كليدواژه :
داده هاي فضايي گمشده , مدل هاي اتورگرسيو فضايي , ماكسيمم درستنمايي برداري شده , اتريس وزن
چكيده فارسي :
گاهي مجموعه داده ها، به عنوان تحقق هاي يك ميدان تصادف فضايي، شامل مقادير گم شده هستند. مقادير مشاهده شده وگم شده فضايي كه در همسايگي يكديگر قرار دارند مي توانند حاوي اطلاعات مفيدي باشند. بازيابي اين اطلاعات از دست رفته به روشي مناسب موجب دستيابي به نتايج معتبر و دقيق تري خواهد شد. به منظور مدل بندي داده هاي فضايي گم شده مي توان از مدل هاي اتورگرسيو استفاده كرد. آنچه كه در استفاده از اين مدل ها حائز اهميت است استفاده از روش مناسب براي دستيابي به برآوردي از پارامترهاي مدل و در نتيجه پيشگويي در موقعيت هاي فاقد مشاهده است. بررسي ها نشان داده است كه در استفاده از اين مدل ها، برآورد ماكسيمم درست نمايي پارامترهاي مدل منجر به محاسبات زمان بر و ماكسيمم هاي موضعي مي شود. در اين مقاله روش جايگزين "ماكسيمم درستنمايي برداري شده" معرفي و نحوه تحليل مدل ها در حضور مقادير گم شده مورد بررسي قرار مي گيرد. همچنين مطالعات شبيه سازي و مثال كاربردي براي ارزيابي عملكرد روش تحت مطالعه ارائه خواهد شد.
چكيده لاتين :
There is often a considerable amount of missingness in spatial data, due to the conditions of
their collection. In spatial data, the intensity of the dependence between observations in the
vicinity is stronger than the dependency of further observations. This feature affects the modeling
and statistical inferences of the data. Thus, missing values that are close to each other or
observations may contain useful information that can be used to lead to more accurate results in
data analysis by reconstructing the data and imputation. It is important to consider the appropriate
hypothesis about the missing data mechanism. Generally missing data mechanisms can be divided
into two categories, ignorable and non-ignorable missingness. Here we continue under the
ignorable missingness hypothesis.
Autoregressive models can be used efficiently to model missing spatial data. Considering the
spatial autoregressive coefficient in the model, these models correct the prediction of the response
variable based on the linear regression model y = Xβ + ε through the weighted mean of the values
in the neighborhood of observations. In this kind of spatial modeling, the spatial dependence of
the observations is considered through an appropriate spatial weight matrix, and the intensity of
the dependence between observations is described through a spatial autoregressive coefficient.
What is essential in using these models is to use a suitable method to obtain an estimate of the
model parameters and thus predict in unobserved situations. Studies have shown that in using
these models, estimating the maximum likelihood of the model parameters leads to timeconsuming calculations and local maximums. An ideal estimator should be able to use highvolume data, also have the ability for fast computations, and should not rely on nonlinear
optimization algorithms that provide only a localized rather than a comprehensive value as the
optimal value. A method was proposed for quickly calculating estimates when the dependent
variable follows a spatial autoregressive process which is known as the "vectorized maximum
likelihood". From a computational point of view, vectoring relative to a parameter of interest
avoids the costs of using nonlinear optimizers, which are typically associated with iteration.
Material and methods
In this paper, we have applied the "vectorized maximum likelihood" to analyze missing spatial
data. Also, simulation studies and a practical example to evaluate the performance of the method
under study are presented.
Results and discussion
According to the results of the simulation study and modeling US election data, modelling data
by spatial autoregressive model, estimating and predicting through the introduced method leads
better results than the conventional ordinary least square method. As an example, in the analysis
of election data via the Spatial Durbin Model (SDM) and vectorized maximum likelihood method,
some coefficient relevant to spatially lagged covariates are significant which are ignored through
linear regression. This can be the weak points of a simple model.
Conclusion
Also, the adjusted determinative coefficient, the sum of square errors, and root mean square
error indicate that the spatial Durbin model, which is fitted by vectorized maximum likelihood
method, leads to less prediction error in comparison with simple linear regression and ordinary
least square method.
