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目录 方法介绍 数学公式 因素求解 案例 计算步骤 应用-预测 应用-异常检测 方法介绍 时间序列分解是时间序列分析的一种方法,思想是将数据分解为不同的因素,以达到解释数据、建立数学模型、数据预测的目的,时间序列分解的方法有很多,较常用的模型有加法模型和乘法模型,这里介绍乘法模型。乘法模型将时间序列值分解为长期趋势因素()、季节变动因素()、循环变动因素()、不规则变动因素(,也可理解为随机因素) 数学公式其中: :第t期的值 :第t期的长期趋势值 :第t期的季节变动因素 :第t期的循环变动因素 :第t期的不规则变动因素(随机因素),服从正态分布 因素求解1、移动平均提取趋势和循环变动: 当n为奇数时:,k是季节长度,例如每周7天,每年12个月等,下同 当n为偶数时: 2、季节性和随机性: 3、季节性:,将季节性和随机性的结果按照季节编号再次计算平均值 4、不规则变动 5、长期趋势:,基于MA数据,趋势外推法(线性回归等)拟合长期趋势 6、循环变动: 案例 以网上的数据为例进行讲解,完整的计算过程数据见下表: 表格说明:1、第二行中f()表示进行了函数运算,例如H=f(B,G)表示H列的值是B列和G列通过运算得到的 序号季节编号k观察值Xt移动平均值T×C长期趋势T循环变动C%S×I比率%S比率%调整后的S比率%I比率%拟合残差ABCD=f(C)E=f(A)F=D/E*100G=C/D*100H=f(B,G)I=f(H)J=G/I*100K=E*F*I*JL=C-K113017.6——2777.16——————————————223043.54——2818.03——————————————332094.352773.482858.9097.0175.5177.6177.5597.372094.350.00442809.842820.602899.7797.2799.62102.38102.3197.372809.840.00513274.82838.062940.6496.51115.39111.23111.15103.813274.800.00623163.282867.402981.5196.17110.32109.07109.00101.213163.280.00732114.312900.833022.3895.9872.8977.6177.5593.992114.310.00843024.572948.693063.2596.26102.57102.38102.31100.263024.570.00913327.483155.663104.11101.66105.44111.23111.1594.873327.480.001023493.483379.643144.98107.46103.37109.07109.0094.843493.480.001133439.933482.623185.85109.3298.7777.6177.55127.373439.930.001243490.793548.293226.72109.9798.38102.38102.3196.163490.790.001313685.083436.573267.59105.17107.23111.23111.1596.483685.080.001423661.233299.983308.4699.74110.95109.07109.00101.793661.230.001532378.433316.643349.3399.0271.7177.6177.5592.472378.430.001643459.553342.203390.2098.58103.51102.38102.31101.183459.550.001713849.633380.193431.0798.52113.89111.23111.15102.463849.630.001823701.183428.933471.9398.76107.94109.07109.0099.033701.180.001932642.383473.313512.8098.8876.0877.6177.5598.102642.380.002043585.523527.673553.6799.27101.64102.38102.3199.353585.520.002114078.663575.423594.5499.47114.08111.23111.15102.634078.660.002223907.063660.423635.41100.69106.74109.07109.0097.933907.060.002332818.463756.043676.28102.1775.0477.6177.5596.762818.460.002444089.53818.853717.15102.74107.09102.38102.31104.674089.500.002514339.613868.793758.02102.95112.17111.23111.15100.924339.610.002624148.63887.933798.89102.34106.70109.07109.0097.904148.600.002732976.453875.183839.76100.9276.8177.6177.5599.042976.450.002844084.643844.153880.6299.06106.26102.38102.31103.864084.640.002914242.423813.343921.4997.24111.25111.23111.15100.094242.420.003023997.583795.363962.3695.79105.33109.07109.0096.643997.580.003132881.013804.054003.2395.0275.7477.6177.5597.662881.010.003244036.233864.164044.1095.55104.45102.38102.31102.104036.230.003314360.333945.924084.9796.60110.50111.23111.1599.424360.330.003424360.534005.764125.8497.09108.86109.07109.0099.874360.530.003533172.184070.474166.7197.6977.9377.6177.55100.493172.180.003644223.764153.484207.5898.71101.69102.38102.3199.404223.760.003714690.484216.504248.4499.25111.24111.23111.15100.084690.480.003824694.484282.004289.3199.83109.63109.07109.00100.584694.480.003933342.354360.614330.18100.7076.6577.6177.5598.843342.350.004044577.634436.434371.05101.50103.18102.38102.31100.864577.630.004114965.464493.854411.92101.86110.49111.23111.1599.414965.460.004225026.054503.364452.79101.14111.61109.07109.00102.405026.050.004333470.144533.554493.66100.8976.5477.6177.5598.703470.140.004444525.944628.154534.53102.0697.79102.38102.3195.594525.940.004515258.714701.924575.40102.77111.84111.23111.15100.625258.710.004625489.584637.214616.27100.45118.38109.07109.00108.615489.580.004733596.76——4657.13——————————————4843881.6——4698.00—————————————— 计算步骤1、移动平均提取趋势和循环变动: 此处季节长度为4,偶数,移动平均计算方式为: D列第三行:2773.48 = ((3017.6+3043.54+2094.35+2809.84)/4+(3043.54+2094.35+2809.84+3274.8)/4)/2;D列其他行类推 2、季节性和随机性 G列第三行:75.51 = 2094.35 / 2773.48 * 100,此处可以乘以100也可以不乘 3、季节性: H列第三行:77.61 = (75.51+72.89+98.77+71.71+76.08+75.04+76.81+75.74+77.93+76.65+76.54)/11 I列第三行:77.55 = 77.61 / (111.23+109.07+77.61+102.38)*400,其他三个类似,四个季节性值的计算如下 行S×I季节1S×I季节2S×I季节3S×I季节4 ————75.5199.62 115.39110.3272.89102.57 105.44103.3798.7798.38 107.23110.9571.71103.51 113.89107.9476.08101.64 114.08106.7475.04107.09 112.17106.7076.81106.26 111.25105.3375.74104.45 110.50108.8677.93101.69 111.24109.6376.65103.18 110.49111.6176.5497.79 111.84118.38————平均值111.23109.0777.61102.38调整后111.15109.0077.55102.314、不规则变动 J列第三行:97.37 = 75.51 / 77.55 5、长期趋势:,此处长期趋势拟合采用线性回归;a、b为参数 D列的数据为Tt,A列的数据为t,进行线性回归拟合,得到拟合公式:Tt = 2736.29 + 40.87 * t 根据上面的公式,E列第一行2777.16 = 2736.29 + 40.87 * 1;第二行2818.03 = 2736.29 + 40.87 * 2以此类推 6、循环变动: F列97.01 = 2773.48 / 2858.90 7、至此,已经将原始数据分解长期趋势(T)、循环变动(C)、季节指数(S)、不规则变动(I),根据乘法模型公式,我们将四种因素相乘,将数据还原 K列第三行:2094.35 = 2858.90 * 97.01 * 77.55 * 97.37/100/100/100 将原始数据与分解后的成分进行可视化: 应用-预测 在求得分解公式后,可以进行预测了,以第49期为例,长期趋势(T)与季节指数(S)是可以通过计算得到;循环变动可以认为在短期内不会变化为100%或者通过其他方法进行预测,比如ARIMA等;不规则变动具有随机性,因此无法进行预测,可以认为值为100%。 T49 = 2736.29 + 40.87 * 49 = 4738.92; S1=111.15%;第49期的季节编号为1 则X49 = T49 * S1 * C * I = 4738.92 * 111.15% * 100% * 100% = 57267.31,其他期的预测如下图: 应用-异常检测此种方法也可以用于异常检测,分解后得到的不规则变动,也就是随机因素,应该是服从正太分布的,从图上也可以看出随机因素的值在100%上下波动,如果离100%相差太远,则可以认为对应的数据点有异常例如不规则变动(I)图的第9个点。
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