Time series

A time series is a series of data points indexed in time order. Most a time series is a sequence taken at successive 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, the daily closing value of the Dow Jones Industrial Average. Time series are frequently plotted via line charts. Time series are used in statistics, signal processing, pattern recognition, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, communications engineering, 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 forecasting is the use of a model to predict future values based on observed values. While regression analysis is employed in such a way as to test theories that the current values of one or more independent time series affect the current value of another time series, this type of analysis of time series is not called "time series analysis", which focuses on comparing values of a single time series or multiple dependent time series at different points in time.

Interrupted time series analysis is the analysis of interventions on a single time series. Time series data have a natural temporal ordering; this makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations. Time series analysis is distinct from spatial data analysis where the observations relate to geographical locations. A stochastic model for a time series will reflect the fact that observations close together in time will be more related than observations further apart. In addition, time series models will make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data. Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods; the former include wavelet analysis.

In the time domain and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain. Additionally, time series analysis techniques may be divided into parametric and non-parametric methods; the parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters. In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure. Methods of time series analysis may be divided into linear and non-linear, univariate and multivariate. A time series is one type of panel data. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel. A data set may exhibit characteristics of both panel data and time series data.

One way to tell is to ask. If the answer is the time data field this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier, unrelated to time it is panel data candidate. If the differentiation lies on the non-time identifier the data set is a cross-sectional data set candidate. There are several types of motivation and data analysis available for time series which are appropriate for different purposes and etc. In the context of statistics, quantitative finance, seismology and geophysics the primary goal of time series analysis is forecasting. In the context of signal processing, control engineering and communication engineering it is used for signal detection and estimation, while in the context of data mining, pattern recognition and machine learning time series analysis can be used for clustering, query by content, anomaly detection as well as forecasting; the clearest way to examine a regular time series manually is with a line chart such as the one shown for tuberculosis in the United States, made with a spreadsheet program.

The number of cases was standardized to a rate per 100,000 and the percent change per year in this rate was calculated. The nearly dropping line shows that the TB incidence was decreasing in most years, but the percent change in this rate varied by as much as +/- 10%, with'surges' in 1975 and around the early 1990s; the use of both vertical axes allows the comparison of two time series in one graphic. Other techniques include: Autocorrelation analysis to examine serial dependence Spectral analysis to examine cyclic behavior which need not be related to seasonality. For example, sun spot activity vari

Trigonometric functions

In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications; the most familiar trigonometric functions are the sine and tangent. In the context of the standard unit circle, where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the y-component of the triangle, the cosine gives the x-component, the tangent function gives the slope. For angles less than a right angle, trigonometric functions are defined as ratios of two sides of a right triangle containing the angle, their values can be found in the lengths of various line segments around a unit circle. Modern definitions express trigonometric functions as infinite series or as solutions of certain differential equations, allowing the extension of the arguments to the whole number line and to the complex numbers.

Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles. In this use, trigonometric functions are used, for instance, in navigation and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates; the sine and cosine functions are commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, average temperature variations through the year. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. With the last four, these relations are taken as the definitions of those functions, but one can define them well geometrically, or by other means, derive these relations; the notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides.

That is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides, it is these ratios. To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A; the three sides of the triangle are named as follows: The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle; the opposite side is the side opposite in this case side a. The adjacent side is the side having both the angles in this case side b. In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a right-angled triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation; the following definitions apply to angles in this range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions.

For example, the figure shows sin for angles θ, π − θ, π + θ, 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π; the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram; the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin sinus for gulf or bay, given a unit circle, it is the side of the triangle on which the angle opens. In that case: sin A = opposite hypotenuse The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle; because the angle sum of a triangle is π radians, the co-angle B is equal to π/2 − A.

In that case: cos A = adjacent hypotenuse The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment tangent to the circle, i.e. the line that touches the circle, from Latin linea tangens or touching line. In our case: tan A = opposite adjacent Tangent may be represented in terms of sine and cosine; that is: tan A = sin A cos A = opposite

Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point, diametrically opposite to it — so situated that a line drawn from the one to the other passes through the center of the sphere and forms a true diameter. This term applies to opposite points on any n-sphere. An antipodal point is sometimes called an antipode, a back-formation from the Greek loan word antipodes, which meant "opposite the feet". In mathematics, the concept of antipodal points is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite through the centre. On a circle, such points are called diametrically opposite. In other words, each line through the centre intersects the sphere in two points, one for each ray out from the centre, these two points are antipodal; the Borsuk–Ulam theorem is a result from algebraic topology dealing with such pairs of points. It says that any continuous function from Sn to Rn maps some pair of antipodal points in Sn to the same point in Rn.

Here, Sn denotes the n-dimensional sphere in -dimensional space. The antipodal map A: Sn → Sn, defined by A = −x, sends every point on the sphere to its antipodal point, it is homotopic to the identity map if n is odd, its degree is n+1. If one wants to consider antipodal points as identified, one passes to projective space. An antipodal pair of a convex polygon is a pair of 2 points admitting 2 infinite parallel lines being tangent to both points included in the antipodal without crossing any other line of the convex polygon. Hazewinkel, Michiel, ed. "Antipodes", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 "antipodal". PlanetMath

Pearson correlation coefficient

In statistics, the Pearson correlation coefficient referred to as Pearson's r, the Pearson product-moment correlation coefficient or the bivariate correlation, is a measure of the linear correlation between two variables X and Y. According to the Cauchy–Schwarz inequality it has a value between +1 and −1, where 1 is total positive linear correlation, 0 is no linear correlation, −1 is total negative linear correlation, it is used in the sciences. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s and for which the mathematical formula was derived and published by Auguste Bravais in 1844.. The naming of the coefficient is thus an example of Stigler's Law. Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations; the form of the definition involves a "product moment", that is, the mean of the product of the mean-adjusted random variables. Pearson's correlation coefficient when applied to a population is represented by the Greek letter ρ and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient.

Given a pair of random variables, the formula for ρ is: where: cov is the covariance σ X is the standard deviation of X σ Y is the standard deviation of Y The formula for ρ can be expressed in terms of mean and expectation. Since cov = E , the formula for ρ can be written as where: σ Y and σ X are defined as above μ X is the mean of X μ Y is the mean of Y E is the expectation; the formula for ρ can be expressed in terms of uncentered moments. Since μ X = E μ Y = E σ X 2 = E = E − 2 σ Y 2 = E = E − 2 E = E = E − E E , the formula for ρ can be written as ρ X, Y = E − E E E − 2 E −

Slope

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is denoted by the letter m. Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between two distinct points on a line. Sometimes the ratio is expressed as a quotient, giving the same number for every two distinct points on the same line. A line, decreasing has a negative "rise"; the line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line; the direction of a line is either increasing, horizontal or vertical. A line is increasing; the slope is positive, i.e. m > 0. A line is decreasing; the slope is negative, i.e. m < 0. If a line is horizontal the slope is zero; this is a constant function. If a line is vertical the slope is undefined.

The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is = Δy. For short distances - where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is = Δx. Here the slope of the road between the two points is described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the slope m of the line is m = y 2 − y 1 x 2 − x 1; the concept of slope applies directly to gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of incline θ by the tangent function m = tan Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1; as a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point.

When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function as an algebraic formula the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve; this generalization of the concept of slope allows complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment; the slope of a line in the plane containing the x and y axes is represented by the letter m, is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.

This is described by the following equation: m = Δ y Δ x = vertical change horizontal change = rise run. Given two points and, the change in x from one to the other is x2 − x1, while the change in y is y2 − y1. Substituting both quantities into the above equation generates the formula: m = y 2 − y 1 x 2 − x 1; the formula fails for a vertical line, parallel to the y axis, where the slope can be taken as infinite, so the slope of a vertical line is considered undefined. Suppose a line runs through two points: P = and Q =. By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line: m = Δ y Δ x = y 2 − y 1 x 2 − x 1 = 8 − 2 13 − 1 = 6 12 = 1 2 {\displaystyle m====={\frac

Oklahoma State University–Stillwater

Oklahoma State University is a public land-grant and sun-grant research university in Stillwater, Oklahoma. OSU was founded in 1890 under the Morrill Act. Known as Oklahoma Agricultural and Mechanical College, it is the flagship institution of the Oklahoma State University System. Official enrollment for the fall 2010 semester system-wide was 35,073, with 23,459 students enrolled at OSU-Stillwater. Enrollment shows. OSU is classified by the Carnegie Foundation as a research university with highest research activity; the Oklahoma State Cowboys and Cowgirls' athletic heritage includes 52 national championships, a total greater than all but three NCAA Division I schools in the United States, first in the Big 12 Conference. Students spend part of the fall semester preparing for OSU's Homecoming celebration, begun in 1913, which draws more than 40,000 alumni and over 70,000 participants each year to campus and is billed by the university as "America's Greatest Homecoming Celebration." On December 25, 1890, the Oklahoma Territorial Legislature gained approval for Oklahoma Territorial Agricultural and Mechanical College, the land-grant university established under the Morrill Act of 1862.

It specified. Such an ambiguous description created rivalry between towns within the county, with Stillwater winning out. Upon statehood in 1907, "Territorial" was dropped from its title; the first students assembled for class on December 14, 1891. Classes were held for two and one-half years in local churches until the first academic building known as Old Central, was dedicated on June 15, 1894, on the southeast corner of campus, which at the time was flat plowed prairie. In 1896, Oklahoma A&M held its first commencement with six male graduates; the first Library was established in Old Central in one room shared with the English Department. The first campus building to have electricity, Williams Hall, was constructed in 1900. With its turreted architecture it was referred to as the "Castle of the Prairies". One of the earliest campus buildings was a barn, used as part of an agricultural experiment station, served by a large reservoir pond created in 1895; the barn burned in 1922, but the pond and remodeled in 1928 and 1943, is now known as Theta Pond, a popular campus scenic landmark.

In 1906, Morrill Hall became the principal building on campus. A fire gutted the building in 1914, but the outside structure survived intact, the interior was reconstructed; the first dormitory for women was completed in 1911. It contained a kitchen, dining hall, some classrooms, a women's gymnasium, it is now houses the Gardiner Art Gallery. By 1919 the campus included Morrill Hall, the Central Building, the Engineering Building, the Women's building, the Auditorium, the Armory-Gymnasium and the Power Plant. At the beginning of World War II, Oklahoma A&M was one of six schools selected by the United States Navy to give the Primary School in the Electronics Training Program known as Naval Training School Elementary Electricity and Radio Materiel. Starting in March 1942, each month a new group of 100 Navy students arrived for three months of 14-hour days in concentrated electrical engineering study. Cordell Hall, the newest dormitory, was used for housing and meals. Professor Emory B. Phillips was the Director of Instruction.

ETP admission required passing the Eddy Test, one of the most selective qualifying exams given during the war years. At a given time, some 500 Navy students were on the campus, a significant fraction of the war-years enrollment; the training activity continued until June 1945, served a total of about 7,000 students. Kamm, a future professor and president of Oklahoma State University. During some of the war years, the Navy had a Yeoman training activity for WAVES and SPARS on the campus. Much of the growth of Oklahoma A&M and the campus architectural integrity can be attributed to work of Henry G. Bennett, who served as the school's president from 1928 to 1950. Early in his tenure Dr. Bennett developed a strategic vision for the physical expansion of the university campus; the plan was adopted in 1937 and his vision was followed for more than fifty years, making the university what it is today, including the Georgian architecture that permeates the campus. The focal point of his vision was a centrally located library building, which became a reality when the Edmon Low Library opened in 1953.

Another major addition to the campus during the Bennett years was the construction of the Student Union, which opened in 1950. Subsequent additions and renovations have made the building one of the largest student union buildings in the world at 611,000 sq ft. A complete renovation and further expansion of the building began in 2010. On May 15, 1957, Oklahoma A&M changed its name Oklahoma State University of Agricultural and Applied Sciences to reflect the broadening scope of curriculum offered. Oklahoma Gov. Raymond Gary signed the bill authorizing the name change passed by the 26th Oklahoma Legislature on May 15, 1957. However, the bill only authorized the Board of Regents to change the name of the college, a measure they voted on at their meeting on June 6. However, the name was shortened to Oklahoma State University for most purposes, the "Agricultural & Applied Sciences" name was formally

Finance

Finance is a field, concerned with the allocation of assets and liabilities over space and time under conditions of risk or uncertainty. Finance can be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance. Matters in personal finance revolve around: Protection against unforeseen personal events, as well as events in the wider economies Transference of family wealth across generations Effects of tax policies management of personal finances Effects of credit on individual financial standing Development of a savings plan or financing for large purchases Planning a secure financial future in an environment of economic instability Pursuing a checking and/or a savings account Personal finance may involve paying for education, financing durable goods such as real estate and cars, buying insurance, e.g. health and property insurance and saving for retirement.

Personal finance may involve paying for a loan, or debt obligations. The six key areas of personal financial planning, as suggested by the Financial Planning Standards Board, are: Financial position: is concerned with understanding the personal resources available by examining net worth and household cash flows. Net worth is a person's balance sheet, calculated by adding up all assets under that person's control, minus all liabilities of the household, at one point in time. Household cash flows total up all from the expected sources of income within a year, minus all expected expenses within the same year. From this analysis, the financial planner can determine to what degree and in what time the personal goals can be accomplished. Adequate protection: the analysis of how to protect a household from unforeseen risks; these risks can be divided into the following: liability, death, disability and long term care. Some of these risks may be self-insurable, while most will require the purchase of an insurance contract.

Determining how much insurance to get, at the most cost effective terms requires knowledge of the market for personal insurance. Business owners, professionals and entertainers require specialized insurance professionals to adequately protect themselves. Since insurance enjoys some tax benefits, utilizing insurance investment products may be a critical piece of the overall investment planning. Tax planning: the income tax is the single largest expense in a household. Managing taxes is not a question of if you will pay taxes, but when and how much. Government gives many incentives in the form of tax deductions and credits, which can be used to reduce the lifetime tax burden. Most modern governments use a progressive tax; as one's income grows, a higher marginal rate of tax must be paid. Understanding how to take advantage of the myriad tax breaks when planning one's personal finances can make a significant impact in which can save you money in the long term. Investment and accumulation goals: planning how to accumulate enough money – for large purchases and life events – is what most people consider to be financial planning.

Major reasons to accumulate assets include purchasing a house or car, starting a business, paying for education expenses, saving for retirement. Achieving these goals requires projecting what they will cost, when you need to withdraw funds that will be necessary to be able to achieve these goals. A major risk to the household in achieving their accumulation goal is the rate of price increases over time, or inflation. Using net present value calculators, the financial planner will suggest a combination of asset earmarking and regular savings to be invested in a variety of investments. In order to overcome the rate of inflation, the investment portfolio has to get a higher rate of return, which will subject the portfolio to a number of risks. Managing these portfolio risks is most accomplished using asset allocation, which seeks to diversify investment risk and opportunity; this asset allocation will prescribe a percentage allocation to be invested in stocks, bonds and alternative investments.

The allocation should take into consideration the personal risk profile of every investor, since risk attitudes vary from person to person. Retirement planning is the process of understanding how much it costs to live at retirement, coming up with a plan to distribute assets to meet any income shortfall. Methods for retirement plans include taking advantage of government allowed structures to manage tax liability including: individual structures, or employer sponsored retirement plans and life insurance products. Estate planning involves planning for the disposition of one's assets after death. There is a tax due to the state or federal government at one's death. Avoiding these taxes means that more of one's assets will be distributed to one's heirs. One can leave one's assets to friends or charitable groups. Corporate finance deals with the sources of funding and the capital structure of corporations, the actions that managers take to increase the value of the firm to the shareholders, the tools and analysis used to allocate financial resources.

Although it is in principle different from managerial finance which studies the financial management of all firms, rather than corporations alone, the main concepts in the study of corporate finance are applicable to the financial problems of all kinds of firms. Corporate f