在MATLAB中，计算回归问题的拟合优度（或判定系数）可用[B,BINT,R,RINT,STATS] = regress(Y,X)指令，其中的STATS的第一个返回值即为R2。

一般来说，R2在0到1的闭区间上取值，但在实验中，有时会遇到R2为inf（无穷大）的情况，这时我们会用到R2的计算公式：

$R^2 = \frac {SSR}{SST} = 1-  \frac {SSE}{SST}$

其中SSR为回归平方和，SSE为残差平方和，SST为总离差平方和，三者还存在下列关系：

$SST =  SSR + SSE$

若用$y_i$表示真实的观测值，用$\bar{y}$表示真实观测值的平均值，用$\hat{y_i}$表示拟合值，则SSR、SSE、SST公式可以写成下列形式：

$SSR = \sum_{i=1}^{n}(\hat{y_i} - \bar{y})^2$

$SSE = \sum_{i=1}^{n}(y_i-\hat{y_i} )^2$

$SST =SSR + SSE= \sum_{i=1}^{n}(y_i - \bar{y})^2$

pandas计算相关系数
在二元变量的相关分析过程中比较常用的有Pearson相关系数、Spearman秩相关系数和判定系数
1. Pearson相关系数
一般用于分析两个连续性变量之间的关系。计算公式如下：
2. Spearman秩相关系数
Pearson线性相关系数要求连续变量的取值分布服从正态分布。

不服从正态分布的变量、分类或等级变量之间的关联性可以采用Spearman秩相关系数，也称等级相关系数来描述。


3. 判定系数
判定系数是相关系数的平方，用r2表示；用来衡量回归方程对y的解释程度。


例子



data.corr() 									 #计算任意两款菜式之间的相关系数
data.corr()['百合酱蒸凤爪']  					 #显示百合酱蒸凤爪与其他菜式的相关系数
data['百合酱蒸凤爪'].corr( data['翡翠蒸香茜饺'])    #计算两个菜品之间的相关系数

数据探索
计算相关系数
为了更加准确地描述变量之间的线性相关程度，可以通过计算相关系统来进行相关分析。
在二元变量的相关分析过程中比较常用的有Pearson相关系数，Spearman秩相关系数和判定系数。
皮尔逊相关系数（Pearson Correlation Coefficient）
一般用于分析两个连续性变量之间的关系，其计算公式如下。

$r = {
\sum_{i=1}^{n}(xi-\overline{x})(yi-\overline{y})\over\sqrt{\sum_{i=1}^{n}(xi-\overline{x})^2\sum_{i=1}^{n}(yi-\overline{y})^2}}$

相关系数r的取值范围：-1 <=  r <= 1

$\begin{cases}
	r &gt; 0 为正相关，r&lt;0为负相关\\
	|r| = 0 表示不存在线性关系\\
	|r| = 1 表示完全线性相关
\end{cases}$

0<|r|<1表示存在不同程度线性相关

$\begin{cases}
	|r|&lt;=0.3为不存在线性相关\\
	0.3&lt;|r|&lt;=0.5为低度线性相关\\
	0.5&lt;|r|&lt;=0.8为显著线性相关\\
	|r|&gt;0.8为高度线性相关
\end{cases}$
Spearman秩相关系数
Pearson线性相关系数要求连续变量的取值服从正太分布。不服从正态分布的变量、分类或等级变量之间的关联性可采用Spearman秩相关系数，也称等级相关系数来描述。
其计算公式如下：

$r={1-{{6\sum_{i=1}^{n}(Ri-Qi)^2}\over{n(n^2-1)}}}$

研究表明，在正态分布假设下，Spearman秩相关系数与Pearson相关系数在效率上是等价的，而对于连续测量数据，更适合用Pearson相关系数来进行分析。
判定系数
判定系数是相关系数的平方，用$r^2$表示；用来衡量回归方程对y的解释程度。
判定系数取值范围：0<=$r^2$<=1, $r^2$越接近于1，表示x与y之间的相关性越强；
$r^2​$越接近于0，表明两个变量之间几乎没有直线相关关系。

数据探索
计算相关系数
为了更加准确地描述变量之间的线性相关程度，可以通过计算相关系统来进行相关分析。
在二元变量的相关分析过程中比较常用的有Pearson相关系数，Spearman秩相关系数和判定系数。
皮尔逊相关系数（Pearson Correlation Coefficient）
一般用于分析两个连续性变量之间的关系，其计算公式如下。

$r = {
\sum_{i=1}^{n}(xi-\overline{x})(yi-\overline{y})\over\sqrt{\sum_{i=1}^{n}(xi-\overline{x})^2\sum_{i=1}^{n}(yi-\overline{y})^2}}$

相关系数r的取值范围：-1 <=  r <= 1

$\begin{cases}
	r &gt; 0 为正相关，r&lt;0为负相关\\
	|r| = 0 表示不存在线性关系\\
	|r| = 1 表示完全线性相关
\end{cases}$

0<|r|<1表示存在不同程度线性相关

$\begin{cases}
	|r|&lt;=0.3为不存在线性相关\\
	0.3&lt;|r|&lt;=0.5为低度线性相关\\
	0.5&lt;|r|&lt;=0.8为显著线性相关\\
	|r|&gt;0.8为高度线性相关
\end{cases}$
近似计算公式
​		上面的公式有一个问题在于算法时可能需要对数据进行多遍扫描。幸运的是，对于算法实现人员而言，还有一个皮尔逊相关系数的近似计算公式：

$r=\frac{\sum_{i=1}^nx_iy_i-\frac{\sum_{i=1}^nx_i\sum_{i=1}^ny_i}{n}}{\sqrt{\sum_{i=1}^nx_i^2-\frac{(\sum_{i=1}^nx_i)^2}{n}}\sqrt{\sum_{i=1}^ny_i^2-\frac{(\sum_{i=1}^ny_i)^2}{n}}}$
Python
def pearson(rating1, rating2):
    sum_xy = 0
    sum_x = 0
    sum_y = 0
    sum_x2 = 0
    sum_y2 = 0
    n = 0
    for key in rating1:
        if key in rating2:
            n += 1
            x = rating1[key]
            y = rating2[key]
            sum_xy += x * y
            sum_x += x
            sum_y += y
            sum_x2 += pow(x, 2)
            sum_y2 += pow(y, 2)
    # now compute denominator
    denominator = sqrt(sum_x2 - pow(sum_x, 2) / n) * sqrt(sum_y2 - pow(sum_y, 2) / n)
    if denominator == 0:
        return 0
    else:
        return (sum_xy - (sum_x * sum_y) / n) / denominator


Spearman秩相关系数
Pearson线性相关系数要求连续变量的取值服从正太分布。不服从正态分布的变量、分类或等级变量之间的关联性可采用Spearman秩相关系数，也称等级相关系数来描述。
其计算公式如下：

$r={1-{{6\sum_{i=1}^{n}(Ri-Qi)^2}\over{n(n^2-1)}}}$

研究表明，在正态分布假设下，Spearman秩相关系数与Pearson相关系数在效率上是等价的，而对于连续测量数据，更适合用Pearson相关系数来进行分析。
判定系数
判定系数是相关系数的平方，用$r^2$表示；用来衡量回归方程对y的解释程度。
判定系数取值范围：0<=$r^2$<=1, $r^2$越接近于1，表示x与y之间的相关性越强；
$r^2$越接近于0，表明两个变量之间几乎没有直线相关关系。
相似度的选择
如果数据受分数贬值（grade-inflation，即不同用户使用不同的评级范围）的影响，则使用皮尔逊相关系数。
如果数据稠密（几乎所有属性都没有零值）且属性值大小十分重要，那么使用诸如欧式距离或者曼哈顿距离。
如果数据稀疏，考虑使用余弦相似度。
相关代码
import codecs 
from math import sqrt

users = {"Angelica": {"Blues Traveler": 3.5, "Broken Bells": 2.0,
                      "Norah Jones": 4.5, "Phoenix": 5.0,
                      "Slightly Stoopid": 1.5,
                      "The Strokes": 2.5, "Vampire Weekend": 2.0},
         
         "Bill":{"Blues Traveler": 2.0, "Broken Bells": 3.5,
                 "Deadmau5": 4.0, "Phoenix": 2.0,
                 "Slightly Stoopid": 3.5, "Vampire Weekend": 3.0},
         
         "Chan": {"Blues Traveler": 5.0, "Broken Bells": 1.0,
                  "Deadmau5": 1.0, "Norah Jones": 3.0, "Phoenix": 5,
                  "Slightly Stoopid": 1.0},
         
         "Dan": {"Blues Traveler": 3.0, "Broken Bells": 4.0,
                 "Deadmau5": 4.5, "Phoenix": 3.0,
                 "Slightly Stoopid": 4.5, "The Strokes": 4.0,
                 "Vampire Weekend": 2.0},
         
         "Hailey": {"Broken Bells": 4.0, "Deadmau5": 1.0,
                    "Norah Jones": 4.0, "The Strokes": 4.0,
                    "Vampire Weekend": 1.0},
         
         "Jordyn":  {"Broken Bells": 4.5, "Deadmau5": 4.0,
                     "Norah Jones": 5.0, "Phoenix": 5.0,
                     "Slightly Stoopid": 4.5, "The Strokes": 4.0,
                     "Vampire Weekend": 4.0},
         
         "Sam": {"Blues Traveler": 5.0, "Broken Bells": 2.0,
                 "Norah Jones": 3.0, "Phoenix": 5.0,
                 "Slightly Stoopid": 4.0, "The Strokes": 5.0},
         
         "Veronica": {"Blues Traveler": 3.0, "Norah Jones": 5.0,
                      "Phoenix": 4.0, "Slightly Stoopid": 2.5,
                      "The Strokes": 3.0}
        }



class recommender:

    def __init__(self, data, k=1, metric='pearson', n=5):
        """ initialize recommender
        currently, if data is dictionary the recommender is initialized
        to it.
        For all other data types of data, no initialization occurs
        k is the k value for k nearest neighbor
        metric is which distance formula to use
        n is the maximum number of recommendations to make"""
        self.k = k
        self.n = n
        self.username2id = {}
        self.userid2name = {}
        self.productid2name = {}
        # for some reason I want to save the name of the metric
        self.metric = metric
        if self.metric == 'pearson':
            self.fn = self.pearson
        #
        # if data is dictionary set recommender data to it
        #
        if type(data).__name__ == 'dict':
            self.data = data

    def convertProductID2name(self, id):
        """Given product id number return product name"""
        if id in self.productid2name:
            return self.productid2name[id]
        else:
            return id


    def userRatings(self, id, n):
        """Return n top ratings for user with id"""
        print ("Ratings for " + self.userid2name[id])
        ratings = self.data[id]
        print(len(ratings))
        ratings = list(ratings.items())
        ratings = [(self.convertProductID2name(k), v)
                   for (k, v) in ratings]
        # finally sort and return
        ratings.sort(key=lambda artistTuple: artistTuple[1],
                     reverse = True)
        ratings = ratings[:n]
        for rating in ratings:
            print("%s\t%i" % (rating[0], rating[1]))
        

        

    def loadBookDB(self, path=''):
        """loads the BX book dataset. Path is where the BX files are
        located"""
        self.data = {}
        i = 0
        #
        # First load book ratings into self.data
        #
        f = codecs.open(path + "BX-Book-Ratings.csv", 'r', 'utf8')
        for line in f:
            i += 1
            #separate line into fields
            fields = line.split(';')
            user = fields[0].strip('"')
            book = fields[1].strip('"')
            rating = int(fields[2].strip().strip('"'))
            if user in self.data:
                currentRatings = self.data[user]
            else:
                currentRatings = {}
            currentRatings[book] = rating
            self.data[user] = currentRatings
        f.close()
        #
        # Now load books into self.productid2name
        # Books contains isbn, title, and author among other fields
        #
        f = codecs.open(path + "BX-Books.csv", 'r', 'utf8')
        for line in f:
            i += 1
            #separate line into fields
            fields = line.split(';')
            isbn = fields[0].strip('"')
            title = fields[1].strip('"')
            author = fields[2].strip().strip('"')
            title = title + ' by ' + author
            self.productid2name[isbn] = title
        f.close()
        #
        #  Now load user info into both self.userid2name and
        #  self.username2id
        #
        f = codecs.open(path + "BX-Users.csv", 'r', 'utf8')
        for line in f:
            i += 1
            #print(line)
            #separate line into fields
            fields = line.split(';')
            userid = fields[0].strip('"')
            location = fields[1].strip('"')
            if len(fields) > 3:
                age = fields[2].strip().strip('"')
            else:
                age = 'NULL'
            if age != 'NULL':
                value = location + '  (age: ' + age + ')'
            else:
                value = location
            self.userid2name[userid] = value
            self.username2id[location] = userid
        f.close()
        print(i)
                
        
    def pearson(self, rating1, rating2):
        sum_xy = 0
        sum_x = 0
        sum_y = 0
        sum_x2 = 0
        sum_y2 = 0
        n = 0
        for key in rating1:
            if key in rating2:
                n += 1
                x = rating1[key]
                y = rating2[key]
                sum_xy += x * y
                sum_x += x
                sum_y += y
                sum_x2 += pow(x, 2)
                sum_y2 += pow(y, 2)
        if n == 0:
            return 0
        # now compute denominator
        denominator = (sqrt(sum_x2 - pow(sum_x, 2) / n)
                       * sqrt(sum_y2 - pow(sum_y, 2) / n))
        if denominator == 0:
            return 0
        else:
            return (sum_xy - (sum_x * sum_y) / n) / denominator


    def computeNearestNeighbor(self, username):
        """creates a sorted list of users based on their distance to
        username"""
        distances = []
        for instance in self.data:
            if instance != username:
                distance = self.fn(self.data[username],
                                   self.data[instance])
                distances.append((instance, distance))
        # sort based on distance -- closest first
        distances.sort(key=lambda artistTuple: artistTuple[1],
                       reverse=True)
        return distances

    def recommend(self, user):
       """Give list of recommendations"""
       recommendations = {}
       # first get list of users  ordered by nearness
       nearest = self.computeNearestNeighbor(user)
       #
       # now get the ratings for the user
       #
       userRatings = self.data[user]
       #
       # determine the total distance
       totalDistance = 0.0
       for i in range(self.k):
          totalDistance += nearest[i][1]
       # now iterate through the k nearest neighbors
       # accumulating their ratings
       for i in range(self.k):
          # compute slice of pie 
          weight = nearest[i][1] / totalDistance
          # get the name of the person
          name = nearest[i][0]
          # get the ratings for this person
          neighborRatings = self.data[name]
          # get the name of the person
          # now find bands neighbor rated that user didn't
          for artist in neighborRatings:
             if not artist in userRatings:
                if artist not in recommendations:
                   recommendations[artist] = (neighborRatings[artist]
                                              * weight)
                else:
                   recommendations[artist] = (recommendations[artist]
                                              + neighborRatings[artist]
                                              * weight)
       # now make list from dictionary
       recommendations = list(recommendations.items())
       recommendations = [(self.convertProductID2name(k), v)
                          for (k, v) in recommendations]
       # finally sort and return
       recommendations.sort(key=lambda artistTuple: artistTuple[1],
                            reverse = True)
       # Return the first n items
       return recommendations[:self.n]