M-estimator

Under aggressive corrosion conditions it is estimated that the maximum corrosion current density in a galvanised steel sheet will be 6 X 10 A m . Estimate the thickness of the galvanised layer needed to give a rust-free life of at least 5 years. The density of zinc is 7.13 Mg m , and its atomic weight is 65.4. Assume that the zinc corrodes to give Zn " ions. 

V atauk, W. M., "Estimating Costs of Air Pollution Control." Lewis, Chelsea, Ml, 1990. 

Guthrie. K. M., Estimating the Cost of High-Pressure Equipment, Chemical Engineering, Dec. 2, 1968, pp. 144-148. 

M - Estimate is based on monitoring data or measurements for the toxic chemical as released to the environment and/or off-site facility. 

Example 2.4 A glass bottle of sparkling water has an acetal cap as shown in Fig. 2.14. If the carbonation pressure is 375 kN/m, estimate the deflection at the centre of the cap after 1 month. The value of Poissons ratio for acetal may be taken as 0.33. 

Example 2.21 A rod of plastic is subjected to a steady axial pull of 50 N and superimposed on this is an alternating axial load of 100 N. If the fatigue limit for the material is 13 MN/m and the creep rupture strength at the equivalent time is 40 MN/m, estimate a suitable diameter for the rod. Thermal effects may be ignored and a fatigue strength reduction factor of 1.5 with a safety factor of 2.5 should be used. 

Example 2.22 A certain grade of acrylic has a Kc value of 1.6 MN and the fatigue crack growth data as shown in Fig. 2.77. If a moulding in this material is subjected to a stress cycle which varies from 0 to 15 MN/m, estimate the maximum internal flaw size which can be tolerated if the fatigue endurance is to be at least 1(P cycles. 

A circular polypropylene plate, 150 nun in diameter is simply supported around its edge and is subjected to a uniform pressure of 40 kN/m. If the stress in the material is not to exceed 6 MN/m, estimate a suitable thickness for the plate and the deflection, 6, after one year. The stress in the plate is given by... 

If the stress in the composite beam in the previous question is not to exceed 7 MN/m estimate the maximum uniformly distributed load which it could carry over its whole length. Calculate also the central deflection after 1 week under this load. The bending moment at the centre of the beam is lVL/24. 

Maxwell and Kelvin-Voigt models are to be set up to simulate the creep behaviour of a plastic. The elastic and viscous constants for the Kelvin-Voigt models are 2 GN/m and 100 GNs/m respectively and the viscous constant for the Maxwell model is 200 GNs/m. Estimate a suitable value for the elastic constant for the Maxwell model if both models are to predict the same creep strain after 50 seconds. 

MN/m estimate Ae maximum defect size which could be tolerated in Ae material if it is to last for at least 1 year. 

In a tensile test on an un-notched sample of acrylic the fracture stress is recorded as 57 MN/m. Estimate Ae likely size of the intrinsic defects in Ae material. 

During flow along a particular channel the temperature drops by 40° C and the pressure drops by SO MN/m. Estimate the overall change in viscosity of the melt in each case. Determine the ratio of the pressure change to the temperature change which would cause no change in viscosity for each of the above materials. 

Kosten, /.pi. costa, expenses, charges, -an-schlag, m. estimate (of cost), -aufwand, m. expenditure. 

Industry or agency Estimated cost ( M) Estimated potential saving ... 

Shay, R. M., Estimating Linear Shrinkage of Semicrystalline Resins from Pressure-Volume Temperature (pvT) Data, SPE-IMD Newsletter 49, Fall 1998. 

FIG. 21 Pull-off force between a hydrophilic Si3N4 tip and mica as a function of the relative humidity (RH) at room temperature. The spring constant of the lever is 0.1 N/m. Estimated tip radius -200 A. Hysteresis is observed between increasing (open circles) and decreasing (closed circles) humidity. (From Ref. 51.)... 

Parameter M Estimate BMV criterion Parameter M Estimate BMV criterion... 

S. W., Melton, C. M. Estimation of gas-liquid chromatographic retention times from molecular structure.. Chromatogr. A 1994, 662, 269-280. 

A storage tank for concentrated nitric acid will be constructed from aluminium to resist corrosion. The tank is to have an inside diameter of 6 m and a height of 17 m. The maximum liquid level in the tank will be at 16 m. Estimate the plate thickness required at the base of the tank. Take the allowable design stress for aluminium as 90 N/mm2. 

Cook, E.M. Estimating Spray Drying Costs, Chemical Engineering Progress, June 1966, p. 93. Block, L.J. Estimating Machinery and Equipment, Erection Costs, Oil and Gas Journal, Aug. 6, 1973, p. 77.. 

D. M., Estimation of aqueous solubility of organic molecules by the group contribution approach. Application... 

One type of common robust estimator is the so-called M-estimator or generalized maximum likelihood estimator, originally proposed by Huber (1964). The basic idea of an M-estimator is to assign weights to each vector, based on its own Mahalanobis distance, so that the amount of influence of a given point decreases as it becomes less and less characteristic. 

In Chen et al. (1997) the following simultaneous estimation strategy is adopted to construct the M-estimator ... 

Iterative calculation. The solution of the M-estimator Eqs. (10.31) and (10.32) is obtained by means of an iterative procedure that, for iteration q, is made up of the following steps ... 

Remark 2 It is worthwhile to mention that the breaking point of the M-estimator, the fraction of outliers that can be tolerated without the estimator potentially breaking down, is about 1 /(g + 1), where g is the dimension of the data (Maronna, 1976). In data reconciliation, outliers refer to the abnormal observations and are a small fraction of the whole data set. In our view, the breaking point property is reasonably good for practical implementations. 

The covariance matrix of measurement errors is a very useful statistical property. Indirect methods can deal with unsteady sampling data, but unfortunately they are very sensitive to outliers and the presence of one or two outliers can cause misleading results. This drawback can be eliminated by using robust approaches via M-estimators. The performance of the robust covariance estimator is better than that of the indirect methods when outliers are present in the data set. 

Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter. Ann. Stat. 4, 51-67. Neudecker, H. (1969). Some theorems on matrix differentiation with special reference to Kronecker matrix products. Am. Stat. Assoc. J. 64,953—963. 

Occasionally it is convenient to refer to the p function in (11.21), but generally the form (11.22) is used in robust M-estimation. The use of the t(r form is due to Hampel s concept of the influence function (Hampel et al., 1986). According to the IF concept, the value of it represents the effect of the residuals on the parameter estimation. If iff is unbounded, it means that an outlier has an infinite effect on the estimation. Thus, the most important requirement for robustness is that iff must be bounded and should have a small value when the residual is large. In fact, the value of the iff function corresponds to the gross error sensitivity (Hampel etal., 1986), which measures the worst (approximate) influence that a small amount of contamination of fixed size can have on the value of the estimator. 

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