Derivative of sine function: A graphical explanationHideo Hirose
The derivative of a sine function can be derived by using the limit for sine function. However, it seems difficult to understand this transformation. Thus, I have drawn a figure expressing the differentiation.
sine関数微分 d sin x / dx = cos x の説明は、sineの差の公式を積に変換して、sin x / x → 1 (x → 0) を使って説明されることが多い。
ここでは、図形的に示してみた。sin x / x → 1 (x → 0) が見えないだけになっているが、結局は、、、
Success/Failure Prediction for Final Examination using the Trend of Weekly On...Hideo Hirose
H. Hirose, Success/Failure Prediction for Final Examination using the Trend of Weekly Online Testing, 7th International Conference on Learning Technologies and Learning Environments (LTLE2018), pp.139-145, July 8-12, 2018.
Attendance to Lectures is Crucial in Order Not to Drop OutHideo Hirose
H. Hirose, Attendance to Lectures is Crucial in Order Not to Drop Out, 7th International Conference on Learning Technologies and Learning Environments (LTLE2018), pp.194-198, July 8-12, 2018.
Derivative of sine function: A graphical explanationHideo Hirose
The derivative of a sine function can be derived by using the limit for sine function. However, it seems difficult to understand this transformation. Thus, I have drawn a figure expressing the differentiation.
sine関数微分 d sin x / dx = cos x の説明は、sineの差の公式を積に変換して、sin x / x → 1 (x → 0) を使って説明されることが多い。
ここでは、図形的に示してみた。sin x / x → 1 (x → 0) が見えないだけになっているが、結局は、、、
Success/Failure Prediction for Final Examination using the Trend of Weekly On...Hideo Hirose
H. Hirose, Success/Failure Prediction for Final Examination using the Trend of Weekly Online Testing, 7th International Conference on Learning Technologies and Learning Environments (LTLE2018), pp.139-145, July 8-12, 2018.
Attendance to Lectures is Crucial in Order Not to Drop OutHideo Hirose
H. Hirose, Attendance to Lectures is Crucial in Order Not to Drop Out, 7th International Conference on Learning Technologies and Learning Environments (LTLE2018), pp.194-198, July 8-12, 2018.
How many times are we tossing coins until we observe head, tail, and head? It's ten. It's not eight. This is an intriguing result against our intuition.
Interesting but difficult problem: find the optimum saury layout on a gridiro...Hideo Hirose
Even though we use a simple but ridiculous problem finding the optimum saury baking layout on a fish gridiron by Joule heat, we can invoke the interest to science by combining electrical engineering, linear algebra and probability viewpoints. These elements are, use of solving linear equation and Poisson's equation, and applying the central limit theorem to this situation. In addition, by removing the constraints, we can create a new problem free from our common sense. Presenting funny but essential problems could be another aspect for active learning using the problem of the interdisciplinary scientific methods.
With 80 steps Galton boards, we can see the binomial distribution approximated to the normal distribution.
Youtube ===>>>
https://www.youtube.com/watch?v=3w4e1RQTAB8
The cumulative exposure model (CEM) is often used to express the failure probability model in the step-up test method; the step-up procedure continues until a breakdown occurs. This probability model is widely accepted in reliability fields because accumulation of fatigue is considered to be reasonable. Contrary to this, the memoryless model (MM) is also used in electrical engineering because accumulation of fatigue is not observed in some cases. We propose here a new model, the extended cumulative exposure model (ECEM), which includes features of both the described models. A simulation study and an application to the actual experimental case of oil insulation test support the validity of the proposed model. The independence model (IM) is also discussed.
Parameter estimation for the truncated weibull model using the ordinary diffe...Hideo Hirose
In estimating the number of failures using the truncated data for the Weibull model, we often encounter a case that the estimate is smaller than the true one when we use the likelihood principle to conditional probability. In infectious disease predictions, the SIR model described by simultaneous ordinary differential equations are often used, and this model can predict the final stage condition, i.e., the total number of infected patients, well, even if the number of observed data is small. These two models have the same condition for the observed data: truncated to the right. Thus, we have investigated whether the number of failures in the Weibull model can be estimated accurately using the ordinary differential equation. The positive results to this conjecture are shown.
A new incomplete data model, the trunsored model, in lifetime analysis is introduced. This model can be regarded as a unified model of the censored and truncated models. Using the model, we can not only estimate the ratio of the fragile population to the mixed fragile and durable populations, but also test a hypothesis that the ratio is equal to a prescribed value. A central point of the paper is that such a test can easily be realized through the newly introduced trunsored model, because it has been difficult to do such a hypothesis test under only the framework of censored and truncated models. Therefore, the relationship of the trunsored model to the censored and truncated models is clarified because the trunsored model unifies the censored and truncated models. The paper also shows how to obtain the estimates of the parameters in lifetime estimation, and corresponding confidence intervals for the fragile population. Typical examples applied to electronic board failures, and to breast cancer data, for lifetime estimation are demonstrated, and successfully worked using the trunsored model.
An accurate ability evaluation method for every student with small problem it...Hideo Hirose
To enhance the chance of use of the item response theory (IRT) in universities, we developed a test evaluation system via the Web for university teachers, and we have been evaluating students' abilities by using the IRT system in midterm and final examinations for two years.
We show a surprising aspect regarding the adoption of the IRT system in university tests. That is, the IRT can not only give us the problem difficulty information but also can provide the accurate student ability evaluation, even if the number of problems is small. Therefore, we can include high and low level test items together so that we can assess a variety of students' abilities accurately and fairly; we do not worry about providing easier problems that will make the lecture level decline; in other words, we do not care about finding the most appropriate problem levels to each student. We can provide all level problems uniformly distributed to all students, and we can still assess the students' abilities accurately. Consequently, students do not raise claims about their scores; they seem to be satisfied with it.
We show these results, in this paper, by a theoretical background, a simulation study, and our empirical results.
A successful maximum likelihood parameter estimation in skewed distributions ...Hideo Hirose
A successful maximum likelihood parameter estimation scheme using
the continuation method (homotopy method) is introduced. This
algorithm is particularly useful for the three-parameter skewed
distributions including thresholds. Such three-parameter
distributions are, for example, Weibull, log-normal, gamma and
inverse Gaussian distributions. As the proposed algorithm can almost
always obtain the local maximum likelihood estimates automatically,
it is of considerable practical value. The Monte Carlo simulation
study shows the effectiveness of the proposed method.
Estimation for the number of fragile samples in the trunsored and truncated m...Hideo Hirose
A method to obtain the estimate and its confidence interval for the number of fragile samples in mixed populations of the fragile and durable samples, i.e., in the trunsored model, is introduced. The confidence interval in the trunsored model is compared with that in the truncated model. Although the maximum likelihood estimates for the parameters in the underlying probability distribution in both models are the same, the confidence interval for the estimated number of samples in the trunsored model is differ from that in the truncated model. When the censoring time goes to infinity, the confidence interval in the truncated model converges to zero, whereas the confidence interval in the trunsored model converges to a positive constant value.
The error for the number of fragile samples in the trunsored model is affected by the two kinds of fluctuation effect due to the censoring time: one is the fluctuation of the parameter estimates, and the other is the ratio of the number of fragile samples to the total number of samples. However, in the truncated model, the fluctuation depends only on the parameter estimates, and the error by this effect will vanish when the censoring time goes to infinity.
A typical example of the method is applied to the case fatality ratio for the infectious diseases such as SARS.
How many times are we tossing coins until we observe head, tail, and head? It's ten. It's not eight. This is an intriguing result against our intuition.
Interesting but difficult problem: find the optimum saury layout on a gridiro...Hideo Hirose
Even though we use a simple but ridiculous problem finding the optimum saury baking layout on a fish gridiron by Joule heat, we can invoke the interest to science by combining electrical engineering, linear algebra and probability viewpoints. These elements are, use of solving linear equation and Poisson's equation, and applying the central limit theorem to this situation. In addition, by removing the constraints, we can create a new problem free from our common sense. Presenting funny but essential problems could be another aspect for active learning using the problem of the interdisciplinary scientific methods.
With 80 steps Galton boards, we can see the binomial distribution approximated to the normal distribution.
Youtube ===>>>
https://www.youtube.com/watch?v=3w4e1RQTAB8
The cumulative exposure model (CEM) is often used to express the failure probability model in the step-up test method; the step-up procedure continues until a breakdown occurs. This probability model is widely accepted in reliability fields because accumulation of fatigue is considered to be reasonable. Contrary to this, the memoryless model (MM) is also used in electrical engineering because accumulation of fatigue is not observed in some cases. We propose here a new model, the extended cumulative exposure model (ECEM), which includes features of both the described models. A simulation study and an application to the actual experimental case of oil insulation test support the validity of the proposed model. The independence model (IM) is also discussed.
Parameter estimation for the truncated weibull model using the ordinary diffe...Hideo Hirose
In estimating the number of failures using the truncated data for the Weibull model, we often encounter a case that the estimate is smaller than the true one when we use the likelihood principle to conditional probability. In infectious disease predictions, the SIR model described by simultaneous ordinary differential equations are often used, and this model can predict the final stage condition, i.e., the total number of infected patients, well, even if the number of observed data is small. These two models have the same condition for the observed data: truncated to the right. Thus, we have investigated whether the number of failures in the Weibull model can be estimated accurately using the ordinary differential equation. The positive results to this conjecture are shown.
A new incomplete data model, the trunsored model, in lifetime analysis is introduced. This model can be regarded as a unified model of the censored and truncated models. Using the model, we can not only estimate the ratio of the fragile population to the mixed fragile and durable populations, but also test a hypothesis that the ratio is equal to a prescribed value. A central point of the paper is that such a test can easily be realized through the newly introduced trunsored model, because it has been difficult to do such a hypothesis test under only the framework of censored and truncated models. Therefore, the relationship of the trunsored model to the censored and truncated models is clarified because the trunsored model unifies the censored and truncated models. The paper also shows how to obtain the estimates of the parameters in lifetime estimation, and corresponding confidence intervals for the fragile population. Typical examples applied to electronic board failures, and to breast cancer data, for lifetime estimation are demonstrated, and successfully worked using the trunsored model.
An accurate ability evaluation method for every student with small problem it...Hideo Hirose
To enhance the chance of use of the item response theory (IRT) in universities, we developed a test evaluation system via the Web for university teachers, and we have been evaluating students' abilities by using the IRT system in midterm and final examinations for two years.
We show a surprising aspect regarding the adoption of the IRT system in university tests. That is, the IRT can not only give us the problem difficulty information but also can provide the accurate student ability evaluation, even if the number of problems is small. Therefore, we can include high and low level test items together so that we can assess a variety of students' abilities accurately and fairly; we do not worry about providing easier problems that will make the lecture level decline; in other words, we do not care about finding the most appropriate problem levels to each student. We can provide all level problems uniformly distributed to all students, and we can still assess the students' abilities accurately. Consequently, students do not raise claims about their scores; they seem to be satisfied with it.
We show these results, in this paper, by a theoretical background, a simulation study, and our empirical results.
A successful maximum likelihood parameter estimation in skewed distributions ...Hideo Hirose
A successful maximum likelihood parameter estimation scheme using
the continuation method (homotopy method) is introduced. This
algorithm is particularly useful for the three-parameter skewed
distributions including thresholds. Such three-parameter
distributions are, for example, Weibull, log-normal, gamma and
inverse Gaussian distributions. As the proposed algorithm can almost
always obtain the local maximum likelihood estimates automatically,
it is of considerable practical value. The Monte Carlo simulation
study shows the effectiveness of the proposed method.
Estimation for the number of fragile samples in the trunsored and truncated m...Hideo Hirose
A method to obtain the estimate and its confidence interval for the number of fragile samples in mixed populations of the fragile and durable samples, i.e., in the trunsored model, is introduced. The confidence interval in the trunsored model is compared with that in the truncated model. Although the maximum likelihood estimates for the parameters in the underlying probability distribution in both models are the same, the confidence interval for the estimated number of samples in the trunsored model is differ from that in the truncated model. When the censoring time goes to infinity, the confidence interval in the truncated model converges to zero, whereas the confidence interval in the trunsored model converges to a positive constant value.
The error for the number of fragile samples in the trunsored model is affected by the two kinds of fluctuation effect due to the censoring time: one is the fluctuation of the parameter estimates, and the other is the ratio of the number of fragile samples to the total number of samples. However, in the truncated model, the fluctuation depends only on the parameter estimates, and the error by this effect will vanish when the censoring time goes to infinity.
A typical example of the method is applied to the case fatality ratio for the infectious diseases such as SARS.