Method, computer program, and storage medium for estimating randomness of function of representative value of random variable by the use of gradient of same function

• US 6,675,126 B2
• Filed: 03/27/2001
• Issued: 01/06/2004
• Est. Priority Date: 03/27/2001
• Status: Expired due to Fees

• Current Assignee: Kabushiki Kaisha Toyota Chuo Kenkyusho (Toyota Motor Corporation)
• Original Assignee: Kabushiki Kaisha Toyota Chuo Kenkyusho (Toyota Motor Corporation)
• Inventors: Roser, Christoph Hermann
• Primary Examiner(s): Hoff, Marc S.
• Assistant Examiner(s): Charioui, Mohamed
• Application Number:
  US09/817,226
• Publication Number:
  US 20020143495A1
• Time in Patent Office: 1,015 Days
• Field of Search: 702/179, 702/181, 702/182, 702/183, 702/FOR.139, 706/13, 706/20, 706/27, 704/236, 704/239, 704/240, 704/255
• US Class Current: 702/179
• CPC Class Codes: G06F 17/18 for evaluating statistical data , e.g. average values, frequency distributions, probability functions, regression analysis forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04

Abstract

A method of estimating a measure of randomness of a function of at least one representative value of at least one random variable is constructed to have the steps of obtaining the at least one random variable; determining the at least one representative value of the obtained at least one random variable; determining a first statistic of the obtained at least one random variable; determining a gradient of the function with respect to the determined at least one representative value; and transforming the obtained first statistic into a second statistic of the function, using the determined gradient. The step of transforming may be adapted to transform the first statistic into the second statistic, such that the second statistic responds to the first statistic more sensitively in the case of the gradient being steep than in the case of the gradient being gentle.

Claims

1. A method of estimating a measure of randomness of a function of at least one representative value of at least one random variable, a plurality of individual data values of which are randomly distributed, the method comprising:

a step of obtaining the at least one random variable;

a step of determining the at least one representative value of the obtained at least one random variable, using a computer;

a step of determining a statistic of the obtained at least one random variable, using the computer;

a step of determining a gradient of the function with respect to the determined at least one representative value, using the computer; and

a step of estimating the measure of randomness of the function, wherein the step of estimating includes transforming by the computer the obtained statistic of the at least one random variable into a statistic of the function, using the determined gradient.

2. The method according to claim 1, further comprising a step of estimating the measure of randomness of the function of the at least one representative value, on the basis of the statistic of the function.

3. The method according to claim 2, applied to a simulation for discrete event, results of which simulation is used to effect the method, wherein the step of estimating comprises estimating the measure of randomness using results of only one execution of the simulation.

4. The method according to claim 3 wherein an accuracy to be satisfied with the statistic of the function is predetermined, and the step of determining a statistic comprises:

(a) determining the statistic of the at least one random variable, on the basis of a sum of individual data values belonging to the at least one random variable;

(b) determining the statistic of the at least one random variable on the basis of the sum, upon adding to the sum at least one new individual data value belonging to the at least one random variable;

(c) determining the statistic of the at least one random variable when at least one new individual data belonging to the at least one random variable becomes available during the simulation;

(d) transforming the determined statistic of the at least one random variable into the statistic of the function; and

(e) terminating the simulation when the predetermined accuracy is satisfied with the statistic of the function.

5. The method according to claim 2 wherein the measure of randomness comprises a range of a confidence interval of the function of the at least one representative value.

6. The method according to claim 1, wherein the function is a function of a plurality of random variables, the step of transforming comprising:

(a) determining a measure of randomness of each one of the plurality of random variables at or in the vicinity of a representative value of each one of the obtained plurality of random variables, as the statistic of each random variable;

(b) determining a measure of dependence between the plurality of random variables; and

(c) transforming the determined measures of randomness of the plurality of random variables into a measure of randomness of the function, using the determined measure of dependence and the determined gradient.

7. The method according to claim 6, wherein the measure of randomness of the each random variables comprises at least one of a maximum likelihood estimator of a variance of the each random variable, an unbiased estimator of the variance, a maximum likelihood estimator of a standard deviation of the each random variables, an unbiased estimator of the standard deviation, a variance of a representative values of the each random variable, a standard deviation of a representative value of the each random variable, a coefficient of variation of the each random variable, a general central moment of the each random variable, a confidence interval of the each random variable, a set of data indicative of the each random variable, a probability density function of the each random variable, and a cumulative density function of the each random variable.

8. The method according to claim 6, wherein the measure of dependence comprises at least one of an unbiased estimator of a covariance of the plurality of random variables, a maximum likelihood estimator of the covariance, and a correlation coefficient of the plurality of random variables.

9. A computer program to be executed by a computer to effect the method according to claim 1.

10. A computer-readable storage medium having stored therein the computer program according to claim 9.

11. The method according to claim 1, wherein the step of transforming comprises transforming the statistic of the at least one random variable into the statistic of the function, such that the statistic of the function responds to the statistic of the at least one random variable more sensitively in the case of the gradient being steep than in the case of the gradient being gentle.

12. The method according to claim 1, wherein each one of the representative value of the each random variable comprises at least one of an average, an arithmetic mean, a geometric mean, a median, a harmonic mean, and a mode, of each one of the at least one random variable.

13. The method according to claim 1, wherein the step of determining the representative value comprises determining the at least one representative value of the at least one random variable upon truncating at least one part of individual data values belonging to the at least one random variable.

14. The method according to claim 1, wherein the statistic of each one of the at least one random variable comprises at least one of a standard deviation, a confidence interval, a set of data, a probability density function, and a cumulative density function, of the each random variable.

15. The method according to claim 1, wherein the statistic of the function comprises at least one of a standard deviation, a confidence interval, a set of data, a probability density function, and a cumulative density function, of the function.

16. The method according to claim 1, wherein the function is a function of a plurality of random variables, the step of transforming comprises transforming the obtained statistic of the plurality of random variables into the statistic of the function, without a calculation of a measure of dependence between the plurality of random variables.

17. The method according to claim 1, applied to an analysis of a plurality of business models to be accepted in realizing a given business, wherein a function of at least one of random variable is predetermined for each one of the plurality of business models, and the function of a representative value of the each random variable for one of the plurality of business models is to be compared with the function of a representative value of the each random variable for another of the plurality of business models.

18. The method according to claim 1, wherein the at least one random variable represents a state of a machine in a manufacturing machine.

19. A method of determining a set of data of a function of a representative value of each one of at least one random variable, which set of data allows an evaluation of a statistic of the function, wherein a plurality of individual data values of the at least one random variable are randomly distributed, the method comprising:

a step of obtaining a set of individual data values belonging to each random variable, which set represents an approximation of a distribution of the each random variable;

a step of determining the representative value of the each random variable, using a computer;

a step of determining a gradient of the function with respect to the determined representative value, using the computer; and

a step of transforming the obtained set of individual data values into the set of data representing the function, using the computer.

20. The method according to claim 19, further comprising a step of estimating a measure of randomness of the function of the representative value, on the basis of the set of data representing the function.

21. The method according to claim 20, wherein the measure of randomness comprises a range of a confidence interval of the function of the representative value.

22. The method according to claim 20, applied to a simulation for discrete event, results of which simulation is used to effect the method, wherein the step of estimating comprises estimating the measure of randomness using results of only one execution of the simulation.

23. The method according to claim 19 wherein the step of transforming the set of individual data values of the each random variable into the set of data representing the function, such that the set of data representing the function responds to the set of individual data values more sensitively in the case of the gradient being steep than in the case of the gradient being gentle.

24. The method according to claim 19, wherein the at least one random variable represents a state of a machine in a manufacturing machine.

25. A method of estimating a measure of randomness of at least one random variable to satisfy a predetermined condition regarding a measure of randomness of a function of at least one representative value of the at least one random variable, wherein a plurality of individual data values of the at least one random variable are randomly distributed, and wherein the predetermined condition is formulated to define a central location and a measure of dispersion, of a distribution of the function, the method comprising:

a step of determining a gradient of the function with respect to the defined central location, using a computer; and

a step of estimating the measure of randomness of the function, wherein the step of estimating includes determining the measure of randomness of the at least one random variable, on the basis of the determined gradient and the defined measure of dispersion, using the computer.

26. The method according to claim 25, wherein the step of determining the measure comprises transforming the defined measure of dispersion into the measure of randomness of the at least one random variable, such that the measure of randomness of the at least one random variable responds to the defined measure of dispersion more sensitively in the case of the gradient being steep than in the case of the gradient being gentle.

27. The method according to claim 25, wherein the measure of dispersion comprises at least one of a standard deviation, a confidence interval, a set of data, a probability density function, and a cumulative density function, of the function.

28. The method according to claim 25, wherein the measure of randomness of each one of the at least one random variable comprises at least one of a standard deviation, a confidence interval, a set of data, a probability density function, and a cumulative density function, of the each random variable.

29. The method according to claim 25, wherein the at least one random variable represents a state of a machine in a manufacturing machine.