تحليل آمارشكل تپه هاي ماسه اي اردستان در حضور خطاي اندازه گيري

Statistical Shape Analysis of The Sand Hills in Ardestan in Presence of Measurement Error

داده ﻫﺎي ﺷﮑﻞ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻌﺪد ﻣﻨﺎﺑﻊ ﺧﻄﺎﻫﺎ اﻏﻠﺐ در ﻣﻌﺮض اﺑﺘﻼ ﺑﻪ ﺧﻄﺎي اﻧﺪازه ﮔﯿﺮي ﻗﺮار دارﻧﺪ. ﻧﺎدﯾﺪه ﮔﺮﻓﺘﻦ ﭼﻨﯿﻦ ﺧﻄﺎﯾﯽ در ﺻﻮرت وﺟﻮد، ﺑﺎﻋﺚ ﺑﺮوز ﻣﺸﮑﻼت زﯾﺎدي از ﻗﺒﯿﻞ ارﯾﺒﯽ ﺑﺮآوردﮔﺮﻫﺎ ﻣﯽ ﺷﻮد. در اﯾﻦ ﺣﺎﻟﺖ ﺑﺮآوردﮔﺮﻫﺎي ﺣﺎﺻﻞ از ﻋﺪم دﺧﺎﻟﺖ ﺧﻄﺎي اﻧﺪازه ﮔﯿﺮي ﺑﺮآوردﮔﺮﻫﺎي ﻧﺎﭘﺨﺘﻪ ﻧﺎﻣﯿﺪه ﻣﯽ ﺷﻮﻧﺪ. ﺑﺮآوردﮔﺮﻫﺎي ﻧﺎﭘﺨﺘﻪ ﺑﺮاي ﭘﺎراﻣﺘﺮﻫﺎي ﻣﻘﯿﺎس و دوران در ﻫﻨﮕﺎم اﺳﺘﻔﺎده از اﻧﻄﺒﺎق ﭘﺮوﮐﺮاﺳﺘﺲ داده ﻫﺎي ﺷﮑﻞ دو ﺑﻌﺪي، ارﯾﺐ ﻫﺴﺘﻨﺪ. ﺑﺮاي ﺗﺼﺤﯿﺢ ارﯾﺒﯽ و ﺑﻬﺒﻮد ﺑﺮآوردﮔﺮﻫﺎي ﻧﺎﭘﺨﺘﻪ، در اﯾﻦ ﻣﻘﺎﻟﻪ روش ﻫﺎي ﮐﺎﻟﺒﯿﺪن رﮔﺮﺳﯿﻮن ﮐﻪ از ﻃﺮﯾﻖ ﺑﮑﺎرﮔﯿﺮي ﻣﺪل ﻫﺎي رﮔﺮﺳﯿﻮﻧﯽ ﻣﺨﺘﻠﻂ و ﺗﻮزﯾﻊ ﻧﺮﻣﺎل ﻣﺨﺘﻠﻂ ﺣﺎﺻﻞ ﻣﯽ ﺷﻮد و ﻫﻤﭽﻨﯿﻦ روش اﻣﺘﯿﺎز ﺷﺮﻃﯽ ﭘﯿﺸﻨﻬﺎد ﻣﯽ ﺷﻮد. ﺑﻪ ﻋﻼوه ، ﺑﺎ اﻧﺠﺎم ﺷﺒﯿﻪ ﺳﺎزي آﻣﺎري ﻋﻤﻠﮑﺮد اﯾﻦ روش ﻫﺎ ﻣﻮرد ﻣﻄﺎﻟﻌﻪ ﻗﺮار ﻣﯽ ﮔﯿﺮﻧﺪ. ﻫﻢ ﭼﻨﯿﻦ، ﺗﺤﻠﯿﻞ آﻣﺎري ﻣﺮﺑﻮط ﺑﻪ ﺷﮑﻞ ﺗﭙﻪ ﻫﺎي ﻣﺎﺳﻪ اي اردﺳﺘﺎن، ﺑﺎ ﻓﺮض وﺟﻮد ﺧﻄﺎي اﻧﺪازه ﮔﯿﺮي در ﻣﺸﺎﻫﺪات، اﻧﺠﺎم ﻣﯽ ﺷﻮد.

According to multiple sources of errors, shape data are often prone to measurement error. Ignoring such error, if does exists, causes many problems including the biasedness of the estimators. The estimators coming from variables without including the measurement errors are called naive estimators. These for rotation and scale parameters are biased, while using the Procrustes matching for two dimensional shape data. To correct this and to improve the naive estimators, regression calibration methods that can be obtained through the complex regression models and invoking the complex normal distribution, as well as the conditional score are proposed in this paper. Moreover, their performance are studied in simulation studies. Also, the statistical shape analysis of the sand hills in Ardestan in Iran is undertaken in presence of measurement errors.