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# 计算机代写|机器学习代写Machine Learning代考|COMP5318 K-Nearest Neighbors

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## 计算机代写|机器学习代写Machine Learning代考|K-Nearest Neighbors

At heart, many learning procedures – especially when our prior knowledge is weak – amount to smoothing the training data. RBF fitting is an example of this. However, many of these fitting procedures require making a number of decisions, such as the locations of the basis functions, and can be sensitive to these choices. This raises the question: why not cut out the middleman, and smooth the data directly? This is the idea behind $K$-Nearest Neighbors regression.

The idea is simple. We first select a parameter $K$, which is the only parameter to the algorithm. Then, for a new input $\mathrm{x}$, we find the $K$ nearest neighbors to $\mathrm{x}$ in the training set, based on their Euclidean distance $\left|\mathbf{x}-\mathbf{x}_i\right|^2$. Then, our new output $\mathbf{y}$ is simply an average of the training outputs for those nearest neigbors. This can be expressed as:
$$\mathbf{y}=\frac{1}{K} \sum_{i \in N_K(\mathbf{x})} \mathbf{y}i$$ where the set $N_K(\mathrm{x})$ contains the indicies of the $K$ training points closest to $\mathrm{x}$. Alternatively, we might take a weighted average of the $K$-nearest neighbors to give more influence to training points close to $\mathrm{x}$ than to those further away: $$\mathbf{y}=\frac{\sum{i \in N_K(\mathbf{x})} w\left(\mathbf{x}i\right) \mathbf{y}_i}{\sum{i \in N_K(\mathbf{x})} w\left(\mathbf{x}_i\right)}, \quad w\left(\mathbf{x}_i\right)=e^{-\left|\mathbf{x}_i-\mathbf{x}\right|^2 / 2 \sigma^2}$$
where $\sigma^2$ is an additional parameter to the algorithm. The parameters $K$ and $\sigma$ control the degree of smoothing performed by the algorithm. In the extreme case of $K=1$, the algorithm produces a piecewise-constant function.

The objective functions used in linear least-squares and regularized least-squares are multidimensional quadratics. We now analyze multidimensional quadratics further. We will see many more uses of quadratics further in the course, particularly when dealing with Gaussian distributions.
The general form of a one-dimensional quadratic is given by:
$$f(x)=w_2 x^2+w_1 x+w_0$$
This can also be written in a slightly different way (called standard form):
$$f(x)=a(x-b)^2+c$$
where $a=w_2, b=-w_1 /\left(2 w_2\right), c=w_0-w_1^2 / 4 w_2$. These two forms are equivalent, and it is easy to go back and forth between them (e.g., given $a, b, c$, what are $w_0, w_1, w_2$ ?). In the latter form, it is easy to visualize the shape of the curve: it is a bowl, with minimum (or maximum) at $b$, and the “width” of the bowl is determined by the magnitude of $a$, the sign of $a$ tells us which direction the bowl points ( $a$ positive means a convex bowl, $a$ negative means a concave bowl), and $c$ tells us how high or low the bowl goes (at $x=b$ ). We will now generalize these intuitions for higher-dimensional quadratics.
The general form for a 2D quadratic function is:
$$f\left(x_1, x_2\right)=w_{1,1} x_1^2+w_{1,2} x_1 x_2+w_{2,2} x_2^2+w_1 x_1+w_2 x_2+w_0$$
and, for an $N$-D quadratic, it is:
$$f\left(x_1, \ldots x_N\right)=\sum_{1 \leq i \leq N, 1 \leq j \leq N} w_{i, j} x_i x_j+\sum_{1 \leq i \leq N} w_i x_i+w_0$$

## 计算机代写|机器学习代写Machine Learning代考|K-Nearest Neighbors

$$\mathbf{y}=\frac{\sum i \in N_K(\mathbf{x}) w(\mathbf{x} i) \mathbf{y}i}{\sum i \in N_K(\mathbf{x}) w\left(\mathbf{x}_i\right)}, \quad w\left(\mathbf{x}_i\right)=e^{-\left|\mathbf{x}_i-\mathbf{x}\right|^2 / 2 \sigma^2}$$ 在哪里 $\sigma^2$ 是算法的附加参数。参数 $K$ 和 $\sigma$ 控制算法执行的平㳑程度。在极端情况下 $K=1$ ，该算法产生一个分段常数函数。

$$f\left(x_1, \ldots x_N\right)=\sum_{1 \leq i \leq N, 1 \leq j \leq N} w_{i, j} x_i x_j+\sum_{1 \leq i \leq N} w_i x_i+w_0$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。