6.1.1 Regression

1. linear regression
   
   • linear regression deals with estimating a variable from another variable.

   \[ Y = Xk + \epsilon\]

   1. The dependent variable (Dep. Variable) states the name of the variable that is fitted.  
      
   2. Model states what model we used in the fit; except OLS, there are several other models such as   weighted least squares (WLS).
      
   3. The number of observations (No. Observations) are listed and the degrees of freedom of the residuals (Df Residuals), that is, the number of observations (59) minus the parameters determined through the fitting 2 (k and   k).  
      
   4. Df Model shows how many parameters were determined (except the constant, that is, intercept).
      
   5. The table to the right of the top table shows you information on how well the model fits the data.
      
   6. R-squared was covered before; here, the adjusted R-square value (Adj. R-squared) is also listed and this is the R-square value corrected for the number of data points and degrees of freedom.
      
   7. The   F-statistic number gives you an estimate of how significant the fit is. Practically, it is the mean squared error of the model divided by the mean squared error of the residuals.
      
   8. The next value, Prob (F-statistic), gives you the probability to get the F-statistic value if the null hypothesis is true, that is, the variables are not related.
      
   9. After this, three sets of   log-likelihood function values follow: the value of the log-likelihood value of the fit, the   Akaike Information Criterion (AIC), and   Bayes Information Criterion (BIC). The AIC and BIC are various ways of adjusting the log-likelihood function for the number of observations and model type.
      
2. multiple regression
   
   • a variable is estimated from two or more others.
3. logistic regression
   

   Logistic regression fits models to one or more discrete variables, which are sometimes binary(that is, can only take the values 0 or 1).

   • One of the main differences between binary logistic and linear regression is that in binary logistic regression, we are fitting the probability of an outcome given a measured (discrete or continuous) variable,
   • while linear regression models deal with characterizing the dependency of two or more continuous variables on each other.
   • Logistic regression gives the probability of an occurrence given some observed variable( s). Probability is sometimes expressed as P( Y | X) and read as   Probability that the value is Y given the variable X.

   \[ln(p/(1-p)) = m + kx\]

6.1.2 Interpolation

6.1.3 Clustering

1. K-means nearest neighbor algorithm for cluster finding.
   

   The K-means algorithm is also referred to as vector quantization. What the algorithm does is finds the cluster (centroid) positions that minimize the distances to all points in the cluster.

2. hierarchical clustering.
   

   Hierarchical clustering is connectivity-based clustering. It assumes that the clusters are connected, or in another word, linked.

   1. Agglomerative clustering starts out with each point in its own cluster and then merges the two clusters with the lowest dissimilarity, that is, the bottom-up approach
      
   2. Divisive clustering is, as the name suggests, a top-down approach where we start out with one single cluster that is divided into smaller and smaller clusters
      

      Magnus Vilhelm Persson. Mastering Python Data Analysis (Kindle Locations 2335-2338). Packt Publishing. Kindle Edition.

6.2.1 Global Optimization

6.2.2 Local Optimization

6.2.3 Constrained Optimization

6.3.1 Numerical Integration

6.3.2 Integration by simulation

7.2.1 Random Variables

7.2.2 Stochastic Processes

7.2.3 Variance Reduction

7.2.4 Valuation

1. European options
   
2. American options
   

7.2.5 Risk Measure

1. VaR
   
2. Credit Value Adjustments
   

8.1.1 Benchmark Case

8.1.2 Real-World data

8.3.1 The DAX index and its 30 stocks

8.3.2 Applying PCA

8.3.3 Constructing a PCA Index

8.4.1 Bayes's formula

8.4.2 PyMC3

9.2.1 modeling and handling dates

9.2.2 constant short rate

9.2.3 Market environment

11.3.1 Least-Square Monte Carlo

13.1.1 VSTOXX Index Data

13.1.2 VSTOXX Futures Data

13.1.3 VSTOXX Options Data

15.1.1 Gradient descent

15.1.2 Conjugate gradient method

15.2.1 Random search

15.2.2 Bayesian optimization

22.2.1 随机森林的构建过程

1. 数据的随机选取：
   
   • 从原始的数据集中采取有放回的抽样，构造子数据集，子数据集的数据量是和原始数据集相同的。不同子数据集的元素可以重复，同一个子数据集中的元素也可以重复。
   • 第二，利用子数据集来构建子决策树，将这个数据放到每个子决策树中，每个子决策树输出一个结果。
   • 最后，如果有了新的数据需要通过随机森林得到分类结果，就可以通过对子决策树的判断结果的投票，得到随机森林的输出结果了。如下图，假设随机森林中有3棵子决策树，2棵子树的分类结果是A类，1棵子树的分类结果是B类，那么随机森林的分类结果就是A类。
2. 待选特征的随机选取
   

   与数据集的随机选取类似，随机森林中的子树的每一个分裂过程并未用到所有的待选特征，而是从所有的待选特征中随机选取一定的特征，之后再在随机选取的特征中选取最优的特征。这样能够使得随机森林中的决策树都能够彼此不同，提升系统的多样性，从而提升分类性能。

3. Random Forest的具体使用-sklearn
   

   以上介绍了随机森林的工作原理，那么在python环境下，我们可以利用python环境下的sklearn包来帮助我们完成任务。举个小例子： 　　特征是通过收盘价数据计算的SMA，WMA，MOM指标，训练样本的特征是从2007-1-4到2016-6-2中截止前一天的SMA，WMA，MOM指标，训练样本的标类别是2007-1-4日到2016-6-2中每一天的涨跌情况，涨了就是True，跌了就是False，测试样本是2016-6-3日的三个指标以及涨跌情况。我们可以判定之后判断结果是正确还是错误，如果通过Random Forest判断的结果和当天的涨跌情况相符，则输出True，如果判断结果和当天的涨跌情况不符，则输出False。

   import talib
   from jqdata import *
   test_stock = '399300.XSHE'
   start_date = datetime.date(2007, 1, 4)
   end_date = datetime.date(2016, 6, 3)
   trading_days = get_all_trade_days()
   start_date_index = trading_days.index(start_date)
   end_date_index = trading_days.index(end_date)
   x_all = []
   y_all = []
   
   for index in range(start_date_index, end_date_index):    # 得到计算指标的所有数据    start_day = trading_days[index - 30]    end_day = trading_days[index]    stock_data = get_price(test_stock, start_date=start_day, end_date=end_day, frequency='daily', fields=['close'])    close_prices = stock_data['close'].values        #通过数据计算指标    # -2是保证获取的数据是昨天的，-1就是通过今天的数据计算出来的指标    sma_data = talib.SMA(close_prices)[-2]     wma_data = talib.WMA(close_prices)[-2]    mom_data = talib.MOM(close_prices)[-2]        features = []    features.append(sma_data)    features.append(wma_data)    features.append(mom_data)        label = False    if close_prices[-1] > close_prices[-2]:        label = True    x_all.append(features)    y_all.append(label) # 准备算法需要用到的数据
    x_train = x_all[: -1]
    y_train = y_all[: -1]
    x_test = x_all[-1]
    y_test = y_all[-1]
    print('data done')
   
    输出：
   data done
   
   from sklearn.ensemble import RandomForestClassifier
   #开始利用机器学习算法计算，括号里面的n_estimators就是森林中包含的树的数目啦
   clf = RandomForestClassifier(n_estimators=10)
   #训练的代码clf.fit(x_train, y_train)
   #
   得到测试结果的代码prediction = clf.predict(x_test)
   # 看看预测对了没print(prediction == y_test)
   print('all done')
   输出：
   [ True]all done