An optimal ordering scheme for the capacitated, finite-horizon, multi-period, multiproduct newsvendor problem was proposed by cite {shao06} using a hedging point policy. This solution requires the calculation of a central curve that divides the different ordering regions and a vector that defines the target inventory levels. The central curve is a nonlinear curve that determines the optimal order quantities as a function of the initial inventory levels. In this paper we propose a method for calculating this curve and vector using spline functions, infinitesimal perturbation analysis (IPA), and convex optimization. Using IPA the derivatives of the cost with respect to the variables that determine the spline function are efficiently calculated. A convex optimization algorithm is used to optimize the spline function, resulting in a optimal policy. We present the mathematical derivations and simulation results validating this solution.

<p>This thesis studies path generation for industrial robots of six degrees of freedom. A path is defined by connection of simple geometrical objects like arcs and straight lines. About each point at which the objects connect, a region, henceforth called a zone, is defined in which deviation from the defined path is permitted. The zone allows the robot to follow the path at a constant speed, but the acceleration needed may vary. </p><p>Some means of calculating the zone path as to make the acceleration continuous will be presented. In joint space the path is described by the use of cubic splines. The transformation of the Cartesian path to paths in joint space will be examined. Discontinuities in the second order derivatives will appear between the splines. </p><p>A few examples of different zone path calculations will be presented where the resulting spline functions are compared with respect to their first and second order derivatives. An investigation of the number of spline functions needed when, given an upper limit of deviation, the transformation back to Cartesian coordinates is made.</p>

This thesis studies path generation for industrial robots of six degrees of freedom. A path is defined by connection of simple geometrical objects like arcs and straight lines. About each point at which the objects connect, a region, henceforth called a zone, is defined in which deviation from the defined path is permitted. The zone allows the robot to follow the path at a constant speed, but the acceleration needed may vary. Some means of calculating the zone path as to make the acceleration continuous will be presented. In joint space the path is described by the use of cubic splines. The transformation of the Cartesian path to paths in joint space will be examined. Discontinuities in the second order derivatives will appear between the splines. A few examples of different zone path calculations will be presented where the resulting spline functions are compared with respect to their first and second order derivatives. An investigation of the number of spline functions needed when, given an upper limit of deviation, the transformation back to Cartesian coordinates is made.

Yes / In engineering design, there are many functional relationships which are difficult to express into a simple and exact mathematical formula. Instead they are documented within a form of line graphs (or plot charts or curve diagrams) in engineering handbooks or text books. Because the information in such a form cannot be used directly in the modern computer aided design (CAD) process, it is necessary to find a way to numerically represent the information. In this paper, a data processing system for numerical representation of line graphs in mechanical design is developed, which incorporates the process cycle from the initial data acquisition to the final output of required information. As well as containing the capability for curve fitting through Cubic spline and Neural network techniques, the system also adapts a novel methodology for use in this application: Grey Models. Grey theory have been used in various applications, normally involved with time-series data, and have the characteristic of being able to handle sparse data sets and data forecasting. Two case studies were then utilized to investigate the feasibility of Grey models for curve fitting. Furthermore, comparisons with the other two established techniques show that the accuracy was better than the Cubic spline function method, but slightly less accurate than the Neural network method. These results are highly encouraging and future work to fully investigate the capability of Grey theory, as well as exploiting its sparse data handling capabilities is recommended.

Despite the potential benefits of larger datasets for crop insurance ratings, pooling yields with similar distributions is not a common practice. The current USDA-RMA county insurance ratings do not consider information across state lines, a politically driven assumption that ignores a wealth of climate and agronomic evidence suggesting that growing regions are not constrained by state boundaries. We test the appropriateness of this assumption, and provide empirical grounds for benefits of pooling datasets. We find evidence in favor of pooling across state lines, with poolable counties sometimes being as far as 2,500 miles apart. An out-of-sample performance exercise suggests our proposed pooling framework out-performs a no-pooling alternative, and supports the hypothesis that economic losses should be expected as a result of not adopting our pooling framework. Our findings have strong empirical and policy implications for accurate modeling of yield distributions and vis-à-vis the rating of crop insurance products.

本研究可分為兩部份，第一部份以CBD為中心，向外擴展之地價空間結構變化情形之測試，主要是方法上的應用。運用較富彈性且能符合多核心都市測試需要之三次雲形函數來測試台北市之地價空間結構。研究發現都市發展方向會影響地價空間結構之變化，而傳統單核心模型之單一核心之假設及無考慮方向性，並不能充份描述現代大都市之真實空間結構。 　　第二部份則為了解各核心對地價影響情形，及在空間上互動關係進行多核心模型之測試。研究發現地價為都市空間中多核心互動的結果。而相同屬性、規模的核心間有排斥效果，不同屬性、規模的核心間則有吸引效果。藉由多核心之分散健全發展，與交通建設之密切配合，將可使都市之地價及都市空間結構更加健全發展。 / This research is divided to two parts. First part is to test the structural variance of the land price which extends by CBD. The method, the cubic spline function, is flexible and it can match the need of the polycentric city. From the test, we find the different directions which will affect the spatial structural variance of the land price. But the factor of the direction isn't included in the hypothesis of the traditional monocentric model, and it can't be described the true spatial structure of modern city sufficiently. 　　Another is the test that to understand the land price variance situation by the different centers and the interactions of them.By this test, it shows that land price is the result of the polycentric interactions of urban spatial structure. We also find there is a repelling effect between the same character and scale centers; vise versa.

La 1ere partie de ce travail traite l'optimisation des tournées sous forme d'un problème d'optimisation nommé Le problème de Voyageur de Commerce. Dans cette partie nous nous intéressons à faire une riche présentation du problème de Voyageur de Commerce, ses variantes, puis nous proposons une stratégie de génération de contrainte pour la résolution du TSP. Ensuite on traite sa version stochastique : le problème de Voyageur de commerce Probabiliste. Nous proposons une formulation mathématique du PTSP et nous présentons des résultats numériques obtenus par résolution exacte pour une série d'instances de petite taille. Dans la seconde partie, nous proposons une méthode d'approximation générale permettant d'approcher différents type de données, d'abord nous traitons l'approximation d'un signal de vent (cas simple, ID), ensuite l'approximation d'un champ de vecteurs avec prise en compte de la topographie qui constitue la principale contribution de cette partie. / The first part of this work deals with route optimization in the form of an optimization problem named The Traveler's Business Problem. In this part we are interested to make a rich presentation of the problem of Traveler Commerce, its variants, then we propose a strategy of constraint generation for the resolution of the TSP. Then we treat its stochastic version : the probabilistic business traveler problem. We propose a mathematical formulation of the PTSP and we present numerical results obtained by exact resolution for a series of small instances. In the second part, we propose a method of general approximation to approximate different type of data, first we treat the approximation of a wind signal (simple case, 1D), then the approximation of a vector field taking into account the topography which is the main contribution of this part.

Dans cette thèse nous avons traité le problème de l'estimation non paramétrique des lois de probabilités. Dans un premier temps, nous avons supposé que la densité inconnue f a été approchée par un mélange de base B-spline quadratique. Puis, nous avons proposé un nouvel estimateur de la densité inconnue f basé sur les fonctions B-splines quadratiques, avec deux méthodes d'estimation. La première est base sur la méthode du maximum de vraisemblance et la deuxième est basée sur la méthode d'estimation Bayésienne MAP. Ensuite, nous avons généralisé notre étude d'estimation dans le cadre du mélange et nous avons proposé un nouvel estimateur du mélange de lois inconnues basé sur les deux méthodes d'estimation adaptées. Dans un deuxième temps, nous avons traité le problème de la segmentation statistique semi supervisée des images en se basant sur le modèle de Markov caché et les fonctions B-splines. Nous avons montré l'apport de l'hybridation du modèle de Markov caché et les fonctions B-splines en segmentation statistique bayésienne semi supervisée des images. Dans un troisième temps, nous avons présenté une approche de fusion basée sur la méthode de maximum de vraisemblance, à travers l'estimation non paramétrique des probabilités, pour chaque pixel de l'image. Nous avons ensuite appliqué cette approche sur des images multi-spectrales et multi-temporelles segmentées par notre algorithme non paramétrique et non supervisé. / In this thesis we are treated the problem of nonparametric estimation probability distributions. At first, we assumed that the unknown density f was approximated by a basic mixture quadratic B-spline. Then, we proposed a new estimate of the unknown density function f based on quadratic B-splines, with two methods estimation. The first is based on the maximum likelihood method and the second is based on the Bayesian MAP estimation method. Then we have generalized our estimation study as part of the mixture and we have proposed a new estimator mixture of unknown distributions based on the adapted estimation of two methods. In a second time, we treated the problem of semi supervised statistical segmentation of images based on the hidden Markov model and the B-sline functions. We have shown the contribution of hybridization of the hidden Markov model and B-spline functions in unsupervised Bayesian statistical image segmentation. Thirdly, we presented a fusion approach based on the maximum likelihood method, through the nonparametric estimation of probabilities, for each pixel of the image. We then applied this approach to multi-spectral and multi-temporal images segmented by our nonparametric and unsupervised algorithm.

共變數的值會隨著時間而改變時，我們稱之為時間相依之共變數。時間相依之共變數往往具有重複測量的特性，也是長期資料裡最常見到的一種共變數形態；在對時間相依之共變數進行重複測量時，可以考慮每次測量的間隔時間相同或是間隔時間不同兩種情形。在間隔時間相同的情形下，我們可以忽略間隔時間所產生的效應，利用分組的Cox模式或是合併的羅吉斯迴歸模式來分析，而合併的羅吉斯迴歸是一種把資料視為“對象 時間單位”形態的分析方法；此外，分組的Cox模式和合併的羅吉斯迴歸模式也都可以用來預測存活機率。在某些條件滿足下，D’Agostino等六人在1990年已經證明出這兩個模式所得到的結果會很接近。 當間隔時間為不同時，我們可以用計數過程下的Cox模式來分析，在計數過程下的Cox模式中，資料是以“對象 區間”的形態來分析。2001年Bruijne等人則是建議把間隔時間也視為一個時間相依之共變數，並將其以B-spline函數加至模式中分析；在我們論文的實證分析裡也顯示間隔時間在延伸的Cox模式中的確是個很顯著的時間相依之共變數。延伸的Cox模式為間隔時間不同下的時間相依之共變數提供了另一個分析方法。至於在時間相依之共變數的預測方面，我們是以指數趨勢平滑法來預測其未來時間點的數值；利用預測出來的時間相依之共變數值再搭配延伸的Cox模式即可預測未來的存活機率。 / It is so called “time-dependent covariates” that the values of covariates change over time. Time-dependent covariates are measured repeatedly and often appear in the longitudinal data. Time-dependent covariates can be regularly or irregularly measured. In the regular case, we can ignore the TEL(time elapsed since last observation) effect and the grouped Cox model or the pooled logistic regression model is employed to anlalyze. The pooled logistic regression is an analytic method using the“person-period”approach. The grouped Cox model and the pooled logistic regression model also can be used to predict survival probablity. D’Agostino et al. (1990) had proved that pooled logistic regression model is asymptotically equivalent to the grouped Cox model. If time-dependent covariates are observed irregularly, Cox model under counting process may be taken into account. Before making the prediction we must turn the original data into“person-interval”form, and this data form is also suitable for the prediction of grouped Cox model in regular measurements. de Bruijne et al.(2001) first considered TEL as a time-dependent covariate and used B-spline function to model it in their proposed extended Cox model. We also show that TEL is a very significant time-dependent covariate in our paper. The extended Cox model provided an alternative for the irregularly measured time-dependent covariates. On the other hand, we use exponential smoothing with trend to predict the future value of time-dependent covariates. Using the predicted values with the extended Cox model then we can predict survival probablity.