Matrices Modeling Contexts · 矩阵建模情境
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| state vector/steɪt ˈvektə/ | 状态向量 | zhuàng tài xiàng liàng |
| state/steɪt/ | 状态 | zhuàng tài |
| transition matrix/trænˈsɪʃn ˈmeɪtrɪks/ | 转移矩阵 | zhuǎn yí jǔ zhèn |
Predicting the future with a matrix
- Every year, some people move from the city to the suburbs, and some move back.
- Given this year's populations, can you predict next year's?
- A matrix can store the movement rates and do the prediction for you.
- This is how matrices model populations, markets, weather, and more.
用矩阵预测未来
- 每一年,有些人从城市搬到郊区,也有些人搬回来。
- 给定今年的人口,你能预测明年的吗?
- 一个矩阵可以储存流动率,替你做这个预测。
- 这就是矩阵为人口、市场、天气等等建模的方式。
State vectors: the current picture
- Collect the current amounts into a state vector 状态向量 — one entry per category.
- For a city and its suburbs: $\begin{bmatrix}\text{city}\\ \text{suburb}\end{bmatrix}$, say $\langle 600, 400\rangle$ (thousands).
- Each state 状态 is one thing being tracked; the vector is a snapshot in time.
- Next year's snapshot will be a new state vector.
状态向量:当前的画面
- 把当前的数量收进一个状态向量(state vector)——每个类别一个元素。
- 对一座城市和它的郊区:$\begin{bmatrix}\text{city}\\ \text{suburb}\end{bmatrix}$,比如 $\langle 600, 400\rangle$(千人)。
- 每个状态(state)是被追踪的一样东西;这个向量是某个时刻的快照。
- 明年的快照将是一个新的状态向量。
A state vector in this kind of model holds… · 此类模型中的状态向量包含…
A state vector lists how much is in each category right now — one entry per state. · 状态向量列出当前每个类别的数量——每种状态对应一个条目。
The transition matrix: the rules
- A transition matrix 转移矩阵 stores the fraction moving between every pair of states.
- Each number is a rate: e.g. $5\%$ of city dwellers move to the suburbs each year.
- The diagram below shows two states with the movement rates on the arrows.
- Those same rates become the entries of the matrix.
转移矩阵:规则
- 转移矩阵(transition matrix)储存每一对状态之间流动的比例。
- 每个数是一个率:例如每年 $5\%$ 的城市居民搬到郊区。
- 下面的图显示两个状态,箭头上标着流动率。
- 那些相同的率就成为矩阵的元素。
The matrix that stores the rates of movement between states is called the ____ matrix. · 存储状态间移动速率的矩阵称为____矩阵。
The transition matrix holds the fraction moving from each state to each other state. · 转移矩阵存储了从每个状态转移到其他每个状态的比例。
Multiplying the transition matrix by the current state vector predicts the next · 次 state. · 将转移矩阵乘以当前状态向量可预测下一个状态。
One matrix-times-vector step advances the model by one time period. · 一次矩阵乘向量的步骤使模型推进一个时间周期。
Stepping forward in time
- Multiply the transition matrix by the current state vector to get next year's.
- One matrix-times-vector = one year forward.
- Want two years? Multiply again. Ten years? Multiply ten times.
- The model marches forward, one step per multiplication.
在时间上向前一步
- 把转移矩阵乘以当前的状态向量,得到明年的。
- 一次"矩阵乘向量" = 向前一年。
- 想要两年?再乘一次。十年?乘十次。
- 模型向前行进,每次相乘一步。
A good use of a matrix? · 矩阵的一个良好用途是什么？
Matrices model transformations, systems and transitions. Sort each task. · 矩阵用于模拟变换、系统和转换。对每项任务进行分类。
To predict the state two periods ahead, you… · 要预测两个周期后的状态，你应…
Each multiplication advances one period, so two periods need two multiplications (or multiply by the matrix squared). · 每次乘法推进一个周期，因此两个周期需要两次乘法（或乘以矩阵的平方）。
Select all · 所有 true statements about matrix models. · 选择关于矩阵模型的所有正确陈述。
The same transition matrix is reused each period — you do not redraw it. The other three are correct. · 每个周期都重复使用相同的转移矩阵——无需重新绘制。其余三项是正确的。
The long run
- Repeated steps often settle toward a steady state that stops changing.
- At the steady state, as many people leave the city as arrive — populations hold constant.
- Matrices let you find this long-run balance without simulating every single year.
- The same idea powers web-page ranking, genetics, and epidemic models.
长期行为
- 反复的步进常常趋向一个不再变化的稳态(steady state)。
- 在稳态,离开城市的人和到达的人一样多——人口保持不变。
- 矩阵让你不必模拟每一年,就能找到这个长期平衡。
- 同样的想法驱动着网页排名、遗传学和流行病模型。
Set the model up consistently: decide once whether the state vector is a column multiplied on the right of the matrix, and make each column's (or row's) rates add to $1$ so no one is lost or invented. A mismatched setup silently gives nonsense predictions.
要一致地建立模型:一次性决定状态向量是乘在矩阵右边的列,并让每一列(或每一行)的率加起来等于 $1$,这样没人被丢失或凭空产生。设置不匹配会悄无声息地给出无意义的预测。
City $= 600$, suburb $= 400$ (thousands). Each year $5\%$ of city → suburb, $2\%$ of suburb → city.
- Leaving the city: $0.05 \times 600 = 30$; arriving: $0.02 \times 400 = 8$.
- New city $= 600 - 30 + 8 = 578$; new suburb $= 400 + 30 - 8 = 422$.
- The matrix does exactly this arithmetic in a single multiplication.
城市 $= 600$,郊区 $= 400$(千人)。每年 $5\%$ 的城市 → 郊区,$2\%$ 的郊区 → 城市。
- 离开城市的:$0.05 \times 600 = 30$;到达的:$0.02 \times 400 = 8$。
- 新城市 $= 600 - 30 + 8 = 578$;新郊区 $= 400 + 30 - 8 = 422$。
- 矩阵在一次相乘里正好做了这个算术。
A state vector holds the current amount in each state, and a transition matrix stores the rates of movement between them. Multiplying the matrix by the state vector steps the model one period forward; repeating it predicts the long-run behaviour. This is how matrices model real-world change.
状态向量装着每个状态当前的数量,而转移矩阵储存它们之间的流动率。把矩阵乘以状态向量让模型向前一个周期;反复做就预测长期行为。这就是矩阵为真实世界的变化建模的方式。