المتوسط المتحرك البسيط

المتوسط المتحرك البسيط

ترتيب المتوسط المتحرك

المجال: العدد( k=a=b) للمشاهدات الماضية المستخدمة لحساب الوسط

• $2k+1$

$X_{t}^{*}={\frac {1}{2k+1}}\sum _{i=t-k}^{t+k}X_{i}\qquad t=k+1,\ldots ,T-k$

مثال:

$\,k$	$\,1$	$\,2$
ترتيب	$2k+1=3$	$2k+1=5$
$\,x_{1}$	$\,x_{1}^{*}$ ---	$\,x_{1}^{*}$ ---
$\,x_{2}$	$\,x_{2}^{*}={\frac {1}{3}}\cdot \sum _{i=1}^{3}x_{i}$	$\,x_{2}^{*}$ ---
$\,x_{3}$	$\,x_{3}^{*}={\frac {1}{3}}\cdot \sum _{i=2}^{4}x_{i}$	$\,x_{3}^{*}={\frac {1}{5}}\cdot \sum _{i=1}^{5}x_{i}$
$\,x_{4}$	$\,x_{4}^{*}={\frac {1}{3}}\cdot \sum _{i=3}^{5}x_{i}$	$\,x_{4}^{*}={\frac {1}{5}}\cdot \sum _{i=2}^{6}x_{i}$
$\vdots$	$\vdots$	$\vdots$
$\,x_{T-2}$	$\,x_{T-2}^{*}={\frac {1}{3}}\cdot \sum _{i=T-3}^{T-1}x_{i}$	$\,x_{T-2}^{*}={\frac {1}{5}}\cdot \sum _{i=T-4}^{T}x_{i}$
$\,x_{T-1}$	$\,x_{T-1}^{*}={\frac {1}{3}}\cdot \sum _{i=T-2}^{T}x_{i}$	$\,x_{4}^{*}$ ---
$\,x_{T}$	$\,x_{T}^{*}$ ---	$\,x_{T}^{*}$ ---

• $2k$

$X_{t}^{*}={\frac {1}{2k}}\left[{\frac {1}{2}}X_{t-k}+{\frac {1}{2}}X_{t+k}+\sum _{i=t-(k-1)}^{t+(k-1)}X_{i}\right]\qquad t=k+1,\ldots ,T-k$

مثال:

$\,k$	$\,1$	$\,2$
ترتيب	$\,2k=2$	$\,2k=4$
$\,x_{1}$	$\,x_{1}^{*}$ ---	$\,x_{1}^{*}$ ---
$\,x_{2}$	$\,x_{2}^{*}={\frac {1}{2}}\left[{\frac {1}{2}}x_{1}+{\frac {1}{2}}x_{3}+x_{2}\right]$	$\,x_{2}^{*}$ ---
$\,x_{3}$	$\,x_{3}^{*}={\frac {1}{2}}\left[{\frac {1}{2}}x_{2}+{\frac {1}{2}}x_{4}+x_{3}\right]$	$\,x_{3}^{*}={\frac {1}{4}}\left[{\frac {1}{2}}x_{1}+{\frac {1}{2}}x_{5}+\sum _{i=2}^{4}x_{i}\right]$
$\,x_{4}$	$\,x_{4}^{*}={\frac {1}{2}}\left[{\frac {1}{2}}x_{3}+{\frac {1}{2}}x_{5}+x_{4}\right]$	$\,x_{3}^{*}={\frac {1}{4}}\left[{\frac {1}{2}}x_{2}+{\frac {1}{2}}x_{6}+\sum _{i=3}^{5}x_{i}\right]$
$\vdots$	$\vdots$	$\vdots$
$\,x_{T-2}$	$\,x_{T-2}^{*}={\frac {1}{2}}\left[{\frac {1}{2}}x_{T-3}+{\frac {1}{2}}x_{T-1}+x_{T-2}\right]$	$\,x_{T-2}^{*}={\frac {1}{4}}\left[{\frac {1}{2}}x_{T-4}+{\frac {1}{2}}x_{T}+\sum _{i=T-3}^{T-1}x_{i}\right]$
$\,x_{T-1}$	$\,x_{T-1}^{*}={\frac {1}{2}}\left[{\frac {1}{2}}x_{T-2}+{\frac {1}{2}}x_{T}+x_{T-1}\right]$	$\,x_{T-1}^{*}$ ---
$\,x_{T}$	$\,x_{T}^{*}$ ---	$\,x_{T}^{*}$ ---