In brightest day, in blackest night, no evil shall escape my sight. Let those who worship evils might beware my power, Green Lantern's light. He's known as the greatest Green Lantern. This is Green Lantern Explained, Hal Jordan. Stay tuned. Hey guys, welcome back to Comic-Ed and TV, the only place on YouTube where all geek culture collides. If you're new to the channel, don't forget to hit that subscribe button so you don't miss out on future videos. Introduced in showcase number 22 in October of 1959, Harold Hal Jordan is the first human to be inducted into the Green Lantern Corps and patrols sector 2814 of the universe. Born in Coast City, California, to Martin and Jessica Jordan with brothers Jack and Jim, Hal faced his worst fear at a young age when he watched as his father, who was a test pilot, die in a plane crash. Against his family's wishes, years later, a grown Hal Jordan joined the United States Air Force. After leaving the Air Force, Hal again followed in his father's footsteps by joining Ferris Aircraft as a test pilot while also pursuing his boss and childhood friend Carol Ferris. Now, something you need to know about Carol is that she and Hal have an on-again-off-again relationship. In fact, at one point, she thought Hal was going to propose to her, but in fact, he actually needed her to co-sign on a loan form. It was after this that she developed a relationship with Hal's Green Lantern successor, Kyle Rainer, but we'll dive more into that in their respective videos. While on patrol, Abinsur was attacked and pursued by the being known as Legion, while on its way to Oa, badly injured, and with his spaceship seriously damaged, he made an emergency landing on the nearest habitable planet, which was Earth, and began using his power ring to search for a successor to wield the ring after he passed on. His ring detected two suitable successors, Hal Jordan and Guy Gardner. Due to being closer to the crash, the ring chose Hal Jordan as the most suitable replacement right before his death. The reason Hal was closer, however, was due to a time-traveling booster gold convincing Guy to visit his dying father to ensure that Hal is chosen over Guy. After this, Jordan was trained by the Green Lantern Corps' drill instructor, Kilowog, and then partnered with Sinestro, a veteran corp'sman who was married to Abinsur's sister and Ser's best friend. Jordan was horrified at his new mentor's totalitarian methods, though Sinestro maintained that his iron-fisted rule was necessary to protect his people from the alien forces. During his training, Jordan helped Sinestro repel an attempted invasion of Coragar by the alien warlords known as the Cuns. When Jordan called for help from the other Green Lantern, Sinestro's dictatorship was exposed and he was forced to appear before the Guardians for Punishment. Kat Matui, the leader of the Corgarian resistance movement, who felt that Sinestro's protection, kept her people from growing as a society through contact with other alien races, was recruited as his replacement in the Corps. Though Kat Matui eventually grew into one of the most respected Green Lanterns, she and the rest of Corgar initially resisted her appointment to the Corps due to Sinestro's actions. Corgar had come to consider the symbol of the Green Lantern Corps an emblem of terror and oppression. Eventually, Guy Gardner became Hal's backup as a Green Lantern, also serving in Sector 2814. Hal would later become a founding member of the Justice League and took it upon himself to assume the leadership role even though the group had secretly voted to make the Flash Barry Allen its team leader. Barry let Hal think he was a leader due to their friendship and his unwillingness to contradict him until another team member revealed it to Hal. Hal eventually developed a strong friendship with fellow Justice League member Green Arrow. Beginning in Green Lantern No. 76, Hal has teamed up with Oliver to tackle the social issues of the late 60s and early 70s. Dennis O'Neill devised an idea of portraying Hal Jordan effectively an intergalactic law enforcement officer as an establishment gradualist liberal figure against Oliver Queen, who O'Neill had characterized as a lusty outspoken anarchist who would stand in for the counterculture movement. The stories continued into the pages of Green Lantern Green Arrow. In Green Lantern No. 151, from April 1982 until January 1984, Jordan is exiled into space for a year by the Guardians in order to prove his loyalty to the Green Lantern Corps, having been accused of paying too much attention to Earth when he had an entire sector of the cosmos to patrol. When he returns to Earth, he finds himself embroiled in a dispute with Carol Ferris, faced with a choice between Love and the Power Ring, Jordan resigns from the Corps. During an earthquake, Guy Gardner was hit by a bus while attempting to rescue one of his students. During his recovery, the Guardians recruited John Stewart to be Jordan's new backup. The Guardians then call Stewart to regular duty as his replacement. During the reign of the Superman storyline, Coast City had been completely destroyed by Cyborg Superman and Mongol. Returning to Earth after the events of Trinity, Green Lantern was horrified and he immediately went after Mongol and beat the Galactic Conqueror in single combat. With everything he cherished, piles of cinder and ash, and after he in the Justice League built a memorial to those lost from the remains of the world engine, Hal tried to use his ring to recreate Coast City completely, down to even creating constructs of the people who had lived there. This was a massive construct that required all of his rings charged. A projection of the Guardians was sent to him furious that he was using his powers for personal reasons and threatening him unless he immediately reported back to them for disciplinary action.  One of the greatest secrets kept by the Guardians was that of the Yellow Impurity which was an ancient monster named Parallax who they had imprisoned in their central power battery. Parallax was a physical embodiment of fear on the emotional electromagnetic spectrum, the opposite of the Green Willpower. Hal became susceptible to fear and the monster gained root in his consciousness through his ring. Through influencing his actions, it would eventually possess him and turn him into one of the greatest villains the universe had ever seen. A team of Green Lantern Corps' greatest warriors were sent to stop him before he could reach Oa. Jordan single-handedly went through Bodica, General Creon, Graff Torn, Hanyu, Jack T Chance, Kehan, Lara, and Tomar II. He left all those he defeated floating in space and took their rings in his possession, adding them to his collection. It was assumed that he left them to die, however it was later revealed that he had left them with just enough charge to protect them in the harsh conditions of space. Sinestro was released from his imprisonment to act as the last line of defense and Hal snapped his neck. It was later revealed this version of Sinestro was simply a construct by the Guardians themselves. Jordan then entered the central power battery and absorbed the energies inside, killing the Guardians and effectively destroying the entire Green Lantern Corps. He emerged in the new identity of Parallax. During his time as Parallax, Hal Baladult has former allies in the JLA as well as his replacement and only Green Lantern left in the universe Kyle Rainer while attempting to restore Coast City. Parallax teamed with Extent, who was actually a future version of the hero Hawk from the duo of Hawk and Dubb who had absorbed the time warping abilities of Wave Rider. Together they planned to alter time as they saw fit, which became known as the Zero Hour crossover event the purpose of which was to set things right, Hal was ultimately defeated by the heroes of Earth. Eventually Hal began to gain lucidity and seek redemption for his crimes, healing the paralyzed Jon Stewart and resurrecting Oliver Queen who had died in an explosion. He extinguished his powers and sacrificed himself to destroy the Sun Eater and reignite Earth's sun during the final night crossover event. During the Day of Judgment crossover event, the Spectre was without a host and demons sought to use it to their advantage. With the help of the Sentinels of Magic, Hal became the new host for the Spectre. Hal attempted to bend the Spectre's mission from vengeance to redemption and his powers to remove the star Sapphire persona from Carol. However the Spectre was the stronger of the two minds and Hal had little success. The Spectre had alternate motives for attaching to Hal Jordan. He explained to Hal that the parasitic fear demon known as Parallax had taken him over while he was trapped by the Guardians in the central power battery having been awoken and directed to Hal by Sinestro. Though Parallax was able to suppress Hal and take over the Spectre Hal was able to fight off Parallax and separate both Parallax and the Spectre from his soul. Ganthe sent a beacon of light to guide Hal's soul back to his body and Hal Jordan was reborn as the Green Lantern. Together with Kyle Rainer, Jon Stewart, Guy Gardner and Killawog they defeated Sinestro re-emprisoned Parallax in the central battery and the Green Lantern Corps was reborn. Jordan then sought to atone for his actions as Parallax and became Green Lantern of Sector 2814 once again and rejoined the Justice League despite Batman's mistrust of him. During the War of Light a story entitled Rage of the Red Lanterns which takes place after the Sinestro Corps war Sinestro kills Lara, the Green Lantern of Sector 112 who eventually became a Red Lantern after killing Amun Sir the son of Abun Sir who was a Yellow Lantern. Hal is overcome with rage which attracts a Red Power Ring inducting him into the Red Lantern Corps as a Red Lantern Hal attacks both Sinestro Corps and the Blue Lanterns. Blue Lantern St. Walker then forces his ring onto Hal in an attempt to free him from the Red Ring's influence. After this he attacks Atrocitus the leader of the Red Lanterns which causes the Red Lantern Ring to explode leaving Atrocitus wounded and Hal wearing a Green, Blue Lantern, Hybrid uniform and rings. Ganthet tells him this was never his intent to make him a Blue Lantern but to become the leader of the Blue Lantern Corps as Green Lantern, reasoning that Hal's willpower would be strong enough to power the whole Blue Lantern Corps. Hal is then called back to Oa by the Guardians so that they can remove the Blue Ring themselves. The Guardians are unable to remove the Blue Ring but continue to blast Hal with Green Energy in an attempt to destroy it. Hal eventually meets Lor Fleaze the only Orange Lantern in the universe whose power comes from Avaris otherwise known as Greed. Lor Fleaze takes an interest in Hal's Blue Ring and decides to take him underground to separate him from the rest of the Lanterns and demands Hal give him the Blue Ring. Realizing that Lor Fleaze's Ring must be constantly recharged by his own Orange Power Battery to create his Orange Lanterns Hal tries to steal the battery while holding the battery. Hal falls under its influence however Lor Fleaze takes his battery back. During Blackest Night Hal allows himself to be possessed by the Fear Entity once more in order to stop the Spectre who has become a Black Lantern. For those of you who don't know the Black Lanterns are fueled by death and feed on emotion. Anyone who's ever died even if they've been resurrected is capable of becoming a Black Lantern. After finding out the Necron who was the Entity of the Black Lanterns and death itself was after the Entity of the White Light Hal discovers that the Entity originated on Earth. Not Oa as the Guardians said. Hal realizes that the Entity needs a host similar to Parallax and Ion. As Hal attempts to leave Parallax, Sinestro tells him it's his destiny to be the host and not Hal. After Necron separates Sinestro from the Entity, Hal bonds with it and uses its power to transform the Resurrected Heroes into White Lantern dubbing them the White Lantern Core. They defeat Necron. During War of the Green Lanterns Hal, John, Guy and Kyle all agree that removing Parallax from the Central Power Battery is the only way to save the Core. They also decide to use different rings. Hal taking the Sinestro Core ring, Kyle taking the Blue Lantern ring, Guy taking the Red Lantern ring, and John taking an Indigo Tribe ring. Hal takes an extra ring, the Orange one, and Guy takes the Violet one to use the full power of the Emotional Spectrum to crack open the Battery and release Parallax. After Hal killed the Rogue Guardian Chrona, the other Guardians believed Hal to be the most dangerous Green Lantern in the Universe. Despite Gantt's attempt to defend him, Hal discharged from the Core, and the Guardians take his ring and send him back to Earth. During DC's New 52, Hal becomes a Green Lantern again after Sinestro, who has been made a Green Lantern, gives him a ring and recruits him for a mission. During this time, the Book of the Black, which is a Black Lantern Book of Prophecies, reveals that Hal joined his destiny to become the greatest Black Lantern. Hal later gets a new look as he goes rogue from the Green Lantern Core in order to create a scapegoat for the Core and be the focus of the Universe's blame and distrust of everything that had taken place in recent issues, such as the Third Army's assault or Relic's attacks. The Core itself unaware of Jordan's intentions to show the Universe that the Green Lanterns are not corrupt and will go after one of their own, believes that he has actually betrayed them when he attacks Kilowog. Along the way, Jordan steals a Green Lantern Prototype Gauntlet, which was used by Krona, and Power Pack from the Armory, allowing him to continue to operate as a hero without the need for a power ring. Although, he is sometimes required to fight other Lanterns to maintain the illusion of independence. In DC, Rebirth, Hal returns as a Green Lantern again, now equipped with his self-constructed power ring, and merging with Will of the Power itself. Searching for the rest of the Green Lanterns and hunting down the Sinestro Core, and returns to Earth, temporarily to assign Simon Baz and Jessica Cruz the task of protecting Earth while he and the rest of the Green Lanterns are away. He takes their power batteries and fuses them together into a single battery to help the two bond as Lantern partners. As a Green Lantern, Hal's powers include projecting hard light constructs, flight, and utilizing various other abilities through his power ring, which are only limited by his imagination and Will power. As Parallax, Hal was one of the most powerful beings in all of the DC Universe. In addition to his normal Green Lantern powers, he was able to manipulate and reconfigure time space to his Will. Manipulate reality at a large scale had vast superhuman strength, which he demonstrated by being able to knock out Superman with one punch. A higher sense of awareness and enhanced durability. As a Yellow Lantern, he has all the same powers as a Green Lantern, only rather than being fueled by Will, it's actually fueled by fear. As a Red Lantern, fueled by the rage, he can perform the same tasks as a Green and Yellow power ring. However, Hal's blood is replaced by a form of corrosive rage energized blood. How can regurgitate this blood? The blood has the effect of bursting into flames of rage. The flames are so potent that they will burn even in space. The blood can melt through the constructs of other ring wielders, penetrate their personal force fields, and corrupt their rings, depleting their energy at an accelerated rate. The Red Ring's blood is one of the few known substances that can destroy a Black Lantern's core faster than even they can regenerate. Since the Red Ring completely takes over the circulatory process of the wearer, removing the heart of the Red Lantern only temporarily incapacitates them. A Red Power Ring operates by expelling the blood of the wearer, replacing it with red light. As a result, removing a Red Lantern's ring will result in the wearer's death. However, a Blue Lantern's ring can reverse this process, freeing the wearer from the ring. The combined powers of blue and green rings can destroy a Red Ring. As an Orange Lantern howls ring uses avarice energy supplied by a power battery, which in most cases takes the form of Orange Light. An Orange Power Ring, possibly one of the most powerful weapons in the known universe, has the ability to effect and use fundamental forces of the known universe, including electromagnetic energies such as gravity, radiation, heat, light, and powerful blasts of concussive force. He can produce orange energy blasts, force fields, energy constructs, energy absorption, identity usurption, energy avatar, and security protocols. However, an Orange Light wielder is susceptible to insatiable greed, hunger, as well as blue light and violet light. As a Blue Lantern howl is fueled by the emotion of hope. Blue Lanterns must be near an active Green Lantern's power ring to tap into their own ring's full power. Otherwise the rings are only capable of the default abilities of flight and a protective aura. This is because the power of hope is nothing without the will power to enact it. While under the influence of a nearby green power ring, a blue power ring has the same abilities as a green ring, plus some unique powers of its own. Blue Lanterns can heal wounds and regenerate lost body parts. As a Black Lantern howl had the power of reanimation, healing, and regeneration, energy spectrum assimilation, aura recognition and infection, and is fueled by death and feeds on the emotional spectrum. As a White Lantern which is fueled by life, he was a white energy conduit. He could dominate the emotional spectrum, white energy blasts, force field generation, energy constructs, environmental playback, flight and create an energy twin, ring duplications, mind alteration, energy absorption, wormhole creation, limited cellular regeneration, electromagnetic scanning, xenolinguistics, material alteration, create an emergency beacon, homing beacon, invisibility, and light refraction, time travel, phasing, thought relay, levitation, overriding rings of other colors, resurrection, and having a pocket dimension directly inside the ring itself. So there you have it geeks on this episode of Green Lantern Explained, the history and origins of the greatest Green Lantern, Hal Jordan. Hey guys if you enjoyed this video make sure you hit that subscribe button right there so you can stay up to date on all things geek culture. Also you can check out one of these two playlists right here on the side for more videos just like the one you just watched.


Good morning, welcome to a new vlog. My name is Sonali and it's a Wednesday at 9 a.m. I'm going to go to a coffee shop at 10 a.m. and meet my friend there. We're gonna go to Greater Goods. I actually went there with Ryan once and I think it's gonna be my new favorite coffee shop because first of all, the matcha was good. And second of all, it was just a really nice big space and they had outlets, so that's definitely most important. But I thought I would show you guys how I make my newest obsession, avocado toast with honey. Last Friday, I went to Blender & Bowls here in Austin. I've heard a lot of good things about that place, so I finally tried it out and they had this avocado toast with honey, peppers, and it was just so good. So I recreated it at home and it's pretty much the same. They use Sardo, but I'm just gonna use Dave's killer bread. I love this one because each slice is just 70 calories. So they are like a little bit smaller and thinner, but I feel like it follows me up. All right, time to pop them into the toaster. I'm gonna go ahead and cut my avocado. Sometimes I like cut it in half and then peel off the skin and then I cut it really aesthetically, but today I'm not really feeling that. So, and this one's brown anyways, I feel like it wouldn't be that pretty. So I'm just gonna mash it up on the toast and then we'll go from there. I've been creating a lot of fashion tech talks recently and it's been so fun to do that. Last night I posted like an Amazon, what I got from Amazon, and I got blocked from like commenting back to anyone. I was trying to comment a link to someone that was asking for a link and it wasn't working like it wouldn't paste it in.  My video is literally going viral. People are actually commenting and like asking questions about like the clothes in the Amazon Fashion Hall and I can't comment back and I'm like literally losing my mind. I searched on Google and it said like it would probably go away in like 24 hours, but like it doesn't even say when it's gonna go away. All right, the toast is ready and I just cut my avocados like this, like literally just as small as I could get it. Okay, now I'm just gonna mash it all down. I feel like this is pretty self-explanatory, but might as well do all the steps while I'm here. I barely have any honey, I'm just gonna heat it up in the microwave so it actually can come out. I'm gonna put some Himalayan sea salt and load this thing up with salt. I love me salty anything. Like I just could literally eat like the salt flake. I don't know if you guys saw that vlog, but I definitely ate like a salt cube and I loved it because I'm crazy. Next we'll do a little bit of garlic powder or a lot because this thing comes out so fast. I'm just gonna mix it in a little. Got our red peppers. Last but not least, we are gonna add some honey to this. Breakfast is served. I'm the type of person that eats the crust first and then saves like the middle for the last part because it's just so good.  Okay, so this sweatshirt thing I got from Amazon. I actually have something similar. Like it from Abercombe and Vich. It's literally pretty much the same exact thing and this thing was like $12, I think maybe 13. It's actually a men's jacket. So it comes in like a bunch of like bigger sizes which honestly I was thinking about returning it and getting one size up but I feel like it's actually a good size. I just think it's so cute to throw on over a matching workout set and then this set is from Set Active. I've been getting a lot of questions about like what bra was do you wear and honestly I wear Set Active's sports bras majority of the week and I love these one-shoulder ones because you can actually wear them underneath like one-shoulder tank tops or like a razor back tank. They go really well with them and they're like hidden really nicely so highly recommend. This color is, I think it's called like OAT or something. I had some money on a matcha and I just need wifi. I wasn't at the coffee shop for too long because the wifi was really, really shitty and my hotspot wasn't working like, you know, amazing. So I was just like, I need to go back home to my own wifi but it was great seeing Natasha. I feel like I always get so motivated when we are just like able to like vent about YouTube and like just everything social media because she also has a YouTube channel and you know is trying to do the whole TikTok thing. So it's just really nice to like get all of it out and I just feel so scatterbrained because I just feel like I could be doing so much more to like grow everything and I just don't really have the time. Like, I don't know, that sounds so stupid but like I do freelance and then I also have my part-time job and then I have YouTube and Instagram and like TikTok and there's just like too many platforms out there right now and I feel like it's just gonna get even worse especially with Instagram, just creating like all of these different features and then TikTok is apparently adding stories now. I'm just like, can we get a break over here? So I'm feeling very overwhelmed in a good way but also not because I feel like my brain's about to explode. So I think I just need to like sit there and like take notes on like things I wanna do and I don't know. I think I just need to write it all down honestly. Like I saw that she had a notebook and she was writing and I was thinking that probably feels really good to just like write it out and like get it out of your head instead of just typing it. You know, I feel like it's different and I have like so many random notebooks. I need to find them though. Also, I need to take a picture of my vaccination card because I'm going to NOLA and apparently they just told the city that some restaurants are requiring proof of vaccination, but I don't wanna like take my card because I don't wanna lose that. So I need to find it and take a picture of it somewhere and here I am. Another thing I need to do is just organize this whole place because I feel like we know where our stuff is but it's just not organized the way I want it to be and I don't know. Okay, as you can tell, I'm very overwhelmed right now so I'm gonna stop vlogging so I can just breathe and figure my shit out and I'll talk to you guys later. It's six 10 and I actually have a shoot tonight. It is at Big Mathias and it is a gender reveal. So I think they're bringing like smoke bombs or like confetti something or another. I've just been sitting on the couch working on organizing my Amazon storefront. So at first I had like just one little section called like fashion items but I actually went in and put like tops in one, bottoms, active wear, all of that stuff so you guys can shop it a lot more easy if you're like actually looking for something specific. I'll definitely have that link down below. I feel like this whole day I've just been eating like random crap, I don't know why but I'm just like not full at all. I think we might get tacos tonight.  Sometimes we just have these weeks where we don't and I didn't go grocery shopping. I was gonna try to just like eat what I had in the fridge but there's really nothing. So yeah, I think tacos sound like a good idea. Actually both went to college station. I had to go there for my work. I did some Matterport tours, super exciting. I literally was there for not even two hours and the drive was like almost four hours total but he got to see college station. What did you think? It was a little. It might be a lot of Boston. Is that a joke? Like I've just been sitting on the couch just like on the computer most nights. Yesterday I had to shoot but most nights I swear. I'm just like sitting on the computer and I just wanted to get outside. So my idea fun is going to the grocery store so we're gonna go to the HEB in Mueller at first. I wanted to go to Whole Foods but then I remembered it's so expensive. Anytime we go there I spend like $40 or like $50 and get like literally two things just because the packaging is cute. So we're gonna go to HEB and apparently they have like food vendors in there but I'm kind of craving like just HEB sushi so that might be a thing. Is this a newer one or just like a big one? I found the best honey mustard in the world. I think I have to get it. I don't even know what I'm gonna eat it with. High school honey mustard. And the HEB parking lot are of a different species. I feel like they are so evil. I have very bad news to report like extremely bad news. So basically in one of my last vlogs I told you guys that Ryan got me serious XM.  What? XM radio. Because I liked the SoulCycle station in his car. So then I go to my favorites and I noticed that it's like not there. Like it doesn't say SoulCycle and it's just like dot, dot, dot. And I was like, where the frick did my SoulCycle radio go? So then we go to like, I guess he hasn't memorized that it's like a TikTok radio. Like what in the world? And literally like we turned it on and it's so funny because it was like the commercial for like, this is TikTok radio like blah, blah, blah, blah. This is what you're gonna hear Doja Cat and then like, oh, no, no, no, no. I was like, yeah, like, oh, no, no, no. Like this cannot be real. They can't be canceling. It's just like the backyard again song. Oh yeah. And she was like, I'm gonna be like singing the backyard again song all day every day. I'm like, please, no.  Like, I feel like radio stations already play TikTok songs, you know? So like, they didn't need a TikTok radio. So I thought someone's better too. They had to fill the mixes. I know they have like such fun mixes, but there's no way that it's like not somewhere in here. I think we just need to find it. Like maybe they just like took over the station, you know? Oh, like that is, that was scary. We just need to find it now though.  Oh, it's a flower instead of a fire fish. Like assortment kind of thing. Looks like that. And then you got a filly roll. I think I just put my finger in the wasabi. And we got some H.E.B. Lime LaCroix. Cheers.  I have two back-to-back trips coming up. I'm going to Noah and Nashville and then he's going home. 31st. Because you're going to a UGA game, right? Hello. Hi, this is Tyler from Maddie's Degree. Four people? Yes, they actually had just emailed me today. So we're good. Oh, you're good? Mm-hmm. I apologize. No, you're good. Thank you so much. Bye. I got a reservation for tomorrow at Maddie's because my friend Hannah is leaving us. She's moving to Noah actually. This weekend is her last weekend here and we're actually going to Noah next weekend and she's like moving then. So we're trying to go all out. We're going to do kind of like a bar crawl situation on Saturday and just do dinner and drinks tomorrow. Who's Georgia playing when you go? We're going to play Clemson in Charlotte. So it'll be at the Carolina Panthers football stadium. What? Georgia's number eight, number four in the nation. Oh wait, is this the first game? Oh, I had a mind blank. I was like, wait, is this a ball game? Like, it's fine. Yeah. Yeah, I'm also gonna get the FedEx Cup I think in East Lake. FedEx Cup? Goal? Yeah. I honestly am kind of sad that I didn't have time to go home this summer because I literally wanted to just swim in my dad's pool, be a baby for like two weeks, have my parents cook for me, but my mom is coming to visit for the first time ever in mid-September, so I'm really excited about that. She is vegan, like, majority of the time though.   Hi. Chopi, she's being an angel. I mean, honestly, she is an angel. Like, we really lucked out with her. I love her so much, but this morning when I was trying to leave for college station, like, pretty early in the morning, I wanted to leave by like 6.30, but we ended up leaving at like 7.20 because first I took her out, she pooped, and she had a poop in her ass, so I had to pull it out and it was like a smooshy poop. This is TMI, but I feel like if you're a dog mom, you would understand. And it got all in her hairs and it had to bathe her. And then, like, literally when we were leaving, she like threw up and I was like, what the hell? She is so dramatic sometimes. Like, I don't even know if she was sick. I think she was just like, they're leaving me. I have to throw up and make it seem like I have to go with them or something, you know? Good morning, it is Friday. I was able to sleep in today because I don't have work and today's just gonna be kind of chill. I'm getting a massage at 11. Don't remember the last time I got a massage. The now actually invited me in for a massage and I got to pick which one I wanted so I'm really freaking excited for that. On Wednesday, I tried to work out like super hard because I thought if I was sore and I got the massage, like it would feel 100 times better, but I'm sore in the wrong places so that was not smart. I'm just like sore on like this part of my leg because I was doing squats but I definitely should have done like back stuff but I think it's too late now because I don't think I'll be sore in time and I don't really wanna go to the gym this morning, honestly. I'm just like kind of in a relaxed mood, kind of setting the vibe for the massage. Before that, I wanted to share a couple recent Amazon finds. I got this bag right here. Actually I have something similar to this.  It's a little bit bigger actually and I thought this was going to be a little bit bigger but I'm fine with it because I don't need anything super big. I really just got it for going out in NOLA because I didn't wanna take like a nicer bag. Well, this is actually pretty nice.  Just in case. So I got this and has this little chain on it and then I put this string on it too that came with it and just tied it here. And obviously you can wear it as a cross body too. I'll probably be wearing it like this in NOLA just so I can have my stuff right here. I filmed a TikTok earlier this week that Herschel sent me like two duffel bags and then I got another box and they sent me something else. I'm like, what the heck?  Oh my goodness. I have a freaking suitcase.  This is what it looks like. It's like a matte tortoise shell kind of color. I think that's so cool. Usually I don't pack this light though. That's the only issue. So I wish it was like a little bit bigger but honestly there are probably some weekend trips that I can use this but I don't know if I'll be able to. Honestly, NOLA is a weekend trip but still you guys know I pack way too much so I don't know.  So I'm definitely gonna use this for more chill weekend trips but I don't know if I'll be able to fit everything in here but I do wanna quickly show you guys the two other duffels that they sent me. Here's the first one. It is a larger one and I love the color that they sent me and here's the baby one. I actually like this more than the bigger one just because I feel like this could be used like in my everyday life. Like if I'm going to a friend's house and I just like wanna pack like food or like I don't know just for bachelor nights you know when I wanna bring my laptop over and like pack a sweater or blanket I feel like this is super cute. Thank you so much to Herschel for sending this over. Seriously it could not have come at a better time so I'm definitely using one of these for my trips coming up. Next up I got some things from Charles and Keith. I actually recently found out about this online website on TikTok and I think it's a European store or just like more popular not in the United States but I saw that they shipped to the US so I wanted to you know try it out because basically they just have like designer looking like shoes, accessories, bags for not designer prices. So it's like pretty much all under $100 and like the shoes are such good prices. I was really wanting to get some new shoes for Nola because I know that I can't really wear my taller heels. So here we go. First pair. Oh my goodness. These are so cute. I am so shook right now. Okay so basically I don't know if you can tell. Let's try to focus. Okay this is like that like hemp string. What's it called? I can't even remember. I used to make bracelets with this stuff but yeah it has this like material. It's a nude shoe and then a low heel which is gonna be great for honestly everything because I don't know I love heels. They just make everything look so elegant but sometimes it's just unrealistic to walk in heels for a lot of events. So I got that pair and then I actually couldn't decide which color I wanted. So I ended up getting them in white too.  So that's why I was like I kind of want the nude too but I freaking love these. I will definitely leave the link down below. I'm pretty sure they were like $53 for a pair of shoes. Like that is unheard of for this quality and like this beautiful shoe. I'm shook honestly. They just have such nice work wear shoes and just like more classic shoes I will say but they have some really nice heels, boots, all of that stuff. Okay let's get into the purse because I'm really excited about it. Okay not that I really need another bag but you know kind of wanted it. All right guys here's the new bag. It's giving me those fall vibes with that quilted texture and it has this gold hardware which you guys know I love gold.  Okay I've been needing like a card holder for so long now and I was really thinking about getting like a designer one just because but I saw them on Charles and Keith and I was like honestly I don't really need a designer one no one's really ever gonna see it. So I just got one from them and this is what it looks like. Got my hair up in a bun.  I just got out of my massage and it was so amazing because first of all I feel like when you walk into a place and the decor is on point and it just like feels vibey in there like you immediately get in that like I don't know you get in that mood to like have a massage if that makes sense. I feel like the AC is pretty loud. But yeah like the decor is stunning so pretty and then when you go like behind when you're like going into your massage I feel like it was in a movie you know like those like meditation retreats. It looked so beautiful in there. I was like can I take a picture of this because it's so cute. So I'll post a picture on the side right here. Before the massage started they asked me like what do I want to get out of the massage and I've never had someone ask me that. So like I didn't know what to really say but I do like to try to manifest so I was just like you know I have like a lot of unknowns so I would like to take this time to like kind of just like think but honestly when I was in there I was just like in a mental zone where I wasn't really thinking about anything like I feel like my mind was blank which I really needed. And then after the massage she told me that I was flat footed and that I had like a lot of tension in my hamstrings and that I needed to do like a certain like like foot raise or like a toe raise kind of thing. And I thought that was really interesting cause I feel like whenever I've gotten massages in the past they're just like okay you're done bye you know so it was kind of cool that she even like told me that and then they gave me a gift so I thought I would open it on camera. Wee! This is so cool. Okay I'm guessing this is a crystal. I've never had crystals but I've always been really intrigued by them because like I said I am kind of intrigued by the whole manifestation thing and I think that's kind of like related to crystals or crystals can like kind of help you do that. I don't know. Okay so this is a black term line. I'm just gonna read the card cause I have no idea what it is. Build a spiritual fortress around you with black term line. Your protective soul sister and personal cheerleader. This stone words off the negative while promoting good luck and happiness. I need this. I got a gua sha. I've actually been wanting to get one but I've never just bit the bullet. So it says upgrade yourself care routine with a gua sha tool. Lift and firm the skin, reduce inflammation and improve circulation, release stagnant energy. Promote lymphatic drainage. It's stunning too. Here is the refresh eye mask and then they gave me the calm balm which I think is more, oh yeah, this is what was in my massage. It's coconut oil based and it has like peppermint oil and just like so much good stuff in it. If you guys have a now location I highly recommend going and treating yourself to a massage because I feel like I don't do this very often and I want to do it more like I want to do more self care and self care can be really anything like it can be going on walks. It can be like doing a freaking eye mask but it can also be treating yourself to experience like a massage or like a facial or something. I'm pretty sure they do have facials too if I'm not mistaken. But anyways, I am about to drive home. I need to shower. It is hair washing day unfortunately. I don't really want to just because I feel like that would save me some time and then I could like create content but I need to because last night my head was itching. I last washed it on Monday and it is Friday now so it definitely needs some cleaning. I just don't want to have to style it again, you know? Just gone out of the shower and I had to do some deep cleaning in the scalp area because I just, I don't know. I feel like it was kind of itchy and I needed a little refresh. So I used the Briogeo scalp shampoo or charcoal shampoo or something. It like comes in a little tub and it feels so good on my head. But now I'm just doing some skincare really needing to just hydrate everything. So I got a few new products from Coco kind. I actually heard about this brand a while ago because I have eczema and I'm pretty sure this brand had like a few products like specifically for eczema but now they have like literally so many products and I've been using this cream lately for my face. Texture, smooth and cream with celery super sieve complex and it just feels so refreshing but also like so moisturizing. It kind of feels like a cream from the brand Fresh that I used to really, really like but obviously it was a little bit more expensive. So I've been really liking this. I had yet to try this so I thought I would go ahead and try it out today. The revitalizing eye cream it comes with this like little silver applicator. I feel like every other eye cream especially like from Bougier brands too not that this isn't a bougie brand but it's just a more affordable brand but a lot of like the bougier eye creams don't even come with like this really nice applicator. This is my first time using this but I feel like I'm definitely going to be using it more. I like it. Feel so good. Just gonna rub that on my upper lids too. I remember when I went into Sephora one day and I was asking like the Sephora consultant like how do people get just like smooth lids and like how their eyeshadows blend so well because I feel like my eyelids were always so like crackly and stuff and she was like you have to be moisturizing your eyelids. Like they have to be you know like a good base to start off with and I was like that makes so much sense. So I definitely took that into account and started moisturizing my eyelids and it honestly did help with like the eyeshadow application. Next I'm just going to put on some Super Group sunscreen just because it feels so good and this week has just been so hot and I don't know if it's just because I'm like missing the California weather because I just went on vacation there or what but I'm just like oh I cannot wait for fall so I don't have to walk outside and immediately start sweating. I'm just going to moisturize my lips with the Coco kind my matcha all over moisture stick but I like to put on my lips. I'm kind of hungry and I have one beef patty left, Jamaican beef patty. I used to love these. My dad would take me and get Jamaican beef patties and ginger beers when I was little and they were so good and I just remembered we went to the grocery store last night and I got some poppy. This is like the best flavor that they have. It's raspberry rose, trust, get it. I feel like I see them on sale now all the time maybe because they knew that they were really expensive but seriously this fricking flavor is so good. Also you can buy like a 12 pack on Amazon or something. I just took the biggest nap. It felt so good but I was not putting in doing that. I wanted to like film TikToks but whatever it felt really, really good. Now I'm picking out outfits and I love this jumpsuit. Like it makes me look fricking snatched. Look at that, amazing. Don't mind the bra, I would change that but I don't know, I'm not feeling this for tonight because the place we're going tonight is at focus, I don't even know. New camera ladies and gents. Figuring out as we go. Tonight we're going to Maddie's. It's pretty much a wedding venue. It's stunning there. So I feel like I should dress up even more. Like I don't know if this is like the vibe because I feel like this is like a daytime thing. So I'm gonna look through my closet and see what else I have. I feel like this is definitely more the vibe. This dress is from Shop Ride Dress. I don't think this is sold on the website anymore but honestly Shop Ride Dress has beautiful maxi dresses that like aren't $100. Like I think they're around like the $60 range which honestly if you're looking for like a wedding dress that's not that bad. Well not like if you're the bride but if you're a wedding guest. $60 is really not that bad because I feel like people buy like revolved dresses now that are like hundreds of dollars. But yeah, this is the full fit. Actually number one, I need to access rise. First time wearing my trousers and keyed heels I decided to go with the white pair and hopefully they're comfortable. I think we're gonna be a little late to our reservation just like a few minutes but I'm on my way to pick up my friends because they live like closer to that restaurant. I was just like, I'll just scoop you guys. Say hi guys. Hi guys. This is stunning. All these peacocks, not that bird. It's a peacock. There was so good and now I'm back home. I think we're gonna go get drinks that kind of tropical. We've been wanting to go to this place for forever. So tonight is the night I saw that they had a frozen pina colada on their menu. So I'm really excited to try it. I got a pina colada. Apparently it's a tumor pina colada and then he got a mango margarita. Is it good? Yeah. Rather this big. Huh? Rather this big. Yeah. How much was it? $25 a bit. Oh damn. Taste test. He has like a resort pina colada but it's still really good. Try mine. Okay. Good morning. It is Saturday and we are about to head out the door to go to Luz which is on the east side. I saw like an Instagram post the other day about like their donuts. They look so good and apparently they only are there on Saturdays and Sundays. So they get there at 8 a.m. It's already nine. So I'm hoping they're not sold out. I'm not even sure if it's that popular that I would even sell out but it looks so good. So I really wanna try it and then Ryan's gonna go golfing and then I'm gonna have a content day because much needed. I literally need to film videos but I just don't have like a filming space because nothing's like decorated. Like yeah, my couch is here but like nothing's above it or anything. So I just feel like it's so plain looking.  But obviously I know you guys love blogs too but I'm just counting down the days till my couch gets here so I can finally hang up all my art and just like figure that yes or no. But the dough is like kind of funky. I wanna show them. This is kind of bitter, this is kind of sad. I was really excited about this. Like it's not easy to mess up donuts but those donuts were actually not good. Like they were really gross. So don't recommend them. They're not actually loose donuts. Like they source them from like an outside bakery. So I definitely so wanna try lose again for like dinner because I know that they have like a happy hour that they do like $5 burgers or something like that. I've only been there twice and I haven't had any. Maybe I shouldn't go back for the burgers. I mean the first time I went I got that like Mexican top with coffee or whatever. Oh yeah. It's like spicy chocolate coffee. Yeah, maybe you didn't know what that was but I don't know. The donuts, gross.  Today I'm going to make it a content day where I just left for golf. So that means I need to be productive and take advantage that he's out of the house. So I'm gonna do a face mask because I feel like whenever I do a face mask before I pump my makeup on, my makeup is flawless throughout the day. So I was like, today's gonna be a pretty long day. Like I said in the beginning of this vlog or middle of this vlog, I feel like this vlog is going to be a very long one. So hope you guys like it. But my friend Hannah is leaving and moving to Nola. So this is her last weekend. So we're gonna do like a bar crawl situation. We're gonna go to Rainey Street and then West and I don't know what else in between. This is a face mask I'm using. I love these face masks from Rael. They are so freaking good. Honestly, I'm not really sure where to get them. I know they have like a website that you can obviously purchase on. I'm not sure if they have them on the Amazon or like Target or where they're sold, but they are so good. Slap this on my face.  Like I said before, I really wanna get back into it, but it definitely does take some planning, especially more than vlogs. So I think I'm gonna just sit down, pull up a word doc and like start typing some stuff out. And also I really want to look for a new like duvet insert. I always see these beds on Instagram that are so plush and fluffy. And I'm like, I don't know what I'm doing wrong.  So if anyone knows how to get that like really plush, fluffy look on a bed, let me know because I'm really wanting to go for like a coastal vibe for our bedroom and just get like blue pillows and blue accents. But like, you know, like more of a muted like light blue. Just finished my makeup, but I still have to do my lips. I recently got this Sephora liquid lipstick. It was so affordable and I actually really like it. It's in the color 77. I don't know if it has like a name or anything, but it's kind of just like this pinky brown color. It was such a struggle for me to find like a brown liquid lipstick that like would go with my skin tone and Sephora. Like everyone was just like pink, like really pink, or just like a pale pink. Any of the brown shades would just be like the color poop, like so dark, nothing in the middle.  So I don't really care, but it's really good actually. So I just put a little on, smack it around, wait till it dries a little bit. And then I do some lip liner. I really like the cork lip liner from MAC. I just do that on top of this lipstick. Also, I wanted to share my new hair product obsession and I definitely want to do like my favorite hair products of all time, like a whole video all on that. So I'm gonna work on that for you guys because I feel like I get a lot of questions about hair products. But anyways, this is the way Matt Pommade, I got this at Sephora when I got that liquid lip. It's a very recent purchase, but I've been trying out like different ways to use it. One way I found to use it is adding texture to your hair. So as you can tell, my hair was freshly curled and it kind of needs to be a little bit separated and just like I want to go for those beachy waves, not this like perfect like ring like curl kind of thing.   Let me show you guys. You can already see a big difference. It really just added a lot of texture to my hair and also like a lot of volume compared to this side. So I'm gonna go ahead and do the other side now. Here's the final product, so much more voluminous and just like effortless I feel like. So again, this is the way Matt Pommade, you need it next time you go to Sephora or even order it online right now. I'll link it down below. I kind of rearranged the guest room to actually have like a filming background. Look how cute this is. Like why did I not do this before? I'm so mad at myself.  I recently got this huge candle from Target for like $15. It smells amazing. I will try to link it down below and I have this wooden bowl that I actually found on Facebook Marketplace but I've seen some on Amazon. So I will link that and then also these beads you can find on Amazon too. And then this vase is from at home. I'm pretty sure I got it when I was living in the sorority house. I think it turned out so cute. I don't know if I should leave it like super simple or add some more stuff, let me know what I should do. It's been a while since I finished filming but I still have the candle burning because ever since I heard that you have to like burn the whole top layer every time you burn a candle, like I literally just can't blow it out if everything's not melted, if that makes sense.  We got a quick late lunch at fly right. It wasn't that good. Like the tater tots are usually really good and they're not good. But anyways, I got this yummy lemonade and I was like, might as well put some vodka in it because my friends are at Matzal Ranch where they were while I was filming. So they were already on their second margarita. So I'm like, how tipsy are you guys? Like I need to be on your level when I see you. They're like, we're definitely tipsy right now. So start drinking. So I'm gonna drink this. It's 5.30 and now I think we're meeting at 6.30. But it's kind of nice because I'm already ready so we'll have to like rush or anything. I might film like a couple more TikToks. I'm really excited to go out tonight because I feel like I have not been out in so long and I've been mentally preparing for it.  But I feel like it was better when you were like mentally prepared and I've been hydrating. I don't know, I sound like a freaking grandma right now but you just gotta do what you gotta do, you know? Shit. Nope. I decided to take it in shot form and I just learned how to take shots so that you don't like taste it afterwards.  So cheers. Wait, no, I'm nervous. Okay, easy.  Let's do an O.T.D. I love this time of day because the apartment just gets lit so nicely. Okay, so shorts are from Topshop. Got it at Nordstrom. They're the A-line short. These are so good. They fit me perfectly. Got a size 31 shirt is Princess Polly. I'm obsessed with like this fit. I don't have to wear bra with it, which is amazing. You guys know that I'm wearing my bar shoes. The Air Force ones, the dirty ones, the OGs, they're nasty. But you know, gotta wear them when we're going to rainy. This night was really fun. I have not been drunk in forever so it was just really fun to be with my friends and have a good night. So I woke up like not hungover but just, you know, not feeling 100%. Actually we woke up at literally 11 and stayed in bed till like pretty much noon and then we got Starbucks and then we took Sophie to Waterloo Park and it's just a really nice area. I think they also built an amphitheater. So there's gonna be like concerts there.  We went midday, which might not have been the smartest decision because it was so hot outside. Then we came home and went to the pool. So we were just having a relaxing day and we're actually on our way to dinner right now. We're going to Suerte, which is a highly recommended restaurant here in Austin. It's been on our bucket list. So tonight is the night. We're finally trying it. Like on the other day, the dress is from Arizona. It has these really cool details, like little dots. And then bag is from Hamilton as well


Hello and welcome to lecture eight of our introduction to machine learning course today We're going to talk about the boosting and the bagging these are these were very hot topics in machine learning and statistics in the 90s and early 2000s but still Remain very popular and very useful algorithms until now So to introduce these things I will I will start with this image that I used a lot in this course So the bias variance tradeoff depending on the model complexity we discussed it several times Imagine some algorithm where you can change its complex the model complexity when you go from the from the very simple models the simple models Underfit the data and show high bias the test error is high and then if you increase the complexity You slowly get to the regime where the model complexity is so high that the model over fits your training data and The very the bias is low, but now the variance is high and the test error is high again And somewhere in between is the is the ideal the sweet spot where you want to be so the boosting and the bagging came the illustrated on these plotters follows The boosting usually starts with a very simple model the model that under fits the data that model that has high bias and Then we apply this boosting procedure. It can be applied to different models increasing the model complexity and bringing the the error down Bagging in contrast starts with a very complex model a model that Overfits the data very very badly and then we back several models together to reduce the variance and bring the performance Improve the performance again So the as I say here the boosting builds complex models out of simple ones It's a sequential procedure that boosts the model to make it more complicated to make it more complex and reduce the bias and the bagging averages complex models hoping to average out the variance and And also reduce the the test error. So we'll start Well, I will start with introducing a particular model called classification trees because in fact, the bagging and the boosting often use The particular kind of particular kind of a classification model calls called the classification tree. So let me introduce it here Imagine you have a very simple data set like that with just two features and it's a binary classification problem So every point can belong to a circle class or the cross class in the classification tree you Try to find one feature That has high predictive performance It can be thresholded such that all points below the threshold Preferentially belong to one class and all points above the threshold preferentially belong to another class and then if The value of this feature is below the threshold We'll just in this case for example classify the point as a circle And if it's above the threshold we can either just classify it as a cross or we can make the model more complicated And then look for the next split. So here we can say if the if this is the case then We will make another split now We are looking at the second feature and if it's above the threshold then it's a cross and if it's below the threshold Then it's a circle and in principle one can keep building this tree. It's a binary tree because it always has two splits So one can keep Adding adding branches to it until you classify all the entire training set correctly and your training performance is 100% You can always achieve that by building such a tree and then it is built in a greedy manner So we first look for the best possible split and this can be done relatively easily Just look at each feature and like scan the possible thresholds really brute force look for the what is the best split Since you're always looking at one feature at a time That's not very costly and then once you found the split you proceed to the next level and then you look for the second best split and so So it's a greedy algorithm to build this tree. Okay, so that's all you need to know about the binary trees Important thing here is that the the size of the tree Regulates model complexity in this case. So you can have the simplest possible Tree that you can have just has one split here and nothing else So it's one split and then everything on the left is classified as one class and everything on the right is another This is has a name. It's called a tree stump It's of course a very very simplistic model on the other side of the spectrum You have a fully grown tree. So that is a tree that achieves 100% training set accuracy You you you cannot build a tree longer than that. So that's a fully grown tree as another other. Sorry as another remark You can use trees for regression that then it's called regression trees as opposed to classification trees which work very similarly But now in each of these Regions, you're just predicting a constant value if it's a regression problem But here we're talking about classification problems today for simplicity. We only have two classes So we'll just be predicting class one or class two okay now in Having introduced that we can start to talk about boosting the boosting Is a method that repeatedly applies classifier usually it's a weak classifier So we will actually later on use tree stumps as An example here, so you just take a tree stump, which is a very very simple classifier with huge bias and You repeatedly apply this classifier to your training set But every time you modify the training set a little bit in particular you change the weights of specific samples. So let me Let me introduce this algorithm to you in the very First like the generic outline of the of the boosting algorithm. So you start with all Samples have the same weight. So this is as this is the same as not having weights at all then you apply your your classifier to this Training data with these weights on the first iteration. You just apply the classifier to the data as As the data set is And then you measure the performance of this data. So the accuracy on the training set and Depending on how well your algorithm performs, you will set the weight of this Of the of the algorithm if it performs very well It will get a high weight if it doesn't perform very well it will get a lower weight and Then the most important thing is that we're going to look for examples that are misclassified by this algorithm So if nothing is misclassified, then we're done, but usually you apply a very weak classifier a Lot of samples will be misclassified We're looking at the ones that are misclassified and we increase their weights Then we proceed to the second boosting iteration. We apply the same algorithm to the training set But now the algorithm is forced to to focus on these samples that were previously misclassified okay So we build another model and then we increase the weights of the samples that are misclassified again And we do keep doing this for a hundred or a thousand boosting iterations So notice here that our models will not necessarily become better like individual Models gm will not become better as we proceed. They will instead they will focus on different parts of the of the Of the training set. So you first fit the model some parts It classifies correctly some parts that misclassifies then the algorithm focuses on these misclassified parts Classifies them correctly, but maybe misclassifies something that was previously classified correctly and so on So you will not using this very weak classifier. You will not achieve a Good performance for each individual model, but you will build a whole bunch of different models that cover different parts of the data so to say and then The final Result of the algorithm the output the clear the resulting classifier is actually the linear combination of all these boosted iterations So we just combine all of these gm Outputs with weights alpha that depend on the individual performance. It's just a linear combination. It's like they all all Iterations vote then If you have a test set on each test case all of these boosted Algorithms classifiers will will vote and the majority vote With these weights will be the final classification. So this is a very generic scheme Notice that you can use any classifier as the input here let me Illustrated on a very simple toy data So here we again have two features and a binary classification problem Let's say we use the tree stumps here now as a weak learner. So here's the iteration one This is the the best tree stump. It performs Not very well, but above chance, right? We classify all of that as circles and all of that as crosses these points here Get misclassified. So on the next iteration, I will draw them larger This means they have an increased weight now the algorithm is forced to classify to pay attention to them So the resulting the resulting Tree stump will be different because the way it's changed so now maybe that is the optimal stump and now these things are all classified correctly together with these circles But all of these crosses are now misclassified So we increase their weight when we go to the third iteration and now fit another stump and it will maybe be like that So now all of that is classified as cross correctly, but these things are misclassified and you can keep going so this will Not this will typically never converge in the sense that you will keep getting different different Models as you go on but then of course you stop After some number of iterations and you do this averaging so in this case I just average the three models and if you look at them You will see that here for example This model votes circle and this also votes circle and these votes cross But it's two to one circle so it's circle and so on and I get the final decision boundary that is not linear And it can be pretty complicated and the longer you boost the more complicated decision boundary you can You can obtain in the final model So it is the number of boosting iterations that regulates model complexity in this case the most popular boosting algorithm was called at a boost for adaptive boosting and It implements exactly the same logic as I had two slides before But what I didn't show on that slide is how we actually choose the alpha and how we choose these double your weights, right? So the the at a boost is a particular way to To To choose those weights In order to to introduce these weights I need first to introduce this weighted accuracy. So on each for each on each given iteration I can define the weighted accuracy of my Classifier as follows. So here this capital I means It is it is one if the point is misclassified So my predicted label is different from the actual label then it's one and if the predicted label it's the same as the True label then is zero and then I would just if I just average The zero one values I get the accuracy, but here I'm taking weights into account So I get I'm getting the weighted accuracy and now I can define the error boost Which just has a particular particular choice for the for the alpha weight Depending on the error rate and a particular choice For the W. So if you if you pay attention to this formula here Let's do a sanity check. So if the error is very large Then what is large L? So large error means close to one, right? Then this will be close to No, sorry, the the very large error is actually 0.5 because you are we're doing binary classification So if you are at a chance level then your error is 50 percent 0.5 So if you plug 0.5 here you get logarithm of one, which is zero. So it's zero weight That makes sense if your error is very low this means it's close to zero Then this inside here will diverge to infinity, which means it will get an infinite weight which makes sense If you are perfect on any one iteration then this should get a very high weight So it's a sensible formula you will get weights between zero and very large depending on your performance and Here you see that everything that it's every sample that is misclassified increase gets gets the weight increased so the formulas Sort of makes sense if you look at them like we did now But they're also rather mysterious like why these choices I can write many different formulas that make sense here And they may perform differently like why this particular choice? So one perspective that I really like on on Erebus that actually appeared only a few years later after the Original algorithm was suggested is that actually we can see this entire thing as a greedy optimization of the exponential loss function. So let me explain what that means I can Introduce this exponential loss function for my binary classification problem so the why eyes here are the true labels that can be minus one or one in this case and These are my predictions. So if I predict the same label This will be minus one and you will get one over e as a contribution to the sum And if I predict a different label, for example, this is one and this is minus one Then I will get one inside the exponent, which means it will be e Contribution to the sum so every time I do a mis-classification. I pay price that is bigger So that makes sense and then I sum over my entire training set and this is my Exponential loss, but g can also predict not necessarily one minus one can in this case it can be predict Any real value and Let's say we want to find the g that is a sum linear sum with some weights of of of my classifiers here and now let's say we want to optimize this exponential loss function by adapting the this this weights and choosing the the individual gm terms in a greedy way so that means first I Find alpha one and g1 and I keep them fixed and now I'm looking for the optimal g2 and Alpha two and then I keep them fixed and I proceed like that So I'm like adding more and more terms to this expansion and every time I try to minimize The exponential loss function so one can prove and I'm not going to prove this in this lecture entirely One can prove that these results in exactly the adabost Procedure so it's it's just a different view of the adabost the only thing I will Show here like I will start the proof But not give it in full here is that let's just take a look at what happens when we use this exponential Loss function and we go from the model from m minus one terms to m terms We do one iteration right so I can write this this model is the sum of everything that I had before Plus this new thing that I'm actually optimizing on this step right and since I have exponent of a sum I can split it in the two terms and This is already fixed so I can just call this to w and this is what is actually allowed to be changed to be to be fit on this iteration so this Thanks to the exponential loss function this weight here appears. That's a sample weight, right? Because it's a different w i for each of the samples so starting with this one can actually Relatively straightforwardly derive the that the optimal ways to choose the to choose these alphas is the formula is the adabost formula and The W's will turn Turn out to be also given by the adabost Formulas so this entire the entire Adaboost Machinery works thanks to the exponential loss function because it allows to split Basically thanks to this trick right so it's very neat In fact, it turns out that one can generalize that one can use other loss function not exponential loss function But something else Anything actually anything you want you say it's it can be in the square error loss function. It can be the The same loss function we used in logistic regression and Then one can you won't have adaboost anymore But it's it gets generalized to something called gradient boosting which is a very also very useful family of algorithms Alright, so some comments on adaboost then To to finish up this part so first of all it is often considered one of the best of the shelf classifier So off the shelf means you have some classification problem that is not like very very domain specific in the sense that maybe if it's a classification problem on the image data then a Convolutional neural network will perform much better than the adaboost if if the if the data has some structure like for example Images then there are other other tools that that may perform better but if you are predicting something like something like I don't know the apartment price based on based on a bunch of different predictors None of which are images Then you will typically do a pretty good job if you use the adaboost with With tree stumps as a classifier So often in a situation where you have some classification problem and you just want to train something Don't want to spend a lot of time on fine-tuning that You can use that a boost. It's one of the one of the good choices. So number of boosting iterations as I Said before controls the model complexity the longer you boost the more complex model you get It can overfit so in principle It's possible that you keep boosting and at some point your test error starts increasing again. However The the the great thing is that it often it overfits very very slowly So you'd like boost for hundred iterations or the thousand and even if it's already started over boost Overfitting it will do this so slowly that you will not lose much in the performance or in many cases It actually does not overfit at all So you keep boosting and the performance just stabilizes the test performance and doesn't go up anymore So that's pretty remarkable We'll get back to that later in the end of the lecture briefly another comment is that if You boost for long enough the training accuracy will get to hundred percent So it can fit any training for any training set you can fit it perfectly if you boost long enough This whole treatment with exponential loss function shows that actually even after the training accuracy is already hundred percent The algorithm won't stop right you can keep boosting even though your This aggregate model is already performing and hundred percent on the training set But the you the rate the boosting iterations can keep going. Nothing will nothing will stop them and the test Error can keep decreasing and the exponential loss on the training set can actually still keep decreasing Even after the classification hits a hundred percent on the training set So this shows that it actually really does optimize the exponential loss function and not just a misclassification All right So let's then start with part two which is the bagging so the bagging stands for bootstrap aggregation and In a way, it's simpler Then then boosting it refers to model averaging where we build a bunch of models and then just average them together But we do not build them One like in this successfully as in as in boosting right whether we need to keep track of the weights and everything But we just completely completely Parallelizable independently build a bunch of models and then average so that's simpler And we build these models on on bootstrapped Datasets so we take our data set then we draw bootstrap copies of it samples of it We build our model and then we average them. So what what does bootstrap data set mean? You might have encountered this this term in statistic. It's very useful concept bootstrapping refers to drawing samples with repetitions Where you draw the same number of samples as you originally had so let's say you had 1,000 samples in your training data if you want to make if you want to do bootstrapping you draw 1,000 samples out of these 1,000 samples, but you allow repetitions So some samples will be selected two times three times four times some samples by chance will not get selected at all So you will get a sample that contains some repetitions But it will have the same size as your original sample, right? It will still have 1,000 1,000 samples so that's often very useful in statistics in different content contexts here We'll just use this bootstrapped samples to build the model so the intuition is that we Have our sample will change it slightly by having like one bootstrap sample another bootstrap sample the third bootstrap sample And every time we'll build a model They will all be a bit different because the samples are a bit different and then we will average So one comment on the bootstrapping procedure one can show As an exercise that on average you will select you will leave out Around one-third of the samples one over e samples will be left out on each bootstrap iteration And this is around 30% so two-thirds of your samples will get on average into Each bootstrap iterations different two-thirds every time of course so The bagging is the idea of bagging is to apply it to the models that have low bias and high variance Okay, for example a fully grown tree. So if you build a fully grown classification tree you will achieve training set 100% you will often have lousy test error because you overfit your training data You are in a high variance situation, but the bias is very low Maybe even zero So the hope here is that if you build a bunch of these models and then average and if the bias was zero for each of them It will stay zero. So you will not you will not the bias will not Change But the variance will go down thanks to this averaging That is the hope of the bagging and then if the variance decreases and the bias stay low our model Will perform better than each individual model in fact if every model were independent Then the variance would even decrease to zero So if you increase the sample size and all all terms are independent Then the variance of the result will just go to zero and you will have perfect model Of course in reality this will not happen because the models are not independent because they are built on the bootstrapped Bootstrapped samples of your data set so it's not new data Right that that that you use to build each successful Model it's it's different data set, but it overlaps with the previous one So they are not independent so the variance will not go to zero, but it will hopefully decrease This is not random for us yet. This is just general bagging procedure random forest makes one important tweak and that is The aim is to decrease this dependency between Between every two model between every pair of Bagged models, so how can we do that? We want the ideas that we want to make every every two models We want to make them a little bit more different than what we get by applying them to two bootstrapped samples And here's how the random forests Do that we are so random forest specifically refers to fully grown To fully grown classification trees It's a forest of trees So now when we build our tree on a given iteration on a given bootstrap sample we start building the tree and then Remember we build the tree greedily by doing these binary splits right so every time we do a split We actually select a subset of variables that can be candidate Candidates for for splitting and then we check those and choose the best one and make the split So we don't scan all variables, but just a subset and it can be a small subset and These introduce additional randomness, so this will make the trees more different from one another and Then we'll hopefully allow the variance to decrease Further, it's just a heuristic, but turns out it works pretty well. This will perform better Sometimes much better than just bagging without this tweak so Important comment is that here Unlike what we had before in at a boost and in boosting the number of trees Do not really regulate model complexity in in any meaningful way. So the number of trees here just If you keep adding trees, you're just keeping the terms into your into your big average The idea is that you choose the number that is that is relatively big to decrease the variance So maybe you choose like a thousand trees for example But by doing this longer your model will not necessarily become more or less complex it will converge to something and then stay there whereas in In boosting you like keep increasing the complexity with every with every subsequent iteration So some comments on random forests now They are also often considered one of the best of the shelf classifiers. They are they are very easy to implement to build to use and often are one of the one of the You know top recommendations to what you can use if you need a classification algorithm they actually in practice Tend to perform similarly well to to add a boost or gradient boosting in in general The good thing is that random forest requires very little tuning. There's almost no free parameters, right? There's a free parameter of how big the tree is but random forest usually uses fully grown trees There's another free parameter of this how many variables you you scan on each split but Usually the performance also Weekly depends on that you you go with the default value and then there's nothing to tune anymore There's no really a regularization parameter to tune. You just let it run and miraculously it performs well Important remark is that typically you will get hundred percent Training set accuracy with a random forest and this is may not be entirely obvious immediately. So why is that and that is because? Remember that two-thirds of your samples will on average get into Each bootstrapped tree each tree is fully grown. So it has hundred percent training set accuracy Now if you look at each training sample, it will by the same logic appear in two-thirds in the training sets of two-thirds of the trees and they all have hundred percent training set accuracy, so the aggregate model the Like the when when these trees vote for every Training set samples two-thirds of them will vote correctly Or more and so you will get you will perfectly classify all training set examples So you're in it seems you are in an overfitting regime, right? You are Whatever the training data is you will reach hundred percent training set accuracy. Nevertheless test set performance can be can be pretty good two more things that are specific for for random forest and Useful often useful in practice. One thing is that it doesn't really require a validation set or a test set or a cross validation procedure and that's because when you're building the forest You naturally have the samples that are left out when you're building each tree, right? This special term in random forest lit which is out of the bag Samples so you have this one third of samples that don't that are not used to train each tree so You can For each sample in your in your data set You can average only trees that came from bootstrap iteration where the where this particular sample was not selected Where this sample was out of bag And then you average the performance only over those trees and this acts as a test set Right, so you don't have a separate test set But for each sample you can only look at trees that didn't use this sample to build the trees And then it acts as if it were a test set so and you don't you you get it for free because you Using this bootstrapping procedure anyway when you build a random forest and once you're done You have this out-of-bag estimate of the performance Which is as good as having a test set so that's pretty convenient and Another thing is that you can use a very similar trick to assess variable importance Which is which is often useful you have a bunch of predictors you're predicting something and then you want to know which which variables Actually contributed a lot to to the performance and you can do this very very easily here when you're using this When you're using this out-of-bag samples to check the performance of the random forest you can Permute one variable at a time so just scramble one variable at a time and then check the performance And if there's a variable that where the performance decreases a lot when you scrambled it Then this means that was an important variable So this is this is pretty standard in random forest literature in applications If you see random forest used and in a scientific paper, you will often see also a plot that shows The variables sorted by importance and this is how the importance is is typically assessed so we're almost done as Closing remarks I want to get back to this Figure that I had in the beginning That shows the bias variance trade off depending on the model complexity, right and then it is It is tempting to think about the boosting as Starting somewhere here and then increasing the model complexity until you get to somewhere here to the sweet spot And the bagging or the random forest is start somewhere here And then you average the models and you actually decrease the model complexity until you also get to the sweet spot So this is I think a wrong image. This is not what is happening here because as we discussed actually both are Examples of interpolating classifiers or classifiers that fit the training set perfectly This means that both of them are in in in in in this interpolating regime So we talked about that earlier in lecture four, I think One can imagine even more complex models behind Behind the interpolation threshold here on the right and remember when we talked about Regression and when we talked in particular about neural networks in the previous lecture. We talked about this double descent phenomenon where the models can be So complex that they feed any training set perfectly, but nevertheless the test error Is good. Thanks to some implicit regularization, this is one property of the neural networks that attracts a lot of attention here, but in in in recent years similar things happen here in both cases. So in random forests your Training set performance is 100% by construction.  So there are relatively recent papers that that study this phenomenon I my impression is that it's not fully understood why and in what Why this happens how this happens in what exact Situations it happens and in what doesn't So this is actually very interesting and active field of research It also nicely shows that there's there's a lot of discussion in literature now about why this happens in neural networks Why the neural networks are in this? atypical Interpolating regime but nevertheless perform well well in fact these two Two models from the 90s the add a boost and the random forest they We see similar phenomenon phenomena there to these are interpolating classifiers that never let us often perform really well Have on a test date I have really good predictive performance. Thank you


Coming to the slides, for SU 2 only one row with 2 S boxes are allowed young diagram. From dimension formula you can show that this one is 2 S plus 1. Incidentally the 2 S could be an integer even integer or odd integer, S is not giving you the the J or S could be half odd integers also. So, 2 S could be not necessarily even integer, it is not even number of boxes. It is even number of boxes for S being integer, but if S is half integer then 2 S can give you half odd integer, 2 S can give you an odd number ok. To come back to your familiar notation, if you take a tensor product of spin 1 with spin 3 by 2, you all know this even in the middle of the sleep if I wake you up, you know how to compose spin 1 with spin 3 by 2. You will start saying J 1 minus J 2 mod and go on right. That is what I have written it on the right hand side. J runs from J 1 minus J 2 mod which is half in steps of 1 goes up to 5 by 2 V J. So, now I am putting it in this notation so that you can map to your tensor product direct sum, right hand side is irrep, left hand side is a reducible representation. What are the dimensions? Spin 1 is dimension is 3, spin 3 by 2 dimension is 4, 3 into 4 is 12 and it has to split up into so many irreps so that the dimensions adds up to 12. Is it happening? For a spin 1, you will put 2 box diagram. For spin 3 by 2, you will put a 3 box diagram because number of box plus 1 is the dimension, 2 plus 1 is the dimension, 3 plus 1 is the dimension. Now you combine and write the diagrams. When you start combining it, there is something more you have to remember. Even here I forgot to mention that if you are looking at SU 3, something probably I said in the earlier lecture, but let me assert it now. We did this, you know the I wrote this. This is in general true as a diagram. If I say that I want to look at SU 3, there are only 3 basic objects with which you can play around. If you put u here, d here and an s here, this box you are forced to put 1 of the 3. This is not allowed of SU 3, but it is allowed for SU 4 if you allow 4 fundamental objects. Up spin, down spin was for SU 2, SU 3, up quark, down quark, s quark, SU 4 will be up quark, down quark, s quark and a charm quark and so on. If you had 4 objects totally antisymmetric with 4 objects is possible. The 3 objects making a totally antisymmetric of 4 objects is impossible ok. So, this diagram will not be there for SU 3. What about this? U d s is forced here and this is just one dimensional. This I explained it that any object with vertical column of n boxes is like as if it is a trivial representation added to things ok. So, I can add any number of them with a trivial unit representation. So, I can call that as a n with I do not know. Let me call it by our conventional one. This is like your trivial unit representation. It does no harm because it is like a identity state anything you multiply with it nothing happens. So, this one also I can remove out and write this as. So, this is like the a 1 which I can ignore and write this as if it is this. These are the two possibilities for SU 3. What is the dimension of this? This is 3 and 3 which is 9. What about this? This is probably 6 and this is 3. Can work it out and see what is the dimension? 3, 4, 2, 3 and then a hoop now ok. So, this is something which you have to remember that whenever you have a vertical 3 box for SU 3 you can ignore it. Whenever you have a vertical 2 box for SU 2, you can ignore it. This is trivial for SU 2. This is trivial for SU 3 ok. So, that is what I have done on the screen that I have removed those trivial representations unwritten in the Young diagram going back to the screen. You see that spin 1 S 2 box, spin 3 by 2 S 3 box you take the product keeping track of things whenever I have a vertical 2 box I can ignore vertical 3 box is not allowed ok for SU 2 doing it for SU 2. So, these are the three diagrams allowed for SU 2 and which one is spin 5 by 2 in this diagram? The one which has 5 boxes corresponds to the spin 5 by 2 representation. The 1 by 3 boxes after removing the trivial representation 2 box became trivial. Here you can make 4 box trivial and remove it out and you get this. This is clear ok. So, this is the tensor product decomposition. It is very beautiful in the context of SU n that this you rigorously I am sure you would have rigorously proved in your quantum mechanics course, but here I am diagrammatically showing that every diagram is an irrep for the SU 2 the diagrams can have only a single row with so many number of boxes each one is an irrep, that you can compose 2 irreps and they can undergo a decomposition into irreducible representation in a systematic way. The next question is about observables. Remember when we did the selection rules I said that every irrep is not only applicable to the states or the basis states primary basis, binary basis or tertiary basis, but you can also apply to electric dipole moment vector magnetic dipole moment vector quadrupole moment vector and so on. So, those are the observables and our aim is to understand selection rules even in the context of the continuous group symmetries. And that is given by your Wigner-Ickard theorem which is not very different from what you saw in your discrete groups. The product of all the 3 states should allow for you a trivial representation. We have all done this right we did this. What is the trivial representation? It should allow me something which gives me this representation for SU 2. If it gives me that then I know that such a transition from an initial tier to a final state triggered by an operator associated with an observable is allowed otherwise it is forbidden. And I am going to assume that the system is respecting such a SU 2 symmetry. If the system is not respecting that symmetry there is no point in talking about it. If the system has a rotational symmetry, a rotational symmetry of spin particles is incorporated in the SU 2 group. For such a group I want to look at whether certain matrix elements are 0 or non-zero. I would not be able to say what is the value, but I can at least say that these matrix elements will be 0 or non-zero and that is Wigner-Ickard theorem. I am sure you have all seen Wigner-Ickard theorem in your quantum mechanics course. I am just rewriting it in the group theory language ok. So, associated with the states j m in the context of SU 2, you will have observables which I am going to write it as ok q which is similar to writing a basis state like k comma q. So, remember your electric dipole moment will be like a vector and if this one is spin j is 1, then the 3 components of that vector should be related to the 3 components of that vector which is what I have shown you here as an example. There is a slight modification writing it similar to your spherical harmonics which you might have seen that you rewrite things so that they are like complex conjugates of each other. The z component is like j is 1 spin 1, but the magnetic quantum number 0 is associated with the z component of the vector.   So, this is why it is called sometimes as a spherical tensors. So, if you would just write Cartesian tensors I should have written it as A x A y and A z to match with the states you convert them into spherical tensors. So, if you compose two rank one spherical tensors rank one means the first k value is 1 ok. Take two vectors it is what I was saying take a tensor product of two vectors in this notation then you will end up getting what are the possibilities this one has j equal to 1 this one has j equal to 1 1 cross 1 will give you 2 spin 2 spin 1 and spin 0 and associated with that you will also have spherical tensors so this is exactly similar to my previous slide. If I replace this three horizontal box by only two boxes. So, whatever is applicable to the states is applicable to the observables also. This double box diagram is for a vector which you are all familiar electric dipole moment is a vector you can start doing the compositions of two vectors and you can find what are the reps using this diagrammatic language. There you use this great orthogonality theorem to write the decomposition here I am not doing that ok. The next question is can I find those explicit states from those fundamental objects which is what you did also if you remember I took a 1 b 1 a 2 b 2 I composed them and then I said that the one which corresponds to a trivial representation is a 1 b 1 plus a 2 b 2 that is the dot product and we derived that using a projector.  Similarly, you will have if you compose two fundamental objects if you take these two as one of those fundamental objects when you compose it what combinations of it will decide the state for this that is not clear here for that you need a projector and that projector is nothing but your Clebsch-Gordon coefficient. Clebsch-Gordon coefficient formally if you see here you have a tensor product of the basis states. So, this right hand side is a reducible representation the projector is your Clebsch-Gordon matrix. Projector on to this will give you a particular state which belongs to an irrep given by spin j exactly like what you did. So, this is you can call the state which you write here in terms of this as a binary basis. Is this clear? This is these are fundamental basis primary basis you compose the two primary basis and get a binary basis to get the binary basis usually you will take a projector. Now the projector role is played by the Clebsch-Gordon matrix and these are the elements of the Clebsch-Gordon ok. Similarly, you can show the generators which was j plus j minus and j z will now for this combined state obtained from composition of two primary states will involve j cross identity plus identity cross j that should operate on this state it is also you can show. You do this mechanically, but this is the way the derivation will go on ok. Similarly, the observables there is always a rhythm between the observables and the states. So, whenever you have an observable you have to look at the commutator bracket with the generators and they have the similar properties. It is like an Eigen state of j z with j plus j minus they increase or decrease by plus or minus 1. Now comes this matrix element earlier we did electric dipole moment. We took an initial state which is in one of the irreps of a discrete group and we took a final state which is another irrep of the same discrete group symmetry and then we ask the question whether that matrix element is 0 or non-zero. Now, in this case SU 2 symmetry tells you that the states have to be j i m i other quantum numbers I put it as alpha we do not need to worry about it. Similarly, the final state is j of m f the question is whether such an observable the operator associated with will it be non-zero or 0 and I am not going to give you the proof there is a rigorous proof of showing it. What I want to stress on this is that to get a unit representation out of it this will be proportional to the Clebsch-Gordon coefficient. What is the Clebsch-Gordon coefficient? You compose k q multiplied with tensor product with j i m i that is what is this composition and you have to get the final state. You all know the properties of the Clebsch-Gordon coefficient when will it be 0? If m f is not equal to m i plus q it is going to be 0 and similarly j i composition with k has to have a j f otherwise it is 0. So, you can trivially rule out which matrix element is 0 which matrix element is non-zero. There is a rigorous proof to show this we cannot determine this element. This element is called reduced matrix element I cannot find the number, but I can definitely say whether this matrix element is 0 or non-zero by looking at the Clebsch-Gordon coefficient that is purely from SU 2 symmetric. So, that is the selection rule for your suppose an experimentalist comes back to me and says that the quadrupole moment answer he can give me the matrix element for the 2 0 component suppose. If he gives me this component k then I know the CG coefficients I can determine this matrix element for the quadrupole moment answer with minus 2 for the q that Wigner-Eckart theorem is the selection rule in the context of SU 2 symmetry and in particle physics many processes decay processes you can rule out such a decay is impossible. You know you can tell from using just the SU 2 symmetry the SU 2 symmetry is what we call it as an isospin symmetry in particle physics. Incidentally you can use the isospins of few particles and look at a process like this where the operator which is the S matrix is a is like spin 0. So, you need to see whether the isospin initial state and the final state are allowed. So, here it is a combination of isospin of proton and the pion and these can be worked out using your Wigner-Eckart theorem in the context of particle physics and this is a very strong indication experimentalists do see that such a decay process if it is the decay ratio between these two is 2 is to 1 you know this is being seen and we can account from purely from our group theory argument.