2020
15. Shekhawati, Rajasthan: The Vibrant Festival!
All packed to explore India in the month of February? Head to Shekhawati region in Rajasthan and attend the Shekhawati festival as this colorful festival will entertain you to the core. Get ready to take part in amusing activities like Folk dance, Mehandi competitions, Matka race and more at this festival.
16. Delhi – The Capital City
Head to the charming capital city of India and explore the majestic monuments, relish the local delicacies and go shop around the city. From Chandni Chowk to Lajpat Nagar, markets in Delhi will make you fall in love with the city.
17. Jodhpur – The Blue City!
Visit Rajasthan's bewitching blue city 'Jodhpur' in February and straight away head to the World Sufi Spirit Festival. Get captivated by the soulful Sufi music played by the best artists from all around the world. Don't forget to book the tickets beforehand or else you will miss out on a grand festival.
Dates: 10 Feb 2020 – 12 Feb 2020
18. Trivandrum, Kerala
A famous festival that is mentioned in the Guinness Book of World Records and guess why is it? In 1997 a world record was made when 1.5 million women attended Attukal Pongala festival held in Trivandrum. The festival includes women cooking food that is offered to Goddess Attukalama and the festival is one of the eminent ones in Kerala. So, get ready to be a part of an Indian festival that is popular all over the world.
Location: Attukal Bhagavathi Temple
19. Bengaluru, Karnataka – The Tree Festival
This one is for the nature lovers. Head to Bengaluru in February and attend the Neralu festival also known as the tree festival. A festival dedicated to trees and their importance in our civilization, Neralu is a festival all nature admirers should definitely attend.
Location: Cubbon Park
20. Ranthambore, Rajasthan – Spot The Tiger!
An abode to many mammals, amphibians, reptiles, and birds, Ranthambore is an ideal destination for all adventure seekers. When you are all set to explore India in February, then you must not miss the most thrilling place in Rajasthan. You can easily spot different species of animals in the month of February at Ranthambore and enjoy your enticing trip at this place.
Further Read: 13 Places In India You Must Visit With Your Family
Already excited? Get packed, book your trip to India with TravelTriangle and leave right away!
Frequently Asked Questions About Places To Visit In February In India
A. Some of the best places to visit in February in India with friends are Goa, Kolkata, Trivandrum, Jaisalmer and Ranthambore.
A. The weather in India in February is quite pleasant. The beginning of the month will be cold and by the month India gets to to prepare for summers. But it is important to note that India is a big country and the weather could be different in different regions.
A. Yes, February actually is one of the best months to see Taj Mahal as the weather is pleasant.
A. Yes, February is a great time to visit India weather is pleasant and you would be able to explore the destination very comfortably.
A. Some of the best places to visit in India in February with family are Darjeeling, Thekkady, Udaipur, Gulmarg and Kodaikanal.
A. February indeed is considered a good time to visit India as most of the regions experience the best weather conditions. Even in the hottest states like Rajasthan, Gujarat and Delhi, the temperatures stay at moderate levels; it is neither too hot nor too cold during this time of the year.
A. Vrindavan is the best place to visit in India in March for the culture-vultures. The city gets drenched in festive spirits as people play Holi, the festival of colors.
A. When it comes to the best place to visit in India in February, it's tough to choose just one. Rajasthan is a lovely place to visit during this time of the year, owing to pleasant weather conditions and various events taking place. Similarly, the Rann Utsav in Gujarat takes place till February. Kerala and Goa are other places to consider if you are a beach-enthusiast.
Check these related links also:
Places To Visit In Winter In India
Unique winter places to visit in India
11 Best Honeymoon Places In India In February 2020


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Comparison of the binding of glucose and glucose 1-phosphate derivatives to T-state glycogen phosphorylase b.Martin, J.L., Johnson, L.N., Withers, S.G.
(1990) Biochemistry 29: 10745-10757
- PubMed: 2125493
- Primary Citation of Related Structures:
- PubMed Abstract:
- Refined Crystal Structure of the Phosphorylase-Heptulose 2-Phosphate-Oligosaccharide-AMP Complex
Johnson, L.N.,Acharya, K.R.,Jordan, M.D.,Mclaughlin, P.J.
(1990) J.Mol.Biol. 211: 645
- Glycogen Phosphorylase B: Description of the Protein Structure 1 1991
Acharya, K.R.,Stuart, D.I.,Varvill, K.M.,Johnson, L.N.
() Glycogen Phosphorylase B: Description of the Protein Structure --: --
- Structural Changes in Glycogen Phosphorylase Induced by Phosphorylation
Sprang, S.R.,Acharya, K.R.,Goldsmith, E.J.,Stuart, D.I.,Varvill, K.,Fletterick, R.J.,Madsen, N.B.,Johnson, L.N.
(1988) Nature 336: 215
- The Allosteric Transition of Glycogen Phosphorylase
Barford, D.,Johnson, L.N.
(1989) Nature 340: 609
- Structural Mechanism for Glycogen Phosphorylase Control by Phosphorylation and AMP
Barford, D.,Hu, S.-H.,Johnson, L.N.
(1991) J.Mol.Biol. 218: 233
The binding of T-state- and R-state-stabilizing ligands to the catalytic C site of T-state glycogen phosphorylase b has been investigated by crystallographic methods to study the interactions made and the conformational changes that occur at the C si ...
The binding of T-state- and R-state-stabilizing ligands to the catalytic C site of T-state glycogen phosphorylase b has been investigated by crystallographic methods to study the interactions made and the conformational changes that occur at the C site. The compounds studied were alpha-D-glucose, 1, a T-state-stabilizing inhibitor of the enzyme, and the R-state-stabilizing phosphorylated ligands alpha-D-glucose 1-phosphate (2), 2-deoxy-2-fluoro-alpha-D-glucose 1-phosphate (3), and alpha-D-glucose 1-methylenephosphonate (4). The complexes have been refined, giving crystallographic R factors of less than 19%, for data between 8 and 2.3 A. Analysis of the refined structures shows that the glucosyl portions of the phosphorylated ligands bind in the same orientation as glucose and retain most of the interactions formed between glucose and the enzyme. However, the phosphates of the phosphorylated ligands adopt different conformations in each case; the stability of these conformations have been studied by using computational methods to rationalize the different binding modes. Binding of the phosphorylated ligands is accompanied by movement of C-site residues, most notably a shift of a loop out of the C site and toward the exterior of the protein. The C-site alterations do not include movement of Arg569, which has been observed in both the refined complex with 1-deoxy-D-gluco-heptulose 2-phosphate (5) [Johnson, L. N., et al (1990) J. Mol. Biol. 211, 645-661] and in the R-state enzyme [Barford, D. & Johnson, L. N. (1989) Nature 340, 609-616]. Refinement of the ligand complexes has also led to the observation of additional electron density for residues 10-19 at the N-terminus which had not previously been localized in the native structure. The conformation of this stretch of residues is different from that observed in glycogen phosphorylase a.
Related Citations:
Organizational Affiliation:
Laboratory of Molecular Biophysics, Oxford, U.K.


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About ACA
The ACA
The ACA Team
When clients choose ACA, they work with knowledgeable, senior professionals that understand both current industry conditions and their historical context. This ensures that clients receive specialized insights that will help them solve today's problems as well as preparing effectively for the future.
Click on the names below to learn more about each ACA team member's exceptional talents and experience.
Donald P. Schenk
President & CEO
Frank L. Rosenberg
Managing Director
Myron Manternach
Senior Advisor
A. R. Moulton, III
Managing Director
Edmund S. Greenslet
Vice Chairman
Sassy N'Diaye
Managing Director Africa
Phillip Bolt
Karl R. Childs
Aaron J. Gellman Ph.D.
Marion S. Griffith, Jr.
Ronald A. Hearst
Jackson L. Hulsey
Peter Jost
Steve Kuykendall
Constance C. Laudenschlager
Arthur F. McMahon
Shashank Nigam
John Oliver
Thomas H. Pace Jr.
David Plavin
Christopher E. Preece
Jerry Van Schaick
Hans J. Weber
Thomas W. Casey
Judith A. Driver
Edward L. Green
William R. Grimes
Harris. Herman
Richard J. Nelson
Joseph W. Klinehamer
James E. Larsen
Myron Manternach
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Last week's seismic decision in the Google Spain case continues to generate many column inches of comment and will no doubt continue to do so for some time. Datonomy's colleagues in Olswang's international privacy team have just published a paper considering the practical implications of this decision in the round. You can access it at this link. The paper considers:
- Google's practical options in terms of next steps
- the implications for individuals' rights
- the implications for online publishers
- what it means for the Right To Be Forgotten under the new EU Regulation
- the impact on wider "data debates" over other technologies such as email scanning and Google Glass
- what it tells us about the workings of Europe's highest commercial court, and tactical tips for bringing referrals on points of EU law.
The paper is also available in PDF here.


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On February 18, 2011, Ugandans went to the polls to elect their nation's president and parliament. These elections represent the second time since 1986 that Uganda has elected its leaders under a multi-party political system.
The International Democrat Union The Presidential and Parliamentary Elections of February 2011 were by all standards not free and fair. Gunilla Carlsson, Swedish Minister for International Development Cooperation, explained that the elections were not up to the standard that she had hoped for.
After lengthy periods of dictatorship and insecurity, the National Resistance Movement (NRM) took power after winning a guerrilla war, and instituted a new system of no-party democracy in 1986 under current President Yoweri Museveni.
A 2005 constitutional referendum restored a multi-party system, and in 2006 the leading opposition candidate won 37 percent of the vote. Whether opposition parties can improve on the gains they made in the 2006 presidential and parliamentary elections remains an unknown. Both the conduct and results of the upcoming elections will be an important marker for determining whether Uganda has continued to make progress in institutionalizing multi-party politics.
Read Gunilla Carlasson's comments to the elections (in Swedish)
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This patented copper fungicide contains both copper hydroxide and copper oxychloride. The biosynergy™ of this combination provides quicker and longer lasting control of a broader spectrum of diseases using less metallic copper. As an OMRI Listed product it is ideal for both conventional and organic production systems. Badge X2 is formulated as a water dispersible granule (WDG) giving it optimal handling and mixing characteristics.
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As per this study, the Global Tire Cord And Tire Fabrics Market was valued USD XX.X MN (by revenue) in 2018 and is anticipated to reach USD XX.X MN by 2025 with a CAGR of X.X%. By volume, the market was estimated XXX KT in 2018 and is expected to reach XXX KT by 2025 with a CAGR of X.X%.
Tire cord fabric is the textile material that offers flex resistance and also aid in tire reinforcement. It is used to keep tires in shape, support vehicle weight and show a positive impact on the tire performance. It is mainly used in personal vehicles, heavy vehicles and off-road vehicles to impart dimensional stability of the tire.
Market Dynamics
Booming automotive industry on account of rising in sales of vehicles are driving the market growth. The growing demand of tire material that easily decomposes, offer better stability and efficiency is again fueling the market growth. Increasing disposable income, growing middle-class population resulting in higher demand for personal vehicles is again boosting the market growth. Ongoing research for the introduction of tires with more stability, high fatigue, and tear resistance is expected to hinder the market growth.
This detailed market study is centered on the data obtained from multiple sources and is analyzed using numerous tools including porter's five forces analysis, market attractiveness analysis and value chain analysis. These tools are employed to gain insights of the potential value of the market facilitating the business strategists with the latest growth opportunities. Additionally, these tools also provide a detailed analysis of each application/product segment in the global market of tire cord and tire fabrics.
Market Segmentation
The broad tire cord and tire fabrics market has been sub-grouped into product and vehicle type. The report studies these subsets with respect to the geographical segmentation. The strategists can gain a detailed insight and devise appropriate strategies to target specific market. This detail will lead to a focused approach leading to identification of better opportunities.
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Inns & Lodges: Lakeside Inn, Michigan
Mention the Lakeside Inn to the residents of this placid bed-and-breakfast hamlet on the southeastern shore of Lake Michigan, and you'll raise eyebrows. They all know the place--it's hard to miss the big, sprawling house perched on a hill overlooking the sandy beach. They've gawked at the strange sculpture garden. They've heard foreign languages wafting out of the windows of its cavernous dining room. They've been told it has no heat, for godssake.
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Lakeside Inn is just off Interstate 94 in Lakeside, Michigan, 90 minutes from Chicago and four hours from Detroit. Double rooms cost $50-$110, continental breakfast (and the occasional lunch or dinner) included. The inn is open Memorial Day through late September; call 616-469-1337 for reservations.


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Entropy 2014, 16(6), 3273-3301; https://doi.org/10.3390/e16063273
Abstract: Clustering sets of histograms has become popular thanks to the success of the generic method of bag-of-X used in text categorization and in visual categorization applications. In this paper, we investigate the use of a parametric family of distortion measures, called the α-divergences, for clustering histograms. Since it usually makes sense to deal with symmetric divergences in information retrieval systems, we symmetrize the α-divergences using the concept of mixed divergences. First, we present a novel extension of k-means clustering to mixed divergences. Second, we extend the k-means++ seeding to mixed α-divergences and report a guaranteed probabilistic bound. Finally, we describe a soft clustering technique for mixed α-divergences.
1. Introduction: Motivation and Background
1.1. Clustering Histograms in the Bag-of-Word Modeling Paradigm
A common task of information retrieval (IR) systems is to classify documents into categories. Given a training set of documents labeled with categories, one asks to classify new incoming documents. Text categorisation [1,2] proceeds by first defining a dictionary of words from a corpus. It then models each document by a word count yielding a word distribution histogram per document (see the University of California, Irvine, UCI, machine learning repository for such data-sets [3]). The importance of the words in the dictionary can be weighted by the term frequency-inverse document frequency [2] (tf-idf) that takes into account both the frequency of the words in a given document, but also of the frequency of the words in all documents: Namely, the tf-idf weight for a given word in a given document is the product of the frequency of that word in the document times the logarithm of the ratio of the number of documents divided by the document frequency of the word [2]. Defining a proper distance between histograms allows one to:
Classify a new on-line document: We first calculate its word distribution histogram signature and seek for the labeled document, which has the most similar histogram to deduce its category tag.
Find the initial set of categories: we cluster all document histograms and assign a category per cluster.
This text classification method based on the representation of the bag-of -words (BoWs) has also been instrumental in computer vision for efficient object categorization [4] and recognition in natural images [5]. This paradigm is called bag-of-features [6] (BoFs) in the general case. It first requires one to create a dictionary of "visual words" by quantizing keypoints (e.g., affine invariant descriptors of image patches) of the training database. Quantization is performed using the k-means [7–9] algorithm that partitions n data into k pairwise disjoint clusters C1,…, Ck, where each data element belongs to the closest cluster center (i.e., the cluster prototype). From a given initialization, batched k-means first assigns data points to their closest centers and then updates the cluster centers and reiterates this process until convergence is met to a local minimum (not necessarily the global minimum) after a provably finite number of steps. Csurka et al. [4] used the squared Euclidean distance for building the visual vocabulary. Depending on the chosen features, other distances have proven useful. For example, the symmetrized Kullback–Leibler (KL) divergence was shown to perform experimentally better than the Euclidean or squared Euclidean distances for a compressed histogram of gradient descriptors [10] (CHoGs), even if it is not a metric distance, since its fails to satisfy the triangular inequality. To summarize, k-means histogram clustering with respect to the symmetrized KL (called Jeffreys divergence J) can be used to quantize both visual words and document categories. Nowadays, the seminal bag-of-word method has been generalized fruitfully to various settings using the generic bag-of-X paradigm, like the bag-of-textons [6], the bag-of-readers [11], etc. Bag-of-X represents each data (e.g., document, image, etc.) as an histogram of codeword count indices. Furthermore, the semantic space [12] paradigm has been recently explored to overcome two drawbacks of the bag-of-X paradigms: the high-dimensionality of the histograms (number of bins) and difficult human interpretation of the codewords due to the lack of semantic information. In semantic space, modeling relies on semantic multinomials that are discrete frequency histograms; see [12].
In summary, clustering histograms with respect to symmetric distances (like the symmetrized KL divergence) is playing an increasing role. It turns out that the symmetrized KL divergence belongs to a 1-parameter family of divergences, called symmetrized α-divergences, or Jeffreys α-divergence [13].
1.2. Contributions
Since divergences D(p : q) are usually asymmetric distortion measures between two objects p and q, one has to often consider two kinds of centroids obtained by carrying the minimization process either on the left argument or on the right argument of the divergences; see [14]. In theory, it is enough to consider only one type of centroid, say the right centroid, since the left centroid with respect to a divergence D(p : q) is equivalent to the right centroid with respect to the mirror divergence D′(p : q) = D(q : p). In this paper, we consider mixed divergences [15] that allow one to handle in a unified way the arithmetic symmetrization of a given divergence D(p : q) with both the sided divergences: D(p : q) and its mirror divergence D′(p : q). The mixed α-divergence is the mixed divergence obtained for the α-divergence. We term α-clustering the clustering with respect to α-divergences and mixed α-clustering the clustering w.r.t. mixed α-divergences [16]. Our main contributions are to extend the celebrated batched k-means [7–9] algorithm to mixed divergences by associating two dual centroids per cluster and to generalize the probabilistically guaranteed good seeding of k-means++ [17] to mixed α-divergences. The mixed α-seedings provide guaranteed probabilistic clustering bounds by picking up seeds from the data and do not require explicitly computing of centroids. Therefore, it follows a fast clustering technique in practice, even when cluster centers are not available in closed form. We also consider clustering histograms by explicitly building the symmetrized α-centroids and end up with a variational k-means when the centroids are not available in closed-form, Finally, we investigate soft mixed α-clustering and discuss topics related to α-clustering. Note that clustering with respect to non-symmetrized α-divergences has been recently investigated independently in [18] and proven useful in several applications.
1.3. Outline of the Paper
The paper is organized as follows: Section 2 introduces the notion of mixed divergences, presents an extension of k-means to mixed divergences and recalls some properties of α-divergences. Section 3 describes the α-seeding techniques and reports a probabilistically-guaranteed bound on the clustering quality. Section 4 investigates the various sided/symmetrized/mixed calculations of the α-centroids. Section 5 presents the soft α-clustering with respect to α-mixed divergences. Finally, Section 6 summarises the contributions, discusses related topics and hints at further perspectives. The paper is followed by two appendices. Appendix B studies several properties of α-divergences that are used to derive the guaranteed probabilistic performance of the α-seeding. Appendix C proves that α-sided centroids are quasi-arithmetic means for the power generator functions.
2. Mixed Centroid-Based k-Means Clustering
2.1. Divergences, Centroids and k-Means
Consider a set of n histograms h1,…, hn, each with d bins, with all positive real-valued bins: , ∀1 ≤ i ≤ d, 1 ≤ j ≤ n. A histogram h is called a frequency histogram when its bins sums up to one: . Otherwise, it is called a positive histogram that can eventually be normalized to a frequency histogram:
The frequency histograms belong to the (d-1)-dimensional open probability simplex Δd:
That is, although frequency histograms have d bins, the constraint that those bin values should sum up to one yields d-1 degrees of freedom. In probability theory, the frequency or counting of histograms either model discrete multinomial probabilities or discrete positive measures (also called positive arrays [19]).
The celebrated k-means clustering [8,9] is one of the most famous clustering techniques that has been generalized in many ways [20,21]. In information geometry [22], a divergence D(p : q) is a smooth C3 differentiable dissimilarity measure that is not necessarily symmetric (D(p : q) ≠ D(q : p), hence the notation ":" instead of the classical "," reserved for metric distances), but is non-negative and satisfies the separability property: D(p : q) = 0 iff p = q. More precisely, let , . Then, we require ∂iD(x : x) = ∂,iD(x : x) = 0 and −∂i∂,jD(x : y) positive definite for defining a divergence. For a distance function D(· : ·), we denote by the weighted average distance of x to a set a weighted histograms:
An important class of divergences on frequency histograms is the f-divergences [23–25] defined for a convex generator f (with f′(1) = f′(1) = 0 and f″(1) = 1):
Those divergences preserve information monotonicity [19] under any arbitrary transition probability (Markov morphisms). f-divergences can be extended to positive arrays [19].
The k-means algorithm on a set of weighted histograms can be tailored to any divergence as follows: First, we initialize the k cluster centers C = {c1,…, ck} (say, by picking up randomly arbitrary distinct seeds). Then, we iteratively repeat until convergence the following two steps:
Assignment: Assign each histogram hj to its closest cluster center:
Center relocation: Update the cluster centers by taking their centroids:
2.2. Mixed Divergences and Mixed k-Means Clustering
Since divergences are potentially asymmetric, we can define two-sided k-means or always consider a right-sided k-means, but then define another sided divergence D′(p : q) = D(q : p). We can also consider the symmetrized k-means with respect to the symmetrized divergence: S(p, q) = D(p : q) + D(q : p). Eventually, we may skew the symmetrization with a parameter λ ∈ [0, 1]: Sλ(p, q) = λD(p : q) + (1 − λ)D(q : p) (and consider other averaging schemes instead of the arithmetic mean).
In order to handle those sided and symmetrized k-means under the same framework, let us introduce the notion of mixed divergences [15] as follows:
Definition 1 (Mixed divergence).
for λ ∈ [0, 1].
A mixed divergence includes the sided divergences for λ ∈ {0, 1} and the symmetrized (arithmetic mean) divergence for .
We generalize k-means clustering to mixed k-means clustering [15] by considering two centers per cluster (for the special cases of λ = 0, 1, it is enough to consider only one). Algorithm 1 sketches the generic mixed k-means algorithm. Note that a simple initialization consists of choosing randomly the k distinct seeds from the dataset with li = ri.
Notice that the mixed k-means clustering is different from the k-means clustering with respect to the symmetrized divergences Sλ that considers only one centroid per cluster.
2.3. Sided, Symmetrized and Mixed α-Divergences
For α ≠ ±1, we define the family of α-divergences [26] on positive arrays [27] as:
with the limit cases D−1(p : q) = KL(p : q) and D1(p : q) = KL(q : p), where KL is the extended Kullback–Leibler divergence:
Divergence D0 is the squared Hellinger symmetric distance (scaled by a multiplicative factor of four) extended to positive arrays:
with the Hellinger distance:
Note that α-divergences are defined for the full range of α values: α ∈ ℝ. Observe that α-divergences of Equation (5) are homogeneous of degree one: Dα(λp : λq) = λDα(p : q) for λ > 0.
When histograms p and q are both frequency histograms, we have:
and the extended Kullback–Leibler divergence reduces to the traditional Kullback–Leibler divergence: .
The Kullback–Leibler divergence between frequency histograms and (α = ±1) is interpreted as the cross-entropy minus the Shannon entropy:
Often, denotes the true model (hidden by nature), and is the estimated model from observations. However, in information retrieval, both and play the same symmetrical role, and we prefer to deal with a symmetric divergence.
The Pearson and Neyman χ2 distances are obtained for α = −3 and α = 3, respectively:
The α-divergences belong to the class of Csiszár f-divergences with the following generator:
Remark 1. Historically, the α-divergences have been introduced by Chernoff [28,29] in the context of hypothesis testing. In Bayesian binary hypothesis testing, we are asked to decide whether an observation belongs to one class or the other class, based on prior w1 and w2 and class-conditional probabilities p1 and p2. The average expected error of the best decision maximum a posteriori (MAP) rule is called the probability of error, denoted by Pe. When prior probabilities are identical, we have. Let S(p, q) = ∫ min(p(x), q(x))dx denote the intersection similarity measure, with 0 < S ≤ 1 (generalizing the histogram intersection distance often used in computer vision [30]). S is bounded by the α-Chernoff affinity coefficient:
for all β ∈ [0, 1]. We can convert the affinity coefficient 0 < Cβ ≤ 1 into a divergence Dβ by simply taking Dβ = 1 − Cβ. Since the absolute value of divergences does not matter, we can rescale appropriately the divergence. One nice rescaling is by multiplying by. This lets coincide the parameterized divergence with the fundamental Kullback–Leibler divergence for the limit values β ∈ {0, 1}. Last, by choosing, it yields the well-known expression of the α-divergences.
Interestingly, the α-divergences can be interpreted as a generalized α-Kullback–Leibler divergence [26] with deformed logarithms.
Next, we introduce the mixed α-divergence of a histogram x to two histograms p and q as follows:
Definition 2 (Mixed α-divergence). The mixed α-divergence of a histogram x to two histograms p and q is defined by:
The α-Jeffreys symmetrized divergence is obtained for :
The skew symmetrized α-divergence is defined by:
2.4. Notations and Hard/Soft Clusterings
Throughout the paper, superscript index i denotes the histogram bin numbers and subscript index j the histogram numbers. Index l is used to iterate on the clusters. The left-sided, right-sided and symmetrized histogram positive and frequency α-centroids are denoted by lα, rα, sα and , respectively.
In this paper, we investigate the following kinds of clusterings for sets of histograms:
Hard clustering. Each histogram belongs to exactly one cluster:
k-means with respect to mixed divergences Mλ,α.
k-means with respect to symmetrized divergences Sλ,α.
Randomized seeding for mixed/symmetrized k-means by extending k-means++ with guaranteed probabilistic bounds for α-divergences.
Soft clustering. Each histogram belongs to all clusters according to some weight distribution: the soft mixed α-clustering.
3. Coupled k-Means++ α-Seeding
It is well-known that the Lloyd k-means clustering algorithm monotonically decreases the loss function and stops after a finite number of iterations into a local optimal. Optimizing globally the k-means loss is NP-hard [17] when d > 1 and k > 1. In practice, the performance of the k-means algorithm heavily relies on the initialization. A breakthrough was obtained by the k-means++ seeding [17], which guarantees in expectation a good starting partition. We extend this scheme to the coupled α-clustering. However, we point out that although k-means++ prove popular and are often used in practice with very good results; it has been recently pointed out that "worst case" configurations exist and even in small dimensions, on which the algorithm cannot beat significantly its expected approximability with a high probability [31]. Still, the expected approximability ratio, roughly in O(log(k)), is very good, as long as the number of clusters is not too large.
Algorithm 2 provides our adaptation of k-means++ seeding [15,17]. It works for all three of our sided/symmetrized and mixed clustering settings:
Pick λ = 1 for the left-sided centroid initialization,
Pick λ = 0 for the right-sided centroid initialization (a left-sided initialization for −α),
with arbitrary λ, for the λ-Jα (skew Jeffreys) centroids or mixed λ centroids. Indeed, the initialization is the same (see the MAS procedure in Algorithm 2).
Our proof follows and generalizes the proof described for the case of mixed Bregman seeding [15] (Lemma 2). In fact, our proof is more precise, as it quantifies the expected potential with respect to the optimum only, whereas in [15], the optimal potential is averaged with a dual optimal potential, which depends on the optimal centers, but may be larger than the optimum sought.
Theorem 1. Let Cλ,α denote for short the cost function related to the clustering type chosen (left-, right-, skew Jeffreys or mixed) in MASand denote the optimal related clustering in k clusters, for λ ∈ [0, 1] and α ∈ (−1, 1). Then, on average, with respect to distribution (14), the initial clustering of MAS satisfies:
Here , g(k) = 2(2 + log k), , ; the min is defined on strictly positive coordinates, and π denotes distribution of Algorithm 2.
Remark 2. The bound is particularly good when λ is close to 1/2, and in particular for the α-Jeffreys clustering, as in these cases, the only additional penalty compared to the Euclidean case [17] is h2(α), a penalty that relies on an optimal triangle inequality for α-divergences that we provide in Lemma 8 below.
Remark 3. This guaranteed initialization is particularly useful for α-Jeffreys clustering, as there is no closed form solution for the centroids (except when α = ±1, see [32]).
Algorithm 3 presents the general hard mixed k-means clustering, which can be adapted also to left- (λ = 1) and right- (λ = 0) α-clustering.
For skew Jeffreys centers, since the centroids are not available in closed form [32], we adopt a variational approach of k-means by updating iteratively the centroid in each cluster (thus improving the overall loss function without computing the optimal centroids that would eventually require infinitely many iterations).
4. Sided, Symmetrized and Mixed α-Centroids
The k-means clustering requires assigning data elements to their closest cluster center and then updating those cluster centers by taking their centroids. This section investigates the centroid computations for the sided, symmetrized and mixed α-divergences.
Note that the mixed α-seeding presented in Section 3 does not require computing centroids and, yet, guarantees probabilistically a good clustering partition.
Since mixed α-divergences are f-divergences, we start with the generic f-centroids.
4.1. Csiszár f-Centroids
The centroids induced by f-divergences of a set of positive measures (that relaxes the normalisation constraint) have been studied by Ben-Tal et al. [33]. Those entropic centroids are shown to be unique, since f-divergences are convex statistical distances in both arguments. Let Ef denote the energy to minimize when considering f-divergences:
When the domain is the open probability simplex , we get a constrained optimisation problem to solve. We transform this constrained minimisation problem (i.e., x ∈ Δd) into an equivalent unconstrained minimisation problem by using the Lagrange multiplier, γ:
Taking the derivatives according to xi, we get:
We now consider this equation for α-divergences and symmetrized α-divergences, both f-divergences.
4.2. Sided Positive and Frequency α-Centroids
The positive sided α-centroids for a set of weighted histograms were reported in [34] using the representation Bregman divergence. We summarise the results in the following theorem:
Theorem 2 (Sided positive α-centroids [34]). The left-sided lα and right-sided rα positive weighted α-centroid coordinates of a set of n positive histograms h1,…, hn are weighted α-means:
Furthermore, the frequency-sided α-centroids are simply the normalized-sided α-centroids.
Theorem 3 (Sided frequency α-centroids [16]). The coordinates of the sided frequency α-centroids of a set of n weighted frequency histograms are the normalised weighted α-means.
Table 1 summarizes the results concerning the sided positive and frequency α-centroids.
4.3. Mixed α-Centroids
The mixed α-centroids for a set of n weighted histograms is defined as the minimizer of:
We state the theorem generalizing [15]:
Theorem 4. The two mixed α-centroids are the left-sided and right-sided α-centroids.
Figure 1 depicts some clustering result with our α-clustering software. We remark that the clusters found are all approximately subclusters of the "distinct" clusters that appear on the figure. When those distinct clusters are actually the optimal clusters—which is likely to be the case when they are separated by large minimal distance to other clusters—this is clearly a desirable qualitative property as long as the number of experimental clusters is not too large compared to the number of optimal clusters. We remark also that in the experiment displayed, there is no closed form solution for the cluster centers.
4.4. Symmetrized Jeffreys-Type α-Centroids
The Kullback–Leibler divergence can be symmetrized in various ways: Jeffreys divergence, Jensen–Shannon divergence and Chernoff information, just to mention a few. Here, we consider the following symmetrization of α-divergences extending Jeffreys J-divergence:
For α = ±1, we get half of Jeffreys divergence:
In particular, when p and q are frequency histograms, we have for α ≠ ±1:
where a symmetric Heinz mean [35,36]:
Heinz means interpolate the arithmetic and geometric means and satisfies the inequality:
(Another interesting property of Heinz means is the integral representation of the logarithmic mean: . This allows one to prove easily that .)
The Jα-divergence is a Csiszár f-divergence [24,25].
Observe that it is enough to consider α ∈ [0, ∞) and that the symmetrized α-divergence for positive and frequency histograms coincide only for α = ±1.
For α = ±1, Sα(p, q) tends to the Jeffreys divergence:
The Jeffreys divergence writes mathematically the same for frequency histograms:
We state the results reported in [32]:
Theorem 5 (Jeffreys positive centroid [32]). The Jeffreys positive centroid c = (c1,…, cd) of a set {h1,…, hn} of n weighted positive histograms with d bins can be calculated component-wise exactly using the Lambert W analytic function:
where denotes the coordinate-wise arithmetic weighted means and the coordinate-wise geometric weighted means.
The Lambert analytic function W [37] (positive branch) is defined by W (x)eW(x) = x for x ≥ 0.
Theorem 6 (Jeffreys frequency centroid [32]). Let denote the Jeffreys frequency centroid and the normalised Jeffreys positive centroid. Then, the approximation factor is such that (with wc ≤ 1)
Therefore, we shall consider α ≠ ±1 in the remainder.
We state the following lemma generalizing the former results in [38] that were tailored to the symmetrized Kullback–Leibler divergence or the symmetrized Bregman divergence [14]:
Lemma 1 (Reduction property). The symmetrized Jα-centroid of a set of n weighted histograms amount to computing the symmetrized α-centroid for the weighted α-mean and −α-mean:
Proof. It follows that the minimization problem reduces to the following minimization:
This is equivalent to minimizing:
Note that α = ±1, the lemma states that the minimization problem is equivalent to minimizing KL(a : x) + KL(x : g) with respect to x, where a = l1 and g = r1 denote the arithmetic and geometric means, respectively. □
The lemma states that the optimization problem with n weighted histograms is equivalent to the optimization with only two equally weighted histograms.
The positive symmetrized α-centroid is equivalent to computing a representation symmetrized Bregman centroid [14,34].
The frequency symmetrized α-centroid asks to minimize the following problem:
Instead of seeking for in the probability simplex, we can optimize on the unconstrained domain ℝd−1 by using a reparameterization. Indeed, frequency histograms belong to the exponential families [39] (multinomials).
Exponential families also include many other continuous distributions, like the Gaussian, Beta or Dirichlet distributions. It turns out the α-divergences can be computed in closed-form for members of the same exponential family:
Lemma 2. The α-divergence for distributions belonging to the same exponential families amounts to computing a divergence on the corresponding natural parameters:
where is a skewed Jensen divergence defined for the log-normaliser F of the family.
The proof follows from the fact that ; see [40].
First, we convert a frequency histogram to its natural parameter θ with ; see [39].
The log-normaliser is a non-separable convex function . To convert back a multinomial to a frequency histogram with d bins, we first set and then retrieve the other bin values as .
The centroids with respect to skewed Jensen divergences has been investigated in [13,40].
Remark 4. Note that for the special case of α = 0 (squared Hellinger centroid), the sided and symmetrized centroids coincide. In that case, the coordinates of the squared Hellinger centroid are:
Remark 5. The symmetrized positive α-centroids can be solved in special cases (α = ±3, α = ±1 corresponding to the symmetrized χ2 and Jeffreys positive centroids). For frequency centroids, when dealing with binary histograms (d = 2), we have only one degree of freedom and can solve the binary frequency centroids. Binary histograms (and mixtures thereof) are used in computer vision and pattern recognition [41].
Remark 6. Since α-divergences are Csiszár f-divergences and f-divergences can always be symmetrized by taking generator, we deduce that symmetrized α-divergences Sα are f-divergences for the generator:
Hence, Sα divergences are convex in both arguments, and the sα centroids are unique.
5. Soft Mixed α-Clustering
Algorithm 4 reports the general clustering with soft membership, which can be adapted to left (λinit = 1), right (λinit = 0) or mixed clustering. We have not considered a weighted histogram set in order not load the notations and because the extension is straightforward.
Again, for skew Jeffreys centers, we shall adopt a variational approach. Notice that the soft clustering approach learns all parameters, including λ (if not constrained to zero or one) and α ∈ ℝ. This is not the case for Matsuyama's α-expectation maximization (EM) algorithm [42] in which α is fixed beforehand (and, thus, not learned).
Assuming we model the prior for histograms by:
the negative log-likelihood involves the α-depending quantity:
because of the concavity of the logarithm function. Therefore, the maximization step for α involves finding:
No closed-form solution are known, so we compute the gradient update in Algorithm 4 with:
The update in λ is easier as:
Maximizing the likelihood in λ would imply choosing λ ∈ {0, 1} (a hard choice for left/right centers), yet we prefer the soft update for the parameter, like for α.
6. Conclusions
The family of α-divergences plays a fundamental role in information geometry: These statistical distortion measures are the canonical divergences of dual spaces on probability distribution manifolds with constant curvature and the canonical divergences of dually flat manifolds for positive distribution manifolds [19].
In this work, we have presented three techniques for clustering (positive or frequency) histograms using k-means:
- (1)
Sided left or right α-centroid k-means,
- (2)
Symmetrized Jeffreys-type α-centroid (variational) k-means, and
- (3)
Coupled k-means with respect to mixed α-divergences relying on dual α-centroids.
Sided and mixed dual centroids are always available in closed-forms and are therefore highly attractive from the standpoint of implementation. Symmetrized Jeffreys centroids are in general not available in closed-form and require one to implement a variational k-means by updating incrementally the cluster centroids in order to monotonically decrease the k-means loss function. From the clustering standpoint, this appears not to be a problem when guaranteed expected approximations to the optimal clustering are enough.
Indeed, we also presented and analyzed an extension of k-means++ [17] for seeding those k-means algorithms. The mixed α-seeding initializations do not require one to calculate centroids and behaves like a discrete k-means by picking up the seeds among the data. We reported guaranteed probabilistic clustering bounds. Thus, it yields a fast hard/soft data partitioning technique with respect to mixed or symmetrized α-divergences. Recently, the advantage of clustering using α-divergences by tuning α in applications has been demonstrated in [18]. We thus expect the computationally fast mixed α-seeding with guaranteed performance to be useful in a growing number of applications.
NICTA is funded by the Australian Government as represented by the Department of Broadband, Communication and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.
Author Contributions
All authors contributed equally to the design of the research. The research was carried out by all authors. Frank Nielsen and Richard Nock wrote the paper. Frank Nielsen implemented the algorithms and performed experiments. All authors have read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interests.
A. Proof Sketch of Theorem 1
We give here the key results allowing one to obtain the proof of the Theorem, following the proof scheme of [15]. In order not to load the notations, weights are considered uniform. The extension to non-uniform weights is immediate as it boils down to duplicate histograms in the histogram set and does not change the approximation result.
Let be an arbitrary cluster of Copt. Let us define and as the uniform and biased distributions conditioned to . The key to the proof is to relate the expected potential of under and to its contribution to the optimal potential.
Lemma 3. Let be an arbitrary cluster of Copt. Then:
where is the uniform distribution over .
Proof. α-coordinates have the property that for any subset , . Hence, we have:
Because Dα(p : q) = D−α(q : p) and lα = r−α, we obtain:
It comes now from (33) and (34) that:
This gives the left-hand side equality of the Lemma. The right-hand side follows from the fact that . □
Instead of , we want a term depending solely on as it is the "true" optimum. We now give two lemmata that shall be useful in obtaining this upper bound. The first is of independent interest, as it shows that any α-divergence is a scaled, squared Hellinger distance between geometric means of points.
Lemma 4. For any p, q and α ≠ 1, there exists r ∈ [p, q], such that (1 − α)2Dα(p : q) = D0(p1−αrα : q1−αrα).
Proof. By the definition of Bregman divergences, for any x, y, there exists some z ∈ [x, y], such that:
and since uα is continuous and strictly increasing, for any p, q, there exists some r ∈ [p, q], such that:
Lemma 5. Let discrete random variable x take non-negative values x1, x2,…, xm with uniform probabilities. Then, for any β > −1, we have var(x1+β/uβ) ≤ var(x), with.
Proof. First, ∀β > −1, remark that for any x, function f(x) = x(uβ − xβ) is increasing for x ≤ u/(1 + β)β. Hence, assuming that the xis are put in non-increasing order without loss of generality, we have f(xi) ≥ f(xj), and so, , ∀i ≥ j, as long as xi ≤ u/(1 + β)β. Choosing u = x1(1 + β)β yields, after reordering and putting the exponent, . Hence:
Dividing by u2β the leftmost and rightmost terms and using the fact that var(λx) = λ2var(x) yields the statement of the Lemma. □
We are now ready to upper bound as a function of .
Lemma 6. For any cluster of Copt,
where z(α), f(λ) and h(α) are defined in Theorem 1.
Proof. The case λ ≠ 0, 1 is fast, as we have by definition:
Suppose now that λ = 0 and α ≥ 0. Because , what we wish to do is upper bound as a function of . We use Lemmatas 4 and 5 in the following derivations, using r(p) to refer to the r in Lemma 4, assuming α ≥ 0. We also note as the variance, under the uniform distribution over , of discrete random variable f(p), for . We have:
We have used the expression of left centroid to simplify the expressions. Now, picking , β = 2α/(1 − α) and in Lemma 5 yields:
Plugging this in (36) yields:
Here, (38) follows the path backwards of derivations that lead to (36). The cases λ = 1 or α < 0 are obtained using the same chains of derivations and achieve the proof of Lemma 6. □
Lemma 6 can be directly used to refine the bound of Lemma 3 in the uniform distribution. We give the Lemma for the biased distribution, directly integrating the refinement of the bound.
Lemma 7. Let be an arbitrary cluster of Copt and C an arbitrary clustering. If we add a random couple (c, c) to C, chosen from with π as in Algorithm 2, then:
where f(λ) and h(α) are defined in Theorem 1.
Proof. The proof essentially follows the proof of Lemma 3 in [15]. To complete it, we need a triangle inequality involving α-divergences. We give it here.
Lemma 8. For any p, q, r and α, we have:
(where the min is over strictly positive values)
Remark: take α = 0; we find the triangle inequality for the squared Hellinger distance.
Proof. Using the proof of Lemma 2 in [15] for Bregman divergence , we get:
Taking x = uα(p), y = uα(q), z = uα(r) yields and the statement of Lemma 8. □
The rest of the proof of Lemma 7 follows the proof of Lemma 3 in [15]. □
We get all of the ingredients to our proof, and there remains to use Lemma 4 in [15] to achieve the proof of Theorem 1.
B. Properties of α-Divergences
For positive arrays p and q, the α-divergence Dα(p : q) can be defined as an equivalent representational Bregman divergence [19,34] over the (uα, υα)-structure [43] with:
where we assume that α ≠ ±1. Otherwise, for α = ±1, we compute Dα(p : q) by taking the sided Kullback–Leibler divergence extended to positive arrays.
In the proof of Theorem 1, we have used two properties of α-divergences of independent interest:
any α-divergence can be explained as a scaled squared Hellinger distance between geometric means of its arguments and a point that belong to their segment (Lemma 4);
any α-divergence satisfies a generalized triangle inequality (Lemma 8). Notice that this Lemma is optimal in the sense that for α = 0, it is possible to recover the triangle inequality of the Hellinger distance.
The following lemma shows how to bound the mixed divergence as a function of an α-divergence.
Lemma 9. For any positive arrays l, h, r and α ≠ ±1, define, with and aη with. Then, we have:
Proof. For all index i, we have:
The arithmetic-geometric-harmonic (AGH) inequality implies:
It follows that (48) yields:
out of which we get the statement of the Lemma. □
C. Sided α-Centroids
For the sake of completeness, we prove the following theorem:
Theorem 7 (Sided positive α-centroids [34]). The left-sided lα and right-sided rα positive weighted α-centroid coordinates of a set of n positive histograms h1,…, hn are weighted α-means:
Proof. We distinguish three cases: α ≠ ±1, α = −1 and α = 1.
First, consider the general case α ≠ ±1. We have to minimize:
Removing all additive terms independent of xi and the overall constant multiplicative factor , we get the following equivalent minimisation problem:
where denote the following aggregation term:
Setting coordinate-wise the derivative to zero of Equation (51), we get:
Thus, we find that the coordinates of the right-sided α-centroids are:
We recognise the expression of a quasi-arithmetic mean for the strictly monotonous generator fα(x):
Therefore, we conclude that the coordinates of the positive α-centroid are the weighted α-means of the histogram coordinates (for α ≠ ±1). Quasi-arithmetic means are also called in the literature quasi-linear means or f-means.
When α = −1, we search for the right-sided extended Kullback–Leibler divergence centroid by minimising:
It is equivalent to minimizing:
where a denotes the arithmetic mean. Solving coordinate-wise, .
When α = 1, the right-sided reverse extended KL centroid is a left-sided extended KL centroid. The minimisation problem is:
Since ∑jwj = 1, we solve coordinate-wise and find log x = ∑jwj log hj. That is, is the geometric mean:
Both the arithmetic mean and the geometric mean are power means in the limit case (and hence quasi-arithmetic means). Thus,
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|Positive centroid
|Frequency centroid
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text:
Anticipation mounts in League's changing landscape
New faces abound for start of domestic campaign with big two still most likely winners
There are straws in the wind already that the 2018 League of Ireland season is being keenly anticipated by the fans. Season ticket sales up 65 per cent at League and Cup winners Cork City is one thing, but the 'sold out' notices going up for Bohemians' Jodi Stand two weeks in advance of Friday's televised clash with arch rivals Shamrock Rovers is another, while best of all is the 600-plus season tickets sold by newly-promoted Waterford United.
Putting these facts into perspective is important, because it shows how vibrant the League has become in recent years. For instance, Cork's double comes only seven years after they emerged from administration, and last year saw them record a healthy €2.7m turnover. Meanwhile, two years ago, Waterford were singing the Blues big time, as they were lucky to top crowds of 150 in the First Division.
No doubt, the Premier Division is where it is at, and nothing illustrates that better than the player-drain from the three clubs relegated at the end of last season. Galway United lost seven of their finest - all to Premier clubs; Finn Harps lost four to Premier clubs, two to the Irish League and one to Doncaster Rovers, while Drogheda lost two to Premier clubs, and five to First Division rivals.
Perhaps the standard of the Premier is best illustrated by the busy export market, with 12 players moving to English or Scottish clubs, and another two departing for clubs on the continent and the USA. All of these players will be a big loss to their clubs, and how well they are replaced will determine the clubs' prospects for 2018.
Top of that list is Seáni Maguire, who moved from Cork to Preston, after one of the most magical seasons - or should that be half-seasons - any player ever enjoyed. His 20 goals in 21 games up to his departure in mid-July wasn't surpassed, and his absence was keenly felt on the run-in to the title as Cork were a depleted team without him.
Manager John Caulfield wasted no time in bulking up his squad, and the addition of Barry McNamee and Aaron Barry from Derry City, Tobi Adebayo-Rowling (Sligo), Colm Horgan (Galway), and the return of Graham Cummins from St Johnstone leaves him with a strong hand. None of them, however, are replacements for Maguire. For that, he will be dependent on a front line of Karl Sheppard, Cummins and Kieran Sadlier clicking and causing havoc.
Dundalk boss Stephen Kenny also had to replace key attackers Dave McMillan and Pat McEleney, and his acquisition of former Lilywhite top-scorer Pat Hoban, Ronan Murray (Galway) and Krisztian Adorjan (Novara, Italy) has the potential to be a really potent strike force, with Dylan Connolly, Jamie McGrath and Michael Duffy providing the service from the wings.
Transfer activity has been most notable among the mid-table clubs who, in a 10-team League, now find themselves closer to the relegation zone while also within touching distance of the four honey-spots that send clubs into European combat, where the rewards are greater than anything the League offers in prize money. For instance, Champions League football - even without winning a game in Europe - will be worth €500,000 to Cork City, whereas the winner's prize in the League is €110,000. Europa League rewards are roughly half of those in the Champions League.
Derry City's return to the Brandywell has been delayed, with their proposed homecoming on February 23 now switched to an away fixture in Sligo. If their ground is not ready by March 2 for the visit of Dundalk, they could find themselves with the unusual situation of playing their first five League games away.
While this would give them a good run of home fixtures later in the season, the old saying "a good start is half the battle" springs to mind, and five away games would not be a good start.
Derry are one of the teams who have been busy in the transfer market. Having lost key players McNamee, Barry and Dean Jarvis to Cork and Dundalk, manager Kenny Shiels has added nine players to his roster, two of them on loan. Bohemians and Bray, the teams who finished just below Derry, each added seven potential first-teamers, while Limerick have gone for a radical overhaul under new manager Tommy Barrett, with 10 newcomers.
Just how well - and how quickly - all these new signings gel, is the key to their fortunes. Liam Buckley is another manager in that situation at St Pat's, where he has eight newcomers to accommodate. Waterford also have eight new signings, having lost six of their promoted squad to clubs as diverse as Ipswich and Larne.
The two Rovers - Shamrock and Sligo - have been reserved in their transfer dealings, with five and six respectively. The Hoops just need to improve their disciplinary record if they are to challenge at the top, while Sligo need a good start, and have every chance of that this time, with their first three games at home. Not that Limerick, Derry and Cork represent easy pickings. But then that's the beauty of this League - there are no longer any easy games.
Picking a winner looks like a toss of the coin between champions Cork and Dundalk, simply because they both know what it takes to win this League - and they each look as strong as ever. Apart from the Maguire Factor, that is, and that might just swing the advantage back to Dundalk.
Opening weekend
Friday February 16 (7.45 unless stated): Bohemians v Shamrock Rovers, Dalymount Park, 7.30 (live, RTé2); Dundalk v Bray Wanderers, Oriel Park; St Patrick's Athletic v Cork City, Richmond Park; Waterford United v Derry City, RSC.
Saturday (7.45): Sligo Rovers v Limerick, The Showgrounds
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