积矩相关系数(PMCC)
积矩相关系数是什么?
- 积矩相关系数(PMCC)是一种给出数值的方法线性相关二元数据
- 样品的PMCC用字母表示
" class="Wirisformula" role="math" alt="尺寸:16px r" style="vertical-align:-4px;height:19px;width:7px">r {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="r" style="vertical-align:-4px;height:19px;width:7px">可以取任意值r {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="负1小于等于r小于等于1" style="vertical-align:-4px;height:19px;width:65px">- 1 ≤ r ≤ 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - r是正的描述正相关
- r的负值描述负相关
- 如果
" class="Wirisformula" role="math" alt="开始mathsize 16px样式r = 0结束样式" style="vertical-align:-4px;height:19px;width:31px">没有相关性r = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="R空间等于1" style="vertical-align:-4px;height:19px;width:35px">表示与完全正相关r = 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="R空间等于- 1" style="vertical-align:-4px;height:19px;width:49px">表示完全负相关r = - 1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 越接近1或-1,相关性越强
- 的值不受梯度的影响
" class="Wirisformula" role="math" alt="尺寸:16px r" style="vertical-align:-4px;height:19px;width:7px">r {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
工作的例子
上面显示了三个散点图,显示了来自不同二元数据集的观测结果。
非线性回归
在AS级别,您学习了如何使用线性回归模型来描述两个变量之间的关系。然而,有可能两个变量之间的关系不符合线性模型,但仍然显示出基于指数增长或衰减的模式。线性回归模型仅适用于PMCC接近1或-1的情况。
非线性回归模型可以采取什么形式?
- 如果一个二元数据集似乎有一个非线性关系它可以符合指数模型
- 非线性回归模型的形式为
" class="Wirisformula" role="math" alt="Y = ax的n次方" style="vertical-align:-4px;height:20px;width:49px" loading="lazy">或y = a x n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Y = kb的x次方" style="vertical-align:-4px;height:20px;width:50px" loading="lazy">在哪里A n k而且b是常数y = k b x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 非线性回归模型的形式为
- 可以用对数来重新排列模型的非线性形式,以得到一个线性回归模型然后用什么来检查数据的趋势呢
- 如果回归模型为
" class="Wirisformula" role="math" alt="Y = ax的n次方" style="vertical-align:-4px;height:24px;width:51px" loading="lazy">数据应该从y = a x n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式x结束样式" style="vertical-align:-4px;height:19px;width:10px" loading="lazy">-值为x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="y" style="vertical-align:-4px;height:19px;width:10px" loading="lazy">-值使用y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X = logx" style="vertical-align:-4px;height:19px;width:63px" loading="lazy">而且X = log x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Y等于log空间Y" style="vertical-align:-4px;height:19px;width:63px" loading="lazy">Y = log y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="大小为16px y大小为16px等于大小为16px a大小为16px x的16px n次方" style="vertical-align:-4px;height:24px;width:51px" loading="lazy">为常量一个而且n,然后y = a x n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="log空间y = log空间a + nlog空间x空间或者空间y = nx + log空间a" style="vertical-align:-4px;height:19px;width:255px" loading="lazy">log y = log a + n log x or Y = n X + log a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 策划
" class="Wirisformula" role="math" alt="开始mathsize 16px style log space x结束风格" style="vertical-align:-4px;height:19px;width:34px" loading="lazy">反对log x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Log空间y" style="vertical-align:-4px;height:19px;width:34px" loading="lazy">会给出一个线性图吗log y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - y截距是
" class="Wirisformula" role="math" alt="日志空间a" style="vertical-align:-4px;height:19px;width:33px" loading="lazy">直线的梯度是nlog a {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这可以通过对两边取对数来表示
- 如果
- 如果回归模型为
" class="Wirisformula" role="math" alt="开始mathsize 16px style y = k b的x次方结束style" style="vertical-align:-4px;height:20px;width:50px" loading="lazy">数据应该从y = k b x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="开始mathsize 16px样式x结束样式" style="vertical-align:-4px;height:19px;width:10px" loading="lazy">值x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="y" style="vertical-align:-4px;height:19px;width:10px" loading="lazy">值使用y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="X = X" style="vertical-align:-4px;height:19px;width:39px" loading="lazy">而且X = x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Y等于log空间Y" style="vertical-align:-4px;height:19px;width:63px" loading="lazy">Y = log y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="Y = kb的x次方" style="vertical-align:-4px;height:24px;width:52px" loading="lazy">为常量k而且b,然后y = k b x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Log空间y = Log空间k + xlog b" style="vertical-align:-4px;height:19px;width:135px" loading="lazy">或log y = log k + x log b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Y等于左括号log空间b右括号X加上log空间Y" style="vertical-align:-5px;height:20px;width:133px" loading="lazy">Y = ( log b ) X + log y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 策划
" class="Wirisformula" role="math" alt="开始mathsize 16px样式x结束样式" style="vertical-align:-4px;height:19px;width:10px" loading="lazy">反对x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Log空间y" style="vertical-align:-4px;height:19px;width:34px" loading="lazy">会给出一个线性图吗log y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - y截距是
" class="Wirisformula" role="math" alt="Log空间k" style="vertical-align:-4px;height:19px;width:34px" loading="lazy">直线的梯度是log k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Log空间b" style="vertical-align:-4px;height:19px;width:34px" loading="lazy">log b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这可以用同样的方法来表示,即对两边取对数
- 例如:
- 如果
- 如果回归模型为
两边取对数
对对数使用加法定律
对数使用幂律
- 以这种方式使用对数来编码数据被称为改变变量
如何使用非线性回归模型?
- 非线性回归模型的使用方法与线性回归模型
- 通过使用对数编码原始数据(改变变量)的回归线Y在X可以找到
- 这可用于预测给定数据范围内的数据值(插值)
- 在给定数据范围之外进行预测被称为外推法而不应该这样做
- 非线性回归模型可以通过代入得到
" class="Wirisformula" role="math" alt="开始mathsize 16px style log space x结束风格" style="vertical-align:-4px;height:19px;width:34px" loading="lazy">而且log x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="Log空间y" style="vertical-align:-6px;height:22px;width:38px" loading="lazy">回的X而且Y回归线中的值并重新排列log y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
工作的例子
下图显示了高度的分布,
数据是用变量的变化来编码的
考试技巧
- 在交替使用原始数据和编码数据时要小心,很容易忘记您正在使用的是哪一个。请记住,如果你的回归线是使用编码数据计算的,那么如果找到预测,你就需要反转它。确保你熟悉对数,指数和它们的法则。请仔细检查用于编码数据的底数,如果
" class="Wirisformula" role="math" alt="开始mathsize 16px style log space x结束风格" style="vertical-align:-4px;height:19px;width:34px" loading="lazy">被使用过,那么它是反向使用的log x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="10的16px大小的幂log大小16px空间大小16px x结束指数" style="vertical-align:-6px;height:26px;width:52px" loading="lazy">,但10 log x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="Ln空间x" style="vertical-align:-6px;height:22px;width:29px" loading="lazy">那么它应该反过来使用吗ln x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"} " class="Wirisformula" role="math" alt="E的ln (x)次方" style="vertical-align:-6px;height:23px;width:31px" loading="lazy">.e ln x {"language":"en","fontFamily":"Times New Roman","fontSize":"18","color":"#FFFFFF"}