The quantization for self-conformal measures with respect to the geometric mean error

Let [Formula: see text] be a Bedford–McMullen carpet associated with a set of affine mappings [Formula: see text] and let [Formula: see text] be the self-affine measure associated with [Formula: see text] and a probability vector [Formula: see text]. We study the asymptotics of the geometric mean error in the quantization for [Formula: see text]. Let [Formula: see text] be the Hausdorff dimension for [Formula: see text]. Assuming a separation condition for [Formula: see text], we prove that the [Formula: see text]th geometric error for [Formula: see text] is of the same order as [Formula: see text].

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Asymptotic of the geometric mean error in the quantization of recurrent self-similar measures

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2015.05.063 ◽

2015 ◽

Vol 431 (2) ◽

pp. 737-751

Author(s):

Mrinal Kanti Roychowdhury ◽

Nina Snigireva

Keyword(s):

Geometric Mean ◽

Mean Error ◽

Self Similar ◽

Geometric Mean Error

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Prediction of unsaturated soil hydraulic conductivity using air permeability: Regression approach

Indian Journal of Agricultural Research ◽

10.18805/ijare.v49i6.6680 ◽

2015 ◽

Vol 49 (6) ◽

Cited By ~ 2

Author(s):

Mohammad Reza Neyshaboury ◽

Mehdi Rahmati ◽

Seyed Alireza Rafiee Alavi ◽

Hosein Rezaee ◽

Amirhosein Nazemi

Keyword(s):

Geometric Mean ◽

Regression Function ◽

Close Correlation ◽

Air Permeability ◽

Geometric Standard Deviation ◽

Water Conductivity ◽

Error Ratio ◽

Relative Water ◽

Variable Head ◽

Geometric Mean Error

A close correlation between water conductivity (<italic>K(θ)</italic>) and air permeability (<italic>K</italic><italic>a</italic>), measured at various water contents, is expected due to tight dependence of water filled porosity to air filled porosity of soils. Finding such a relation will greatly facilitate the prediction of unsaturated water conductivity (<italic>K(θ)</italic>). So, the purpose of the current investigation was to find out if a reliable relation or function between the two permeabilities can be established. In this regard, <italic>K(θ)</italic> and <italic>K</italic><italic>a</italic> were measured by pressure plate outflow and variable head methods, respectively, at the range of 0 to -100 kPa matric potential (<italic>ψ</italic><italic>m</italic>). A linear regression function between relative water conductivity (<italic>K</italic><italic>r</italic>(<italic>θ</italic>)) and <italic>K</italic><italic>a</italic> (<italic>LogK</italic><italic>r</italic> (<italic>θ</italic>)=<italic>a</italic>+<italic>bLogK</italic><italic>a</italic>) with the correlation coefficient (<italic>R</italic>) from 0.884 to 0.999 were established for the 22 examined soils. The overall <italic>R</italic> for 128 data pairs (<italic>K</italic><italic>r</italic>(<italic>θ</italic>) and <italic>K</italic><italic>a</italic>) became 0.821 (being significant at <italic>P</italic><0.01) with the slope (<italic>b</italic>) of -2.54 and intercept (<italic>a</italic>) of -10.93. For the comparison propose <italic>K</italic><italic>r</italic>(<italic>θ</italic>) were also predicted from RETC using experimental SMC data and van Genuchten and Brooks-Corey models. The reliability of the <italic>K</italic><italic>r</italic>(<italic>θ</italic>) prediction from <italic>K</italic><italic>a</italic> based on root mean square error (RMSE), geometric mean error ratio (GMER), and geometric standard deviation of error ratio (GSDER) criteria became considerable greater than those predicted from the two mentioned models.

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