Patterns and Points

Since reflectometry is a major metrology tool in CMP processes, another important issue is the number of measurement points on the wafer that are required to determine the film thickness and uniformity without sacrificing cycle time. Table II presents a comparison of number of data points vs the measurement efficiency and accuracy using the polar map pattern on the 

Number of Measurement Points versus Nonuniformity Accuracy and Time Consumed for the Same Wafer 

Number of measurements (points) Time (min) to measure Nonuniformity across the wafer (%) 

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Fig. 3. Different types of measurement patterns and points (a) 49-point diameter scan, upper left (b) 49-point polar map, upper right (c) 21-point contour map, lower left and (c) 9-point contour map, lower right.

The explorative analysis of data sets by visual data mining applications takes place in a three-step process During the first step (overview), the user can obtain an overview of the data and maybe can identify some basic relationships between specific data points. In the second step (filtering), dynamic and interactive navigation, selection, and query tools will be used to reorganize and filter the data set. Each interaction by the user will lead to an immediate update of the data scene and will reveal the hidden patterns and relationships. Finally, the patterns or data points can be analyzed in detail with specific detail tools. 

Close examination of these areas under a low-power microscope revealed smoothly rippled, spherical surfaces in the weld region and a chevron pattern that pointed back to the weld in the plate. Cross sections cut through the weld revealed substantial subsurface porosity and regions where oxidized surfaces prevented metallurgical bonding of the weld (Fig. 

Sensitivity to process gas inlet temperature. Figure 7-13b shows TTE variation with changes in process gas inlet temperature. The sensitivity of variable speed machines to temperature variation is less than constant speed machines. The pattern and symmetry around the design point, however, are the same for constant speed machines. For example, note that a 3% decrease in the inlet temperature causes TTE to drop to 98% and that the same percentage drop in TTE occurs when gas inlet temperature rises by 3%. 

In rooms where energy is introduced primarily by supply air jets, air distribution methods are referred to as mixing type. With a perfect mixing-type air distribution, airflow pattern and air velocity at any point in the room are... 

From a practical point of view, power consumption is perhaps the most important parameter in the design of stirred vessels. Because of the very different flow patterns and mixing mechanisms involved, it is convenient to consider power consumption in low and high viscosity systems separately. 

I thank Dr D. Schechtman for cooperation in providing me with the X-ray diffraction pattern and for other information, and Professor Barclay Kamb for pointing out to me that an icosahedron becomes chiral when it shares its faces. This investigation was supported in part by a grant from the Japan Shipbuilding Industry Foundation. 

CBs, like OPs, can cause a variety of sublethal neurotoxic and behavioral effects. In one study with goldfish Carrasius auratus), Bretaud et al. (2002) showed effects of carbofuran on behavioral end points after prolonged exposure to 5 pg/L of the insecticide. At higher levels of exposure (50 or 500 pg/L), biochemical effects were also recorded, including increases in the levels of norepinephrine and dopamine in the brain. The behavioral endpoints related to both swimming pattern and social interactions. Effects of CBs on the behavior of fish will be discussed further in Chapter 16, Section 16.6.1. 

As a result, a considerable amount of effort has been expended in designing various methods for providing difference approximations of differential equations. The simplest and, in a certain sense, natural method is connected with selecting a, suitable pattern and imposing on this pattern a difference equation with undetermined coefficients which may depend on nodal points and step. Requirements of solvability and approximation of a certain order cause some limitations on a proper choice of coefficients. However, those constraints are rather mild and we get an infinite set (for instance, a multi-parameter family) of schemes. There is some consensus of opinion that this is acceptable if we wish to get more and more properties of schemes such as homogeneity, conservatism, etc., leaving us with narrower classes of admissible schemes. 

In the illustration of Fig. 29.4 we regard the matrix X as either built up from n horizontal rows of dimension p, or as built up from p vertical columns x,.of dimension n. This exemplifies the duality of the interpretation of a matrix [9]. From a geometrical point of view, and according to the concept of duality, we can interpret a matrix with n rows and p columns either as a pattern of n points in a p-dimensional space, or as a pattern of p points in an n-dimensional space. The former defines a row-pattern P" in column-space 5, while the latter defines a column-pattern P in row-space S". The two patterns and spaces are called dual (or conjugate). The term dual space also possesses a specific meaning in another... 

Fig. 29.7. Illustration of a pattern of points with rank of 2. The pattern is represented by a matrix X with dimensions 5x4 and a linear dependence between the three columns of X is assumed. The rank is shown to be the smallest number of dimensions required to represent the pattern in column-space 5 and in row-space S".

Fig. 29.8. (a) Pattern of points in column-space S (left panel) and in row-space S" (right panel) before column-centering, (b) After column-centering, the pattern in 5 is translated such that the centroid coincides with the origin of space. Distances between points in S are conserved while those in S" are not. (c) After column-standardization, distances between points in S and 5" are changed. Points in 5" are located on a (hyper)sphere centered around the origin of space. 

After preprocessing of a raw data matrix, one proceeds to extract the structural features from the corresponding patterns of points in the two dual spaces as is explained in Chapters 31 and 32. These features are contained in the matrices of sums of squares and cross-products, or cross-product matrices for short, which result from multiplying a matrix X (or X ) with its transpose ... 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

The matrix-to-vector product can be interpreted geometrically as a projection of a pattern of points upon an axis. As we have seen in Section 29.4 on matrix products, if X is an nxp matrix and if v is a p vector then the product of X with v produces the n vector s ... 

The matrix X defines a pattern P" of n points, e.g. x, in which are projected perpendicularly upon the axis v. The result, however, is a point s in the dual space S". This can be understood as follows. The matrix X is of dimension nxp and the vector V has dimensions p. The dimension of the product s is thus equal to n. This means that s can be represented as a point in S". The net result of the operation is that the axis v in 5 is imaged by the matrix X as a point s in the dual space 5". For every axis v in 5 we will obtain an image s formed by X in the dual space. In this context, we use the word image when we refer to an operation by which a point or axis is transported into another space. The word projection is reserved for operations which map points or axes in the same space [11]. The imaging of v in S into s in S" is represented geometrically in Fig. 29.9a. Note that the patterns of points P" and P are represented schematically by elliptic envelopes. 

Fig, 29.10. Geometrical interpretation of multiple linear regression (MLR). The pattern of points in S representing a matrix X is projected upon a vector b, which is imaged in 5" by the point y. The orientation of the vector b is determined such that the distance between y and the given y is minimal. 

In a general way, we can state that the projection of a pattern of points on an axis produces a point which is imaged in the dual space. The matrix-to-vector product can thus be seen as a device for passing from one space to another. This property of swapping between spaces provides a geometrical interpretation of many procedures in data analysis such as multiple linear regression and principal components analysis, among many others [12] (see Chapters 10 and 17). 

Orthogonal rotation produces a new orthogonal frame of reference axes which are defined by the column-vectors of U and V. The structural properties of the pattern of points, such as distances and angles, are conserved by an orthogonal rotation as can be shown by working out the matrices of cross-products ... 

In Chapter 29 we introduced the concept of the two dual data spaces. Each of the n rows of the data table X can be represented as a point in the p-dimensional column-space S . In Fig. 31.2a we have represented the n rows of X by means of the row-pattern F. The curved contour represents an equiprobability envelope, e.g. a curve that encloses 99% of the points. In the case of multinormally distributed data this envelope takes the form of an ellipsoid. For convenience we have only represented two of the p dimensions of SP which is in reality a multidimensional space rather than a two-dimensional one. One must also imagine the equiprobability envelope as an ellipsoidal (hyper)surface rather than the elliptical curve in the figure. The assumption that the data are distributed in a multinormal way is seldom fulfilled in practice, and the patterns of points often possess more complex structure than is shown in our illustrations. In Fig. 31.2a the centroid or center of mass of the pattern of points appears at the origin of the space, but in the general case this needs not to be so. 

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