Add note

content/math/MOC.md

@@ -8,10 +8,14 @@ date: 2023-12-03
 
 # Statistics
 
-## Basic concept
+## Basic Concept
 
 * [Quantile](math/Statistics/Basic/Quantile.md)
 
+## Significance Test
+
+* [Basic about significance test](math/Statistics/significance_test/whats_the_significance_test.md)
+
 # Discrete mathematics
 
 ## Set theory

content/math/Statistics/significance_test/whats_the_significance_test.md

@@ -0,0 +1,111 @@
+---
+title: What's the significance test
+tags:
+  - advanced
+  - statistics
+  - math
+date: 2024-04-15
+---
+⭐⭐The significance test tells us **whether or not what we observe in the sample is expected to be true in the populatio**n, and can be conducted through a **hypothesis test**.⭐⭐
+
+# Goal⭐
+
+• The sample correlation coefficient ($r$) between x and y is known (can be computed using samples)
+• The population correlation coefficient ($\rho$) between x and y is unknown (because we only have sample data)
+• Goal: We want to make an inference about the value of $\rho$ based on $r$
+
+
+# Hypothesis Test
+
+The hypothesis test will let us infer whether the value of the population correlation coefficient $\rho$ is close to 0 or significantly different from 0. We decide this based on the sample correlation coefficient $r$ and the sample size $n$
+
+Situation:
+
+• **$\rho$ close to 0**: means there is not a significant linear correlation between x and y in the population.
+• **$\rho$ significantly different from 0**: means there is a significant correlation between x and y in the population.
+
+For hypothesis test, we have two types - the **null** and **alternative** hypotheses.
+
+• The **alternative hypothesis** is always what we are trying to **prove**.
+• The **null hypothesis** is the hypothesis that we are trying to provide evidence **against**.
+
+### Detailed Explaination
+
+1. **零假设（Null Hypothesis）** - 通常表示为$H_0$​。 零假设是一个默认的假设，它通常表示没有效应、没有差异或变量之间没有关系。换句话说，零假设通常提出了一个“无效果”的声明，比如两组数据的均值没有差异、两个变量之间没有相关性等。在假设检验中，研究者试图通过收集和分析数据来寻找拒绝零假设的证据。
+    
+    例如，如果研究者想要检验两种药物对血压的影响是否有差异，零假设可能是“两种药物对降低血压的效果没有差异”。
+    
+2. **备择假设（Alternative Hypothesis）** - 通常表示为$H_1$或$H_a$​。 备择假设与零假设相对立，它提出了一个“有效果”的声明，即存在效应、差异或关系。备择假设通常是研究者真正感兴趣的假设，它反映了研究者希望证明的效应或关系。
+    
+    在上述药物对血压影响的例子中，备择假设可能是“两种药物对降低血压的效果存在差异”。
+
+
+### Classic Test Method
+
+Here's some classic test method:
+
+* **T-Test** (Student's T-Test)
+* **ANOVA** (Analysis of Variance)
+* **Mann-Whitney U Test** - MWU (Mann-Whitney U Test)
+* **Kruskal-Wallis H Test** - KW (Kruskal-Wallis H Test)
+* **Friedman Test** - FT (Friedman Test)
+* **ANCOVA** - ANCOVA (Analysis of Covariance)
+* **Pearson Correlation Coefficient** - PCC (Pearson Correlation Coefficient)
+* **Factor Analysis** - FA (Factor Analysis)
+* **Cluster Analysis** - CA (Cluster Analysis)
+* **Time Series Analysis** - TSA (Time Series Analysis)
+* ... ...
+
+## T-test
+
+Here we introduce a very classic test method, called T-test, also called Student’s T-test, which is an inferential statistic that allows to test an assumption applicable to a population, or simply, it allows to use sample data to generalize an assumption to an entire population.
+
+
+> [!info] 
+> T检验是由威廉·戈塞特（William Sealy Gosset）在20世纪初提出的，他使用笔名“Student”发表了相关论文，因此这种检验有时也被称为“Student's t-test”。它通常用于样本量较小的情况 
+### T-value
+
+Equation:
+
+$$
+t=\frac{r\times\sqrt{n-2}}{1-r^2}
+$$
+$$
+\text{n is the sample size, r is the correlation coefficient}
+$$
+
+T值（t-value）是从T检验中得到的统计量，它是**样本均值差异与样本内变异性的比率**。T值越大，意味着样本均值之间的差异相对于样本内的变异性越大，这通常会导致更低的p值，从而增加了**拒绝零假设**（即两组均值没有差异）的证据。换句话说，T值越大，我们越有理由相信两组之间的均值差异是真实存在的，而不是由随机变异引起的。
+
+T值的大小并不直接影响相关性的可重复性。然而，如果我们在讨论的是重复测量的均值差异，那么在某种程度上，较大的T值可能会使得研究者更有信心认为这种差异是可靠的，因为较大的T值意味着较小的p值，这可能**会使得研究结果在统计上更显著**。但是，这并**不意味着相关性的可重复性得到了提高**，因为**相关性的可重复性涉及到的是变量之间关系的一致性，而不是均值差异的显著性。**
+
+### P-value
+
+![](math/Statistics/significance_test/attachments/Pasted%20image%2020240415174359.png)
+
+
+P值（P-value），全称为概率值（Probability value），是统计假设检验中的一个重要概念。**它用于帮助我们决定是否拒绝零假设**（null hypothesis）。**P值衡量的是，在零假设为真的情况下，观察到的统计量（如T值、Z值等）或更极端情况出现的概率**。
+
+在进行假设检验时，我们通常设定一个显著性水平（alpha level），常用的值有0.05、0.01等。这个显著性水平是我们事先决定的拒绝零假设的阈值。如果计算出的P值小于这个显著性水平，我们就有理由拒绝零假设，认为观察到的效应或差异是统计上显著的，即不太可能是由随机变异引起的。
+
+例如，假设我们进行一个T检验，零假设是两组数据的均值没有差异。如果我们得到的P值为0.03，而我们设定的显著性水平为0.05，那么P值小于0.05，我们就拒绝零假设，认为两组数据的均值存在显著差异。
+
+需要注意的是，**P值并不直接告诉我们零假设是真还是假，也不提供效应大小的信息**。它只是表示**在零假设成立的前提下，得到当前统计结果或更极端结果的概率**。此外，P值也不是与发现效应的大小相关联的概率。**一个很小的P值**并不意味着效应很大，它**只意味着结果不太可能是偶然发生的**。
+
+在使用P值时，应该**避免一些常见的误解和滥用**，如将P值解释为支持备择假设的概率，或者将P值与研究的结论重要性等同起来。正确理解和解释P值对于统计分析的有效性和科学性的结论至关重要。
+
+# Conclusion
+
+1、小概率原理：小概率事件在一次试验中是几乎不可能发生的，假若在一次试验中小概率事件事实上发生了。那只能认为该事件不是来自我们假设的总体，也就是认为我们对总体所做的假设不正确。
+
+2、观察到的显著水平：由样本资料计算出来的检验统计量观察值所截取的尾部面积。这个概率越小，反对原假设，认为观察到的差异表明真实的差异存在的证据便越强，观察到的差异便越加理由充分地表明真实差异存在。
+
+3、检验所用的显著水平：针对具体问题的具体特点，事先规定这个检验标准。
+
+4、在检验的操作中，把观察到的显著性水平与作为检验标准的显著水平标准比较，小于这个标准时，得到了拒绝原假设的证据，认为样本数据表明了真实差异存在。大于这个标准时，拒绝原假设的证据不足，认为样本数据不足以表明真实差异存在。
+
+5、检验的操作可以用稍许简便一点的作法：根据所提出的显著水平查表得到相应的值，称作临界值，直接用检验统计量的观察值与临界值作比较，观察值落在临界值所划定的尾部内，便拒绝原假设；观察值落在临界值所划定的尾部之外，则认为拒绝原假设的证据不足。
+
+
+# Reference
+
+* https://kimi.moonshot.cn/chat

content/signal_processing/advanced_statistic/autocorrelation/autocorrelation.md

@@ -0,0 +1,223 @@
+---
+title: Autocorrelation in Signal Processing
+tags:
+  - advanced
+  - signal
+  - statistics
+  - signal-processing
+date: 2024-04-15
+---
+# Background
+## Covariance
+
+
+Covariance can classify three types of relationships between two random variables.
+
+1. **Relationships with positive trends**
+2. **Relationships with negative trends**
+3. **Times when there is no relationship because there is no trend**
+
+Also very importantly, covariance is a **computational stepping stone** to more interesting thing, like **correlation.**
+
+
+Here's the equation to calculate the covariance:
+
+$$
+\begin{equation}
+\begin{split} 
+cov(X, Y) & = E[(X-E[X])(Y - E[Y])] \\ 
+& = E[XY] - E[X]E[Y] \\
+cov(X, Y) & = \frac{\sum(x-\overline{x})(y-\overline{y})}{n-1} \\
+\end{split}
+\end{equation}
+$$
+![](signal_processing/advanced_statistic/attachments/Pasted%20image%2020240415171344.png)
+
+![](signal_processing/advanced_statistic/attachments/Pasted%20image%2020240415171351.png)
+
+
+Covariance is hard to **interpret** because it is sensitive to the **scale**
+
+
+To solve the scale effect, here's the correlation:
+
+## Correlation
+
+![](signal_processing/advanced_statistic/attachments/Pasted%20image%2020240415171510.png)
+
+
+We can quantify the strength of the relationship with correlation (**Pearson’s correlation**)
+
+$$
+r= corr(X, Y) = \frac{cov(X, Y)}{\sqrt{var(X)}\sqrt{var(Y)}}
+$$
+
+$corr(X, Y)$ is between -1 to 1
+
+
+> [!tip] 
+>  Correlation can still be 1 even we have so little data, even only 2 random dots In general, more data, **the more confidence we believe the correlation** 
+>  
+>  NOTE: When we’re talking about correlation, we’re only talking about using **straight line**
+
+
+![](signal_processing/advanced_statistic/attachments/Pasted%20image%2020240415171736.png)
+
+For correlation, we usually use **p-value** to **quantify the confidence** of the straight line relationship. **The more samll p-value, the more confident we say they are straight line relationship**; Like the figure:
+
+![](signal_processing/advanced_statistic/autocorrelation/attachments/Pasted%20image%2020240415171834.png)
+
+![](signal_processing/advanced_statistic/autocorrelation/attachments/Pasted%20image%2020240415171855.png)
+
+
+About P-value, you have better know what's [significance test](math/Statistics/significance_test/whats_the_significance_test.md)
+
+
+## Random Signal
+
+- 随机信号(Random Signals)在任何时间的取值都是不能先验确定的随机变量
+- 虽然随机信号的取值不能先验确定，但这些取值却服从某种统计规律，换言之，随机信号或过程可以用概率分布特性统计地描述
+- 随机变量 $X=x(t)$，离散状态为随机序列 $x(n)$,$x_k(n)$是随机序列$x(n)$的一个样本序列
+
+### Statistics for Random Signal
+
+$$
+\mu_x(n)=E\{x(n)\}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^N x_k(n)
+$$
+$$
+E\{x^2(n)\}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^N x^2_k(n)
+$$
+$$
+\sigma^2_x(n)=E\{[x(n)-\mu_x(n)]^2\}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^N[x_k(n)-\mu_x(n)]^2
+$$
+$$
+\sigma^2_x(n)=E\{x^2(n)\}-\mu^2_x(n)
+$$
+$$
+R_x(n_1,n_2)=E\{x(n_1)x(n_2)\}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^N x_k(n_1)x_k(n_2)
+$$
+
+# Autocorrelation
+
+## Equation
+
+
+$$
+R_x(n_1,n_2) = E\{x(n_1)x(n_2)\}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}x_k(n_1)x_k(n_2)
+$$
+若令时移m=n_2-n_1，有：
+
+$$
+R_x(n,n+m) = E\{x(n)x(n+m)\}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}x_k(n)x_k(n+m)
+$$
+
+
+## Properties
+
+1. **Symmetry**
+
+$$
+R_x(m)=R_x(-m)
+$$
+
+2. **Maximum at zero**
+
+$$
+R_x(0) \geq |R_x(m)|
+$$
+
+$$
+R_x(0)=E\{x(n)x(n)\}=E\{x^2(n)\} \geq 0
+$$
+
+$$
+E\{[x(n)\pm x(n+m)]^2\} \geq 0
+$$
+
+So,
+
+$$
+E\{x^2(n)\} \pm 2E\{x(n)x(m+n)\} + E\{x^2(m+n)\} \geq 0
+$$
+
+Depending on stable signal,
+
+$$
+E\{x^2(n)\} = E\{x^2(m+n)\} = R_x(0)
+$$
+
+So,
+
+$$
+2R_x(0)\pm2R_x(m) \geq 0
+$$
+
+$$
+R_x(0) \geq |R_x(m)|
+$$
+
+3. **Relation between Autocovariance and Autocorrelation**
+
+$$
+C_x(m)=R_x(m)-\mu_x^2
+$$
+
+$$
+C_x(0)=R_x(0)-\mu_x^2=E\{x^2(n)\}-\mu_x^2 = \sigma_x^2
+$$
+
+4. 在**非周期平稳序列**上，随机变量在时间越来越远后，相关性变得越来越弱，$m \rightarrow \infty$，可以认为不相关
+
+$$
+R_x(\infty)=\lim_{m\rightarrow\infty}R_x(m)=\lim_{m\rightarrow\infty}E\{x(n)x(m+n)\} = E\{x(n)\}E\{x(n+m)\}=\mu_x^2
+$$
+
+5. 平稳信号**不含相位信息**
+
+$$
+y(n)=x(n-l)
+$$
+
+$$
+R_y(m)=R_x(m),C_y(m)=C_x(m)
+$$
+
+6. 本质性质，**正定性(Positive Definiteness)**
+
+**Positive Definiteness Matrix**
+
+$$
+A\in R^{n\times n},\text{ so, it have any vector } \alpha\in R^n:  \\
+\alpha^T A \alpha \geq 0
+$$
+
+将正定性推广到相关函数，有随机变量$X=[x(1),x(2),\cdots,x(n)]^T,$ 则存在相关矩阵$R$,
+
+$$
+R=\begin{bmatrix}
+R(0) & \cdots & R(-n) \\
+\vdots & \ddots & \vdots \\
+R(n) & \cdots & R(0)
+\end{bmatrix}
+$$
+
+存在任意向量$\alpha \in R^n$
+
+$$
+\alpha^T R \alpha \geq0
+$$
+
+从频域角度来看，Bochner：
+
+若$f(x)$正定，则等价于：
+
+$$
+\mathcal{F}(f(x)) = \int_{-\infty}^{\infty}f(x)e^{-j\omega x}dx \geq 0
+$$
+
+其傅里叶变换始终大于0
+
+# Reference
+
+* https://pinkr1ver.notion.site/Autocorrelation-Analysis-Power-Spectral-Density-330755770347472989062c6b31f18a21?pvs=4
+* https://www.youtube.com/watch?v=qtaqvPAeEJY&t=24s

content/signal_processing/advanced_statistic/autocorrelation/period_detect.md

@@ -0,0 +1,9 @@
+---
+title: Autocorrelation to Detect Period
+tags:
+  - statistics
+  - math
+  - advanced
+  - signal-processing
+date: 2024-04-15
+---

content/signal_processing/device_and_components/oscilloscope/oscilloscope.md

@@ -65,3 +65,5 @@ Let's see the vertical subsystem of the oscilloscope. The basic form of an analo
 # Reference
 
 * https://www.linkedin.com/pulse/working-principle-indicators-oscilloscope-1-na-wang-6pkmc/
+* https://zhuanlan.zhihu.com/p/672667254
+* https://zhuanlan.zhihu.com/p/393063086

content/signal_processing/signal_processing_MOC.md

@@ -41,3 +41,16 @@ date: 2024-03-18
 
 * [Gaussian Impulse Generating](signal_processing/impulse_generating/gaussian_impulse.md)
 
+
+## Autocorrelation
+
+* [Autocorrelation in Signal Processing](signal_processing/advanced_statistic/autocorrelation/autocorrelation.md)
+* [Autocorrelation to Detect Period](signal_processing/advanced_statistic/autocorrelation/period_detect.md)
+
+# Software
+
+## CST MWS
+
+* [CST MWS Basic](signal_processing/software/simulation/CST/basic.md)
+
+

content/signal_processing/software/simulation/CST/basic.md

@@ -0,0 +1,9 @@
+---
+title: Basic about CST Microwave Studio (MWS)
+tags:
+  - software
+  - "#signal-processing"
+  - "#signal"
+date: 2024-04-15
+---
+CST WMS是一款专业的三维电磁仿真软件，专门用于提供**高频**问题的快速和精确的三维电磁仿真。