统计机器学习-Multinoulli分布、多项式分布

Multinoulli分布(多元伯努利分布)：

模型：       $Mu(p)$

        d面获得每一面的概率: $p_1,p_2,...,p_d$

分布函数：
$$p(x|p)=\prod _{k=1}^d{p}_k^{x_k}$$
$$E(X)=p$$
似然函数：
$$L=log(\prod _{n=1}^N\prod _{k=1}^d{p}_k^{x_{nk}})=log(\prod _{k=1}^d{p}_k^{m_k})$$$$m_k=\sum _n{x}_{nk}$$
极大似然估计：
$$L=ln(\prod ^{n=1\to N}\prod ^{k=1\to d}{p}_k^{x_{nk}})=ln(\prod ^{k=1\to d}{p}_k^{m_k})=\sum ^{k=1\to d}{m}_{k}ln{p}_k+\lambda (\sum ^{k=1\to d}{p}_k-1)$$
              $p_k=\frac{m_k}{\lambda }$   $\lambda =-N$

其中   $\lambda ({\sum }_{k=1}^d{p}_k-1)$   的由来
是因为   ${\sum }_{k=1}^d{p}_k=1$   ,
（概率密度函数和为1）,在做极大似然估计时候，必须满足这一条件。对于带有约束的优化问题，常用拉格朗日乘子法，  $\lambda >0$  表示拉格朗日乘数，表示约束条件的强度。

多项式分布：

模型：       $Mult(n,p)$
        d面获得每一面的概率: $p_1,p_2,...,p_d$
        掷了n次，每面出现的次数：$(x_1,x_2,...,x_d)$
        满足条件：$x_1+x_2+...+x_d=n$
             $x_i\ge 0$
分布函数：
$$C_n^{x_1}{C}_{n-x_1}^{x_2}...C_{n-x_1-x_2+...x_{d-1}}^{x_d}{p}_1^{x_1}...p_d^{x_d}$$
$$f(x)=\frac{n!}{x^{(1)}!...x^{(d)}!}(p_1{)}^{x^{(1)}}...(p_d{)}^{x^{(d)}}$$
多项式展开定理：
$$(p_1+...+p_d{)}^n=\sum _{x\in \Delta d,n}\frac{n!}{x^{(1)}!...x^{(d)}!}(p_1{)}^{x^{(1)}}...(p_d{)}^{x^{(d)}}$$
矩生成函数：

$$E(x^j)=n{p}_j$$
$$Cov[x^{(j)},x^{(j^{\prime })}]=\{\begin{array}{l}n{p}_j(1-p_j)(j=j^{\prime })\\ -n{p}_j{p}_{j^{\prime }}(j\ne j^{\prime })\end{array}$$

一个服从多项式分布的例子：

$$将这个基因碱基序列可视化$$

Matplotlib:

import xlrd as xl
import numpy as np
from collections import Counter
import matplotlib.pyplot as plt
import pandas as pd

data = xl.open_workbook("等位基因.xlsx")
table = data.sheets()[0]
if data.sheet_loaded(sheet_name_or_index=0):
    cols = table.ncols  # 列数
    lists = [table.col_values(_) for _ in range(cols)]
    list_x = [_ for _ in range(1, len(lists) + 1)]
    list_A = []
    list_G = []
    list_C = []
    list_T = []
    for item in lists:
        dicts = dict(Counter(item))
        list_A.append(dicts.get('A', 0))
        list_G.append(dicts.get('G', 0))
        list_C.append(dicts.get('C', 0))
        list_T.append(dicts.get('T', 0))
    columns = ('A', 'G', 'C', 'T')
    data = []
    data.append(list_A)
    data.append(list_G)
    data.append(list_C)
    data.append(list_T)
    data = np.array(data)
    data = data.T
    df = pd.DataFrame(data, columns=columns, index=[_ for _ in range(1, cols + 1)])
    df.plot(kind='bar', stacked=True,colormap="cool_r",legend="reverse")
    print(df)
    ax=plt.gca()
    ax.spines['right'].set_color('none')
    ax.spines['top'].set_color('none')
    plt.xlabel("Sequence Position")
    plt.ylabel("Bits")
    plt.show()

else:
    print("打开文件失败")

Pyecharts:

import xlrd as xl
import numpy as np
from pyecharts.charts import *
from collections import Counter
from pyecharts import options as opts
from pyecharts.render import make_snapshot
from snapshot_selenium import snapshot
from pyecharts.globals import ThemeType

data = xl.open_workbook("等位基因.xlsx")
# table=data.sheet_by_name('Sheet1')
# table=data.sheet_by_index(0)
table = data.sheets()[0]
if data.sheet_loaded(sheet_name_or_index=0):
    rows = table.nrows  # 行数
    cols = table.ncols  # 列数
    lists = [table.col_values(_) for _ in range(cols)]
    list_x = [_ for _ in range(1, len(lists) + 1)]
    list_A = []
    list_G = []
    list_C = []
    list_T = []
    for item in lists:
        dicts = dict(Counter(item))
        list_A.append(dicts.get('A', 0))
        list_G.append(dicts.get('G', 0))
        list_C.append(dicts.get('C', 0))
        list_T.append(dicts.get('T', 0))
    bar = (
        Bar(init_opts=opts.InitOpts(theme=ThemeType.LIGHT))
            .add_xaxis(list_x)
            .add_yaxis("A", list_A, stack='stack1')
            .add_yaxis("G", list_G, stack='stack1')
            .add_yaxis("C", list_C, stack='stack1')
            .add_yaxis("T", list_T, stack='stack1')
            .set_series_opts(label_opts=opts.LabelOpts(is_show=False))
            .set_global_opts(title_opts=opts.TitleOpts(pos_left="10%"),
                             yaxis_opts=opts.AxisOpts(name="Bits"),
                             xaxis_opts=opts.AxisOpts(name="Sequence Position")))
    make_snapshot(snapshot, bar.render(), "111.png")
else:
    print("打开文件失败")