Moving_Average_수식_유도

Moving Average의 개념

Moving Average의 개념은 지난 post에서 언급한 글(https://purumir.github.io/2019/10/06/지수평활법-exponential-smoothing/ )의 내용을 참조하시기 바랍니다.

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Moving Average(1) 수식유도

moving average(1)의 수식은 아래와 같이 정의될 수 있습니다.

$${X}_{t} = {a}_{t} - {\theta}{a}_{t-1}\\ a_{t} \simeq WN(0, \delta_{a}^2)일때$$

White Noise성 데이터는 다음과 같은 성질을 가집니다.

$$E({a}_{t}) = 0\\V({a}_{t})={\sigma{a}^2}\\ Cov({a}_{t},{a}_{t}) = {\sigma{a}^2} / ~~t{\neq}s이면~Cov({a}_{t}, {a}_{s})=0~\\ {\rho}_{x}(h)=\frac{Cov({X}_{t},{X}_{t+h})}{\sqrt{V(x_{t})}\sqrt{V(x_{t+h})}}관계에~의해\\ Corr({a}_{t}, {a}_{s})=0, t{\neq}s\\ Cov(aX, Y) = aCov(X,Y)\\\\ \\$$

Moving Average(1)의 수식 전개는 아래와 같습니다.

1) 기대값(Expectation) 전개

$$E({X}_{t}) = E[{a}_{t}-{\theta}{a}_{t-1}] = E({a}_{t})-{\theta}E({a}_{t-1}) = 0 - 0 =0$$

2) 분산(Variance) 전개

자기 자신에 대한 covariance는 분산(Variance)와 동일합니다.

$$V({X}_{t}) = V({a}_{t} -{\theta}{a}_{t-1}) = V({a}_{t}) + {\theta}^2V({a}_{t-1})~\\ ~//Variance에서는~{\theta가~나가면서~제곱이~됩니다.}\\ = {\sigma{a}^2} + {\theta}^2{\sigma{a}^2} = (1+{\theta}^2){\sigma{a}^2}$$

3) 공분산(Covariance) 전개

$$Cov({x}_{t}, {x}_{t-h})=Cov({a}_{t}-{\theta}{a}_{t-1}, {a}_{t-h}-{\theta{a}_{t-h-1}})\\= Cov({a}_{t}, {a}_{t-h}) - {\theta}Cov({a}_{t-1}, {a}_{t-h})\\-{\theta}Cov({a}_{t}, {a}_{t-h-1} )+{\theta}^2Cov({a}_{t-1}, {a}_{t-h-1})$$

4)

$${\gamma}_{x}(h)~에~대한~전개\\ h=0~일경우~~~\\ {\gamma}_{x}(0)=Cov({x}_{t}, {x}_{t}) = Cov({a}_{t}, {a}_{t}) - {\theta}Cov({a}_{t-1}, {a}_{t}) \\- {\theta}Cov({a}_{t}, {a}_{t-1})+{\theta}^2Cov({a}_{t-1}, {a}_{t-1})\\={\sigma{a}^2}+ {\theta}^2{\sigma{a}^2} = (1+{\theta}^2){\sigma{a}^2}\\$$ $$h=1~일경우~~~\\{\gamma}_{x}(1)=Cov({x}_{t}, {x}_{t-1}) = Cov({a}_{t}, {a}_{t-1}) -{\theta}Cov({a}_{t-1}, {a}_{t-1})\\-{\theta}Cov({a}_{t}, {a}_{t-2})+{\theta}^2Cov({a}_{t-1}, {a}_{t-2})=-{\theta}{\sigma{a}^2}$$ $$h=2~일~경우\\ {\gamma}_{x}(2)=Cov({x}_{t},{x}_{t-2})=Cov({a}_{t},{a}_{t-2})-{\theta}Cov({a}_{t-1}, {a}_{t-2})\\-{\theta}Cov({a}_{t},{a}_{t-2})+{\theta}^2Cov({a}_{t-1},{a}_{t-2}) = 0$$

5) 자기상관계수(Auto Correlation Function) 전개

자기공분산(auto covariance function)을 통해서 자기상관계수(auto correlation function)을 구하게 됩니다.

$${\rho}_{x}(h)= \frac{\gamma_{x}(h)}{\sqrt{\gamma_{x}(0)}\sqrt{\gamma_{x}(h)}} = \frac{\gamma_{x}(h)}{\sqrt{\gamma_{x}(0)}\sqrt{\gamma_{x}(0)}} =\frac{\gamma_{k}}{\gamma_{0}}=\frac{Cov({X}_{t},{X}_{t+h})}{\sqrt{V(x_{t})}\sqrt{V(x_{t+h})}}$$

위 수식에 따라서

$${\rho_{x}(h)}에~대해서\\ h=0일때~\frac{\gamma_{x}(0)}{\gamma_{x}(0)}=1\\ h=1일때~\frac{\gamma_{x}(1)}{\gamma_{x}(0)}=\frac{-{\theta}{\sigma{a}^2}}{(1+{\theta}^2){\sigma{a}^2}}=\frac{-\theta}{(1+\theta^2)}\\ h=2일때~\frac{\gamma_{x}(2)}{\gamma_{x}(0)}=\frac{0}{(1+{\theta}^2){\sigma{a}^2}}=0$$

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Moving Average(2) 수식유도

moving average(2)의 수식은 아래와 같이 정의될 수 있습니다.

$${X}_{t}={a}_{t}-{\theta}_{1}{a}_{t-1}-{\theta}_{2}{a}_{t-2}$$

Moving Average(2)의 수식 전개는 아래와 같습니다.

1) 기대값(Expectation) 전개

$$E(X_{t})=E({a}_{t}-{\theta}_{1}{a}_{t-1}-{\theta}_{2}{a}_{t-2})=E({a}_{t})-{\theta_{1}}E({a}_{t-1})-{\theta_{2}E(a_{t-2})}=0-0-0=0$$

2) 분산(Variance) 전개

$$V(X_{t})=V({a}_{t}-{\theta}_{1}{a}_{t-1}-{\theta}_{2}{a}_{t-2})\\ =V(a_{t})+\theta_{1}^2V(a_{t-1})+\theta_{2}^2V(a_{t-2})\\ ={\sigma{a}^2} + \theta_{1}^2{\sigma{a}^2}+\theta_{2}^2{\sigma{a}^2}\\ =(1+\theta_{1}^2+\theta_{2}^2){\sigma{a}^2}$$

3) 공분산(Covariance) 전개

$$Cov({x}_{t}, {x}_{t-h}) = Cov({a}_{t}-{\theta}_{1}{a}_{t-1}-{\theta}_{2}{a}_{t-2},~{a}_{t-h}-{\theta}_{1}{a}_{t-h-1}-{\theta}_{2}{a}_{t-h-2})\\ =Cov(a_{t}, a_{t-h})-\theta_{1}Cov(a_{t}, a_{t-h-1})-\theta_{2}Cov(a_{t}, a_{t-h-2})\\ -\theta_{1}Cov(a_{t-1}, a_{t-h})+\theta_{1}^2Cov(a_{t-1}, a_{t-h-1})+\theta_{1}\theta_{2}Cov(a_{t-1}, a_{t-h-2})\\ -\theta_{2}Cov(a_{t-2}, a_{t-h})+\theta_{1}\theta_{2}Cov(a_{t-2},a_{t-h-1})+\theta_{2}^2Cov(a_{t-2}, a_{t-h-2})$$

4) 자기공분산(Auto Covariance Function) 전개

$${\gamma}_{x}(h)~에~대한~전개\\ \gamma_{x}(0) = Cov(x_{t}, x_{t}) = Cov(a_{t}, a_{t})-\theta_{1}Cov(a_{t}, a_{t-1})-\theta_{2}Cov(a_{t}, a_{t-2})\\ -\theta_{1}Cov(a_{t-1}, a_{t})+\theta_{1}^2Cov(a_{t-1}, a_{t-1})+\theta_{1}\theta_{2}Cov(a_{t-1}, a_{t-2})\\ -\theta_{2}Cov(a_{t-2}, a_{t})+\theta_{1}\theta_{2}Cov(a_{t-2}, a_{t-1})+\theta_{2}^2Cov(a_{t-2}, a_{t-2})\\ ={\sigma{a}^2}+\theta_{1}^2{\sigma{a}^2}+\theta_{2}^2{\sigma{a}^2}\\ =(1+\theta_{1}^2+\theta_{2}^2){\sigma{a}^2}$$ $$\gamma_{x}(1) = Cov(x_{t}, x_{t-1})=Cov(a_{t}, a_{t-1})-\theta_{1}Cov(a_{t}, a_{t-2})-\theta_{2}Cov(a_{t}, a_{t-3})\\ -\theta_{1}Cov(a_{t-1}, a_{t-1}) + \theta_{1}^2Cov(a_{t-1}, a_{t-2})+\theta_{1}\theta_{2}Cov(a_{t-1}, a_{t-3})\\ -\theta_{2}Cov(a_{t-2}, a_{t-1})+\theta_{1}\theta_{2}Cov(a_{t-2}, a_{t-2})+\theta_{2}^2Cov(a_{t-2}, a_{t-3})\\ = -\theta_{1}{\sigma{a}^2}+\theta_{1}\theta_{2}{\sigma{a}^2}=(-\theta_{1}+\theta_{1}\theta_{2}){\sigma{a}^2}$$ $$r_{x}(2) = Cov(x_{t}, x_{t-2}) = -\theta_{2}Cov(a_{t-2}, a_{t-2}) = -\theta_{2}{\sigma{a}^2}\\ r_{x}(3) = 0(~t-3이~없으므로~모두~0이~된다.)$$

5) 자기상관계수(Auto Correlation Function) 전개

자기공분산(auto covariance function)을 통해서 자기상관계수(auto correlation function)을 구하게 됩니다.

$${\rho}_{x}(h)= \frac{\gamma_{x}(h)}{\sqrt{\gamma_{x}(0)}\sqrt{\gamma_{x}(h)}} = \frac{\gamma_{x}(h)}{\sqrt{\gamma_{x}(0)}\sqrt{\gamma_{x}(0)}} =\frac{\gamma_{k}}{\gamma_{0}}=\frac{Cov({X}_{t},{X}_{t+h})}{\sqrt{V(x_{t})}\sqrt{V(x_{t+h})}}$$

위 수식에 따라서

$${\rho_{x}(h)}에~대해서\\ h=0일때~\frac{\gamma_{x}(0)}{\gamma_{x}(0)}=1\\ h=1일때~\frac{\gamma_{x}(1)}{\gamma_{x}(0)}=\frac{(-\theta_{1}+\theta_{1}\theta_{2}){\sigma{a}^2}}{(1+\theta_{1}^2+\theta_{2}^2){\sigma{a}^2}}=\frac{(-\theta_{1}+\theta_{1}\theta_{2})}{(1+\theta_{1}^2+\theta_{2}^2)}\\ h=2일때~\frac{\gamma_{x}(2)}{\gamma_{x}(0)}=\frac{ -\theta_{2}{\sigma{a}^2}}{(1+\theta_{1}^2+\theta_{2}^2){\sigma{a}^2}}=\frac{-\theta_{2}}{1+\theta_{1}^2+\theta_{2}^2}$$