**PROBLEM STATEMENT (*HOMEWORK 5 POINTS)**

Quite a few ways exist for learning more about the Champions in Genesis. First, you can play the game. Nothing better than playing the game with friends to learn the capabilities of each champion and the associated cards. Second, you can read about the champions and watch videos. For example, the Chronowalkers on Youtube have a terrific series on the champions. But what else can you do after you’ve exhausted the first two options? You can do mathematical analysis. Lots of options here, one of which is Cluster Analysis.

What is cluster analysis? It’s looking at naturally occurring groupings of champions based on their stats. OK, that definition is accurate but doesn’t help much. To better explain it, let’s say you have an Olympic size swimming pool and all 24 champions are ready to dive in. Before they dive in, you ask them to walk along one side of the pool (everyone on the same side.) You tell them to start at one corner and walk one meter for every point in health. OK, after walking, the champions are lined up by health ready to dive into the pool. Next, you say, “Jump in and swim away from this side of the pool. One meter for every 10 points of aura.” Great now the champions are swimming in the pool at different locations. Next, you ask them to dive down – one meter for every point of energy reduction. You also ask them to hold their breath as long as they can – come on, they are champions, they can do it.

Quick take a picture of the pool showing where the champions are. Are the champions evenly spread out in the pool? Uh, somewhat. They probably clumped into groups a bit. Where are these clumps/groups? Who is in each clump? Actually, we don’t call them groups or clumps. We call them clusters.

Now you know what cluster analysis is.

**CHOICE OF VARIABLES (*HOMEWORK 10 POINTS)**

In order to learn more about the champions, I took Health, Aura, Energy Reduction, Damage, Awareness, and Speed. I reduced these to a range from zero to one. For example, Aura ranges from 80 to 150. I took each champion’s Aura, subtracted 80 and then divided by 70. Malik’s aura became 0 and Nubia’s Aura became 1. I did this so that each attribute has the same equal weight in the analysis. It was a bit boring to calculate them all, but I got them all done. Oh, and I coded Action speed attacks, such as Idiris' Fireball, as 0, and Swift speed as 1.

Next, I had them jump in the pool. Now, this particular pool has six dimensions, not three like a regular swimming pool (depth, length, and width), but the computer doesn’t care. After looking at where they all landed, I used a little more math to determine the “correct” number of clusters. This is a bit subjective. I looked at cluster density which is how close the cluster members to each other and distance between clusters, which is the empty space between clusters. (Visualize galaxies in the universe.)

**LABELING THE CLUSTERS (*HOMEWORK 10 POINTS)**

I believe the math came up with 5 distinct groups: Fighters, Summoners, Casters, Rangers, and Sluggers. These are nicknames I came up with to quickly describe them. The first three are pretty familiar, which tells me the math is right. The last two are interesting in their own right. Further below we will take a closer look at each one.

**CENTROIED TABLE (*HOMEWORK 5 POINTS)**

The next two graphics, show where these champions ended up in the pool, or more accurately where the "centers" of each cluster is in the pool. Think of each cluster as being a group of champions that like to socialize together and hang in the pool together. Where is the center of each clique? We could express the location of each center as coordinates. A table of centers is called, appropriately enough, a Centroid Table.

**CENTROIED CHART (*HOMEWORK 5 POINTS)**

We can also show these centers as lines in a graph. One line per cluster. This is a little tougher to visualize, but take the values in the Centroid Table and express them as a Line Graph. Again, each line is a centroid of champions. This is called a Centroid Plot.

**CLUSTER DESCRIPTIONS (*HOMEWORK 40 POINTS)**

It helps to describe each cluster with some kind of label. Something catchy and descriptive. As already stated above, I chose to call these five clusters: Fighters, Summoners, Casters, Rangers, and Sluggers. Here are short descriptions of each one.

Fighters

High in:	Health and Energy Reduction
Low in:	Aura and Awareness
Speed:	Swift
Members (5):	Feng, Korr, Long, Malik, Essa
Affiliations:	Most are in Thymos. Some are in Bellum and Chaos.

This is the typical, well-rounded fighters. Standard strategy is to seek out opposing champion, rush in, bash. High health is used to last just long enough to do a lethal amount of damage without worrying about losing. Low Aura means more techniques that deal damage directly to the front. The typical weakness here is getting hit multiple times on the flanks by summons.

Summoners

High in:	Aura
Low in:	Health, Energy Reduction, and Awareness
Speed:	Swift
Members:	Erika, Kendra, Nubia, Odara, Suttir, Alta, Bazin, Cato, Sara
Affiliations:	Most in Axon and Chaos

These are the champions that use Aura to put summons out front to do the fighting. Usually most effective against Fighters because they can’t kill all the summons all the time. Have a difficult time with Casters. Summoners typically have a special ability that interacts with beckoning or summons, such as Bazin and Odara. Summoners use summons as a wall of protection.

Casters

High in:	Aura
Low in:	Health, Energy Reduction, and Damage
Speed:	Action
Members (3):	Idiris, Manot, Kasha
Affiliations:	Most are in Bellum and Chaos

Casters love Area of Effect spells. When the summons are attacking, casters can wipe them out with one spell, such as Landslide and Tsunami. Therefore, they need to be high in Aura. Good casters have greater Awareness than summoners because a caster may need to get in front to cast spells, but still will beckon when needed.

Rangers

High in:	Energy Red Damage Awareness
Low in:	Aura
Speed:	Swift
Members (4):	RaHa, Rayne, Tashir, Shoon
Affiliations:	Spread out across all the affiliation, except Phage.

Rangers can be considered an offshoot of Fighters with the chief difference of Awareness and Health. Rangers typically have higher Awareness, which could help in keep a distance and taking less damage. Rangers can call up cheap summons as roadblocks. However, Rangers also have less Health. So, watch out.

Sluggers

High in:	Energy Reduction and Damage
Low in:	Aura and Awareness
Speed:	Action
Members (3):	Kunn, Iblis, Qureen
Affiliations:	Most are in Bellum and Axon.

Sluggers throw lots of high damage punches. They love techniques that cost Energy and love to keep pounding away at opposing champions and summons. Having high damage means they are typically dishing more damage when trading blows.

**IMPLICATIONS (*HOMEWORK 25 POINTS)**

These are the five clusters. What did we learn from this? If you can keep up with who is in which cluster and have a general approach in how to deal with each cluster, you could develop a winning strategy, an effective Forethoughts, and key tactics for the arena for whatever champion you face.

For me, I'm thinking I will pick a Ranger. I'm leaning toward Tashir. What if I play a long game with Tashi? Take cards that build up more health or lessen damage, and then use Tashir's greater awareness to get in more hits with the relatively high health. Typical fighters fear summoners, so I need an advantage over summoners. Maybe I can use the swift speed to deal with summons faster than they be summoned. OK, maybe not, but I will keep thinking.

*Note - Homework weights are a specific college level course in data analytics.