STATISTICAL RESOLVING POWER OF THE APN METHOD

A. A. Kaplin, A. A. Zheltonozhko, A. I. Ruban

Resolving power (RP) is one of the principal characteristics of physical and physicochemical methods of analysis. Questions of RP in polarography have been considered mainly as applied to alternating-current variants /1,2/. The criterion of static RP is taken to be the minimum difference between the potentials of half-waves or peaks at which, for a fixed concentration ratio, one of the elements can be determined with a specified error. Similarly, as the measure of RP with respect to concentration, at fixed $\varphi_{1/2}$ or $\varphi_n$, the limiting concentration ratio is taken at which determination of the elements can be carried out with a specified error. As applied to the APN method, an analogous approach, which may be called deterministic, was developed in works /3,4/. As the value of the limiting RP in the parametric theory of the APN method /5/, the ratio of the anodic oxidation currents of two elements was proposed, corresponding to the registration of a horizontal plateau near the peak of the element being determined; the anodic peaks are approximated by a Gaussian-type function. This approach is partly deterministic, since the calculations use a number of experimentally determined mean values $\delta_I$, $\delta_2$, $\varphi_{n_1}$, $\varphi_{n_2}$. In both approaches the cathodic or anodic current of one of the elements is regarded as the source of systematic error in measuring the other. A fully deterministic approach to the estimation of static RP makes it possible to calculate the theoretical RP as a function of a number of parameters of the anodic oxidation stage $(D, W, \tau, K_s, \ldots)$; however, the theoretical limit of RP is, as a rule, not attained in experiment because of random fluctuations of the anodic currents of the elements being determined under experimental conditions.

We propose a statistical approach to the estimation of limiting RP, based on solving the problem of detecting a signal against a background of interference. The current $i_{1(\varphi_2)}$ of the interfering element (1) and the total current $I_{1,2(\varphi_2)}$, recorded at the potential $\varphi_2$ of the element being determined (2), after introduction of the element being determined (2), are regarded as random variables. It is required to determine the mi-

the minimum difference \(I^{o}_{2}(\varphi_2)\) between the two signals, taking into account their statistical character. After calculating \(I^{o}_{2}(\varphi_2)\), the value of the statistical RP for the tone \(\rho_{\mathrm{э}}\) can be estimated from the following relations:

\[ \rho_{\mathrm{э}}=\frac{I_{1}(\varphi_{1})}{I^{o}_{2}(\varphi_{2})}, \tag{1} \]

or

\[ \rho_{\mathrm{э}}=C_{1}\frac{K}{I^{o}_{2}(\varphi_{2})}, \tag{2} \]

where \(C_{1}\) is the fixed concentration of element (I) in the solution. Taking into account the expression

\[ C_{0}=I^{o}_{2}K^{-1} \tag{3} \]

the relation of the RP in terms of concentrations has the form

\[ \rho_{c}=C_{1}\frac{K}{I^{o}_{2}(\varphi_{2})}, \tag{4} \]

where, in accordance with the parametric theory,

\[ K_{i}=K_{a,i}\cdot S\gamma Vv^{-1}, \tag{5} \]

\[ \gamma=1-e^{-B}, \tag{6} \]

\[ B=\frac{Ke}{ZF}\cdot S_{0}\cdot\frac{\tau}{V}. \tag{7} \]

Thus, the estimation of \(\rho_{c}\) and \(\rho_{\mathrm{э}}\) reduces to determining the detection limit \(I^{o}_{2}(\varphi_2)\). For estimating the detection limit, a large number of relations have been proposed in the literature\(^{1)}\), especially for the spectral method, which to one degree or another make use of the statistical characteristics of the useful signal and of the interference; as the interference, a “blank experiment” is often considered. Practically, apparently, all relations for the detection limit can be used to estimate \(I^{o}_{2}(\varphi_2)\) and subsequently calculate the limit of statistical RP. However, these relations for \(I^{o}_{2}(\varphi_2)\), as we have already noted earlier /7/, have a number of shortcomings: the use of large samples for estimating variances, the absence in a number of relations of such variables as the specified probability of correct detection and the probability of a “false alarm,” allowance for the statistical characteristics of both the useful signal and the interference, and the absence of a formulation of problems for the search for optimal conditions for detecting the useful signal. Naturally, these shortcomings are also retained when using

DERIVATION OF RELATIONS FOR ESTIMATING RP IN THE APN METHOD

In the present work it is proposed to derive relations for

\[ I_2^c(\psi_2)=\Delta m, \]

free of a number of the enumerated shortcomings, and relations for statistical RP.

Let there be two random independent samples²:

\[ \{x_1,x_2,\ldots,x_{n_1}\}=\{x_i\} \quad \text{and} \quad \{y_1,y_2,\ldots,y_{n_2}\}=\{y_j\}, \]

with mathematical expectations, respectively, \(m_x, m_y\), and variances \(\sigma_x^2\) and \(\sigma_y^2\). The unknown parameters are \(m_x\) and \(m_y\). It is necessary to establish whether \(m_x\) and \(m_y\) are equal and, if they are not equal, to estimate the detection limit—the minimum value of their difference

\[ \Delta m=m_x-m_y, \]

recognized with a prescribed guarantee \(P\).

We shall perform preliminary processing of the samples by forming the random variable

\[ Z=\bar{x}-\bar{y}, \qquad \bar{x}=\frac{1}{n_1}\sum_{i=1}^{n_1} x_i, \qquad \bar{y}=\frac{1}{n_2}\sum_{j=1}^{n_2} y_j, \tag{8} \]

where \(Z\) has the following first moments:

\[ m_z=m_x-m_y, \qquad \sigma_z^2=\frac{\sigma_x^2}{n_1}+\frac{\sigma_y^2}{n_2}. \tag{9} \]

To solve the problem it is necessary to use the following two simple hypotheses: \(H_0\)—the hypothesis that \(m_z=0\);

\(H_I\)—the hypothesis that \(m_z\), with the prescribed probability \(P\), is equal to the minimally recognizable value \(\Delta m\).

The variances \(\sigma_x^2\) and \(\sigma_y^2\) in formula (9) were assumed unknown. Otherwise we have their estimates

\[ S_x^2=\frac{1}{n_1-1}\sum_{i=1}^{n_1}(x_i-\bar{x})^2, \qquad S_y^2=\frac{1}{n_2-1}\sum_{j=1}^{n_2}(y_j-\bar{y})^2. \tag{10} \]

Taking into account the unbiasedness of the estimates \(S_x^2, S_y^2\), an unbiased estimate for \(\sigma_z^2\) is

\[ S_z^2=\frac{S_x^2}{n_1}+\frac{S_y^2}{n_2}. \tag{11} \]

The subsequent solution of the problem of recognizing the two hypotheses \(H_0\) and \(H_I\) should be carried out having only the estimates \(S_x^2\) and \(S_y^2\) available. The solution depends essentially on whether the hypothesis of equality of the variances \(\sigma_x^2\) and \(\sigma_y^2\) is satisfied.

Some information from the statistical theory of recognition of two hypotheses /9/. Choice of recognition rules

Introduce the following notation:

\(f(Z/H_i)=\varphi_i(Z)\) is the probability density of the random variable \(Z\), if hypothesis \(H_i\) is true \((i=0; I)\);

\(\rho_i\) is the a priori probability that hypothesis \(H_i\) is realized \((i=0; I)\);

\(P_0+P=I;\quad \vec C=\begin{vmatrix} C_{00} & C_{01}\\ C_{10} & C_{11}\end{vmatrix}\) is the payoff matrix (loss matrix);

\(C_{ij}\) are the losses in the case of accepting hypothesis \(H_j\), if hypothesis \(H_i\) is true; \(C_{ii}\le\) — negative losses are regarded as gains under a correct decision; \(Z\) is the domain of variation of the random variable \(Z\); \(Z_i\) is the subdomain of variation \(Z\), upon falling into which a conclusion is drawn that hypothesis \(H_i\) is correct.

The value of the mean risk /8/, characterizing the process of recognizing two simple hypotheses,

\[ R= \int_{Z_0} C_{00}\rho_0\varphi_0(Z)\,dZ + \int_{Z_1} C_{01}\rho_0\varphi_0(Z)\,dZ + \int_{Z_1} C_{11}\rho_1\varphi_1(Z)\,dZ + \int_{Z_0} C_{10}\rho_1\varphi_1(Z)\,dZ \tag{12} \]

For fixed \(n\), from the condition of the minimum of the mean risk with respect to the decision rule, we obtain the equation of the surface that divides the domain \(Z\) into two parts: \(Z_0\) and \(Z_1\),

\[ \varphi_{01}(z)=(C_{00}-C_{01})\rho_0\varphi_0(z) +(C_{10}-C_{11})\rho_1\varphi_1(z)=0, \tag{13} \]

and the optimal decision rule takes the form:

hypothesis \(H_0\) is accepted (or \(Z\in Z_0\)) if \(\varphi_{01}(Z)\ge 0\); hypothesis \(H_I\) is accepted (or \(Z\in Z_1\)) if \(\varphi_{10}(Z)<0\) (14).

In the equation for the mean risk (12), \(\int_{Z_1}\varphi_0(Z)\,dZ=\alpha\) is the error of the first kind (the probability that hypothesis \(H_I\) is accepted while hypothesis \(H_0\) is true);

\(\int_{Z_0}\varphi_0(Z)\,dZ=1-\int_{Z_1}\varphi_1(Z)\,dZ=1-\alpha\) is the probability of a correct decision with respect to hypothesis \(H_0\);

\(\int_{Z_1}\varphi_1(Z)\,dZ=\beta\) is the error of the second kind (the probability of erroneous acceptance of hypothesis \(H_0\) while hypothesis \(H_I\) is true);

\(\int_{Z_1}\varphi_1(Z)\,dZ=1-\beta\) is the probability of a correct decision with respect to \(H_I\).

The decision rule (14) can be represented in the form of a threshold procedure: if \(l(z)\ge h\), then hypothesis \(H_0\) is accepted; if

\[ \ell(z) < h, \]

then hypothesis \(H_1\) is accepted. Here

\[ \ell(z)=\frac{\varphi_0(z)}{\varphi_1(z)} \quad\text{is the likelihood ratio,} \]

\[ h=\frac{C_{01}-C_{11}}{C_{01}-C_{00}}\cdot\frac{P_1}{P_0} \quad\text{is the threshold.} \tag{16} \]

In the decision rule (15) it is sometimes more convenient to use not \(\ell(z)\) and \(h\), but \(\ln \ell(z)\) and \(\ln h\). The Bayesian procedure given above can be applied if the a priori probabilities \(P_0\) and \(P_1\) are known. The payoff matrix is specified by the investigator or is adjusted in the process of training /8,9/.

From the general approach presented, based on minimizing average losses, follow most of the existing rules for recognizing two hypotheses /6,7/. The recognition of several hypotheses is carried out analogously.

The minimax rule. If the a priori probabilities \(P_0\) and \(P_1\) are unknown, then the minimax rule is used. The maximum of the average risk with respect to the a priori probability \(P_0\) is computed (where \(P_1=1-P_0\)), i.e., orientation is toward the worst case. Then the obtained value of the average risk is minimized with respect to the decision rule. As a result, the recognition procedure remains the threshold one (15), but the threshold satisfies an additional relation obtained from the condition of the maximum of the average risk with respect to \(P_0\). Suppose that \(C_{11}=C_{22}=0\), i.e., only losses caused by an incorrect decision are of interest. Then

\[ R=C_{01}P_0\alpha+C_{10}(1-P_0)\beta \tag{17} \]

one chooses such a value \(P_0\) that ensures the maximum of \(R\). In this case \(P_0\) satisfies

\[ C_{21}\beta=C_{12}\alpha . \tag{18} \]

Next we apply rule (15), where

\[ h=\frac{C_{21}}{C_{12}}\cdot\frac{1-P_0^*}{P_0^*}. \tag{19} \]

Recognition of hypotheses by the minimax rule under a normal distribution law for samples

Let us turn to the solution of the hypothesis-recognition problems posed earlier, taking \(C_{12}=C_{21}\). We assume that the sample \(\{x_i\}\)

has a normal distribution law¹ with mathematical expectation \(m_x\) and variance \(\delta_x^2-N(m_x,\delta_x^2)\). We make an analogous assumption with respect to the sample \(\{y_i\}-N(m_y,\delta_y^2)\). In this case the random variable \(Z\) has the normal distribution law \(N(m_z;\delta_z^2)\) with parameters (9). From condition (18) it follows that \(P_0=P_I=1/2\). As a result, the threshold recognition procedure (15) takes the form:

if \(Z \le \dfrac{\Delta m}{2}\), then hypothesis \(H_0\) is accepted;

if \(Z > \dfrac{\Delta m}{2}\), then hypothesis \(H_I\) is accepted, where the value \(\Delta m\), minimally recognizable with a prescribed probability \(P\), is determined from the expression for the probability of the correct decision with respect to \(H_0\):

\[ P=1-\alpha=\int_{-\infty}^{\Delta m/2}\varphi_n(z)\,dz =\Phi\!\left(\frac{\Delta m}{2\delta_z}\right)=\Phi(u_{1-\alpha}). \tag{20} \]

In relation (20),

\[ \Phi(j)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{j} e^{-t^2/2}\,dt \]

is the tabulated distribution function of the normal law; \(u_{1-\alpha}\) is the quantile determined from tables of the function \(\Phi(j)\) for the prescribed probability \(P\) (or \(\alpha\)). Taking (20) and (9) into account,

\[ \Delta m=u_{1-\alpha}\cdot 2\delta_z =2u_{1-\alpha}\sqrt{\frac{\delta_x^2}{n_1}+\frac{\delta_y^2}{n_2}}. \tag{21} \]

Let us consider the practically important case when the variances \(\delta_x^2\) and \(\delta_y^2\) are unknown, but estimates for them are known. In this case the solution of the problem of recognizing the hypotheses depends on whether or not \(\delta_x^2\) and \(\delta_y^2\) are equal.

1. Suppose that \(\delta_x^2=\delta_y^2=\delta^2\). The unbiased estimate of \(\delta^2\) is determined as

\[ S^2= \frac{\displaystyle\sum_{i=1}^{n_1}(x_i-\bar{x})^2+ \displaystyle\sum_{j=1}^{n_2}(y_j-\bar{y})^2} {n_1+n_2-2}. \tag{22} \]

and the random variable

\[ T=\frac{\bar{x}-\bar{y}}{S\sqrt{n_1^{-1}+n_2^{-1}}} \tag{23} \]

has Student’s distribution law (independent of the unknown parameter \(\delta^2\)) with \(\nu=(n_1+n_2-2)\) degrees of freedom. Usually the tabulated value of Student’s distribution law (distribution density) is given for a random variable having

mathematical expectation equal to zero. Denote this density by \(\varphi'(t,\nu)\). As a rule, the a priori probabilities \(P_0, P_I\) are unknown in practical cases. Therefore we again turn to the minimax procedure, considering instead of \(Z\) the random variable \(T\) /see (23)/. In this case the ratio \(\varphi_0(z)\) is replaced by \(\varphi'(t,\nu)\), and \(\varphi_1(z)\) by \(\varphi'\left(t-\dfrac{\Delta m}{S\sqrt{n_1^{-1}+n_2^{-1}}},\nu\right)\). The decision rule takes the form:

hypothesis \(H_0\) is accepted if \(t \le \dfrac{\Delta m}{2S\sqrt{n_1^{-1}+n_2^{-1}}}\);

hypothesis \(H_I\) is accepted if \(t > \dfrac{\Delta m}{2S\sqrt{n_1^{-1}+n_2^{-1}}}\). \(\tag{24}\)

The magnitude \(\Delta m\), minimally recognizable with a given probability \(P=1-\alpha\), is determined from the relation

\[ P = 1-\alpha = \int_{-\infty}^{\dfrac{\Delta m}{2S\sqrt{n_1^{-1}+n_2^{-1}}}} \varphi'(t,\nu)\,dt, \tag{25} \]

i.e.,

\[ \Delta m \doteq 2t_{1-\alpha,\nu}\cdot S\left(n_1^{-1}+n_2^{-1}\right)^{1/2}, \tag{26} \]

where \(t_{1-\alpha,\nu}\) is the quantile of Student’s distribution with \(\nu=(n_1+n_2-2)\) degrees of freedom.

If there are two samples \(\{x_i\};\ \{y_i\}\), whose variances are not equal \(\sigma_x^2 \ne \sigma_y^2\) (this hypothesis is checked beforehand), then the recognition procedure will be approximate. Without discussing the question of the degree of approximation, we give the final formula for calculating the magnitude of the useful signal minimally recognizable with a given probability \((1-\alpha)\):

\[ \Delta m \simeq 2t_{1-\alpha,\nu}\cdot S_z;\qquad \nu=n_1+n_2-2, \tag{27} \]

where \(S_z\) is determined by expressions (10) and (11). The obtained relations for \(\Delta m\) are used to estimate the statistical resolving power in terms of current and concentration.

For example, the basic relations for the resolving power in terms of current are

\[ \rho_J = C_i\cdot \frac{K_{a,1}(\varphi_1)S_0\nu V} {2t_{1-\alpha,\nu}\cdot \vartheta} \cdot \left( \frac{\sigma_x^2}{n_1}+\frac{\sigma_y^2}{n_2} \right)^{-1/2}, \tag{28} \]

\[ \rho_J = C_i\cdot \frac{K_{a,1}(\varphi_1)S_0\nu V} {2t_{1-\alpha,\nu}S\vartheta} \cdot \left( \frac{1}{n_1}+\frac{1}{n_2} \right)^{-1/2}, \tag{29} \]

\[ \rho_J = C_i\cdot \frac{K_{a,1}(\varphi_1)S_0\nu V} {2t_{1-\alpha,\nu}\vartheta} \cdot \left( \frac{S_x^2}{n_1}+\frac{S_y^2}{n_2} \right)^{-1/2}, \tag{30} \]

where \(S_x\), \(S_y\), \(S\) are determined from (10), (11), and (22). The corresponding relations for \(\rho_c\) can be obtained by substituting in (4) the values \(\Delta m\) and \(K_{a,2}(\psi_2)\). Estimates of the resolving power with respect to the potential \(\varphi\) are cumbersome and are not given in the present work.

Thus, the relations for \(\rho_I\) and \(\rho_c\) make it possible to calculate the limiting statistical resolving power as a function of the deterministic characteristics of the anodic peaks of the interfering and determined elements \(\bigl[K_{a,1}(\psi_1); K_{a,2}(\psi_2)\bigr]\), the conditions under which concentration is carried out (\(S\)), the concentration of the interfering element in the solution (\(c_I\)), and the statistical characteristics of the anodic currents of both elements at the peak potential \((\psi_2)\) of the determined element.

For calculation of the limiting statistical resolving power in a specified interval of variation of the parameters, the following approach is possible. By means of experimental design, regression dependences \(K_{a,1}(\psi_1)\); \(K_{a,2}(\psi_2)\); \(\sigma_x\); \(\sigma_y\); \(S\); \(S_x\); \(S_y\) on the parameters under study are constructed. These dependences are substituted into the corresponding relations for \(\rho_y\) or \(\rho_c\). The subsequent optimization problems with respect to the parameters reduce either to finding the maximum resolving power, or to finding parameters at which the resolving power is equal to a specified value. For the case of specified values of \(c_I\) and \(I_2^{0}(\psi_2)\), the optimization reduces to finding the maximum of the function \(K_{a,1}(\psi_1)\) or \(K_{a,2}(\psi_2)\) from the anodic-current equations.

The approach described in this work makes it possible to compare the theoretical and experimental resolving power of various variants of the polarographic method.

Literature

1. S. B. Tsfasman. Electronic Polarographs. Moscow, Metallurgizdat, 1960.

2. V. V. Senkevich. ZhAKh, 26, 461, 1971.

3. V. E. Gorodovykh. Izv. TPI, 164, 104, 1967.

4. A. G. Stromberg, V. F. Yankauskas. Izv. TPI, 164, 99, 1967.

5. A. G. Stromberg, A. A. Zheltonozhko, A. A. Kaplin. ZhAKh, 6, 1045, 1973.

6. N. I. Tarasevich et al. Methods of Spectral and Chemical-Spectral Analysis. Moscow, Khimiya, 1973.

7. A. A. Kaplin, A. I. Ruban. Advances in Polarography with Accumulation. Tomsk, TPI Press, 1973.

1. Ya. Z. Tsypkin. Fundamentals of the Theory of Learning Systems. Moscow, Nauka, 1970.
2. Collected volume, Signal Reception in the Presence of Noise. Moscow, Foreign Literature Publishing House, 1960.
3. A. A. Kashin, A. N. Pokrovskaya, B. A. Kubrak, M. V. Krasheninnikov. Chemistry and Chemical Technology. Tomsk, Tomsk Polytechnic Institute Publishing House, 1973.

On the Theory of Pelletizing Wet Disperse Materials

V. M. Vityugin

The broad introduction of granulated disperse materials into metallurgy and chemical technology urgently requires the creation of a solid theoretical basis for the purposeful improvement of this process. The possibility of application and the effectiveness of the granulation process are determined first of all by the tendency of each specific disperse material to pelletization, i.e., by its pelletizability. Pelletizability ultimately also determines two principal technical-and-economic indicators of the process: the specific productivity of the equipment and the quality of the granulate. Thus, identifying the causes that determine the pelletizability of disperse materials, and developing a scientifically grounded method for assessing pelletizability, are primary tasks in the theory of granulation.

Criteria for assessing pelletizability must take into account both the properties of the disperse material and the specific features of the pelletizing process in mechanical granulators of the drum and pan types. In this sense, the water-physical characteristics of the disperse material and the structural-mechanical properties of the pelletized system acquire the greatest importance, respectively. The water-physical properties, expressed through characteristic moisture capacities, predetermine the strength of the raw granulate /1/. The plasticity of the material being pelletized, in turn, has a decisive influence on the rate of pelletizing, since the main feature of this process consists in the great freedom of development of bulk deformations of the forming granules; a stable process will be effective only in the case of predominant development of plastic deformations /2/.

Naturally, the water-physical and structural-mechani-

1. The correspondence of the distribution of the quantities of anodic peaks in the APN method to the normal law was shown in /10/. ↩

2. In the APN and PPN method, sample (2) refers to measurements of the current \(i_l(\psi_n)\) of element (I), and sample (1) to measurements of the total current after the introduction of the determining element (2). ↩