用多因子策略构建强大的加密资产投资组合 #大类因子分析：因子合成篇#

多重共线性导致的后果：

1.完全共线性下参数估计量不存在

2.近似共线性下OLS估计量非有效

二、步骤一：同类型因子的相关性检验

检验新求出的因子与已入库因子的相关性。通常来说，有两类数据求相关性：

1.根据所有token在回测期间的因子值求相关

2.根据所有token在回测期间的因子超额收益值求相关

三、步骤二：因子取舍、因子合成

对于相关性较高的因子集合，可以采取两种方式处理：

（1）因子取舍

根据因子本身的ICIR值、收益率、换手率、Sharpe 比率，挑选某维度下最有效的因子进行保留，删除其他因子。

（2）因子合成

对因子集合中的因子进行合成，截面上尽可能多的保留有效信息

假设当前有3个待处理的因子矩阵：

synthesis = pd.concat([a,b,c],axis = 1) 
synthesis
a    b   c
BTC.BN    0.184865    -0.013253 -0.001557
ETH.BN    0.185691    0.022708    0.031793
BNB.BN    0.242072    -0.180952 -0.067430
LTC.BN    0.275923    -0.125712 -0.049596
AAVE.BN    0.204443    -0.000819 -0.006550
...    ...  ...  ...
SOC.BN    0.231638    -0.095946 -0.049495
AVAX.BN    0.204714    -0.079707 -0.041806
DAO.BN    0.194990    0.022095    -0.011764
ETC.BN    0.184236    -0.021909 -0.013325
TRX.BN    0.175118    -0.055077 -0.039513

2.1 等权加权

各因子权重相等（w=1/因子个数），综合因子=各因子值加总求平均。

Eg.动量类因子，一个月收益率、两个月收益率、三个月收益率、六个月收益率、十二个月收益率，这六个因子的因子载荷各占1/6的权重，合成新的动量因子载荷，然后再重新进行标准化处理。

synthesis1 = synthesis.mean(axis=1) # 按行求均值

2.2 历史IC加权、历史ICIR、历史收益加权

用回测期的IC值（ICIR值、历史收益值）对因子进行加权。过去有很多期，每一期都有一个IC值，所以用它们的均值作为因子的权重。通常使用回测期IC的均值（算数平均值）作为权重。

# 权重归一化（后文中的因子加权方式也基本都需要进行权重归一化）
w_IC = ic.mean() / ic.mean().sum()
w_ICIR = icir.mean() / icir.mean().sum()
w_Ret = Return.mean() / Return.mean().sum()
synthesis2 = (synthesis * w_IC).sum(axis=1)
synthesis2 = (synthesis * w_ICIR).sum(axis=1)
synthesis2 = (synthesis * w_Ret).sum(axis=1)

2.3历史IC半衰加权、历史ICIR半衰加权

2.1与2.2都是计算算数平均值，回测期的每一次IC、ICIR对于因子的作用被默认为相同。

但现实中，回测期的每一期对于当期的影响程度不完全相同，存在时间上的衰减。越接近当前期的时期，影响越大，越远影响越小。在此原理，求IC权重前首先定义一个半衰权重，距离当期越近的权重值越大、越远权重越小。

半衰权重数学推导

2.4 最大化ICIR加权

通过求解方程，计算最优因子权重w使得ICIR最大化

协方差矩阵的估计问题：协方差矩阵用于衡量不同资产之间的关联性。统计学中常以样本协方差矩阵代替总体协方差矩阵，但在样本量不足时，样本协方差矩阵与总体协方差矩阵的差异会很大。所以有人提出了压缩估计的方法，原理是使估计协方差矩阵与实际协方差矩阵之间的均方误差最小

方式：

1.样本协方差矩阵

<span class="token comment" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;"># 最大化ICIR加权(样本协方差)</span>
ic_cov <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> np<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>array<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>ic<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>cov<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
inv_ic_cov <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> np<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>linalg<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>inv<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>ic_cov<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
ic_vector <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> np<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>mat<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>ic<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>mean<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
w <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> inv_ic_cov <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">*</span> ic_vector<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>T
w <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> w <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">/</span> w<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span><span class="token builtin" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">sum</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span> 
synthesis4 <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> <span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>synthesis <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">*</span> pd<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>DataFrame<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>w<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">,</span>index<span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span>synthesis<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>columns<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">[</span><span class="token number" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">0</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">]</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span><span class="token builtin" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">sum</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>axis<span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span><span class="token number" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">1</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>

2.Ledoit-Wolf收缩：引入一个缩小系数，将原始的协方差矩阵与单位矩阵进行混合，以减少噪音的影响。

<span class="token comment" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;"># 最大化ICIR加权(Ledoit-Wolf压缩估计协方差)</span>
<span class="token keyword" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">from</span> sklearn<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>covariance <span class="token keyword" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">import</span> LedoitWolf
model<span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span>LedoitWolf<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
model<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>fit<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>ic<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
ic_cov_lw <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> model<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>covariance_
inv_ic_cov <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> np<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>linalg<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>inv<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>ic_cov_lw<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
ic_vector <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> np<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>mat<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>ic<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>mean<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
w <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> inv_ic_cov<span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">*</span>ic_vector<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>T
w <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> w<span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">/</span>w<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span><span class="token builtin" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">sum</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>
synthesis4 <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span> <span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>synthesis <span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">*</span> pd<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>DataFrame<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>w<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">,</span>index<span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span>synthesis<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span>columns<span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">[</span><span class="token number" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">0</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">]</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">.</span><span class="token builtin" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">sum</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">(</span>axis<span class="token operator" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">=</span><span class="token number" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">1</span><span class="token punctuation" style="font-size: inherit; font-family: PingFang SC,Helvetica Neue,Helvetica,Arial,Hiragino Sans GB,Heiti SC,Microsoft YaHei,WenQuanYi Micro Hei,sans-serif;">)</span>

3.Oracle近似收缩：对Ledoit-Wolf收缩的改进，目标是通过对协方差矩阵进行调整，从而在样本大小较小的情况下更准确地估计真实的协方差矩阵。（编程实现与Ledoit-Wolf收缩同理）

2.5 主成分分析PCA

主成分分析（Principal Component Analysis，PCA）是一种用于降维和提取数据主要特征的统计方法。其目标是通过线性变换，将原始数据映射到一个新的坐标系，使得数据在新坐标系下的方差最大化。

具体而言，PCA首先找到数据中的主成分，也就是数据中方差最大的方向。然后，它找到与第一个主成分正交（无关）且具有最大方差的第二个主成分。这个过程一直重复，直到找到数据中所有的主成分。

# 主成分分析(PCA)
from sklearn.decomposition import PCA
model1 = PCA(n_components=1)
model1.fit(f)
w=model1.components_
w=w/w.sum()
weighted_factor=(f*pd.DataFrame(w,columns=f.columns).iloc[0]).sum(axis=1)