| 總結: | One of the main tasks in the problems of machine learning and curve fitting is to develop suitable models for given data sets. It requires to generate a function to approximate the data arising from some unknown function. The class of kernel regression estimators is one of main types of nonparametric curve estimations. On the other hand, fractal theory provides new technologies for making complicated irregular curves in many practical problems. In this paper, we are going to investigate fractal curve-fitting problems with the help of kernel regression estimators. For a given data set that arises from an unknown function <i>m</i>, one of the well-known kernel regression estimators, the Nadaraya–Watson estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>m</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula>, is applied. We consider the case that <i>m</i> is Hölder-continuous of exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>β</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and the graph of <i>m</i> is irregular. An estimation for the expectation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><mover accent="true"><mi>m</mi><mo stretchy="false">^</mo></mover><msup><mrow><mo>−</mo><mi>m</mi><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula> is established. Then a fractal perturbation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mrow><mo>[</mo><mover accent="true"><mi>m</mi><mo stretchy="false">^</mo></mover><mo>]</mo></mrow></msub></semantics></math></inline-formula> corresponding to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>m</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> is constructed to fit the given data. The expectations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>f</mi><mrow><mo>[</mo><mover accent="true"><mi>m</mi><mo stretchy="false">^</mo></mover><mo>]</mo></mrow></msub><mo>−</mo><mover accent="true"><mi>m</mi><mo stretchy="false">^</mo></mover><msup><mrow><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>f</mi><mrow><mo>[</mo><mover accent="true"><mi>m</mi><mo stretchy="false">^</mo></mover><mo>]</mo></mrow></msub><msup><mrow><mo>−</mo><mi>m</mi><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula> are also estimated.
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