Derivative of sine function: A graphical explanation

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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、、、

Derivative of sine function: A graphical explanation

Derivative of sine function: A graphical explanation

Derivative of sine function: A graphical explanation

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More from Hideo Hirose

　極値を求める問題は工学系に限らずあらゆる面で利用価値のある数学の応用分野である。高校生のときには1変数の場合の極値を取り扱い視覚的にもわかりやすく理解しやすかった。しかし、大学で多変数関数の微分に入ると急に複雑に感じ始める。2変数までは視覚的に捉えられても、3変数以上になるともう感覚的に捉えることができなくなり、理解しにくくなる。極値問題を取り扱うときにも同様で、2変数まではなんとか説明されている大学の教科書も3変数以上になると省略されていることが多い。しかし、実用面から考えると3変数以上の極値を求める応用分野はとても広い。重要なテーマなのである。　ここでは主に3変数以上の極値問題をできるだけわかりやすく概観することを試みる。解析だけの知識だけでは2変数の極値問題に限られてしまうので線形代数の知識も使いながら3変数以上に汎用性を広げていく。線形代数の、行列式、固有値、固有ベクトル、直行変換、2次形式、正値（負値）などの概念を使うことになるが、逆に解析と線形代数のつながりが見えてくる重要な場面にもなっている。

多変数の極値問題は解析と線形代数の融合だ

多変数の極値問題は解析と線形代数の融合だ

多変数の極値問題は解析と線形代数の融合だ

非線形方程式の解法にはNewton法が定番。しかし、収束は初期値に敏感であることはあまり教科書に載っていない。非常に簡単な問題でもいいので試しに指数関数が入った方程式をNewton法で解いてみるとイライラしてくるくらい初期値設定が難しいことがわかる。現実問題とはそんなもの。初期値設定を手助けする意味でHomotopy法を使ってみる。数学的には美しい理論。おもしろいのはHomotopy法を使うということは微分方程式を解いていることにつながるということ。エンジョイして欲しい。

Homotopy法による非線形方程式の解法

Homotopy法による非線形方程式の解法

Homotopy法による非線形方程式の解法

Different classification results under different criteria, distance and proba...

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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.

Interesting but difficult problem: find the optimum saury layout on a gridiro...

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Central Limit Theorem & Galton Board

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Central Limit Theorem & Galton Board

サンマが一尾。魚焼きの網はあるがコンロと炭がない。電気はある。電気でサンマを焼いてみよう。回路方程式, ポアソン方程式, 中心極限定理を用いて魚焼き網の両端から電流を流してジュール熱で魚を焼く。さて、魚をどのように置いたらうまく焼けるだろうか。最適な焼き方について考えてみよう。最後にはあっと驚く発想の転換と意外な結果が。 ….. 電気学会教育フロンティア研究会, FIE-16-015, pp.71-78（February 29-30, 2016）

回路方程式, ポアソン方程式, 中心極限定理による最適なサンマの焼き方

回路方程式, ポアソン方程式, 中心極限定理による最適なサンマの焼き方

回路方程式, ポアソン方程式, 中心極限定理による最適なサンマの焼き方

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.

Extended cumulative exposure model, ecem

Extended cumulative exposure model, ecem

Extended cumulative exposure model, ecem

「科学=予測」Science cafe 2011 june 10

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「科学=予測」Science cafe 2011 june 10

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.

Parameter estimation for the truncated weibull model using the ordinary diffe...

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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.

Trunsored data analysis

Trunsored data analysis

Trunsored data analysis

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.

An accurate ability evaluation method for every student with small problem it...

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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.

A successful maximum likelihood parameter estimation in skewed distributions ...

A successful maximum likelihood parameter estimation in skewed distributions ...

A successful maximum likelihood parameter estimation in skewed distributions ...

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.

Estimation for the number of fragile samples in the trunsored and truncated m...

Estimation for the number of fragile samples in the trunsored and truncated m...

Estimation for the number of fragile samples in the trunsored and truncated m...

In difficult classification problems of the z-dimensional points into two groups having 0-1 responses due to the messy data structure, it is more favorable to search for the denser regions for the response 1 assigned points than to find the boundaries to separate the two groups. To such problems often seen in customer databases, we have developed a bump hunting method using probabilistic and statistical methods. By specifying a pureness rate in advance, a maximum capture rate will be obtained. Then, a trade-off curve between the pureness rate and the capture rate can be constructed. In finding the maximum capture rate, we have used the decision tree method combined with the genetic algorithm. We first explain a brief introduction of our research: what the bump hunting is, the trade-off curve between the pureness rate and the capture rate, the bump hunting using the tree genetic algorithm, the upper bounds for the trade-off curve using the extreme-value statistics. Then, the assessment for the accuracy of the trade-off curve is tackled from the genetic algorithm procedure viewpoint. Using the new genetic algorithm procedure proposed, we can obtain the upper bound accuracy for the trade- off curve. Then, we may expect the actually attain- able trade-off curve upper bound. The bootstrapped hold-out method is used in assessing the accuracy of the trade-off curve, as well as the cross validation method.

Bump hunting とその応用

Bump hunting とその応用

Bump hunting とその応用

Suppose that we are interested in classifying n points in a z-dimensional space into two groups having response 1 and response 0 as the target variable. In some real data cases in customer classification, it is difficult to discriminate the favorable customers showing response 1 from others because many response 1 points and 0 points are closely located. In such a case, to find the denser regions to the favorable customers is considered to be an alter- native. Such regions are called the bumps, and finding them is called the bump hunting. By pre-specifying a pureness rate p in advance a maximum capture rate c could be obtained; the pureness rate is the ratio of the number of response 1 points to the total number of points in the target region; the capture rate is the ratio of the number of response 1 points to the total number of points in the total regions. Then a trade-off curve between p and c can be constructed. Thus, the bump hunting is the same as the trade-off curve constructing. In order to make future actions easier, we adopt simpler boundary shapes for the bumps such as the union of z-dimensional boxes located parallel to some explanation variable axes; this means that we adopt the binary decision tree. Since the conventional binary decision tree will not provide the maximum capture rates because of its local optimizer property, some probabilistic methods would be required. Here, we use the genetic algorithm (GA) specified to the tree structure to accomplish this; we call this the tree GA. The tree GA has a tendency to provide many local maxima of the capture rates unlike the ordinary GA. According to this property, we can estimate the upper bound curve for the trade-off curve by using the extreme-value statistics. However, these curves could be optimistic if they are constructed using the training data alone. We should be careful in assessing the accuracy of these curves. By applying the test data, the accuracy of the trade-off curve itself can easily be assessed. However, the property of the local maxima would not be preserved. In this paper, we have developed a new tree GA to preserve the property of the local maxima of the capture rates by assessing the test data results in each evolution procedure. Then, the accuracy of the trade-off curve and its upper bound curve are assessed.

Accuracy assessment for the trade off curve and its upper curve in the bump h...

Accuracy assessment for the trade off curve and its upper curve in the bump h...

Accuracy assessment for the trade off curve and its upper curve in the bump h...

When we want to grasp the characteristics of the time series signals emitted massively from electric power apparatuses or electroencephalogram, and want to decide some diagnoses about the apparatuses or human brains, we may use some statistical distribution functions. In such cases, the generalized normal distribution is frequently used in pattern analysts. In assessing the correctness of the estimates of the shape of the distribution function accurately, we often use a Monte Carlo simulation study; thus, a fast and efficient random number generation method for the distribution function is needed. However, the method for generating the random numbers of the distribution seems not easy and not yet to have been developed. In this paper, we propose a random number generation method for the distribution function using the the rejection method. A newly developed modified adaptive rejection method works well in the case of log-convex density functions.

Random number generation for the generalized normal distribution using the re...

Random number generation for the generalized normal distribution using the re...

Random number generation for the generalized normal distribution using the re...

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Central Limit Theorem & Galton Board

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