The daily time series Flare Index (FI) data of Northern Hemisphere, Southern Hemisphere and Total Disk for Solar Cycle 21- 23 and 24 up to Dec. 2014 has been pre-processed using a 2nd order exponential smoothing algorithm to remove orthogonal noise. The smoothed data in each case is processed for scaling analysis using Rescaled-Range Analysis as well as Finite Variance Scaling Method in order to search for the Hurst exponent. As the value of H obtained from our analysis lies in between 0 and 1, so it can be said that the signal may behave like Fractional Brownian Motion. Also, it is observed that H is less than 0.5 which indicates the data is anti-persistent in nature and it has a strong negative correlation within the signal. The value of H also indicates the oscillating features of the signal which might have some fundamental periodicities in the Suns atmosphere.

We study the uniform ergodicity of Markov processes (Zn, n ≥ 1) of order 2 with a general state space (Z, 𝒵). Markov processes of order higher than 1 were defined in the literature long ago, but scarcely treated in detail. We take as the basis for our considerations the natural transition probability Q of such a process. A Markov process of order 2 is transformed into one of order 1 by combining two consecutive variables Z2n–1 and Z2n into one variable Yn with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Yn, n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Zn, n ≥ 1) is defined in terms of the same property for (Yn, n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Zn, n ≥ 1). We apply the general results to study the uniform ergodicity of Markov processes of order 2 which arise in some nonlinear time series models and as sequences of smoothed values in sequential smoothing procedures of Markovian observations. As for the time series models, Markovian noise sequences are covered.

The classical exponential smoothing procedure to be applied to a sequence of observations is modified into some adaptive variants and also generalized to the recursive scheme

This means that given the history of observations and smoothed values up to time n, the updated smoothed value Wn +1 is given by a function u of Wn and Xn, which satisfies only certain structural properties. We study these procedures within the framework of random systems with complete connections and exploit results from Markov process theory to study the sequence . A broad class of ‘smoothing functions' u, which induce sequences with nice statistical properties, is presented. A method to estimate the limit of the expected smoothed values is also developed.,

We consider exponential smoothing Yn = αXn + (1 − α)Yn−1, 0 < α < 1, in experience rating. Here the premium Yn is determined by the policy's own claims history (Xn). In order to uniformize the fluctuation of premiums, it is appropriate to use a bigger α for the big policies than for the small ones. When the size of the policy changes with time, a need arises to change α correspondingly. It has recently been shown that changing based on the size of the premiums Yn may lead to too low a tariff level. This result is presented here and illustrated by means of simulation. Further, some general results are given how the changing can be made without a decline in the tariff level. The results are applied to a tariff system in which the linking of the smoothing parameter to the size of the policy is particularly motivated.

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.

Some inadequacies of both the traditional (exponential smoothing) and Box-Jenkins approaches to time series forecasting of economic data are investigated. An approach is suggested which integrates these two methodologies. It is based on smoothing the data using straight line segments instead of differencing to obtain stationarity, and forecasting using an autoregressive-moving-average model for the residuals from the most recent linear segment. The efficiency of this approach is calculated theoretically using a series comprising integrated white noise.