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Sequences and Series of Functions
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Here c = 0 and an = 1/n!. When studying the convergence of a power series, most of the time the winning strategy is to use the Ratio Test: Dn=|an+1xn+1||anxn|=|x|1(n+1)!1n!=|x|n+1→0,n→∞,
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
This test always works. The ratio test does not always apply, because the limit of the ratios of successive terms might not exist. When this limit does exist, we have 1R=limn→∞|an+1an|.
Exploring students’ consistency in their notions: the case of comparing infinite series
Published in International Journal of Mathematical Education in Science and Technology, 2021
The participants of the study were 93 male students aged 16–17 from five classes in a governmental high-school in the Hamedan province of Iran. Their knowledge of infinity was based more on working with infinite sets and sequences. To achieve the purpose of the study, we need to study students without formal knowledge of convergence of series, because in the light of related formal knowledge, by considering known convergent or divergent series or applying d’Alembert’s ratio test or some other standard ones or recalling and using definitions to examine convergence of series even without a full understanding and a visual reasoning of the concept, students may answer correctly (Alcock & Simpson, 2004, 2005). Therefore, in the absence of formal knowledge, their first responses were intuitive, and the provided answers can show their imaginations of the concept. Facing other responses, each of which can be somewhat justified, creates a cognitive conflict. Attempts to address this challenge may lead participants to change their previous response, if another approach seems more justifiable to them. Otherwise, keeping their first notions, participants are considered as consistent in their notions.
Identification of simplified energy performance models of variable-speed air conditioners using likelihood ratio test method
Published in Science and Technology for the Built Environment, 2020
Maomao Hu, Fu Xiao, Howard Cheung
The balance between model accuracy and model simplicity is a critical issue in model identification. This article presents a model selection approach to identify the simplified energy performance model of variable-speed ACs for building energy performance assessment, as well as for model-based FDD and optimal control. The likelihood ratio test, a statistical hypothesis testing method, is applied to compare the full model with a series of candidate submodels. Maximum likelihood estimation is used to estimate the parameters of each candidate model. It has been shown that the approach is able to effectively select the cooling capacity and COP models for variable-speed ACs with reasonable complexity and satisfactory accuracy. The identified models have three variables and six parameters in total. The item of air temperature is first order, while the item of compressor speed is third order. The RMSE and R2 of the most suitable Q* model are 0.0188 and 0.9954, respectively. The RMSE and R2 of the most suitable COP* model are 0.0463 and 0.9867, respectively.
Monitoring of count data time series: Cumulative sum change detection in Poisson integer valued GARCH models
Published in Quality Engineering, 2019
O. Arda Vanli, Rupert Giroux, Eren Erman Ozguven, Joseph J. Pignatiello
In this article, we propose a likelihood ratio test formulation to model the time series behavior in count data and detect, as quickly as possible, changes in the time series model parameters. The process under statistical control is assumed to follow a seasonal INGARCH(1,1) time series model with Poisson deviates, and a new cumulative sum (CUSUM) monitoring statistic is developed to detect changes from the assumed model. The approach considers a Phase II statistical process control (SPC) study, that is, it is assumed that the in-control process parameters, Poisson rate and the time series model parameters are known (Montgomery 2009). The proposed INGARCH(1,1) model-based CUSUM is compared to existing INARCH(1) model-based CUSUM and a standard CUSUM that assume an i.i.d. process. We present a case study to illustrate the proposed methodology on surveillance of traffic crash events. Main contributions of this article are (i) to quantify the benefits of using a seasonal INGARCH(1,1) model-based CUSUM compared to existing CUSUM techniques in monitoring under various seasonality and autocorrelation conditions, (ii) to provide a unified framework for estimation of seasonal integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models using the glm function in R, and (iii) to study the goodness-of-fit (GOF) of models with various seasonality and moving average type autocorrelation structures.