数学-1.FIT)

# R语言 ARMA 参数模型数学公式在时间序列分析中，自回归移动平均模型（ARMA模型）是一种常见的方法。

ARMA模型结合了自回归（AR）和移动平均（MA）部分来拟合时间序列数据。

## 数学公式一个ARMA(p, q)模型可以表示为：Xt=c+∑i=1pϕiXt−i+∑j=1qθjεt−j+εtXt = c + \sum_{i=1}^{p} \phi_i X_{t-i} + \sum_{j=1}^{q} \theta_j \varepsilon_{t-j} + \varepsilon_tXt=c+∑i=1pϕiXt−i +∑j=1qθjεt−j+εt其中：* XtXXt是时间序列在时刻 ttt 的值。

* ccc 是常数项。

* ϕi\phi_iϕi 是自回归部分的参数，表示时间序列对过去值的依赖程度。

* θj\theta_jθj是移动平均部分的参数，表示时间序列对当前和过去噪声项（误差）的依赖程度。

* εt\varepsilon_tεt是白噪声过程，通常假设为独立同分布（iid）的正态分布，均值为0，方差为σ2\sigma^2σ2。

* ppp 是自回归部分的阶数，表示模型考虑的过去值的数量。

* qqq 是移动平均部分的阶数，表示模型考虑的过去噪声项的数量。

## ARMA模型的特性* **平稳性**：ARMA模型通常应用于平稳时间序列，即时间序列的统计特性（如均值和方差）不随时间变化。

* **预测**：ARMA模型可用于预测时间序列的未来值。

通过拟合模型参数，我们可以使用过去的观测值来预测未来的点。

* **自相关函数（ACF）和偏自相关函数（PACF）**：这些函数用于诊断ARMA模型的阶数。

自相关函数衡量时间序列与其自身过去值之间的相关性，而偏自相关函数衡量在给定中间值时这种相关性的程度。

## 在R中实现ARMA模型在R语言中，可以使用`forecast`或`TSA`包来拟合ARMA模型。

下面是一个简单的例子，展示如何使用`arima()`函数来拟合一个ARMA(1, 1)模型：```R# 加载必要的包install.packages("TSA")library(TSA)# 生成一些模拟数据set.seed(123) # 设置种子以保证结果可复现data <- arima.sim(n = 100, list(ar = c(0.6), ma = c(0.4))) # 模拟ARMA(1, 1)数据# 拟合ARMA(1, 1)模型fit <- arima(data, order = c(1, 0, 1))# 输出模型结果fit```这将拟合一个ARMA(1, 1)模型到模拟数据，并输出模型的参数估计和其他统计信息。

实验报告一·实验指导书解读本次实验是通过两个变量的多组记录数据利用最小二乘法寻求两个变量之间的函数关系！两个变量之间的函数关系要紧有两种：一是线性关系（一次函数）；二是非线性关系（非一次的其它一元函数）。

因此本实验做两件事：一是线性拟合（练习1）；二是非线性拟合（练习2、3、4）。

练习2是用多项式函数拟合，练习3是用指数函数、对数函数、双曲函数、三角函数、分式有理多项式函数等初等函数拟合，练习4是用分段函数（非初等函数）拟合。

二、实验打算1.用线性函数拟合程序线性拟合曲线ft1可由如下mathematica程序求出：lianxi1biao= { {100,45} , {110,51} , { 120,54} , {130,61} , {140,66} , {150,70} , {160,74} , {170,78} , {180,85} , {190,89} }ft1=Fit[lianxi1biao,{1,x},x]gp = Plot [ ft1 , {x,100,190} , PlotStyle -> { RGBColor[1,0,0]} ]fp = ListPlot [ lianxi1biao,PlotStyle->{PointSize[],RGBColor[0,0,1]} ]Show[fp,gp]a= ;b= ;f[x_]=a*x+b;dareta=Sum[(lianxi1biao[[i,2]]-f[lianxi1biao[[i,1]]])^2,{i,1,10}]修改、补充程序:要说明拟合成效，要紧从形（大多数散点是不是在拟合曲线上或周围）与量（残差是不是小）！计算残差的程序：假设对两个变量的多组记录数据已有程序biao={{x1,y1},{x2,y2},…,{xn,yn}}而且通过Fit取得线性拟合函数y=ax+b咱们能够先概念函数（程序）f[x_]:=a*x+b再给出计算残差的程序dareta=Sum[(biao[[i ,2]]-f[biao[[i ,1]]])^2,{i ,1, n}]程序说明：biao[[i]]是提取表biao的第i行，即{xi,yi}biao[[i ,1]] 是提取表biao的第i行的第一个数, 即xibiao[[i ,2]] 是提取表biao的第i行的第一个数, 即yibiao[[i ,2]]-f[biao[[i ,1]]] 即yi-(a*xi+b)实验思路1、先对练习1的十组数据线性拟合，并从形与量看拟合成效；2、对练习1的十组数据中的九组数据线性拟合，并从形与量看拟合成效；3、对练习1的十组数据中的八组数据线性拟合，并从形与量看拟合成效；4、对练习1的十组数据中的七组数据线性拟合，并从形与量看拟合成效；5、对练习1的十组数据中的六组数据线性拟合，并从形与量看拟合成效。

2023新高考一卷一、语文试题1. 阅读理解（1）阅读下面一则短文，完成后面的题目。

哈密瓜是农田中非常常见的瓜果之一，它营养丰富，含有大量水分和多种维生素，被誉为“水果之王”。

然而，哈密瓜产量普遍较低，这主要是由于其生长期较长，对环境要求较高的缘故。

近年来，一位名叫李明的农民在千方百计提高哈密瓜产量方面取得了很大的突破。

李明首先对哈密瓜的生长过程进行了仔细研究，发现哈密瓜需要充分的阳光照射才能生长得好。

因此，他在种植哈密瓜的地方设置了遮阳网，避免日照过强对哈密瓜的伤害。

同时，他还根据不同生长阶段的哈密瓜的需求，科学浇水，合理施肥，为哈密瓜提供了良好的生长环境。

此外，李明还认真研究了哈密瓜的病虫害防治方法。

他采用了物理防治、生物防治和化学防治相结合的方式，选择合适的方法和药品来防治，并且及时对哈密瓜进行病种鉴定，以便更准确地进行防治措施。

经过多年的努力和实践，李明终于培育出了一种新品种的哈密瓜，产量达到了原来的两倍以上。

他的成功经验也成了其他农民种植哈密瓜的参考，帮助更多的人提高了经济收入。

1）李明是如何提高哈密瓜产量的？2）请简要介绍李明是如何进行哈密瓜病虫害的防治的。

（2）阅读下面一首诗，完成后面的题目。

静夜思李白床前明月光，疑是地上霜。

举头望明月，低头思故乡。

1）这首诗作者感受到了什么？2）请简要概括整首诗的意境。

2. 写作你喜欢阅读吗？请以“我的阅读之旅”为题，写一篇不少于800字的作文，记叙你自小到大的阅读经历和收获。

二、数学试题1. 选择题（1）1/4加1/6等于（）。

A. 5/24B. 7/24C. 5/12D. 7/12（2）下列数中，最小的是（）。

A. -3B. -2C. 2D. 3（3）一部小说书的原价是120元，打8折后的价格是（）。

A. 12元B. 40元C. 96元D. 112元（4）已知2x-8=4，求x的值。

A. -2B. -1C. 1D. 22. 解答题（1）把千分之一改成百分数，它的值是多少？（2）一条细铁丝长17米，要把它锯成53段，每段长度相等。

SAMPLE QUESTION FOR 3 POINTSWhich digits are missing on the right?A) 3 and 5 B) 4 and 8 C) 2 and 0 D) 6 and 9 E) 7 and 1SAMPLE QUESTION FOR 4 POINTSGeorge has 2 cats of the same weight. What is the weight of one cat if George weighs 30 kilograms?A) 1 kilogram B) 2 kilograms C) 3 kilogramsD) 4 kilograms E) 5 kilogramsSAMPLE QUESTION FOR 5 POINTSIn a certain game it is possible to make the following exchanges:Adam has 6 pears. How many strawberries will Adam have after he trades all his pears for just strawberries?A) 12 B) 36 C) 18 D) 24 E) 6SAMPLE QUESTION FOR 3 POINTSWhich digits are missing on the right?A) 3 and 5 B) 4 and 8 C) 2 and 0 D) 6 and 9E) 7 and 1SAMPLE QUESTION FOR 4 POINTSGeorge has 2 cats of the same weight. What is the weight of one cat if George weighs 30 kilograms?A) 1 kilogram B) 2 kilograms C) 3 kilogramsD) 4 kilograms E) 5 kilogramsSAMPLE QUESTION FOR 5 POINTSIn a certain game it is possible to make the following exchanges:Adam has 6 pears. How many strawberries will Adam have after he trades all his pears for just strawberries?A) 12 B) 36 C) 18D) 24 E) 6SAMPLE QUESTION FOR 3 POINTSIn which figure is the number of black kangaroos larger than the number of white kangaroos?SAMPLE QUESTION FOR 4 POINTSEach time Pinocchio lies, his nose gets 6 cm longer. Each time he tells the truth, his nose gets 2 cm shorter. After his nose was 9 cm long, he told three lies and made two true statements. How long was Pinocchio's nose afterwards?A) 14 cm B) 15 cm C) 19 cm D) 23 cm E) 31 cmSAMPLE QUESTION FOR 5 POINTSJoining the midpoints of the sides of the triangle in the drawing we obtain a smaller triangle. We repeat this one more time with the smaller triangle. How many triangles of the same size as the smallest resulting triangle fit in the original drawing?A) 5 B) 8 C) 10 D) 16 E) 32SAMPLE QUESTION FOR 3 POINTSIn which figure is the number of black kangaroos larger than the number of white kangaroos?SAMPLE QUESTION FOR 4 POINTSEach time Pinocchio lies, his nose gets 6 cm longer. Each time he tells the truth, his nose gets 2 cm shorter. After his nose was 9 cm long, he told three lies and made two true statements. How long was Pinocchio's nose afterwards?A) 14 cm B) 15 cm C) 19 cm D) 23 cm E) 31 cmSAMPLE QUESTION FOR 5 POINTSJoining the midpoints of the sides of the triangle in the drawing we obtain a smaller triangle. We repeat this one more time with the smaller triangle. How many triangles of the same size as the smallest resulting triangle fit in the original drawing?A) 5 B) 8 C) 10 D) 16E) 32SAMPLE QUESTION FOR 3 POINTSNathalie wanted to build the same cube as Diana had (Figure 1). However, Nathalie ran out of small cubes and built only a part of the cube, as you can see in Figure 2. How many small cubes must be added to Figure 2 to form Figure 1?A) 5 B) 6 C) 7 D) 8 E) 9SAMPLE QUESTION FOR 4 POINTSMary shades various shapes on square sheets of paper, as shown.How many of these shapes have the same perimeter as the sheet of paper itself?A) 2 B) 3 C) 4 D) 5 E) 6SAMPLE QUESTION FOR 5 POINTSThere are four buttons in a row as shown below. Two of them show happy faces, and two of them show sad faces. If we press on a face, its expression turns to the opposite (e.g. a happy face turns into a sad face). In addition to this, the adjacent buttons also change their expressions to the opposite. What is the least number of times you need to press the buttons in order to get all happy faces?A) 2 B) 3 C) 4 D) 5 E) 6SAMPLE QUESTION FOR 3 POINTSNathalie wanted to build the same cube as Diana had (Figure 1). However, Nathalie ran out of small cubes and built only a part of the cube, as you can see in Figure 2. How many small cubes must be added to Figure 2 to form Figure 1?A) 5 B) 6 C) 7D) 8 E) 9SAMPLE QUESTION FOR 4 POINTSMary shades various shapes on square sheets of paper, as shown.How many of these shapes have the same perimeter as the sheet of paper itself?A) 2 B) 3 C) 4D) 5 E) 6SAMPLE QUESTION FOR 5 POINTSThere are four buttons in a row as shown below. Two of them show happy faces, and two of them show sad faces. If we press on a face, its expression turns to the opposite (e.g. a happy face turns into a sad face). In addition to this, the adjacent buttons also change their expressions to the opposite. What is the least number of times you need to press the buttons in order to get all happy faces?A) 2 B) 3C) 4 D) 5 E) 6SAMPLE QUESTION FOR 3 POINTSIn the picture, the big triangle is equilateral and has an area of 9. The lines are parallel to the sides and divide the sides into three equal parts. What is the area of the shaded part?A) 1 B) 4 C) 5 D) 6 E) 7SAMPLE QUESTION FOR 4 POINTSVasya wrote down several consecutive integers. Which of the following could not be the percentage of odd numbers among them?A) 40 B) 45 C) 48 D) 50 E) 60SAMPLE QUESTION FOR 5 POINTSThe diagram shows a shaded quadrilateral KLMN drawn on a grid. Each cell of the grid has sides of length 2 cm. What is the area of KLMN?A) 96 cm2B) 84 cm2C) 76 cm2D) 88 cm2E) 104 cm2SAMPLE QUESTION FOR 3 POINTSIn the picture, the big triangle is equilateral and has an area of 9. The lines are parallel to the sides and divide the sides into three equal parts. What is the area of the shaded part?A) 1 B) 4 C) 5 D) 6E) 7SAMPLE QUESTION FOR 4 POINTSVasya wrote down several consecutive integers. Which of the following could not be the percentage of odd numbers among them?A) 40 B) 45C) 48 D) 50 E) 60SAMPLE QUESTION FOR 5 POINTSThe diagram shows a shaded quadrilateral KLMN drawn on a grid. Each cell of the grid has sides of length 2 cm. What is the area of KLMN?A) 96 cm2B) 84 cm2C) 76 cm2D) 88 cm2E) 104 cm2SAMPLE QUESTION FOR 3 POINTSThe number 200013 – 2013 is not divisible byA) 2. B) 3. C) 5. D) 7. E) 11.SAMPLE QUESTION FOR 4 POINTSThe points P and Q are opposite vertices of a regular hexagon and the points R and S are the midpoints of opposite edges, as shown. The area of the hexagon is 60 cm2. What is the product of the lengths of PQ and RS?A) 40 cm2B) 50 cm2C) 60 cm2D) 80 cm2E) 100 cm2SAMPLE QUESTION FOR 5 POINTSHow many positive integers are multiples of 2013 and have exactly 2013 divisors (including 1 and the number itself)?A) 0 B) 1 C) 3 D) 6 E) other answerSAMPLE QUESTION FOR 3 POINTSThe number 200013 – 2013 is not divisible byA) 2. B) 3. C) 5. D) 7.E) 11.SAMPLE QUESTION FOR 4 POINTSThe points P and Q are opposite vertices of a regular hexagon and the points R and S are the midpoints of opposite edges, as shown. The area of the hexagon is 60 cm2. What is the product of the lengths of PQ and RS?A) 40 cm2B) 50 cm2C) 60 cm2D) 80 cm2E) 100 cm2SAMPLE QUESTION FOR 5 POINTSHow many positive integers are multiples of 2013 and have exactly 2013 divisors (including 1 and the number itself)?A) 0 B) 1 C) 3 D) 6E) other answerLEVELS 11 AND 12SAMPLE QUESTION FOR 3 POINTSWhich of the following numbers is the largest?A) 2013 B) 20+13C) 2013 D) 2013E) 20 ·13SAMPLE QUESTION FOR 4 POINTSRadu has identical plastic pieces in the shape of a regular pentagon. He glues them edge to edge to complete a circle, as shown in the picture. How many pieces are there in this circle?A) 8 B) 9 C) 10 D) 12 E) 15SAMPLE QUESTION FOR 5 POINTSHow many pairs (x, y) of integers with x < y exist such that their product equals 5 times their sum?A) 4 B) 5 C) 6 D) 7 E) 8LEVELS 11 AND 12 ANSWERSSAMPLE QUESTION FOR 3 POINTSWhich of the following numbers is the largest?A) 2013 B) 20+13C) 2013 D) 2013E) 20 ·13SAMPLE QUESTION FOR 4 POINTSRadu has identical plastic pieces in the shape of a regular pentagon. He glues them edge to edge to complete a circle, as shown in the picture. How many pieces are there in this circle?A) 8 B) 9 C) 10D) 12 E) 15SAMPLE QUESTION FOR 5 POINTSHow many pairs (x, y) of integers with x < y exist such that their product equals 5 times their sum?A) 4B) 5 C) 6 D) 7 E) 8。

比较两个列数据变化协调变化是否一致的数学方法It can be difficult to compare the changes in two sets of data and determine if they are moving in sync. One mathematical method that can be used to assess this is correlation analysis. Correlation analysis quantifies the strength and direction of the relationship between two variables. By calculating the correlation coefficient, we can determine the degree to which changes in one variable are associated with changes in another.在比较两组数据的变化并确定它们是否同步变化时可能会感到困难。

用来评估这一点的一种数学方法是相关性分析。

相关性分析量化了两个变量之间关系的强度和方向。

通过计算相关系数，我们可以确定一个变量的变化程度与另一个变量的变化相关程度。

The correlation coefficient ranges from -1 to 1. A correlation of 1 indicates a perfect positive relationship, where an increase in one variable is associated with an increase in the other. Conversely, a correlation of -1 indicates a perfect negative relationship, where an increase in one variable is associated with a decrease in the other. Ifthe correlation coefficient is close to 0, it suggests that there is no linear relationship between the two variables.相关系数的取值范围是-1到1。

SAMPLE QUESTION FOR 3 POINTS6. The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which tower is the tallest?(A) pink tower (B) green tower (C) red tower (D) silver tower(E) We don’t know.SAMPLE QUESTION FOR 4 POINTS15. The picture shows the five houses of five friends and their school. The school is the largest building in the picture. To go to school, Doris and Ali walk past Leo’s house. Eva walks past Chole’s house. Which is Eva’s house?(A) (B) (C) (D) (E) SAMPLE QUESTION FOR 5 POINTS18. Every time the witch has 3 apples, she turns them into 1 banana. Every time she has3 bananas, she turns them into 1 apple. What will she end up with if she starts with4 apples and5 bananas?(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 3 POINTS6. The pink tower is taller than the red tower but shorter than the green tower. The silver tower is taller than the green tower. Which tower is the tallest?(A) pink tower (B) green tower (C) red tower (D) silver tower(E) We don’t know.SAMPLE QUESTION FOR 4 POINTS15. The picture shows the five houses of five friends and their school. The school is the largest building in the picture. To go to school, Doris and Ali walk past Leo’s house. Eva walks past Chole’s house. Which is Eva’s house?(A) (B)(C) (D) (E) SAMPLE QUESTION FOR 5 POINTS18. Every time the witch has 3 apples, she turns them into 1 banana. Every time she has3 bananas, she turns them into 1 apple. What will she end up with if she starts with4 apples and5 bananas?(A)(B) (C) (D) (E)SAMPLE QUESTION FOR 3 POINTS2. How many fish will have their heads pointing towards the ring when we straighten the line?(A) 3 (B) 5 (C) 6 (D) 7 (E) 8SAMPLE QUESTION FOR 4 POINTS15. On a tall building there are 4 fire escape ladders, as shown. The heightsof 3 ladders are at their tops. What is the height of the shortest ladder?(A) 12 (B) 14 (C) 16 (D) 20 (E) 22SAMPLE QUESTION FOR 5 POINTS23. Elena wants to write the numbers from 1 to 9 in the squares shown. The arrows always point from a smaller number to a larger one. She has already written 5 and 7. Which number should she write instead of the question mark?(A) 2 (B) 3 (C) 4 (D) 6 (E) 8SAMPLE QUESTION FOR 3 POINTS2. How many fish will have their heads pointing towards the ring when we straighten the line?(A) 3 (B) 5 (C) 6 (D) 7 (E) 8SAMPLE QUESTION FOR 4 POINTS15. On a tall building there are 4 fire escape ladders, as shown. The heightsof 3 ladders are at their tops. What is the height of the shortest ladder?(A) 12 (B) 14 (C) 16 (D) 20 (E) 22SAMPLE QUESTION FOR 5 POINTS23. Elena wants to write the numbers from 1 to 9 in the squares shown. The arrows always point from a smaller number to a larger one. She has already written 5 and 7. Which number should she write instead of the question mark?(A) 2 (B) 3 (C) 4 (D) 6 (E) 8SAMPLE QUESTION FOR 3 POINTS10. There is a square with line segments drawn inside it. The line segments are drawn either from the vertices or the midpoints of other line segments. We colored 18 of the large square. Which one is our coloring?(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 4 POINTS13. Rosa wants to start at the arrow, follow the line, and get out at the other arrow. Which piece, if placed in the middle, cannot produce this? Note: The piece can be rotated.(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 5 POINTS29. 10 elves and trolls each were given a token with a different number from 1 to 10 written on it. They were each asked what number was on their token and all answered with a number from 1 to 10. The sum of the answers was 36. Each troll told a lie and each elf told the truth. What is the smallest number of trolls there could be in the group?(A) 1 (B) 3 (C) 4 (D) 5 (E) 7SAMPLE QUESTION FOR 3 POINTS10. There is a square with line segments drawn inside it. The line segments are drawn either from the vertices or the midpoints of other line segments. We colored 18 of the large square. Which one is our coloring?(A) (B) (C) (D)(E)SAMPLE QUESTION FOR 4 POINTS13. Rosa wants to start at the arrow, follow the line, and get out at the other arrow. Which piece, if placed in the middle, cannot produce this? Note: The piece can be rotated.(A) (B) (C) (D) (E)SAMPLE QUESTION FOR 5 POINTS29. 10 elves and trolls each were given a token with a different number from 1 to 10 written on it. They were each asked what number was on their token and all answered with a number from 1 to 10. The sum of the answers was 36. Each troll told a lie and each elf told the truth. What is the smallest number of trolls there could be in the group?(A) 1 (B) 3 (C) 4 (D) 5 (E) 7SAMPLE QUESTION FOR 3 POINTS5. When the five pieces shown fit together correctly, the result is a rectangle with a calculationwritten on it. What is the result of this calculation?(A) –100 (B) –8 (C) –1 (D) 199 (E) 208SAMPLE QUESTION FOR 4 POINTS13. The area of the large square is 16 cm 2 and the area of each small square is 1 cm 2. What is the total area of the black flower?(A) 3 cm 2(B) 72 cm 2(C) 4 cm 2(D) 112 cm 2(E) 6 cm 2SAMPLE QUESTION FOR 5 POINTS26. 2021 colored kangaroos are arranged in a row and are numbered from 1 to 2021. Eachkangaroo is colored red, gray, or blue. Among any three consecutive kangaroos, there are always kangaroos of all three colors. Bruce guesses the colors of five kangaroos. These are his guesses: Kangaroo 2 is gray; Kangaroo 20 is blue; Kangaroo 202 is red; Kangaroo 1002 is blue; Kangaroo 2021 is gray. Only one of his guesses is wrong. What is the number of the kangaroo whose color he guessed incorrectly?(A) 2 (B) 20 (C) 202 (D) 1002 (E) 2021SAMPLE QUESTION FOR 3 POINTS5. When the five pieces shown fit together correctly, the result is a rectangle with a calculation written on it. What is the result of this calculation?(A) –100 (B) –8 (C) –1 (D) 199 (E) 208SAMPLE QUESTION FOR 4 POINTS13. The area of the large square is 16 cm2 and the area of each smallsquare is 1 cm2. What is the total area of the black flower?(A) 3 cm2(B) 72 cm2(C) 4 cm2(D) 112 cm2(E) 6 cm2SAMPLE QUESTION FOR 5 POINTS26. 2021 colored kangaroos are arranged in a row and are numbered from 1 to 2021. Each kangaroo is colored red, gray, or blue. Among any three consecutive kangaroos, there are always kangaroos of all three colors. Bruce guesses the colors of five kangaroos. These are his guesses: Kangaroo 2 is gray; Kangaroo 20 is blue; Kangaroo 202 is red; Kangaroo 1002 is blue; Kangaroo 2021 is gray. Only one of his guesses is wrong. What is the number of the kangaroo whose color he guessed incorrectly?(A) 2 (B) 20 (C) 202 (D) 1002 (E) 2021SAMPLE QUESTION FOR 3 POINTS1. Each year, the third Thursday in March is named Kangaroo Day. The dates of Kangaroo Day for the next few years are shown below, with one error. Which date is wrong?(A) March 17, 2022 (B) March 16, 2023 (C) March 14, 2024(D) March 20, 2025 (E) March 19, 2026SAMPLE QUESTION FOR 4 POINTS13. The numbers from 1 to 6 are placed in the circles at the intersections of three rings.The position of number 6 is shown. The sums of the numbers on each ring are the same. What number is placed in the circle with the question mark?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5SAMPLE QUESTION FOR 5 POINTS25. The smaller square in the picture has an area of 16 and the gray triangle has an area of 1. What is the area of the larger square?(A) 17 (B) 18 (C) 19 (D) 20 (E) 21SAMPLE QUESTION FOR 3 POINTS1. Each year, the third Thursday in March is named Kangaroo Day. The dates of Kangaroo Day for the next few years are shown below, with one error. Which date is wrong?(A) March 17, 2022 (B) March 16, 2023 (C) March 14, 2024(D) March 20, 2025 (E) March 19, 2026SAMPLE QUESTION FOR 4 POINTS13. The numbers from 1 to 6 are placed in the circles at the intersections of three rings.The position of number 6 is shown. The sums of the numbers on each ring are the same. What number is placed in the circle with the question mark?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5SAMPLE QUESTION FOR 5 POINTS25. The smaller square in the picture has an area of 16 and the gray triangle has an area of 1. What is the area of the larger square?(A) 17 (B) 18 (C) 19 (D) 20 (E) 21LEVELS 11 AND 12SAMPLE QUESTION FOR 3 POINTS1. Paula’s weather app shows a diagram of the predicted weather and maximum temperatures for the next seven days, as shown. Which of the following represents the corresponding graph of maximum temperatures?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 4 POINTS19. A naughty puppy grabs the end of a roll of toilet paper and walks away at a constant speed. Which of the functions below best describes the thickness y of the roll as a function of the unrolled part x?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 5 POINTS30. A certain game is won when one player gets 3 points ahead. Two players A and B are playingthe game and at a particular point, A is 1 point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game?(A) 12(B) 23(C) 34(D) 45(E) 56LEVELS 11 AND 12 ANSWERSSAMPLE QUESTION FOR 3 POINTS1. Paula’s weather app shows a diagram of the predicted weather and maximum temperatures for the next seven days, as shown. Which of the following represents the corresponding graph of maximum temperatures?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 4 POINTS19. A naughty puppy grabs the end of a roll of toilet paper and walks away at a constant speed. Which of the functions below best describes the thickness y of the roll as a function of the unrolled part x?(A) (B) (C)(D) (E)SAMPLE QUESTION FOR 5 POINTS30. A certain game is won when one player gets 3 points ahead. Two players A and B are playingthe game and at a particular point, A is 1 point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game?(A) 12(B)23(C) 34(D) 45(E) 56。

结构方程模型指标介绍结构方程模型（Structural Equation Modeling, SEM）是一种统计分析方法，用于检验观测或测量指标与潜在变量之间的关系。

通过构建数学模型，结构方程模型可以帮助研究者了解变量之间的因果关系和测量指标之间的关联性。

本文将介绍结构方程模型的基本概念和常用指标。

一级标题2二级标题1在结构方程模型中，常用的指标包括： 1. 参数估计指标 - 确定性系数（R-squared） - 标准化系数（Standardized Coefficients） - 因素载荷（Factor Loadings） 2. 模型拟合度指标 - 卡方检验（Chi-Square Test） - 拟合优度指数（Goodness of Fit Index, GFI） - 修正拟合指数（Adjusted Goodness ofFit Index, AGFI） - 根均方误差逼近指数（Root Mean Square Error of Approximation, RMSEA） - 比较拟合指数（Comparative Fit Index, CFI） 3.判定系数（Coefficient of Determination） 4. 相对拟合指数（Relative Fit Index, RFI）二级标题2三级标题1参数估计指标可以帮助研究者了解变量之间的关系及其强度。

其中，确定性系数（R-squared）可以衡量模型解释因变量方差的能力，取值范围为0到1，越接近1表示模型解释能力越强。

标准化系数（Standardized Coefficients）可以衡量变量之间的关系强度，也可以用于比较不同变量对因变量的影响程度。

因素载荷（Factor Loadings）表示观测指标与潜在变量之间的关联性，取值范围为-1到1，绝对值越大表示关联程度越强。

三级标题2模型拟合度指标用于评估结构方程模型的拟合程度。

卡方检验（Chi-Square Test）可以检验实际观测数据与模型拟合数据之间的差异，一般情况下，卡方值越小越好。

小学数学必备数值在小学数学的学习中，有一些数值是必须掌握的。

这些数值包括整数、小数、分数、百分数、比例和代数表达式等。

这些数值不仅在日常生活和商业活动中广泛使用，也是进一步学习数学和其他科学知识的基础。

1、整数：整数包括正整数、0和负整数。

例如，1、2、3是正整数，0和-1、-2、-3是整数。

整数在日常生活中非常常见，如人的数量、物体的数量等。

2、小数：小数是一种十进制数，它由整数部分和小数部分组成。

例如，3.14、-0.5和2.78都是小数。

小数在日常生活中也经常出现，如测量、重量和价格等。

3、分数：分数是一种有理数，它由分子和分母组成。

例如，1/2、2/3和3/4都是分数。

分数可以用来表示部分与整体的关系，如一个苹果可以分成两半，一半就是1/2个苹果。

4、百分数：百分数是一种表达比例的方式，它通常以一个小数表示。

例如，50%可以写作0.5。

百分数在商业和统计中广泛使用，可以用来比较不同事物的相对大小。

5、比例：比例是两个数的比值，通常用分数表示。

例如，如果一个物体的长度是另一个物体长度的两倍，那么它的长度与另一个物体的长度的比例就是2:1。

6、代数表达式：代数表达式是用字母表示的数学表达式。

例如，x+3、2x-1和3y+z都是代数表达式。

代数表达式在解决数学问题和理解变量之间的关系时非常有用。

以上这些数值是小学数学的基本内容，必须熟练掌握。

通过理解这些数值的概念和用法，我们可以更好地理解数学的基础知识，为以后的学习和工作打下坚实的基础。

三个内角和等于180度；三角形的一个外角等于和它不相邻的两个内角的和；三角形的一个外角大于任何一个和它不相邻的内角。

任意多边形至少有一组对边平行，只有等边多边形才有两组对边平行。

任意多边形的内角和定理：n边形的内角和等于(n-2)180°。

任意多边形的外角和定理：任意多边形的外角和等于360°。

用一个内角的度数可以表示任意多边形的外角。

一个多边形去掉一个顶点，形成的多边形可能是三角形，一定不是多边形。

英文的数学名言有哪些(精选2篇)英文的数学名言有哪些「篇一」摘要：数学是一门固定、精确和逻辑严谨的学科，其中蕴藏着一些令人深思的名言。

本文将介绍不低于30句的英文数学名言，这些名言涵盖了数学的重要原理和概念，展现了数学的美妙和智慧。

正文：1. "Mathematics is the language with which God has written the universe." - Galileo Galilei2. "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater." - Albert Einstein3. "Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." - William Paul Thurston4. "Mathematics is the music of reason." - James Joseph Sylvester5. "Mathematics is the key and door to the sciences." - Roger Bacon6. "Mathematics, like music, is a universal language." - Carl Friedrich Gauss7. "Mathematics is the most beautiful and most powerful creation of the human spirit." - Stefan Banach8. "Mathematics is the art of giving the same name to different things." - Henri Poincaré9. "Pure mathematics is, in its way, the poetry of logical ideas." - Albert Einstein10. "The only way to learn mathematics is to do mathematics." - Paul Halmos11. "In mathematics, the art of proposing a question must be held of higher value than solving it." - Georg Cantor12. "Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost." - W.S. Anglin13. "The study of mathematics, like the Nile, begins in minuteness but ends in magnificence." - Charles Caleb Colton14. "Mathematics is the science of patterns." - Richard Hamming15. "The essence of mathematics is not to make simple things complicated, but to make complicated things simple." - Stan Gudder16. "Mathematics is the door and key to the sciences." - Roger Bacon17. "The mathematician's patterns, like the pXXnter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way." - G.H. Hardy18. "In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths.And it's not always at the top of the mountXXn where the experts live." - Yoko Ogawa19. "Mathematics is a game played according to certXXn simple rules with meaningless marks on paper." - David Hilbert20. "Mathematics is the queen of the sciences and number theory is the queen of mathematics." - Carl Friedrich Gauss21. "Mathematics is the science that draws necessary conclusions." - Benjamin Peirce22. "Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers." - Shakuntala Devi23. "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." - John von Neumann24. "The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." - Henri Poincaré25. "Mathematics is about as stable as a toothache." - Harold Marston Morse26. "The study of mathematics, like the Nile, begins in minuteness, and ends in magnificence." - Charles Caleb Colton27. "Mathematics is the supreme judge." - Gottfried Wilhelm Leibniz28. "Mathematics is the most perfect language of the mind." - Carl Friedrich Gauss29. "Mathematics is the language in which God has written the universe." - Galileo Galilei30. "Mathematics is an easy, game-like description of the world." - Jean Piaget结论：这些英文数学名言展示了数学的重要性、美妙性和智慧性。