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ence of intermodulation products. If two signals with frequencies F1 and F2 are passed through a nonlinear device or medium, the result will contain IM products that are spurious frequency energy components. These components may be present either inside and/ or outside the frequency band of interest for a particular device or system. IM products may be produced from harmonics of the desired signal in question, either as products between harmonics, or as one of the basic signals and the harmonic of the other basic signal, or between both signals themselves.7 The products result when two (or more) signals beat together or mix. These products can be sums and/ or differences. Look at the mixing possibilities when passing F1 and F2 through a nonlinear device. The coef cients indicate the rst, second, or third harmonics. Second-order products F1 F2 ; Third-order products 2F1 F2 ; 2F2 F1 ; and Fourth-order products 2F1 2F2 ; 3F1 F2 . . . .
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where m is the cardinality of the support set of Q. We can expand the other mutual informations similarly. For simplicity in notation, we consider a rate pair as a vector and denote a pair satisfying the inequalities in (15.58) for a speci c input product distribution p1q (x1 )p2q (x2 ) as Rp1 ,p2 as Rq . Speci cally, let Rq = (R1q , R2q ) be a rate pair satisfying R1q < I (X1 ; Y |X2 )p1q (x1 )p2q (x2 ) , R2q < I (X2 ; Y |X1 )p1q (x1 )p2q (x2 ) , R1q + R2q < I (X1 , X2 ; Y )p1q (x1 )p2q (x2 ) . (15.92) (15.93) (15.94)
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MANAGING GROUNDWATER SYSTEMS
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(a) The (m x m) unit matrix 1m is unitary.
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silica Silicon dioxide (Si O2 ). A common natural component of many minerals and a common uncharged solute in groundwaters. slug test A technique for measuring the hydraulic conductivity (q.v.) of an aquifer (q.v.) by monitoring of the decline of water levels in a well back to their original state, following the sudden addition of a known volume ( slug ) of water to the top of the water column in a well. soakaways Chambers (1 3 m deep) deliberately intended to discharge water into the subsurface, used for disposal of surface drainage and/or wastewaters. soil moisture Subsurface water (q.v.) above the water table (q.v.). soil moisture de cit The amount of water which would need to be added to a given body of dried soil in order to bring its moisture content up to eld capacity (q.v.). soil zone The uppermost layer of unconsolidated earth materials, which will normally support plant life if suf cient moisture is present. solute A dissolved chemical constituent (q.v.) in a water.
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The multiple regression model is a straightforward extension of the simple linear regression model, with analogous assumptions about the error term. We examined ve inferential methods in this chapter: (1) the t-test for the relationship between the response variable y and a particular predictor variable xi , in the presence of the other predictor variables, x(i) , where x(i) = x1 , x2 , . . . , xi 1 , xi+1 , . . . , xm denotes the set of all predictors not including xi ; (2) the F-test for the signi cance of the regression as a whole; (3) the con dence interval, i , for the slope of the ith predictor variable; (4) the con dence interval for the mean of the response variable y given a set of particular values for the predictor variables x1 , x2 , . . . , xm ; and (5) the prediction interval for a random value of the response variable y given a set of particular values for the predictor variables x1 , x2 , . . . , xm . One may apply a separate t-test for each predictor x1 , x2 ,or x3 , examining whether a linear relationship exists between the target variable y and that particular predictor. On the other hand, the F-test considers the linear relationship between the target variable y and the set of predictors (e.g., { x1 , x2 , x3 }) taken as a whole. Categorical predictor variables may also be used as inputs to regression models, through the use of indicator variables (dummy variables). For use in regression, a categorical variable with k categories must be transformed into a set of k 1 indicator variables. An indicator variable, also known as a dummy variable, is a binary 0/1 variable, which takes the value 1 if the observation belongs to the given category, and takes the value 0 otherwise. These indicator variables de ne a set of parallel (hyper-) planes. The vertical distance between these parallel planes, as measured by the coef cient for the indicator variable, represents the estimated effect of the particular indicator variable on the target variable with respect to the reference category. In the interests of parsimony, we should nd some way to penalize the R 2 measure for models that include predictors that are not useful. Fortunately, such a 2 penalized form for R 2 does exist, and is known as the adjusted R 2 . If Radj is much 2 less than R , this is an indication that at least one variable in the model may be extraneous, and the analyst should consider omitting that variable from the model. 2 When one is building models in multiple regression, one should use Radj and s rather 2 than the raw R . The sequential sums of squares partitions the SSR into the unique portions of the SSR that are explained by the particular predictors given any earlier predictors. Thus, the values of the sequential sums of squares depends on the order in which the variables are entered into the model. Multicollinearity is a condition where some of the predictor variables are correlated with each other. Multicollinearity leads to instability in the solution space, possibly leading to incoherent results. The high variability associated with the estimates means that different samples may produce coef cient estimates with widely different values. Variance in ation factors may be used to detect the presence of multicollinearity. Depending on the task confronting the analyst, multicollinearity may not in fact present a fatal defect. Multicollinearity does not degrade the accuracy of the response predicted. However, the data miner must therefore strictly limit the use of a multicollinear model to estimation and prediction of the target variable. Interpretation of the model would not be appropriate, since the individual coef cients may not make sense in the presence of multicollinearity.
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The following subroutine reads the text file created in the previous example, and stores the values beginning at the active cell. The code reads each character and essentially parses the line of data, ignoring quote characters and looking for commas to delineate the columns.
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B. Exposure to the phobic stimulus almost invariably provokes an immediate anxiety response, which may take the form of a situationally bound or situationally predisposed panic attack. Note: in children, the anxiety may be expressed by crying, tantrums, freezing, or clinging. C. The person recognizes that the fear is excessive or unreasonable. Note: in children, this feature may be absent. 1 = absent or false
us. In such a case, the fixed and variable costs have to be estimated in the best possible way. Suppose the Model Motel was considering renting the adjacent premises and converting the space into a 50-seat coffee shop to better serve the needs of its motel customers. The owner of the motel and the accountant have developed the cost projections shown in Exhibit 8.6. With this information we can answer the question: What must the minimum sales be to earn the return on investment desired This can be answered by using the basic CVP equation. Required sales level Fixed expenses Return on investment (profit) 100% Variable cost % $85,600 100% $115,750 45% $257,222 Assuming the estimates of costs are reasonably accurate, the owner of the Model Motel would have to decide whether the projected required sales revenue of $257,222 could be attained from motel customers and other potential customers in the area. If the volume could be reached, then the new venture would be profitable. $30,150 55%
FIGURE 15.5 slot.
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