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Modeling Change

Proportionality:

If \(y = kx\) , then we remember \(y\propto x\).我們稱 \(x\) 和 \(y\) 成比例關係. \(k\) 為非零常數. 如果 \(x\) 和 \(y\) 成比例關係那麼它們的影象將呈一條直線狀.

Modeling Change:

A powerful paradigm (正規化) to use in modeling change is

\[\mathrm{future\ value} = \mathrm{present\ value} + \mathrm{change}. \]

Difference equations:

對一數列 \(A = \{a_{1}, \dots , a_{n}\}\), 它的 \(n\) 階差分方程定義為

\[\Delta a_{n} = a_{n+1} - a_{n}. \]

Sequence:

A sequence is a function whose domain is the set of all non-negative integers and whose range is a subset of the real numbers.

Dynamical system model:

A dynamical system is a relationship among terms in a sequence

.

\[a_{n+1} = ra_{n},\quad n =0, 1, 2, \ldots,\quad a_{0} = c. \]

Numerical Solutions:

A numerical solution is a table of values satisfying the dynamical system.

The Method of Conjecture (猜測法)

The method of conjecture is a powerful mathematical technique to hypothesize (假設) the form of a solution to a dynamical system and then to accept or reject the hypothesis. It is based on exploration (探索) and observation (觀察) from which we attempt to discern a pattern to the solution.

The solution of the linear dynamical system

\(a_{n+1} = ra_{n}\) for \(r\) any nonzero constant is \(a_{k} = r^{k} a_{0},\) where \(a_{0}\) is a given initial value.

Equilibrium value (平衡值) or Fixed point (不動點)

對形為 \(a_{n+1} = ra_{n}+b\) 的動力系統, 如果對所有的 $k = 0, 1, 2, \ldots $ 有 \(a_{k} = a\), 則將 \(a\) 稱為該動力系統 \(a_{n+1} = f(a_{n})\) 的平衡值或不動點. 即, \(a_{k} = a\) 是該動力系統的常數解

The equilibrium value for the dynamical system

動力系統 $$a_{n+1} = ra_{n}+b,\ r \neq1 $$ 的平衡點為 $$a = \frac{b}{1-r}.$$ 若 \(r=1\) 且 \(b=0\), 那麼每個數都是平衡點. 若 \(r = 1\) 且 \(b\neq0\), 那麼不存在平衡點.

The solution of the dynamical system

\[a_{n+1} = ra_{n} + b,\ r \neq 1 \]

is $$ a_{k} = r^{k}c +\frac{b}{1-r}$$ for some constant c (which depends on the initial condition).

The Modeling Process, Proportionality, and Geometric Similarity

System :

A system is an assemblage (集合體) of objects joined in some regular interaction or interdependence.

Mathematical model

is a mathematical construct designed to study a particular real-world system or phenomenon. We include graphical, symbolic, simulation, and experimental constructs.

Properties of a model :

Fidelity : The preciseness of a model’s representation of reality.
Costs : The total cost of the modeling process.
Flexibility : The ability to change and control conditions affecting the model as required data are gathered.

Construction of Models

Step 1 : Identify the problem. What is the problem you would like to explore?

Step 2 : Make assumptions.

1. Classify the variables.
2. Determine interrelationships among the variables selected for study.

Step 3 : Solve the model.

Step 4 : Verify the model.

1. Does it address the problem?
2. Does it make common sense?
3. Test it with real-world data.

Step 5 : Implement the model.

Step 6 : Maintain the model.

Describing models.

1. A model is said to be robust (穩定性) when its conclusions do not depend on the precise satisfaction of the assumptions. (positive term)
2. A model is fragile (脆弱的) if its conclusions do depend on the precise satisfaction of some sort of conditions.
3. The term sensitivity (敏感性) refers to the degree of change in a model’s conclusions as some condition on which they depend is varied.

Geometric Similarity

Two objects are said to be geometrically similar (幾何相似性) if there is a one-to-one correspondence (一一對應關係**) between points of the objects such that the ratio of distances between corresponding points is constant for all possible pairs of points.

Geometrically similar is a simplification in certain computations, all the function arguments can be expressed as a proportionality in terms of some selected characteristic dimension.

Model Fitting

在分析一個數據集合 (a collection of data points) 時, 有三個可能需要解決的任務 :

1. 按照一個或一些選出的模型型別對資料進行擬合.
2. 從一些已經擬合的型別中選取最合適 (the most appropriate) 的模型.
3. 根據收集的資料做出預報 (Making predictions).

Sources of Error in the Modeling Process (建模過程中的誤差來源)

1. Formulation error (公式化的誤差/模型誤差) result from the assumption that certain variables are negligible or from simplifications in describing interrelationships among the variables in the various sub-models.
2. Truncation error (截斷誤差) are attributable to the numerical method used to solve a mathematical problem.
3. Round-off error (舍入誤差) are caused by using a finite digit machine for computation.
4. Measurement error (測量誤差) are caused by imprecision in the data collection.

Chebyshev approximation criterion

給定某種函式型別 \(y = f(x)\) 和 \(m\) 個數據點 \((x_{i},y_{i})\) 的一個集合, 對整個集合極小化最大絕對偏差 \(|y_{i} - y(x_{i})|.\) 即確定函式型別 \(y = f(x)\) 的引數從而極小化數量$$\max {|y_{i} - f(x_{i})|},\quad i= 1, 2, \ldots , m.$$ 這一重要的準則稱為 Chebyshev 近似準則.

Minimizing the Sum of the Absolute Deviations (極小化絕對偏差之和)

給定某種函式型別 \(y = f(x)\) 和 \(m\) 個數據點 \((x_{i},y_{i})\) 的一個集合, 極小化絕對偏差 (minimize the sum of the absolute deviations) $$|y_{i} - y(x_{i})|$$ 的和, 也就是確定函式型別 \(y = f(x)\) 的引數, 極小化 $$\sum_{i=1}^{m}|y_{i} - y(x_{i})|.$$

Least-Squares Criterion (最小二乘準則)

給定某種函式型別 \(y = f(x)\) 和 \(m\) 個數據點 \((x_{i},y_{i})\) 的一個集合, 確定函式型別 \(y = f(x)\) 的引數, 極小化和數 $$\sum_{i=1}^{m}|y_{i} - y(x_{i})|^2.$$

Relating the Criteria

• 極小化絕對偏差和賦予每一資料點相等的均值 (with equal weight) 來平均這些偏差.
• Chebyshev 準則對潛在有極大偏差的單個點給予更大的權值 (weight).
• 最小二乘準則根據與中間某處的遠近來加權, 其權與單個點具有的顯著偏差有關.

Normal Equations (正規方程/法方程)

\[\begin{cases} a \sum_{i=1}^{m} x_{i}^2 + b \sum_{i=1}^{m} x_{i} = \sum_{i=1}^{m} x_{i}y_{i} \\ a \sum_{i=1}^{m} x_{i} +mb = \sum_{i=1}^{m} y_{i} \\ \end{cases} \]

應用消去法解得
The slope (斜率) $$a = \frac{m\sum_{i=1}^{m} x_{i}y_{i} - \sum_{i=1}^{m} x_{i} \sum_{i=1}^{m} y_{i}}{m\sum_{i=1}^{m} x_{i}^2 - (\sum_{i=1}^{m} x_{i})^2}$$
The intercept (截距) $$b = \frac{\sum_{i=1}^{m} x_{i}^2 \sum_{i=1}^{m} y_{i} - \sum_{i=1}^{m} x_{i}y_{i} \sum_{i=1}^{m} x_{i}}{m\sum_{i=1}^{m} x_{i}^2 - (\sum_{i=1}^{m} x_{i})^2}$$