8.3 局部加权回归 8.3 Locally Weighted Regression 前一节描述的最近邻方法可以被看作在单一的查询点x=xq上逼近目标函数f(x)。局部加权 回归是这种方法的推广。它在环绕xq的局部区域内为目标函数f建立明确的逼近。局部加 权回归使用附近的或距离加权的训练样例来形成这种对f的局部逼近。例如，我们可以使 用线性函数、二次函数、多层神经网路或者其他函数形式在环绕xq的邻域内逼近目标函数 。“局部加权回归”名称中，之所以叫“局部”是因为目标函数的逼近仅仅根据查询点附 近的资料，之所以叫“加权”是因为每一个训练样例的贡献是由它与查询点间的距离加权 的，之所以叫“回归”是因为统计学习界广泛使用这个术语来表示逼近实数值函数的问题 。 The nearest-neighbor method described in the previous chapter can be seen as a method of approximation to the target function f(x) at a single point x = xq. Locally weighted regression is an extension of this method. It establishes explicit approximation to target function f in a local area around xq. Locally weighted regression achieves local approximation to f using neighboring or distance-weighted training examples. For example, we can use linear functions, quadratic functions, multi-layer neural networks, or other types of functions to approximate the target function in the vicinity of xq. Regarding the name of "locally weighted regression," why it is called locally is because that the target function is approximated as to only the data surrounding the query point; it is called "weighted" because the contribution of each training example is weighted by its distance to the query point; it is called "regression" because because this term is widely used in statistician circles to formulate problems about approximation of real-valued functions. 给定一个新的查询实例xq，局部加权回归的一般方法是建立一个逼近f hat，使f hat拟合 环绕xq的邻域内的训练样例。然後用这个逼近来计算f hat (xq)的值，也就是为查询实例 估计的目标值输出。然後f hat的描述被删除，因为对於每一个独立的查询实例都会计算 不同的局部逼近。 For a given new query example xq, the general method of locally weighted ^ regression is to construct a f for approximation and fit the training examples ^ in the neighborhood around xq. Subsequently, compute the value of f using this approximation; that is, estimate an output value for the training example. Then ^ the description of f is removed, because a local approximation for each query example is computed independently. 8.3.1 局部加权线性回归 8.3.1 Locally-Weighted Linear Regression 下面，我们先考虑局部加权回归的一种情况，即使用如下形式的线性函数来逼近xq邻域的 目标函数f： Below, we first consider one situation in locally-weighted regression, which is the use of the following linear function to approximate the neighborhood of xq in target function f: ^ f (x)=w0+w1a1(x)+...+wnan(x) 和前面一样，ai(x)表示实例x的第i个属性值。 回忆第4章中我们讨论的梯度下降方法，在拟合以上形式的线性函数到给定的训练集合时 ，它被用来找到使误差最小化的系数w0 ... wn。在那一章中我们感兴趣的是目标函数的 全域逼近。所以当时我们推导出的权值选择方法是使训练集合D上的误差平方和最小化， 即： E = ....... （8.5） As above, ai(x) denotes the ith attribute value of example x. Remember the gradient descent method we discussed in chapter 4. When fitting linear functions of the above form to a given training set, the method is used to find coefficients w0 to wn that minimize the error. In that chapater, we were interested in the global approximation of the target function. Thus, our weighting method was derived by minimizing the sum of square errors on training set D. E = ....... （8.5） http://tinyurl.com/ntos6ca --

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