\[ \lg K_{S_1}=\lg \frac{\beta W}{S a_{\infty}\gamma RT} -\frac{\beta nF}{2.3RT}\left(\varphi_{n,a}-\varphi^0\right)-q_1\lg C_R; \]

\[ K_{S_2}=1\cdot 10^{-10}\ \text{cm/sec};\qquad K_{S_1}=7\cdot 10^{-8}\ \text{cm/sec}. \]

The calculated values of the constants indicate the irreversibility of the arsenic discharge–ionization processes under the conditions studied.

Literature

1. A. G. Stromberg, L. N. Popova. Electrochemistry, 4, 39, 1968.
2. N. A. Koshakova. Dissertation, TPI, Tomsk, 1968.
3. H. Matsuda, Y. Ayabe, Z. Elektrochem., 63, 1164, 1959.
4. L. Z. Brainina. Inversion Voltammetry of Solid Phases, Moscow, “Khimiya,” 1972.
5. E. A. Zakharova. Izv. TPI, 128, 53, 1964.
6. E. K. Spirin. Dissertation, TPI, Tomsk, 1965.
7. Pourbaix M., Atlas d’équilibres électrochimiques, Paris, 1963.

SMOOTHING OF EXPERIMENTAL RESULTS ON THE BASIS OF THE USE OF A NONPARAMETRIC APPROACH

A. I. Ruban

In problems of investigating chemical-technological objects, an indispensable part is the smoothing of experimental results, i.e., the construction of regression relationships between variables. For these purposes the method of least squares (MLS) is usually used. However, it has a number of substantial limitations that narrow the scope of its application.

In the present work a general approach to smoothing experimental results is proposed. It is more universal than MLS and leads to comparatively simple computational algorithms.

1. Statement of the Problem

Assume that there is a system of random variables \(Y, X_1,\ldots,X_k\), connected with one another by an unknown static dependence. It is necessary, on the basis of experimental data

\[ \begin{gathered} y_1,x_{11},\ldots,x_{1k},\\ y_2,x_{21},\ldots,x_{2k},\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ y_n,x_{n1},\ldots,x_{nk}, \end{gathered} \tag{I.1} \]

where \(n\) is the measurement number, to determine an estimate \(\hat{\mathcal M}\{Y \mid x_1,\ldots,x_k\}\) of the regression surface \(\mathcal M\{Y \mid x_1,\ldots,x_k\}\)—the mean dependence between

It is known /1/ that the conditional mathematical expectation \(\mathcal M\{Y \mid x_1,\ldots,x_k\}\) satisfies the criterion of least mean-square deviations

\[ \mathcal M\left\{\left[Y-\mathcal M\{Y \mid x_1,\ldots,x_k\}\right]^2\right\} = \min_{\eta}\mathcal M\left\{\left[Y-\eta(x_1,\ldots,x_k)\right]^2\right\}. \tag{I.2} \]

Therefore, in practice the following approach is usually used.

A form is specified for the estimate of the mean dependence of \(Y\) on \(X_1,\ldots,X_k\),

\[ \hat{\mathcal M}\{Y \mid x_1,\ldots,x_k\} = \hat{\eta}(x_1,\ldots,x_k,\alpha) \tag{I.3} \]

up to the vector of parameters \(\alpha\). In doing so, a priori information about \(\mathcal M\{Y \mid x_1,\ldots,x_k\}\) is used. If such information is absent, then a segment of a Taylor series is taken as (I.3). In the case of a nonlinear dependence, one must either take into account a considerable number of terms or divide the domain \((X_1,\ldots,X_k)\) into parts, in each of which a linear approximation is admitted. In both cases the dimension of the vector \(\alpha\) is large and greatly exceeds \(K\), the dimension of the domain of the “input” variables.

After the dependence (I.3) has been chosen, the parameters \(\alpha\) are determined from the criterion (I.2), in which, instead of the mathematical expectation operator \(\mathcal M\), its estimate is taken:

\[ J(\alpha) = \frac{1}{n}\sum_{i=1}^{n} \left(y_i-\hat{\eta}(x_{i1},\ldots,x_{ik},\alpha)\right)^2 = \min_{\alpha} \tag{I.4} \]

This is the simplest form of the least-squares criterion.

The least-squares method has its own special features, which substantially restrict its field of application.

1. In the analysis of results in the LSM it is assumed that \((X_1,\ldots,X_k)\) are measured exactly. Although at the present time there exists a solution of the problem in the presence of small additive measurement errors in the variables \((X_1,\ldots,X_k)\), the results are sufficiently complicated and are not used in practice.

2. All disturbances are reduced to additive measurement errors in \(Y\).

3. These errors do not depend on the “inputs” \((X_1,\ldots,X_k)\).

4. The parametric specification of the regression surface (I.3) is known.

5. The best properties of the estimates \(\alpha\) are obtained when they

enter linearly into the estimate of the regression surface (1.3), and when the additive measurement errors of \(y\) have a normal distribution law.

The main advantage of this approach lies in the possibility of determining optimal (in the least-squares sense) estimates for small sample sizes.

If conditions I–5 are not satisfied, then, for constructing an estimate of the regression curve \(\hat{\mathcal U}\{y/x_1,\ldots,x_k\}\), one may apply the approach described below, based on the use of estimates of joint probability densities, which have found wide application in nonparametric statistics. For the one-dimensional case (one “input,” one “output”) some results are contained in papers /2–4/.

2. Method of solution.

The regression curve is uniquely determined by the joint probability density \(f(y,x_1,\ldots,x_k)=f(y,\vec{x})\):

\[ \rho(\vec{x})=\mathcal U\{y/\vec{x}\} = \int_{-\infty}^{\infty} y f(y,\vec{x})\,dy \bigg/ \int_{-\infty}^{\infty} f(y,\vec{x})\,dy . \tag{2.1} \]

Replacing the density \(f(y,\vec{x})\) by its estimate \(\hat f_n(y,\vec{x})\), we obtain an estimate for the desired regression curve /4/:

\[ \hat{\rho}_n(\vec{x}) = \int_{-\infty}^{\infty} y \hat f_n(y,\vec{x})\,dy \bigg/ \int_{-\infty}^{\infty} \hat f_n(y,\vec{x})\,dy . \tag{2.2} \]

Here the index \(n\) means that the estimates \(\hat f_n(y,\vec{x})\) and \(\hat{\rho}_n(\vec{x})\) are constructed on the basis of the sample (I.1) of size \(n\).

The estimates \(\hat f_n(y,\vec{x})\) may be chosen in various forms. However, when prior information is insufficient, it is desirable to have estimates with properties stable with respect to changes in this information. We shall dwell on the use of Parzen estimates

\[ \hat f(y,\vec{x}) = n^{-\frac{4}{k+5}} \sum_{i=1}^{n} \prod_{j=1}^{k} d_j K_j\!\left[ d_j n^{\frac{1}{k+5}}(x_j-x_{ij}) \right] d_0 K_0\!\left[ d_0 n^{\frac{1}{k+5}}(y-y_i) \right] \tag{2.3} \]

Here the smoothing coefficients and kernels \(K_0,K_1,\ldots,K_k\) are chosen optimally /5/:

\[ K_j[\xi]=K[\xi]= \begin{cases} \dfrac{3}{4\sqrt{5}}\left(1-\dfrac{1}{5}\xi^2\right), & |\xi|\le \sqrt{5},\\[6pt] 0, & |\xi|\ge \sqrt{5}, \end{cases} \tag{2.4} \]

\[ j=0,1,\ldots,k . \]

The constant positive coefficients \(\{\alpha_j\}\) are determined by the form of the unknown density \(f(y,\vec{x})\). We shall retain the uncertainty with respect to these coefficients, so that, by choosing them, we can solve in the best way the basic problem of constructing a smoothing regression line. Note that for any \(0<\alpha_j<\infty,\ j=1,\ldots,K\), the estimate \(\hat f_n(y,\vec{x})\) converges as \(n\to\infty\) to the true density \(f(y,\vec{x})\), if the sample values (1.1) are independent for different \(n\).

Substituting (2.3) into (2.2), we obtain an estimate of the regression surface

\[ \hat{\rho}_n(\vec{x},\alpha)= \frac{ \displaystyle \sum_{i=1}^{n} y_i \prod_{j=1}^{K} K_j\left[\alpha_j n^{\frac{1}{K+3}}(x_i-x_{ij})\right] }{ \displaystyle \sum_{i=1}^{n} \prod_{j=1}^{K} K_j\left[\alpha_j n^{\frac{1}{K+3}}(x_i-x_{ij})\right] }. \tag{2.5} \]

Then the parameters \(\alpha=(\alpha_1,\ldots,\alpha_K)\) are chosen from the requirement of best smoothing (1.4).

The advantages of the described method for constructing nonlinear regression dependences.

1. The need to carry out a parametrization of the regression curve disappears. The parametrization procedure is rather complicated, requires the availability of considerable a priori information, and therefore in many cases researchers usually confine themselves to a linear or quadratic approximation.

2. With an optimal choice of the structure of the sparsity coefficients, the number of adjustable parameters \(\alpha_1,\ldots,\alpha_K\) has the dimension of the space of the “input” variables—\(K\), i.e., the problem of adjusting \(\alpha\) by criterion (1.4) coincides in dimension with the problem of constructing a linear regression equation. However, the nonlinear dependence on \(\alpha\) somewhat complicates the procedure of finding them. Below a nonparametric algorithm for their adjustment will be given, based on the use of an equation of type (2.5). It should be noted that, as \(n\) grows, the sensitivity of the quality function \(J(\alpha)\) with respect to the parameters \(\alpha\) decreases, and for any \(0<\alpha<\infty\) the conditions for convergence of the estimate \(\hat{\rho}_n(\vec{x},\alpha)\) to \(\rho(\vec{x})\) are satisfied. That is, if we have a large sample size \(n\), then the choice of \(\alpha\) becomes noncritical, and for any of their sets we obtain good quality of smoothing of the experimental values.

3. The resulting estimate of the regression curve always is—

is asymptotically unbiased and consistent (even in the presence of measurement errors of the “inputs”).

Let us now consider the algorithm for computing the \(K\) parameters \(\alpha_1,\ldots,\alpha_k\) from criterion (1.4). These parameters are random variables, since the sample size is finite, and the parameters \(\alpha\) themselves depend on the sample values (1.1)—random variables. \(J(\alpha)\) is also a random variable. Then, using the general scheme for constructing adaptive algorithms /6/, we obtain the iterative algorithm for adjusting \(\alpha\)

\[ \alpha_j[\ell]= \frac{ \sum_{i=1}^{\ell-1}\alpha_j[i]\, K\left[\beta(\ell-1)^{1/6}\left(J^*-J(\alpha[i])\right)\right] }{ \sum_{i=1}^{\ell-1} K\left[\beta(\ell-1)^{1/6}\left(J^*-J(\alpha[i])\right)\right] } +\Delta\alpha_j[\ell] \tag{2.6} \]

\[ \ell=2,3,4\ldots\ldots \]

Here: \(\ell\) is the iteration number; \(K(\xi)\) is the kernel (2.4); \(\beta\) is a positive constant, one way of choosing which is given below; \(J(\alpha[i])\) is the quality function \(J\), calculated at \(\alpha_1=\alpha_1[i]\), \(\alpha_2=\alpha_2[i]\), \(\ldots\), \(\alpha_k=\alpha_k[i]\);

\[ J^*=\min\{J(\alpha[1]),\ldots,J(\alpha[\ell-1])\}. \tag{2.7} \]

In algorithm (2.6), in the first operations the quantity \(J^*\) may be set equal to zero.

We describe the initial stage of the computation by (2.6). We choose \(\alpha[1]\) arbitrarily and compute \(J(\alpha[1])=J_1\). We choose \(\beta\) from the condition that the denominator of (2.6) at \(\ell=2\) not be equal to zero:

\[ 0<\beta<\frac{\sqrt{5}}{J_1} \tag{2.8} \]

Then, in the following iterations, the selected \(\beta\) is kept constant. We set \(J^*=0\), compute \(\alpha[2]\), etc.

3. Example. There is a series of measured values of a certain time-varying variable

\[ y(t_1),\,y(t_2),\,\ldots,\,y(t_n). \tag{3.1} \]

The dependence on time is unknown, and it becomes necessary to filter out the measurement errors, i.e., to obtain a smoothed function of time \(\hat y(t)\). We obtain the solution by means of the algorithm (2.5)

\[ \hat y(t)= \frac{ \sum_{i=1}^{n} y(t_i)\, K\left[\alpha n^{1/6}(t-t_i)\right] }{ \sum_{i=1}^{n} K\left[\alpha n^{1/6}(t-t_i)\right] }, \tag{3.2} \]

where $\alpha$ is a scalar parameter whose calculation can be carried out by formula (2.6).

4. Conclusion

The general approach to the construction of algorithms for smoothing experimental results described in this work makes it possible to build computational schemes that are easily implemented on a computer. The method is universal; therefore, the author recommends using it in the study of complex, poorly studied processes in objects of chemistry and chemical technology.

At present a standard procedure for the smoothing algorithm is being developed. Its implementation will enable chemical engineers and chemical technologists, within a short time, to process large arrays of experimental data on a computer in order to calculate the mathematical relationships between parameters.

References

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2. E. A. Nadaraya. Remarks on nonparametric estimates of probability density and regression curve. Theory of Probability and Its Applications, vol. 15, no. 1, 1970.

3. V. P. Zhivoglyadov, A. V. Medvedev. Nonparametric algorithms for identification, pattern recognition, and dual control. In: Investigation and Optimization of Stochastically Distributed Systems. ILIM Publishing House, Frunze, 1971.

4. N. N. Aprusheva, V. D. Konakov. The use of nonparametric estimates in regression analysis. Factory Laboratory, 24, no. 5, 1973.

5. A. P. Sorin. On the use of nonparametric density estimates in pattern-recognition problems. Proceedings of SFTI, issue 63, Mathematical Statistics and Its Applications, 1973.

6. V. P. Zhivoglyadov, A. V. Medvedev. Nonparametric stochastic algorithms for control and extremum search. Automation and Computer Engineering, no. 2, 1973.