On the monotonic property of the probability of undetected error for a shortened code

1991 ◽  
Vol 37 (5) ◽  
pp. 1409-1411 ◽  
Author(s):  
T. Fujiwara ◽  
T. Kasami ◽  
S.-p. Feng
Keyword(s):  
Monotonic Property

scholarly journals Estimation accuracy of covariance matrices when their eigenvalues are almost duplicated

2018 ◽  
Vol 7 ◽  
Author(s):  
Kantaro Shimomura ◽  
Kazushi Ikeda
Keyword(s):  
Covariance Matrix ◽  
Estimation Method ◽  
Estimation Accuracy ◽  
Essential Information ◽  
The Matrix ◽  
The Difference ◽  
Processing Techniques ◽  
Monotonic Property ◽  
Signal Processing Techniques ◽  
Special Case

The covariance matrix of signals is one of the most essential information in multivariate analysis and other signal processing techniques. The estimation accuracy of a covariance matrix is degraded when some eigenvalues of the matrix are almost duplicated. Although the degradation is theoretically analyzed in the asymptotic case of infinite variables and observations, the degradation in finite cases are still open. This paper tackles the problem using the Bayesian approach, where the learning coefficient represents the generalization error. The learning coefficient is derived in a special case, i.e., the covariance matrix is spiked (all eigenvalues take the same value except one) and a shrinkage estimation method is employed. Our theoretical analysis shows a non-monotonic property that the learning coefficient increases as the difference of eigenvalues increases until a critical point and then decreases from the point and converged to the distinct case. The result is validated by numerical experiments.

On monotonicity of the relaxation functions of viscoelastic materials

1970 ◽  
Vol 67 (2) ◽  
pp. 503-508 ◽  
Author(s):  
W. A. Day
Keyword(s):  
Closed Path ◽  
Viscoelastic Materials ◽  
One Dimensional ◽  
Relaxation Functions ◽  
Completely Monotone ◽  
Strain Space ◽  
Work Done ◽  
Decreasing Functions ◽  
The Given ◽  
Monotonic Property

This note is concerned with one-dimensional viscoelastic materials. Experiments show that for linear materials the relaxation functions are monotone decreasing functions of the time. This monotonic property has not, as far as I am aware, been characterized and in this note I provide a characterization formulated in terms of an assertion about the work done on the material over certain closed paths in strain space. More explicitly, consider any path in strain space which starts from equilibrium and arrives at the strain e at time t. We can now take the material around a closed path by either retracing the given path immediately or we can do it by holding the strain fixed at value e for a time r and then retracing the given path. Let us call the total work done on the material on the first closed path w(0) and on the second w(r). In general, because of the memory of the material, w(t) ≠ w(0). If it is true that w(r) ≥ w(0), whatever value r ≥ 0 has and whatever the initial strain path starting from equilibrium, we say that work is always increased by delay on retraced paths. The characterization proved here is that if G(·) is the relaxation function, with equilibrium elastic modulus G(∞), then G(·) −G(∞) is completely monotone if and only if work is always increased by delay on retraced paths.†

scholarly journals Discovering Periodic Patterns in Time Series from Twitter Data Set

2020 ◽  
Vol 9 (4) ◽  
pp. 92-102
Keyword(s):  
Time Series ◽  
Periodic Pattern ◽  
Periodic Patterns ◽  
Time Intervals ◽  
Data Set ◽  
Periodic Behavior ◽  
Expensive Process ◽  
Monotonic Property ◽  
Pruning Technique ◽  
Regular Patterns

The important class of regularities that exist in a time series is nothing but the Partial periodic patterns. These patterns have key properties such as starting, stopping, and restartinganywhere− within a series. Partial periodic patterns areclassifiedinto two types: (i) regular patterns− exhibiting periodic behavior throughout a series with some exceptions and( ii) periodic patterns exhibiting periodic behavior only for particular time intervals within a series. We have focused primarily on finding regular patterns during past studies on partial periodic search. The knowledge pertaining to periodic patterns cannot be ignored. This is because useful information pertaining to seasonal or time-based associations between events is provided bythem. Because of the foll o wi n g two main reasons, finding periodic patterns is a non-trivial task. (i) Each periodic pattern is associated with time-based information pertaining to its durations of periodic appearances in a series. Since the information can vary within and across patterns, obtaining this information ischallenging. (ii) As they do not satisfy the anti-monotonic property, finding all periodic patterns is a computationally expensive process. In this paper, periodic pattern model is proposed by addressing the above issues. Periodic Pattern growth algorithm along with an efficient pruning technique is also proposed to discover these patterns. The results through Experimentation have shown that Periodic patterns canbe really useful and it has also proven that our algorithm isnoteworthy.

scholarly journals Expansion of Generalized Hukuhara Differentiable Interval Valued Function

2019 ◽  
Vol 15 (03) ◽  
pp. 553-570
Author(s):  
Priyanka Roy ◽  
Geetanjali Panda
Keyword(s):  
Higher Dimension ◽  
Interval Valued Function ◽  
Generalized Hukuhara Differentiability ◽  
Monotonic Property ◽  
Theoretical Results ◽  
Interval Valued ◽  
Theoretical Developments

In this paper, the concept of [Formula: see text]-monotonic property of interval valued function in higher dimension is introduced. Expansion of interval valued function in higher dimension is developed using this property. Generalized Hukuhara differentiability is used to derive the theoretical results. Several examples are provided to justify the theoretical developments.

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