A Better Alternative to Piecewise Linear Time Series Segmentation

Time series are difficult to monitor, summarize and predict. Segmentation organizes time series into few intervals having uniform characteristics (flatness, linearity, modality, monotonicity and so on). For scalability, we require fast linear time algorithms. The popular piecewise linear model can determine where the data goes up or down and at what rate. Unfortunately, when the data does not follow a linear model, the computation of the local slope creates overfitting. We propose an adaptive time series model where the polynomial degree of each interval vary (constant, linear and so on). Given a number of regressors, the cost of each interval is its polynomial degree: constant intervals cost 1 regressor, linear intervals cost 2 regressors, and so on. Our goal is to minimize the Euclidean (l_2) error for a given model complexity. Experimentally, we investigate the model where intervals can be either constant or linear. Over synthetic random walks, historical stock market prices, and electrocardiograms, the adaptive model provides a more accurate segmentation than the piecewise linear model without increasing the cross-validation error or the running time, while providing a richer vocabulary to applications. Implementation issues, such as numerical stability and real-world performance, are discussed.Comment: to appear in SIAM Data Mining 200

Seřizování regulátorů PID pro lineární dynamické systémy metodou “Hrubé síly a od oka”

For PID controllers in control loops, which in practice still prevails, it is necessary to determine controller parameter values which assure sufficient quality and robustness of control. A lot of PID tuning methods is described and well established in daily use. They are often based either on behavior of closed control loop or on a mathematic model of the controlled plant (and its L-transfer function), which allow to determine controller parameters by direct calculation. With increasing computing power of HW and SW aids it could be beneficial to use a method for determining the optimal setting of PID controller by simulation of control loop behavior for various meaningful combinations of controller parameter values (state space search). Proposed method “brute force and visual” of PID controller tuning is based on repeated simulation of control process as a reaction to Heaviside step of desired value and disturbing value for varying parameters of controller’s P, I, D components. For discrete controller also impact of sampling period T was considered. For each of the individual simulation runs there is kept a diagram as an image file as well as the control optimality criterion value. After execution of all the simulation runs for parameter values from the considered searched state space the simulation outputs are sorted by value of the selected control optimality criterion. The parameters of the individual simulation which has scored best are used for setting of the real controller and evaluated in the real control loop. From the simulations available for wide combination of parameter values it is possible to estimate position of isles of stability and choose the PID controller values by an expert choice.Při návrhu parametrů PID regulátoru (které jsou v praxi dosud nejobvyklejší) pro regulační obvody je zapotřebí stanovit parametry regulátoru tak, aby zajišťovaly dostatečnou kvalitu a robustnost regulace. Je definováno a v praxi zavedeno mnoho seřizovacích metod pro nastavení PID regulátorů vycházejících z chování uzavřeného regulačního obvodu či naopak z matematického modelu regulované soustavy (a jejího L-přenosu), které umožňují stanovit parametry regulátoru přímým výpočtem. S rostoucím výkonem HW a SW nástrojů může ale být výhodné použít přístup, který nalezne optimální hodnoty nastavení regulačního procesu na základě simulací a analýzy chování regulačního obvodu pro různé kombinace možných parametrů regulátoru. Navržená metoda seřízení PID regulátoru lineárního dynamického systému „hrubou silou a od oka“ je založena na opakované simulaci regulačního pochodu při skoku řídící veličiny a poruchové veličiny pro měnící se konstanty regulátoru P, I, D, (a v případě číslicového regulátoru i vzorkovací periodu T) pro smysluplné rozpětí hodnot – odtud „hrubou silou“ v názvu metody. Z každého běhu simulace je uložen diagram průběhů zajímavých veličin (žádaná veličina, regulovaná veličina, porucha) jako obrazový soubor a jsou pro tuto simulaci vypočteny hodnoty ukazatelů kvality regulace. Po provedení simulací pro všechny hodnoty parametrů z uvažovaného stavového prostoru jsou výstupy simulací setříděny podle ukazatelů kvality regulace; simulace s parametry regulace, která skórovala podle ukazatelů kvality regulace nejlépe, je pak podkladem pro nastavení parametrů regulátoru a ověření v reálném regulačním obvodu. Protože máme k dispozici vizualizované podoby průběhů regulačního procesu pro mnoho kombinací vstupů, můžeme odhadnout i polohu a podobu „ostrovů stability“ a pro dosažení robustnější regulace volit parametry regulátoru i expertní volbou, tedy „od oka“

How people make friends in social networking sites - A microscopic perspective

We study the detailed growth of a social networking site with full temporal information by examining the creation process of each friendship relation that can collectively lead to the macroscopic properties of the network. We first study the reciprocal behavior of users, and find that link requests are quickly responded to and that the distribution of reciprocation intervals decays in an exponential form. The degrees of inviters/accepters are slightly negatively correlative with reciprocation time. In addition, the temporal feature of the online community shows that the distributions of intervals of user behaviors, such as sending or accepting link requests, follow a power law with a universal exponent, and peaks emerge for intervals of an integral day. We finally study the preferential selection and linking phenomena of the social networking site and find that, for the former, a linear preference holds for preferential sending and reception, and for the latter, a linear preference also holds for preferential acceptance, creation, and attachment. Based on the linearly preferential linking, we put forward an analyzable network model which can reproduce the degree distribution of the network. The research framework presented in the paper could provide a potential insight into how the micro-motives of users lead to the global structure of online social networks.Comment: 10 pages, 12 figures, 2 table