---
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title: 'math model :population'
tags: [' Talk', Templates]

---

---
title: math model :population
tags: Templates, Talk
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description: View the slide with "Slide Mode".
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# math model :population



---


##  人口金字塔
<style>
 image{
        margin-top: 0px;
        margin-bottom: 0px;
    }
    h3{
        margin-bottom: 0px;
    }
</style>



![](https://hackmd.io/_uploads/r1qfGR4w3.png =600x250)
### ⬇️
![](https://hackmd.io/_uploads/rkBHG04Ph.png =600x250)

---


## 模擬方法
將整個社會分成不同種人群,不同族群有不同的屬性,不同屬性會影響的生育能理和死亡率和狀態

---

## data format
<style>
    .ss{
        margin-left: auto;
        margin-right: auto;
        text-align: center
    }
    th,td{
    border-bottom: 1px solid !important;
    border-right: 1px solid!important;
    border-left: 1px solid!important;
    border-top: 1px solid!important;
    text-align: center
    }
</style>
[國家發展委員會](https://pop-proj.ndc.gov.tw/pyramid.aspx?uid=64&pid=60)

 <table class="ss">
<tr>
    <th>index</th>
    <th >age</th>
    <th >sex</th>
    <th >religion</th>
    <th >density</th>

<tr style="border=10px">
    <td>0</td>
    <td rowspan="4">5</td>
    <td rowspan="2">0</td>
    <td>0</td>
    <td>0.4</td>
  </tr>
  <tr>
    <td>1</td>
    <td>1</td>
    <td>0.3</td>
  </tr>

<tr>
    <td>2</td>
	<td rowspan="2">1</td>
	<td>0</td>
    <td>0.2</td>
</tr>
  <tr>
    <td>3</td>
    <td>1</td>
    <td>0.1</td>
  </tr>

</table>

---

## age

shift people every 5 year

 <table class="ss">
<tr>
    <th>index</th>
    <th >age</th>
    <th >sex</th>
    <th >religion</th>
    <th >density</th>
    <th>every 5 year</th>
    <th >density</th>
    

<tr style="border=10px">
    <td>0</td>
    <td rowspan="4">5</td>
    <td rowspan="2">0</td>
    <td>0</td>
    <td>0.4</td>
    <td rowspan="8">➡️</td> 
    <td >0</td>
    
  </tr>
  <tr>
    <td>1</td>
    <td>1</td>
    <td>0.3</td>
    <td>0</td>
      
  </tr>

<tr>
    <td>2</td>
	<td rowspan="2">1</td>
	<td>0</td>
    <td>0.2</td>
    <td>0</td>
</tr>
  <tr>
    <td>3</td>
    <td>1</td>
    <td>0.1</td>
    <td>0</td>
  </tr>
<tr style="border=10px">
    <td>4</td>
    <td rowspan="4">10</td>
    <td rowspan="2">0</td>
    <td>0</td>
    <td>0.1</td>
    <td>0.4</td>
  </tr>
  <tr>
    <td>5</td>
    <td>1</td>
    <td>0.1</td>
    <td>0.3</td>
  </tr>

<tr>
    <td>6</td>
	<td rowspan="2">1</td>
	<td>0</td>
    <td>0.1</td>
    <td>0.2</td>
</tr>
  <tr>
    <td>7</td>
    <td>1</td>
    <td>0.1</td>
    <td>0.1</td>
  </tr>
</table>

---

## born algothm data
$$
A=\begin{bmatrix}
0.4\\
0.3\\
0.2\\
0.1\\
\vdots\\
\end{bmatrix}
A^T=\begin{bmatrix}
0.4&
0.3&
0.2&
0.1&
\cdots\\
\end{bmatrix}
$$

---

## born algothm matrix
在兩人相遇為獨立式事件的前提下
$$A\cdot A^T=
\begin{bmatrix}
0.4\\
0.3\\
0.2\\
0.1\\
\vdots\\
\end{bmatrix}\cdot
{\begin{bmatrix}
0.4&
0.3&
0.2&
0.1&
\cdots\\
\end{bmatrix}}\\=\begin{bmatrix}
0.4*0.4 & 0.4*0.3 & 0.4*0.2 & 0.4*0.1 &\cdots\\
0.3*0.4 & 0.3*0.3 & 0.3*0.2 & 0.3*0.1 &\cdots\\
0.2*0.4 & 0.2*0.3 & 0.2*0.2 & 0.2*0.1 &\cdots\\\
0.1*0.4 & 0.1*0.3 & 0.1*0.2 & 0.1*0.1 &\cdots\\\
\vdots & \vdots&  \vdots& \vdots&  	\ddots\\
\end{bmatrix}$$

---

## born mask

男男 或 女女 無法生育
$$
\begin{bmatrix}
0.4*0.4 & 0.4*0.3 & 0.4*0.2 & 0.4*0.1 &\cdots\\
0.3*0.4 & 0.3*0.3 & 0.3*0.2 & 0.3*0.1 &\cdots\\
0.2*0.4 & 0.2*0.3 & 0.2*0.2 & 0.2*0.1 &\cdots\\\
0.1*0.4 & 0.1*0.3 & 0.1*0.2 & 0.1*0.1 &\cdots\\\
\vdots & \vdots&  \vdots& \vdots&  	\ddots\\
\end{bmatrix}\times \begin{bmatrix}
0& 0& 1& 1&\cdots\\
0& 0& 1& 1&\cdots\\
1& 1& 0& 0&\cdots\\
1& 1& 0& 0&\cdots\\
\vdots& \vdots&  \vdots&  \vdots& 	\ddots\\
\end{bmatrix}\\=\begin{bmatrix}
0 & 0 & 0.4*0.2 & 0.4*0.1 &\cdots\\
0 & 0 & 0.3*0.2 & 0.3*0.1 &\cdots\\
0.2*0.4 & 0.2*0.3 & 0 & 0 &\cdots\\\
0.1*0.4 & 0.1*0.3 & 0 & 0 &\cdots\\\
\vdots & \vdots&  \vdots& \vdots&  	\ddots\\
\end{bmatrix}$$

---

### 特徵影響生育
同宗教生育率較高
$$\begin{bmatrix}
0 & 0 & 0.08 & 0.04 &\cdots\\
0 & 0 & 0.06 & 0.03 &\cdots\\
0.08 & 0.06 & 0 & 0 &\cdots\\\
0.04 & 0.03 & 0 & 0 &\cdots\\\
\vdots & \vdots&  \vdots& \vdots&  	\ddots\\
\end{bmatrix} \times \begin{bmatrix}
0.9& 0.5& 0.9& 0.5&\cdots\\
0.5& 0.9& 0.5& 0.9&\cdots\\
0.9& 0.5& 0.9& 0.5&\cdots\\
0.5& 0.9& 0.5& 0.9&\cdots\\
\vdots& \vdots&  \vdots&  \vdots& 	\ddots\\
\end{bmatrix}$$

---

### 特徵混合
```graphviz
digraph google{
    // layout=neato;
    // layout=circo;
	// layout=twopi;
    fontname="Helvetica,Arial,sans-serif"
    fontcolor=white
    bgcolor="black"
    //bgcolor="transparent"
    node[shape=re,style=filled,color=white,colorscheme=ylgnbu7]
    edge[color=white,fontcolor=white]
	"0.01"->"男 0.005"[label="0.5"]
	"0.01"->"女 0.005"[label="0.5"]
    "男 0.005"->"男 父親宗教 0.0025"[label="0.5"]
    "男 0.005"->"男 母親宗教 0.0025"[label="0.5"]
    "女 0.005"->"女 父親宗教 0.0025"[label="0.5"]
    "女 0.005"->"女 母親宗教 0.0025"[label="0.5"]
}
```

---

## dead?


$$
\begin{bmatrix}
0.4\\
0.3\\
0.2\\
0.1\\
\vdots\\
\end{bmatrix}\times\begin{bmatrix}
1-0.05\\
1-0.05\\
1-0.02\\
1-0.02\\
\vdots\\
\end{bmatrix}=
\begin{bmatrix}
0.380\\
0.285\\
0.196\\
0.980 \\
\vdots\\
\end{bmatrix}
$$

---

## overview?
$t$: 當前時間
$x_t$: 在t時的所有組群向量
$D_t$: 在t時的組群死亡權重向量
$W_t$: 在t時的組群生育權重向量
### $$x_{t+1} = (1-D_t)  x_t+W_t(x_t\cdot x_t^T)$$

優點:可以擴展族群
缺點:非線性

---

## 結果比較

* 基督教：生育率降低
* 伊斯蘭教：生育率提高
* 佛教：死亡率降低

| ![](https://cdn.discordapp.com/attachments/890087003847282709/1119654640426942464/image.png) | ![](https://cdn.discordapp.com/attachments/890087003847282709/1119654701038833776/image.png) | ![](https://hackmd.io/_uploads/r1eSJFaDn.png) |
| -------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------- |
| ![](https://hackmd.io/_uploads/SkAZu8iP3.png)| ![](https://hackmd.io/_uploads/B1l0P8sDn.png)  | ![](https://hackmd.io/_uploads/Sy1X_UiPn.png)|


---

## 結果比較

![](https://hackmd.io/_uploads/SksuLLsDn.png)
