Multivariate function derivatives

The final example described in this chapter is that done by Schriebej and co-workers. They used a scaffold with multivariant sites derived from shikimic acid. However, the end products of their synthesis were compact, highly functionalized structures reminiscent of alkaloids.34 An objective of this study was to produce a very large number of compounds for miniaturized cell-based assays. In fact, about 2 million compounds were made. 

A single-variable function is convex if the second derivative is strictly positive over the range of the dependent variable, as shown in Fig. 4(1). As show in Fig. 4(2), a function is concave when the second derivative is negative over the dependent variable range. For a multivariable function, the matrix of second derivatives (termed the Hessian, H(x)) is used to check the convexity (or concavity) of the function ... 

There are two types of unconstrained multivariable optimization techniques those requiring function derivatives and those that do not. An example of a technique that does not require function derivatives is the sequential simplex search. This technique is well suited to systems where no mathematical model currently exists because it uses process data directly. 

Derivative Methods.—The most well developed of the derivative methods are univariate in nature, that is, they approach the minimum of the multivariate function along a sequence of lines (directions) in the many-dimensional space, and the problem is then to determine an algorithm for the choice of these directions. Usually (but not always) it is required that the current direction be followed until a minimum of the function in that direction is found. One may say that these methods are based on a sequence of onedimensional searches. 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

To respect the orientation of frames, we will make also x = y and X3 = x. By applying the chain rule for multivariate functions to the derivatives of Eqn. (4), we obtain ... 

Theorem A-2-7 Mean value theorem of differential calculus for multivariable functions Let/(x, y, z) be continuous and have continuous first partial derivatives in a domain D. Furthermore, let (x0, y0> zo) and (x0 4 h, y0 + k, z0 + /) be points in D such that the line segment joining these points lies in D. Then... 

A. Guzman. Derivatives and Integrals of Multivariable Functions. Birkhauser, Boston, MA, 2003. 

Partial Derivatives Are the Slopes of the Tangents to Multivariate Functions... 

The Extrema of Multivariate Functions Occur Where the Partial Derivatives Are Zero... 

How do we find an extremum of a multivariate function First, recall how to find an extremum when there is only one variable. Figure 5.4 shows a singlevariable function, fix) = X +b, where h is a constant. To locate the extremum, find the point at which the first derivative equals zero ... 

Some functions depend on more than a single variable. To find extrema of such functions, it is necessary to find where all the partial derivatives are zero. To find extrema of multivariate functions that are subject to constraints, the Lagrange multiplier method is useful. Integrating multivariate functions is different from integrating single-variable functions multivariate functions require the concept of a pathway. State functions do not depend on the pathway of integration. The Euler reciprocal relation relation can be used to distinguish state functions from path-dependent functions. In the next three chapters, we will combine the First and Second Laws with multivariate calculus to derive the principles of thermodynamics. 

Differentiating multivariate functions. Compute the partial derivatives, (dfldx)v and idf/dy)x, for the following functions. 

Then the derivative of P is related to the derivatives of the three variables, because small changes in n, in T, and in V will each contribute to the overall change in P. In general, derivatives of multivariable functions require knowing how all the variables depend on each other. In these instances, we will use the partial derivative, represented by the symbol d, which is simply the derivative of the function with respect to one variable treating all the other variables as though they are constants. The expression... 

Multivariable RTO of nonlinear objective functions using function derivatives is recommended with more than two variables. In particular, the conjugate gradient... 

Listing 5.5. Code for partial derivative of multivariable function. 

Many other optimization problems involve calculating the minimum of a multivariable function subject of certain constraints on the variables. One such example will be shown here to illustrate the use of the partial derivative code with such problems. The problem to be considered is finding the minimum of a function /(x) subject to constraints on the x = x, x, ..x variables given by equations of the form ... 

We consider an nxn table D of distances between the n row-items of an nxp data table X. Distances can be derived from the data by means of various functions, depending upon the nature of the data and the objective of the analysis. Each of these functions defines a particular metric (or yardstick), and the graphical result of a multivariate analysis may largely depend on the particular choice of distance function. 

We also make a distinction between parametric and non-parametric techniques. In the parametric techniques such as linear discriminant analysis, UNEQ and SIMCA, statistical parameters of the distribution of the objects are used in the derivation of the decision function (almost always a multivariate normal distribution... 

The method of steepest descent uses only first-order derivatives to determine the search direction. Alternatively, Newton s method for single-variable optimization can be adapted to carry out multivariable optimization, taking advantage of both first- and second-order derivatives to obtain better search directions1. However, second-order derivatives must be evaluated, either analytically or numerically, and multimodal functions can make the method unstable. Therefore, while this method is potentially very powerful, it also has some practical difficulties. 

Another class of methods of unidimensional minimization locates a point x near x, the value of the independent variable corresponding to the minimum of /(x), by extrapolation and interpolation using polynomial approximations as models of/(x). Both quadratic and cubic approximation have been proposed using function values only and using both function and derivative values. In functions where/ (x) is continuous, these methods are much more efficient than other methods and are now widely used to do line searches within multivariable optimizers. 

Another direction of development of the data set is to strengthen the in vitro-in vivo correlations and develop multivariate models to predict in vivo endpoints, such as therapeutic effects and adverse events. In this respect, it will be interesting to examine which data (among in silico descriptors, in vitro primary and secondary data, in vitro functional data, etc.) are most appropriate to derive robust and predictive models. 

---