Variables Than Objects

The situation in which the data matrix has more variables (columns) than objects (rows) was touched on earlier, but we will now consider it in more detail as it is an important problem we encounter with large datasets. Most people are aware that with two data points, it is possible to construct a line, a one-dimensional object, and with three data points, a plane, a two-dimensional object. This process can be continued so that four data points allows a three-dimensional object, five points, four dimensions, and so on. Thus, the maximum dimensionality of an object, and hence the maximum number of dimensions, in a dataset is — 1, where n is the number of data points. For dimensions, we can substitute independent pieces of information, and thus, the maximum that any dataset may contain is — 1. This, however, is a maximum and in reality the true dimensionality, where dimension means information, is often much less than n 1. 

From experience, we know that some researchers need a bit more convincing that with p n, at least p — n variables are not contributing any 

To compare analyses based on these three sets of a -values, it is necessary to generate the corresponding y-values. This process is performed here by adopting the linear model in Eq. [7] 

In each case, the model was fitted by least squares and the results summarized for (1) sequential partitions of the regression sums of squares (Table 1) (2) the estimated regression coefficients and their standard errors (Table 2). The main points of the analyses are as follows  

For dataset 1, the contribution of each variable to the regression sum of squares is the same no matter what order the variables are added when fitting the model. As a consequence, the regression sum of squares can be 

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The group means and covariances can also be estimated robustly, for example, by the minimum covariance determinant (MCD) estimator (see Section 2.3.2). The resulting discriminant rule will be less influenced by outlying objects and thus be more robust (Croux and Dehon 2001 He and Fung 2000 Hubert and Van Driessen 2004). Note that Bayes discriminant analysis as described is not adequate if the data set has more variables than objects or if the variables are highly correlating, because we need to compute the inverse of the pooled covariance matrix in Equation 5.2. Subsequent sections will present methods that are able to deal with this situation. 

Most used methods for typical data in chemometrics, containing more variables than objects per group and highly correlating variables, are... 

It is a common misunderstanding that in a statistical analysis the number of objects must exceed the number of variables. This is true of regression analysis but it is not true of PLS. Since PLS is based upon projections, we can easily have more variables than objects. Any object included in the analysis can be characterized by a large number of descriptors, and the outcome of the experiment can be... 

It cannot handle more variables than objects. 

Color is basically a perception that is developed in the mind of a given individual, and consequently different people can perceive a particular color in various fashions. Such variability in interpretation causes great difficulty in the evaluation of color-related phenomena, and it leads to subjective rather than objective judgements. For obvious reasons, the development of a quantitative method foi... 

A significant part of developing a model used for other than determining static sets of heat and material balances (which are sufficient for some model objectives, such as providing the basis for new plant design) is specifying which variables are independent and which are dependent. Far more variables are dependent variables than are independent in essentially all models. For simulation and optimization... 

The number of variables (e.g. chemical compounds) should be lower than objects (e.g. samples). 

Overfitting is the commonest problem in multivariate statistical procedures when the number of variables is greater than objects (samples) one can fit an elephant with enough variables. Tabachnick and Fidell (1983) have suggested minimum requirements for some multivariate procedures to avoid the overfitting or underfitting that can occur in a somewhat unpredictable manner, regardless of the multivariate procedure chosen. 

Control engineers know that it takes one manipulated variable for each measured variable we wish to control to setpoint. When the number of controlled variables equals the number of manipulated variables we pair up the different variables and use PI controllers for regulation. Sometimes we are fortunate to have more manipulated variables than control specifications. We can then optimize the use of the manipulators while controlling to setpoint (e.g., valve position control). Sometimes, however, the number of control objectives exceeds the number of available manipulators and we cannot control all variables to setpoint. This is when the concept of partial control is useful. 

The results of a PLS analysis can be transformed to regression coefficients of the X block variables, most often leading to the curious result that more regression coefficients than objects are obtained. It should be mentioned that in the case of one dependent variable and a number of X variables that equals the number of PLS components, the results from regression analysis and, after appropriate transformation, from PLS analysis are numerically identical. 

The second reason for variable removal is if they are redundant. Redundant variables develop in a dataset for two reasons (1) because more variables (p) exist than objects ( ) and (2) multicolinearity. 

If a reaction involves the use of a very expensive catalyst, then it is sensible to develop a reactor structure that minimizes reactor volume (or catalyst mass), for a desired conversion, rather than objective functions involving concentration. In this way, reactor volume is the primary variable that must be optimized rather than concentration or conversion. 

All else being equal, virtually aU processes have fewer control degrees of freedom than objectives for the control system. Hence the only way to ensure proper satisfaction of all control objectives is to use the process design itself to make sure that the process can be operated effectively and that a sufficient number of manipulated variables are available (Ref. 

Picture any process as consisting of a block with a number of input (manipulated) variables and an equal number of output (controlled) variables. The object is to control a given process output by manipulating the one input that will have the greatest influence on it. If this is not done, another input will have more influence on the controlled variable than the one which it manipulates through the controller. To assess all the possibilities, the gain of each controlled variable to each manipulated variable must be determined. It is extremely desirable that these gain terms be normalized to eliminate dimensions and to place them all on the same basis. 

Step 16 In most cases, the reactor temperature profile obtained from step 15 will show similar trend as plant measurement rather than perfect agreement. To make model s prediction on reactor temperature profile matches plant measurement well, we select all of the temperature rise variables as objective functions, assign new tuning ranges to and tune all of the global reaction activities . 

Budgeted income statements are identical in form to ac tual income statements. However, the budgeted numbers are objectives rather than achievements. Budgetary models based on mathematical equations are increasingly being used. These may be used to determine rapidly the effect of changes in variables. Variance analysis is discussed in the treatment of manufacturing-cost estimation. 

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