Error control

One of the important methods for studying the mineralogical properties of ore processes is the content of the main useful and harmful components in the ore under the microscope, the state of existence, the mineral grain size, the mosaic relationship and the continuous and dissociation of the ore during the crushing process. The situation quickly makes reliable conclusions. One method often used to answer the above questions is to observe a certain amount of mineral particles under a microscope. From the most prudent assumption, in order to make a complete and accurate judgment, the ideal method is to observe all (mineral) particles of the ore sample studied. It is only this practice that is difficult to achieve in practical work. Because in a certain sense, for any ore sample, on the issue of "number of particles", it is really a "infinity". Therefore, it is usually the case that a limited amount of particles in the sample are observed with a microscope. The number of particles can be determined in two ways. The empirical approach is to take 1000 to 1500 observation points; the other is to obtain a reasonable sample observation based on mathematical statistics.
First, the distribution function of the parameter mark quantity characterizes the process mineralogy parameter of a certain nature of ore, which is the final representative value after the mirror observation data is sorted and sorted. For example, the average particle size and particle size distribution histogram is the result of mathematical processing of the mineral particle size after each test. The particle size value of the particles can be referred to as the mark amount of the last two parameters. For a completely similar understanding, the number of mineral particles in a monomer is naturally the sign of the monomer dissociation parameter.
The minerals formed by various mineralizations on the earth's crust are the products of a combination of various factors in time and space. Its own elemental composition and its crystal structure have strong statistical regularity. Therefore, the markings of various process mineralogical parameters determined by it also appear without exception: the continuous change in size, the significant difference, and the inevitable contingency of whether a certain marker value can become an observation value. . Therefore, the parameter flag is actually a continuous random variable, which is now represented by x.
To this end, let ω 1 , ω 2 ,..., ω m be a sequence of independent random variables with specific mathematical expectations and variances.


The growth of minerals in ore is greatly affected by various factors as follows. These factors are the concentration of useful elements of the ore-forming solution, the temperature at which it is crystallized, the pressure, the cooling rate, the type of medium surrounding the element, the amount, and the pH of the solution. All of these factors affect minerals independently of each other. The process mineralogy of the ore that is assembled from minerals is the sum of the effects of many of these accidental factors.
If z 1 , z 2 ,...,z m ... respectively represent the respective parts of the parameter quality that can be affected by the above factors. then:
X = z 1 z 2 ...z m ...
Take the logarithm on both sides:
Lnx = lnz 1 +lnz 2 +...lnz m +...
Let lnx=y,lnz 1 =ω 1 ,lnz 2 =ω 2 ...
Therefore: y = ω 1 + ω 2 +... ω n +...
We know that ω 1 , ω 2 ..., ω n ... do not contain factors that have a significant influence on the amount of the mark. Thus, the conditions of the above conclusions are satisfied. So y obeys a normal distribution. It goes without saying that the antilog x of y will inevitably obey the log-normal distribution.
It can be concluded that the process mineralogical parameter mark x of the ore obeys a lognormal distribution. The distribution function of this distribution is:

Where σlnx is the variance of the logarithm of the marker value of the process mineralogical parameter;
Lnxg - the logarithm of the geometric mean of the process mineralogy parameter.

Second, the theoretical basis of the program design
Mathematical statistics is the science of studying the statistical laws of things or phenomena. Through the study of limited observation data, it finds out the statistical laws, so as to infer the whole of the things studied and quantitatively reflect the basic characteristics of things. The problem now is to make the actual mineralogy characteristics of the whole (ore sample), such as mineral chemical composition, particle size, shape, monomer dissociation degree, etc., based on the sample (microscopic observation) data. judgment. But the sample is only a part of the whole, from which to infer the overall characteristics, of course, it is impossible to have 100% confidence. Conversely, there may be erroneous judgments of one kind or another. For example, if the ore dissociation degree of the ore sample screening product has passed, we may make a judgment based on the sample. Such errors are mathematically referred to as "first type of errors"; on the other hand, it is also possible to sift products that are in fact unqualified by the dissociation of the monomer, and accept them as Qualified products by mistake. It is called "the second type of error" in mathematical statistics. The probability of occurrence of two types of errors is represented by a, β, respectively.

a = p(u- â–³ u>b>u+ â–³ u ), actual H+ â–³ lnx >h>H- â–³ lnx

β = p(u- △ u>b>u+ △ u), actual H+ △ lnx >h>H- △ lnx [next]

If L(H; m, u) is expressed , the acceptance probability for the case (u- Δ u Δu , actual H- Δ lnx > h > H + Δ lnx ) is expressed . then:


β = L(H; m, u)

For the problem studied, it can be proved:

Where H is a certain mineralogical parameter of the ore;
â–³ lnx - the sampling error range;
H——the best observed value of the parameter;
u - the best value of the sampling indicator;
△u——index tolerance;
B——sampling indicators;
m - sample size (number of mineral particles observed under the microscope);
t - the probability of guaranteeing the degree.

Since L(H;m,u) is a monotonic decreasing function of H, if a scheme can be found, the probability of occurrence of two kinds of errors at two control points is equal to the required a 0 and β 0 respectively . And the two observation points a, β of the scheme are controlled, then the probability of occurrence of the two types of errors in the scheme is controlled.
In order to make the observation data under one sampling mirror more effective and reliable (that is, to improve the accuracy of the estimator as much as possible, and to reduce the types of errors that may be made when the observation under the microscope is observed), in addition to requiring the sample to be independent and representative. There is also a requirement for sample size. To improve the accuracy and reliability of observational data, on the one hand, it is necessary to improve the statistical analysis method so that it can make full use of the information provided by the sample to estimate and judge the sample as a whole; on the other hand, how to be in certain human and material conditions. Get more useful information. Adding the front and the back together is to choose the most reasonable sample size (the best observation of mineral particles) for a specific problem.
III. Scheme Design For the sample to be analyzed, M can be used to represent the total number of particles of a certain mineral in the ore sample, then (M; m, u) constitutes a complete sampling observation scheme. For the ore sample, some mineral particles are M. The m particles are randomly observed, and the sampling index of a certain parameter is tested as b. If u-â–³u b>u+â–³u, the observation data under the microscope is considered to be Letter, the results are not acceptable. At M
In the case of fixed, m, u change, one program (M; m, u) will have --- a scheme to choose from.
In the total M particles of the sample, the total and index of a certain mineralogical parameter of a process is B, then B/M=H. When among the M particles, m are randomly observed under the microscope, and the sampling index is tested as b. At this time, b/m=h is the observed value of the parameter in the sample. The question is, can the observed value h=b/m be used as an estimate of the overall H=B/M of the sample? If so, how large is the sampling error range Δ lnx for this estimate? For the former problem, it is necessary to specifically study the mathematical expectation E(H)=E(b/m); and the solution of the latter problem needs to be calculated for the mean square error σh.
Here b can be regarded as a random variable (b=0, 1, 2, ..., m), and can also be regarded as the sum of m independent random variables x 1 , x 2 , x 3 ,..., x m :
b= x 1 +x 2 +x 3 +...+x m

The value of the random variable xi(i=1,2,3,...,m) depends on the i-th observed mineral particle condition, and B is the total and index of the sample population:

The estimated total values ​​of the samples and the indicators that may be obtained from the sample are: [next]
Estimated amount of B = ——b

In fact, the mathematical expectation of this estimate is the total of the sample and the indicator B:

E(——) = B = MH

So E(h) = E(--) = H
b b B
That is, the mathematical expectation of h = - is the parameter H. Thus h = - can be seen as an estimate of the parameter H = -. On the other hand, the sample capacity m m M
The amount m divided by the sum of the x's (sampling index b) can also be regarded as the average value of each x.

— 1 b
x = -—(x 1 +x 2 +x 3 +...+x m ) = — = h
m m