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# 线性代数代考_Linear Algebra代考_MAST10007 Trend Analysis of Data

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## 线性代数代考_Linear Algebra代考_Trend Analysis of Data

Let $f$ represent an unknown function whose values are known (perhaps only approximately) at $t_0, \ldots, t_n$. If there is a “linear trend” in the data $f\left(t_0\right), \ldots, f\left(t_n\right)$, then we might expect to approximate the values of $f$ by a function of the form $\beta_0+\beta_1 t$. If there is a “quadratic trend” to the data, then we would try a function of the form $\beta_0+\beta_1 t+\beta_2 t^2$. This was discussed in Section 6.6, from a different point of view.
In some statistical problems, it is important to be able to separate the linear trend from the quadratic trend (and possibly cubic or higher-order trends). For instance, suppose engineers are analyzing the performance of a new car, and $f(t)$ represents the distance between the car at time $t$ and some reference point. If the car is traveling at constant velocity, then the graph of $f(t)$ should be a straight line whose slope is the car’s velocity. If the gas pedal is suddenly pressed to the floor, the graph of $f(t)$ will change to include a quadratic term and possibly a cubic term (due to the acceleration). To analyze the ability of the car to pass another car, for example, engineers may want to separate the quadratic and cubic components from the linear term.

If the function is approximated by a curve of the form $y=\beta_0+\beta_1 t+\beta_2 t^2$, the coefficient $\beta_2$ may not give the desired information about the quadratic trend in the data, because it may not be “independent” in a statistical sense from the other $\beta_i$. To make what is known as a trend analysis of the data, we introduce an inner product on the space $\mathbb{P}_n$ analogous to that given in Example 2 in Section 6.7. For $p, q$ in $\mathbb{P}_n$, define
$$\langle p, q\rangle=p\left(t_0\right) q\left(t_0\right)+\cdots+p\left(t_n\right) q\left(t_n\right)$$
In practice, statisticians seldom need to consider trends in data of degree higher than cubic or quartic. So let $p_0, p_1, p_2, p_3$ denote an orthogonal basis of the subspace $\mathbb{P}_3$ of $\mathbb{P}_n$, obtained by applying the Gram-Schmidt process to the polynomials $1, t, t^2$, and $t^3$. By Supplementary Exercise 11 in Chapter 2, there is a polynomial $g$ in $\mathbb{P}_n$ whose values at $t_0, \ldots, t_n$ coincide with those of the unknown function $f$. Let $\hat{g}$ be the orthogonal projection (with respect to the given inner product) of $g$ onto $\mathbb{P}_3$, say,
$$\hat{g}=c_0 p_0+c_1 p_1+c_2 p_2+c_3 p_3$$

## 线性代数代考_Linear Algebra代考_Fourier Series (Calculus required)

Continuous functions are often approximated by linear combinations of sine and cosine functions. For instance, a continuous function might represent a sound wave, an electric signal of some type, or the movement of a vibrating mechanical system.

For simplicity, we consider functions on $0 \leq t \leq 2 \pi$. It turns out that any function in $C[0,2 \pi]$ can be approximated as closely as desired by a function of the form
$$\frac{a_0}{2}+a_1 \cos t+\cdots+a_n \cos n t+b_1 \sin t+\cdots+b_n \sin n t$$
for a sufficiently large value of $n$. The function (4) is called a trigonometric polynomial. If $a_n$ and $b_n$ are not both zero, the polynomial is said to be of order $\boldsymbol{n}$. The connection between trigonometric polynomials and other functions in $C[0,2 \pi]$ depends on the fact that for any $n \geq 1$, the set
$${1, \cos t, \cos 2 t, \ldots, \cos n t, \sin t, \sin 2 t, \ldots, \sin n t}$$
is orthogonal with respect to the inner product
$$\langle f, g\rangle=\int_0^{2 \pi} f(t) g(t) d t$$
This orthogonality is verified as in the following example and in Exercises 5 and 6.

## 线性代数代考Linear Algebra代考_Trend Analysis of Data

$6.6$ 节中从不同的角度进行了讨论。 新车的侏能, 并且 $f(t)$ 表示当时车与车之间的距离梠一些参考。点。如果汽车以但定速度行驶，则图形为 $f(t)$ 应该是一条直线，其 $了 p, q$ 在 $\mathbb{P}{n \text { ，定义 }}$
$$\langle p, q\rangle=p\left(t_0\right) q\left(t_0\right)+\cdots+p\left(t_n\right) q\left(t_n\right)$$

$$\hat{g}=c_0 p_0+c_1 p_1+c_2 p_2+c_3 p_3$$

## 线性代数代考_Linear Algebra代考_Fourier Series (Calculus required)

$$\frac{a_0}{2}+a_1 \cos t+\cdots+a_n \cos n t+b_1 \sin t+\cdots+b_n \sin n t$$
$C[0,2 \pi]$ 取圩这样一个事实，即对于任何 $n \geq 1$, 集合
$$1, \cos t, \cos 2 t, \ldots, \cos n t, \sin t, \sin 2 t, \ldots, \sin n t$$

$$\langle f, g\rangle=\int_0^{2 \pi} f(t) g(t) d t$$

## MATLAB代写

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