A comparison of the ordinary and a varying parameter exponential smoothing

1989 ◽  
Vol 26 (4) ◽  
pp. 784-792 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Finite Interval ◽  
Exponential Smoothing ◽  
Uniform Ergodicity ◽  
Geometric Convergence ◽  
Monotonically Increasing Function ◽  
Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn –1)Xn +(1 – ß (Zn–1))Zn–1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.Together with (Zn), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn –1 where α is a constant, 0 < α < 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

A comparison of the ordinary and a varying parameter exponential smoothing

1989 ◽  
Vol 26 (04) ◽  
pp. 784-792 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Finite Interval ◽  
Exponential Smoothing ◽  
Uniform Ergodicity ◽  
Geometric Convergence ◽  
Monotonically Increasing Function ◽  
Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn – 1)Xn +(1 – ß (Zn – 1))Zn – 1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 &lt; c &lt; d &lt; 1. Together with (Zn ), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn – 1 where α is a constant, 0 &lt; α &lt; 1. We show that (Yn ) and (Zn ) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

scholarly journals A monotonicity in reversible Markov chains

2006 ◽  
Vol 43 (2) ◽  
pp. 486-499 ◽  
Author(s):  
Robert Lund ◽  
Ying Zhao ◽  
Peter C. Kiessler
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
State Space ◽  
Hazard Rate ◽  
Probability Generating Function ◽  
Countable State Space ◽  
Return Times ◽  
Geometric Convergence ◽  
Reversible Markov Chains ◽  
Countable State

In this paper we identify a monotonicity in all countable-state-space reversible Markov chains and examine several consequences of this structure. In particular, we show that the return times to every state in a reversible chain have a decreasing hazard rate on the subsequence of even times. This monotonicity is used to develop geometric convergence rate bounds for time-reversible Markov chains. Results relating the radius of convergence of the probability generating function of first return times to the chain's rate of convergence are presented. An effort is made to keep the exposition rudimentary.

RENEWAL CONVERGENCE RATES FOR DHR AND NWU LIFETIMES

2002 ◽  
Vol 16 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Kenneth S. Berenhaut ◽  
Robert Lund
Keyword(s):  
Power Series ◽  
Convergence Rate ◽  
Markov Chains ◽  
Generating Functions ◽  
Hazard Rate ◽  
Convergence Rates ◽  
Renewal Sequence ◽  
Geometric Convergence ◽  
Power Series Methods

This article studies the geometric convergence rate of a discrete renewal sequence with decreasing hazard rate or, more generally, new worse than used lifetimes. Several variants of these structural orderings are considered. The results are derived from power series methods; roots of generating functions are the prominent issue. Optimality of the rates are considered. Examples demonstrating the utility of the results, as well as applications to Markov chains, are presented.

scholarly journals A monotonicity in reversible Markov chains

2006 ◽  
Vol 43 (02) ◽  
pp. 486-499 ◽  
Author(s):  
Robert Lund ◽  
Ying Zhao ◽  
Peter C. Kiessler
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
State Space ◽  
Hazard Rate ◽  
Probability Generating Function ◽  
Countable State Space ◽  
Return Times ◽  
Geometric Convergence ◽  
Reversible Markov Chains ◽  
Countable State

In this paper we identify a monotonicity in all countable-state-space reversible Markov chains and examine several consequences of this structure. In particular, we show that the return times toeverystate in a reversible chain have a decreasing hazard rate on the subsequence of even times. This monotonicity is used to develop geometric convergence rate bounds for time-reversible Markov chains. Results relating the radius of convergence of the probability generating function of first return times to the chain's rate of convergence are presented. An effort is made to keep the exposition rudimentary.

Extinction in the Pigeon after Continuous Reinforcement: Effect of Number of Reinforced Responses

1971 ◽  
Vol 28 (1) ◽  
pp. 331-338 ◽  
Author(s):  
Laurel Furumoto
Keyword(s):  
Animal Studies ◽  
Food Pellet ◽  
Key Peck ◽  
Continuous Reinforcement ◽  
Gm Food ◽  
Free Operant ◽  
Time To Extinction ◽  
Monotonically Increasing Function ◽  
Increasing Functions ◽  
Increasing Function

Number of responses and time to extinction were measured after 3, 10, 1000, 3000, 5000, and 10,000 reinforced key-peck responses during conditioning. Each response was reinforced with a 045-gm. food pellet. The number of responses in extinction was a monotonically increasing function which became asymptotic beyond 1000 reinforced responses. Number of reinforced responses during conditioning significantly affected the number of responses in extinction ( p < .001) but not the time to extinction. The results support the findings of previous free-operant bar-press studies with rats. Free-operant animal studies of extinction after continuous reinforcement have consistently produced monotonically increasing functions and have typically employed relatively small amounts of reinforcement. Amount of reward may be an important parameter determining the shape of the extinction function in the free-operant studies.

Judgment of Temporal Duration of Stimulus Components in a Spatial Sequence

1973 ◽  
Vol 37 (2) ◽  
pp. 619-623
Author(s):  
Suchoon S. Mo ◽  
Michael D. Blaszcszack ◽  
Kathleen Ward
Keyword(s):  
Stimulus Presentation ◽  
Temporal Duration ◽  
Monotonically Increasing Function ◽  
Increasing Function

Judgment of the duration of the stimulus components of tri-grams consisting of consonants was a monotonically increasing function of the letter positions in the sequence of left to right. This tendency was more clearly demonstrated when the frequency of the stimulus presentation exceeded the frequency of the presentation of the stimulus components.

Thermodynamic Performance of an Actual Thermo-Acoustic Refrigeration Micro-Cycle

2012 ◽  
Vol 614-615 ◽  
pp. 64-68
Author(s):  
Tuo Wang ◽  
Feng Wu ◽  
Jin Hua Fei ◽  
Ming Fang Liu
Keyword(s):  
Coefficient Of Performance ◽  
Pressure Amplitude ◽  
Cooling Load ◽  
Acoustic Effect ◽  
Thermodynamic Performance ◽  
Monotonically Increasing Function ◽  
Analytic Expressions ◽  
New Type ◽  
The Relationship ◽  
Increasing Function

Thermo-acoustic refrigerator is a new type of engine, which is based on the thermo-acoustic effect. A new model which expresses as an ellipse in pressure-volume diagram is established to investigate the thermodynamic performance of an actual thermo-acoustic refrigeration micro-cycle. The demarcation points of endothermic processes and exothermic processes in the actual micro-cycle are found. The analytic expressions of the dimensionless cooling load and the coefficient of performance (COP) are deduced. The relationship between the dimensionless cooling load and the COP are investigated by numerical examples. The results show that the dimensionless cooling load is a monotonically increasing function of the COP and the pressure amplitude.

Full-Spectrum k-Distribution Model for Radiation in Gas Based on Multi-Group Methods

2014 ◽  
Vol 1008-1009 ◽  
pp. 839-845
Author(s):  
Yue Zhou ◽  
Qiang Wang ◽  
Hai Yang Hu
Keyword(s):  
Narrow Band ◽  
Wide Band ◽  
Distribution Model ◽  
Full Spectrum ◽  
Distribution Method ◽  
One Dimensional ◽  
Group Methods ◽  
Monotonically Increasing Function ◽  
Homogeneous Media ◽  
Increasing Function

The k-distribution method applied in narrow band and wide band is extended to the full spectrum based on spectroscopic datebase HITEMP, educing the full-spectrum k-distribution model. Absorption coefficents in this model are reordered into a smooth,monotonically increasing function such that the intensity calculations are performed only once for each absorption coefficent value and the resulting computations are immensely more efficent.Accuracy of this model is examined for cases ranging from homogeneous one-dimensional carbon dioxide to inhomogeneous ones with simultaneous variations in temperature. Comparision with line-by-line calculations (LBL) and narrow-band k-distribution (NBK) method as well as wide-band k-distribution (WBK) method shows that the full-spectrum k-distribution model is exact for homogeneous media, although the errors are greater than the other two models. After dividing the absorption coefficients into several groups according to their temperature dependence, the full-spectrum k-distribution model achieves line-by-line accuracy for gases inhomogeneous in temperature, accompanied by lower computational expense as compared to NBK model or WBK model. It is worth noting that a new grouping scheme is provided in this paper.

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