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Whittaker–Henderson Smoothing
Whittaker–Henderson smoothing or Whittaker–Henderson graduation is a digital filter that can be applied to a set of digital data points for the purpose of smoothing the data, that is, to increase the precision of the data without distorting the signal tendency. It was first introduced by Georg Bohlmann (for order 1). E. T. Whittaker, E.T. Whittaker independently proposed the same idea in 1923 (for order 3). Robert_Henderson_(mathematician), Robert Henderson contributed to the topic by his two publications in 1924 and 1925. Whittaker–Henderson smoothing can be seen as P-Splines of degree 0. The special case of order 2 also goes under the name Hodrick–Prescott filter. Mathematical Formulation For a signal y_i, i=1, \ldots, n, of equidistant steps, e.g. a time series with constant intervals, the Whittaker–Henderson smoothing of order p is the solution to the following penalized least squares problem: : \hat = \operatorname_ \sum_^n (y_i - x_i)^2 + \lambda \sum_^ (\Delta^p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Filter
In signal processing, a digital filter is a system that performs mathematical operations on a Sampling (signal processing), sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is typically an electronic circuit operating on continuous-time analog signals. A digital filter system usually consists of an analog-to-digital converter (ADC) to sample the input signal, followed by a microprocessor and some peripheral components such as memory to store data and filter coefficients etc. Program Instructions (software) running on the microprocessor implement the digital filter by performing the necessary mathematical operations on the numbers received from the ADC. In some high performance applications, an FPGA or ASIC is used instead of a general purpose microprocessor, or a specialized digital signal processor (DSP) with specific paralleled architecture for expedi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Bohlmann
Georg Bohlmann (23 April 1869 – 25 April 1928) was a German mathematician who specialized in probability theory and actuarial mathematics. Life and career Georg Bohlmann went to school in Berlin and Leipzig and took his ''Abitur'' at the ''Wilhelms-Gymnasium'' in Berlin in 1888. After that, he began studying mathematics at the University of Berlin under Leopold Kronecker, Lazarus Fuchs, and Wilhelm Dilthey. As he advanced in his studies, Lie groups became the focus of his interest. Since this area was poorly represented at Berlin, he moved to the University of Halle, where he obtained his doctorate in 1892 under Albert Wangerin with a dissertation on the topic ''Ueber eine gewisse Klasse continuierlicher Gruppen und ihren Zusammenhang mit den Additionstheoremen'' ("On a certain class of continuous groups and their relation to addition theorems"). After that, he worked at the Meteorological Institute of Berlin, where presumably his interest in applied mathematics developed. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Henderson (mathematician)
Robert Henderson (24 May 1871, Russell, Ontario – 16 February 1942, Crown Point, New York) was a Canadian-American mathematician and actuary. Education and career Robert Henderson matriculated at age 16 at the University of Toronto and graduated there in 1891 with a bachelor's degree in mathematics. He spent a year as a fellow of the University of Toronto and then in 1892 was employed at the Government Insurance Department in Ottawa until he left Canada in 1897. From 1897 until his retirement in 1936 he worked for the Equitable Life Assurance Society of the United States. There he was from 1903 to 1911 an assistant actuary, from 1911 to 1920 an actuary, from 1920 to 1929 the second vice-president, and from 1929 to 1936 the vice-president. Henderson worked in the field of actuarial science, including life insurance and its history, mortality tables, interpolation, cumulative frequency analysis, and moments. In 1914 he was a member of the Committee of the Census appointed by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodrick–Prescott Filter
The Hodrick–Prescott filter (also known as Hodrick–Prescott decomposition) is a mathematical tool used in macroeconomics, especially in real business cycle theory, to remove the cyclical component of a time series from raw data. It is used to obtain a smoothed-curve representation of a time series, one that is more sensitive to long-term than to short-term fluctuations. The adjustment of the sensitivity of the trend to short-term fluctuations is achieved by modifying a multiplier \lambda. The filter was popularized in the field of economics in the 1990s by economists Robert J. Hodrick and Nobel Memorial Prize winner Edward C. Prescott, though it was first proposed much earlier by E. T. Whittaker in 1923., see Whittaker-Henderson smoothing. The Hodrick–Prescott filter is a special case of a smoothing spline. The equation The reasoning for the methodology uses ideas related to the decomposition of time series. Let y_t\, for t = 1, 2, ..., T\, denote the logarithms of a time ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements. Time series ''analysis'' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothing Spline
Smoothing splines are function estimates, \hat f(x), obtained from a set of noisy observations y_i of the target f(x_i), in order to balance a measure of goodness of fit of \hat f(x_i) to y_i with a derivative based measure of the smoothness of \hat f(x). They provide a means for smoothing noisy x_i, y_i data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where x is a vector quantity. Cubic spline definition Let \ be a set of observations, modeled by the relation Y_i = f(x_i) + \epsilon_i where the \epsilon_i are independent, zero mean random variables. The cubic smoothing spline estimate \hat f of the function f is defined to be the unique minimizer, in the Sobolev space W^2_2 on a compact interval, of : \sum_^n \^2 + \lambda \int \hat^(x)^2 \,dx. Remarks: * \lambda \ge 0 is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate. This is oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frederick Macaulay
Frederick Robertson Macaulay (August 12, 1882 – March 1970) was a Canadian economist of the Institutionalist School. He is known for introducing the concept of bond duration. Macaulay's contributions also include a mammoth empirical study of the time series behavior of interest rates published in 1938 and a study of short selling on the New York Stock Exchange (Macaulay and Durand, 1951). The term "Macaulay duration" is named after him. Macaulay was born in Montreal to a family influential in Montreal business; his father, Thomas Bassett Macaulay, was a well-known actuary He obtained his bachelor's and master's degrees from the |
