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Forecast, comparsion, initial values, between zero and one, 最小二乘法：选择 a 和…

- - - - MAPE（平均绝对百分比误差）：消除量纲影响，适合跨数据集比较
    - - MAD（平均绝对偏差）：衡量平均绝对误差
    - - MSE（均方误差）：衡量误差的平方均值，对异常值敏感。
  - - - big number
        
        biased
      - close to zero
        
        unbiased
  - - - random
        
        unbiased
      - make a trend
        
        biased
- - - - smoothing constact α
        
        declining set of weights to all past data.
        
        A small value is desirable
        
        Catch the rapidly changing pattern in the data
        
        A larger value is desirable
        
        The series is stable and want to smooth out random variations.
        
        Iterative procedure to decide α
        
        Pick α minimizing MSE for test data set
      - initial value
        
        set it equal to the first observation
        
        数据波动小，首期有代表性
        
        use it equla to the average of the first 5 or 6 observations
        
        存在短期波动，需要平滑初始值，减少异常值对初始预测的影响
    - - the mean of all relevant historical observations
  - - - 预测值y（y hat）：预测的因变量值
        
        a：截距（Intercept），表示当 X=0 时的预测值
        
        调整模型在纵轴上的位置，使预测线与实际数据匹配。
        
        b：斜率（Slope），表示 X 每增加1单位，Y 的预期变化量
        
        反映趋势的强度与方向（正/负相关）
    - - The Current Level estimate (alpha )
        
        the estimate of the average value
        
        one estimate of the level comes from your most recent observations
        
        another estimate comes from previous value of level plus previous value of trend
      - The Trend Estimate ( beta)
        
        y intercept / the slope
        
        zero
        
        no trend
        
        change in mean value
        
        previous trend
      - Forecast T periods into the future
        
        T-step-ahead forecast made in period t
  - - - seasonal factors
        
        Ct (multiplier)
        
        how the demand above / below the overall average
        
        如何为无趋势的季节性时间序列计算季节性因子
        
        step1: Compute the sample mean of all the data.
        
        计算所有数据的平均值
        
        step2: Divide each observation by the sample mean
        
        逐个计算，观测值/均值=因子
        
        step3 ：Average the factors for like periods within each season
        
        不同季节的同一期取均值，例如：将每年的春天的因子加总求均值，得到春天的季节性因子
        
        step4：The resulting averages are the N
        seasonal factors
        
        . They will always add to exactly N.
        
        归一化，加总所有的季节性因子，等于季节性长度
        
        如何对同时含有季节性变化和趋势的时间序列进行去季节化处理
        
        step1: form the deseasonalized series
        by removing the seasonal variation from the original series
        
        利用公式，以及季节性因子，去季节化
        
        step2: Suppose that we fit a simple linear regression to this data where time is the independent variable.
        
        利用线性回归，预测数据
        
        step3 : To forecast, one first applies the regression to forecast the deseasonalized series, and then re-seasonalizes
        by multiplying by the appropriate factor
        
        将预测的数据，乘以季节性因子，重新季节化
      - Triple Exponential Smoothing
        
        Winters's method for seasonal problems
        
        The series
        
        the current level of the seasonalized series
        
        update
        
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        initialize
        
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        The Trend
        
        update
        
        G t：当前时刻t的趋势估计，表示需求随时间变化的斜率（如每月增长10单位）。
        
        S t −S t−1：当前水平与上一期水平的差值，反映短期趋势变化。
        
        β（Beta）：趋势平滑参数（0 ≤ β ≤ 1），控制趋势更新的速度。
        
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        initialize
        
        The seasonal factors
        
        Ct：当前时刻t的季节性因子，反映该周期在季节内的需求波动（如12月因子为1.5，表示需求比平均水平高50%）
        
        Dt/St：实际观测值与去季节化水平的比值，表示当前季节的强度。
        
        γ（Gamma）：季节性平滑参数（0 ≤ γ ≤ 1），控制季节性因子更新的速度。
        
        γ ≈ 1：优先使用最新季节数据（如促销活动后）
        
        γ ≈ 0：保留历史季节模式（如稳定的节假日效应）
        
        initialize
        
        the forecast made in period t for any future period t+T
        
        Initialization Procedure
        
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      - the length of the season
        
        N
- - - - 具有相似的准确性