_{1}

One confusing question over a long period of time is how transfer the discrete function transfers into continuous function. Recently the issue has been resolved but some details of the transformation process will be introduced in the paper. The correlation coefficients of 100,000 values are established from the two groups of data with the range between -1 and 1, creating a histogram from these correlation coefficient values known as “the probability mass function.” The coefficient values are brought into the discrete distribution function, so that transfers into the discrete cumulative function, next converted into a continuous cumulative function, next which is differentiated to get the density function, so that it is easy to being research analysis. A model will be established during the process of the conversion what the medium is “the least squares algorithm.” Finally, when the integral of the area within the range of the density function equals to 1, this implies that the transformation complete succeeds from the discrete function to the continuous function.

The author has spent some time investigating the reason for the discrete function transferring into continuous function. Due to the method that is what discrete functions convert into continuous functions which are not easy to find in literature [

The data of two groups are composed of normal 100 sets and normal 1000 sets from historical data, each of which consists of 6 elements (H_{2}, C_{2}H_{4}, C_{2}H_{2}, CH_{4}, C_{2}H_{6}, CO). The correlation coefficient of the 100,000 cases have values fallen between −1 and +1 according to the correlation coefficient theory [

For the method of generating of the correlation coefficient value, please refer to the literature [

In order to find the most similar continuous accumulation function of the above discrete cumulative function, this article refers to Gaussian, Weibull, and others distrbution. The figure of the cumulative continuious function of Weibull distribution and which is widely employed as a model in testing. Maximum likelihood equations are derived for estimating the distribution parameters from complete samples, singly censored samples and progressively (multiple) censored samples. Asymptotic variance-covariance matrices are given for each of these sample types. An illustrative example is included [

F ( r ; λ ) = { ( e B ∗ r − 1 ) − 1 ≤ r ≤ 1 } (1)

The curves of cumulative for the actual data (red line) and exponential of standard (blue line) are shown in

F ( r ; A ; B ) = { A ∗ ( e B ∗ r − 1 ) − 1 ≤ r ≤ 1 } (2)

After the method of the least squaresalgorithm [

After the function was converted from discrete function to continuous, the function still needs verification by the integration of continuous function. First, the continuous accumulation function is differentiated into density function with adjacent re-integration from −1 to 1 of variables the value of the area equals to 1 (or approaching 1) and has been verified correct in the process, which is known as a successful conversion. The formula of the mathematical was been differentiated from continuous function, as shown the formula 3. The Y (A) stands for the area of the integration of f ( r ) that was shown below.

f ( r ) = 0.195 ∗ e 2.78 ∗ r (3)

Y ( A ) = ∫ − 1 1 f ( r ) d r = ∫ − 1 1 0.195 ∗ e 2.69 ∗ r d r ≅ 1

The discrete function transferring into continuous function is an important work in the process of analyzing of data and investigation in all fields. There are variously of ways for transfer but the paper only introduces which is called “the least square algorithm”. This method yields the corresponding coefficient of the Equation (2), while the conversion work is performed by density function. In this paper, the area value of the coverted probability density functin by integration from −1 to 1 is not meet with theoretical value 1, but approaching, found in the process of calculation. Nonetheless, the result of the transformation was not influential. To conclude briefly, the paper will provide the graphical of the program for reference below [

The author declares no conflicts of interest regarding the publication of this paper.

Lin, M.-J. (2018) Techniques of Discrete Function Transfers into Continuous Function in Practice. Engineering, 10, 680-687. https://doi.org/10.4236/eng.2018.1010049