[PYTHON] State space model using Pystan -Deleste event attack format 2001 rank border-

Overview

I learned how to analyze a state-space model using Stan in a book, so I practiced it with actual data.
This time I tried to move it with the following model for the time being
Local level model
Smoothing trend model
Local linear trend model
Time-varying coefficient model (explanatory variable is only the length of the event period)
As a result, it still doesn't fit well at all
Because trends and event lengths alone cannot explain
I would like to make other features and analyze them with the time-varying coefficient model.
The scripts created this time are c1 ~ c4 of here.

pip install pystan

Data acquired by here extracted only from the format of Attapon

A model that thinks that "the state follows a random walk, noise is added to that state, and output comes out."
Formulas are [here](https://qiita.com/kazuya_minakuchi/items/09b010927688b322df9d#%E3%83%AD%E3%83%BC%E3%82%AB%E3%83%AB%E3%83 See% AC% E3% 83% 99% E3% 83% AB% E3% 83% A2% E3% 83% 87% E3% 83% AB)

Impressions

It is not in the range of 5% -95% at all, but for the time being, I feel that I can see the overall trend by passing through the center.
The peak is around the latter half of 2018, and is it gradually decreasing?

A model that thinks that "the amount of change in the state follows a random walk, noise is added to that state, and the output comes out."
Amount of change in state: Amount of change from the previous state to the current state
Formulas are [here](https://qiita.com/kazuya_minakuchi/items/09b010927688b322df9d#%E5%B9%B3%E6%BB%91%E5%8C%96%E3%83%88%E3%83 % AC% E3% 83% B3% E3% 83% 89% E3% 83% A2% E3% 83% 87% E3% 83% AB)

Impressions

Add "drift component that changes with time" to the local level model
Something like a combination of a local level model and a smoothing trend model
When the change of the level component is almost 0, it is the same as the smoothing trend model.
Formulas are [here](https://qiita.com/kazuya_minakuchi/items/09b010927688b322df9d#%E3%83%AD%E3%83%BC%E3%82%AB%E3%83%AB%E7%B7 % 9A% E5% BD% A2% E3% 83% 88% E3% 83% AC% E3% 83% B3% E3% 83% 89% E3% 83% A2% E3% 83% 87% E3% 83% AB )

Impressions

Again, this is not in the 5% -95% range at all.
It is between the smoothness of the smoothing trend and the rattling of the local trend.

Add explanatory variables that may affect the objective variable. It is assumed that the coefficients of the explanatory variables also change over time.
Formulas are [here](https://qiita.com/kazuya_minakuchi/items/09b010927688b322df9d#%E6%99%82%E5%A4%89%E4%BF%82%E6%95%B0%E3%83 % A2% E3% 83% 87% E3% 83% AB)

Impressions

Not yet in the 5% -95% range at all.
The period coefficient is closer to minus
Thinking normally, that shouldn't be the case (I don't think "the shorter the period, the more points you can earn")
Seems to be pulled by changes due to other factors

Not fit well yet
Because the factors that affect the border are not fully reflected
A list of things that came to mind as elements that seemed to be related
Elements for each event
Excitement of the event ≒ Popularity of idols who are ranking rewards ** (Everyone aims to be within 2000th place for this purpose, so it seems to have the strongest influence) **
Elements that are likely to be included in the trend (It seems good to use a local linear trend because there are some that change gradually and some that change suddenly)
Changes in the number of active users as the popularity of the game changes
As the number of characters increases, the number of people who can form a neglected formation increases
Increased time efficiency with grand live implementation
I chose 2001 as the expected value, but it seems easier to predict the 1st place w
1st place is likely to be the value when you use your time to push the limits of human beings.
However, since 2001 is the border of the highest reward for the event, there should be more demand for Kochi (I have never run the event properly, but I also want to know)