Squares of difference

The sum of squares of differences between points on the regression line yi at Xi and the arithmetic mean y is called SSR... 

Square of Difference between points at distance h apart... 

Fig. 30. Snapshot of particle volume fraction fields obtained while solving a kinetic theory-based TFM. Fluid catalytic particles in air. Simulations were done over a 16 x 16 cm periodic domain. 128 x 128 cells (shown in the figure). The average particle volume fraction in the domain is 0.05. Dark (light) color indicates regions of high (low) particle volume fractions. Squares of different sizes illustrate regions (i.e., filters) of different sizes over which averaging over the cells is performed. Source Andrews and Sundaresan (2005).

Fig. 12.6 (A) Calculated (solid curves) and measured (points) mean bromide concentrations, normalized by the mass of appUed solute per unit soil surface, at times r = 18 and 32 days (B) Mean bromacd concentrations, normalized by the mass of apphed solute per unit soU surface, at times r = 18 and 32 days (solid curves best-fit based on least squares of differences between computed and measured mean concentrations, = 0.14 mL/g, X = 0.022dashed curves based on sum of squares of differences, = 0.16 mL/g, day X = 0.012 daysdotted-dashed curves ...

Figure 5.6 Mean square of differences between consecutive Raman spectra as a function of mixing time of a binary system. Reprinted from Vergote etal. (2004)62 with permission from Elsevier.

Sum of Square of Differences Between Experimental Data and Predicted Results (SSE) for Flowthrough and Batch Reactors Without Acid Addition at 180°C... 

Figure 2.3 Hydrogen storage capacity at RT (diamonds) and 77 K (squares) of different carbon materials versus their BET specific surface area [52]. The dotted line represents the theoretical curve according to [50],...

The scattering intensity is proportional to the square of difference of electron density between the scattering heterogeneities and their surrounding. In porous materials the pores may be assumed as heterogeneities for which electron density differs from that of materials constituting the porous skeleton. There are many porous materials for which power law, I(q)=Io-q (o Iq are constants), is fulfilled in a certain q region. 

A very clear presentation of this two-dimensional data collection describing peptide binding to DR1 and to DRB1 1501 is given in Figure 11.4. Relative competition-values of the 220 sublibraries are indicated by squares of different black intensities resulting in a fingerprint of the studied class II molecule. 

The most widely used objective function involves the sum of squares of differences between observed and calculated parameter values... 

Coaxial parallel squares of different edge length... 

Temp. = 0. Volume of gas = r., Difference between Calculated and Observed. Square of Difference between Calculated and Observed. 

Figure 1. An oxide mask of square type a), b) and c) - rows of oxide squares of different area with side orientation like [110], [100] and [100] + 30", respectively.

SOLUTION. This is an example of linear least-squares analysis (LLSA), where the objective function is continuous. Typically, LLSA is performed on a discrete set of data points and one seeks to minimize the sum of squares of differences between the data and a continuous model function. In this case, we seek to minimize the square of the difference between two continuous functions over the complete range of reactant conversions that are possible (i.e., 0 < x < 1 for irreversible reactions). Hence, the sum of squares in the objective function to be... 

This technique fits a straight line to data as dependent vtJues related to a single set of values of an independent parameter. The set of equations that are used minimizes the sum of the squares of differences between the dependent vtdues and the line. 

If all force constants are being refined simultaneously with different, carefully selected weights, W,, attached to them, then the differences between the observed and calculated values, = Xi(obs) — A./(calc), and Jij are calculated for each datum Xi. The corresponding contributions are added to the scalar sum of weighted squares of differences e We, to the vector J We, and to the matrix J WJ. The normal equations are formed as... 

Eor the overdetermined system. Ax does not equal b. This is because the solution minimizes the sum of squares of differences between Ax and b. The least squares solution is useful in several applications, such as regression, which is covered in Chapter 7. 

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