formulasearchengine - User contributions [en]

Bernstein's problem

2013-12-25T10:12:17Z

151.76.142.58: /* References */

{{Orphan|date=February 2011}}

'''Adaptive Gabor representation''' ('''AGR''') is a [[Gabor representation]] of a signal where its variance is adjustable. There's always a trade-off between time resolution and frequency resolution in traditional [[short-time Fourier transform]] (STFT). A long window leads to high frequency resolution and low time resolution. On the other hand, high time resolution requires shorter window, with the expense of low frequency resolution. By choosing the proper elementary function for signal with different spectrum structure, adaptive Gabor representation is able to accommodate both narrowband and wideband signal.

==Gabor expansion==
In 1946, [[Dennis Gabor]] suggested that a signal can be represented in two dimensions, with time and frequency coordinates. And the signal can be expanded into a discrete set of Gaussian elementary signals.

===Definition===
The Gabor expansion of signal s(t) is defined by this formula:

: <math>s(t)=\sum_{m=-\infty}^\infty \sum_{n=-\infty}^\infty C_{m,n}h(t-mT)e^{jnt\Omega}</math>

where ''h''(''t'') is the Gaussian elementary function:

: <math>h(t)=\left( \frac{\alpha}{\pi} \right)^\frac{1}{4}e^{\left( -\frac{\alpha}{2}t^2 \right)}</math>

Once the Gabor elementary function is determined, the Gabor coefficients <math>C_{m,n}</math>can be obtained by the inner product of s(t) and a dual function <math>\gamma(t)</math>

:<math>C_{m,n}=\int s(t)\gamma^*(t-mT)e^{-jnt\Omega} \, dt.</math>

<math>T</math> and <math>\Omega</math> denote the sampling steps of time and frequency and satisfy the criteria

:<math>T\Omega\leqq2\pi \,</math>

====Relationship between Gabor representation and Gabor transform====
Gabor transform simply computes the Gabor coefficients <math>C_{m,n}</math> for the signal s(t).

==Adaptive expansion==
Adaptive signal expansion is defined as

:<math>s\left( t \right)=\sum_p B_p h_p(t)</math>

where the coefficients <math>B_p</math> are obtained by the inner product of the signal s(t) and the elementary function <math>h_p</math>

:<math> B_p =\left \langle s,h_p\right \rangle \, </math>

Coeffients <math>B_p</math> represent the similarity between the signal and elementary function.<br/>
Adaptive signal decomposition is an iterative operation, aim to find a set of elementary function <math> \left\{ h_p(t) \right\} </math>, which is most similar to the signal's time-frequency structure.<br/>
First, start with w=0 and <math>s_0\left( t \right)=s\left( t \right)</math>. Then find <math>h_0\left( t \right)</math> which has the maximum inner product with signal <math>s_0\left( t \right)</math> and

: <math>\left| B_p \right|^2 = \max_h \left| \left \langle s_p (t),h_p(t) \right \rangle \right|^2</math>

Second, compute the residual:

: <math>s_1\left(t\right) =s_0\left(t\right)-B_0 h_0\left(t\right),</math>

and so on. It will comes out a set of '''residual''' (<math>s_p\left(t\right)</math>), '''projection''' (<math>B_p=\left \langle s_p(t),h_p(t) \right \rangle</math>), and '''elementary function''' (<math>h_p\left(t\right)</math>) for each different p. The energy of the residual will vanish if we keep doing the decomposition.

=== Energy conservation equation ===
If the elementary equation (<math>h_p\left(t\right)</math>) is designed to have a unit energy. Then the energy contain in the residual at the pth stage can be determined by the residual at p+1th stage plus (<math>B_p</math>). That is,

:<math>\left \| s_p(t) \right \|^2=\left \| s_{p+1}(t) \right \|^2 + \left| B_p \right|^2,</math>
:<math>\left \| s(t) \right \|^2=\sum_{p=o}^\infty \left| B_p \right|^2,</math>

similar to the [[Parseval's theorem]] in Fourier analysis.

==Adaptive Gabor representation==
The selection of elementary function is the main task in adaptive signal decomposition. It is natural to choose a Gaussian-type function to achieve the lower bound for the inequality:

: <math>h_p(t)=\left( \frac{\alpha}{\pi} \right)^\frac{1}{4}e^{ -\frac{\alpha}{2}(t-T_p)^2}e^{jt\Omega_p},</math>

where <math>\left(T_p,\Omega_p\right)</math> is th mean and <math>\alpha_p^{-1}</math> is the variance of Gaussian at <math>\left(T_p,\Omega_p\right)</math>. And

: <math>s\left( t \right) = \sum_p B_p h_p(t) = \sum_p B_p\left( \frac{\alpha}{\pi} \right)^\frac{1}{4} e^{-\frac{\alpha}{2}(t-T_p)^2 }e^{jt\Omega_p}</math>

is called the adaptive Gabor representation.

Changing the variance value will change the duration of the elementary function (window size), and the center of the elementary function is no longer fixed. By adjusting the center point and variance of the elementary function, we are able to match the signal's local time-frequency feature. The better performance of the adaptation is achieved at the cost of matching process. The trade-off between different window length now become the trade-off between computation time and performance.

==See also==
*[[Gabor transform]]
*[[Short-time Fourier transform]]

== References ==
*M.J. Bastiaans, "Gabor's expansion of a signal into Gaussian elementary signals", Proceedings of the IEEE, vol. 68, Issue:4, pp. 538–539, April 1980
*Shie Qian and Dapang Chen, "Signal Representation using adaptive normalized Gaussian functions," ''Signal Processing'', vol. 42, no.3, pp. 687–694, March 1994
*Qinye Yin, Shie Qian, and Aigang Feng, "A Fast Refinement for Adaptive Gaussian Chirplet Decomposition," IEEE Transactions on Signal Processing, vol. 50, no.6, pp. 1298-1306, June 2002
*Shie Qian, ''Introduction to Tiem-Frequency and Wavelet Transforms'', Prentice Hall, 2002
{{DEFAULTSORT:Adaptive Gabor Representation}}
[[Category:Time–frequency analysis]]

Baire category theorem

2013-12-25T10:09:40Z

151.76.142.58: /* External links */

:''For discounting in the sense of downplaying or dismissing, see [[Minimisation (psychology)]]. For the band of the same name, see [[Discount (band)]]. See also: [[Discounts and allowances]]''

'''Discounting''' is a financial mechanism in which a [[debtor]] obtains the right to delay payments to a [[creditor]], for a defined period of time, in exchange for a charge or fee.<ref name="Finance_Discount">See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Efficient Market", "Market Value" and "Opportunity Cost" in Downes, J. and Goodman, J. E. ''Dictionary of Finance and Investment Terms'', Baron's Financial Guides, 2003.</ref> Essentially, the party that owes money in the present purchases the right to delay the payment until some future date.<ref name="Economics_Discount">See "Discount", "Compound Interest", "Efficient Markets Hypothesis", "Efficient Resource Allocation", "Pareto-Optimality", "Price", "Price Mechanism" and "Efficient Market" in Black, John, ''Oxford Dictionary of Economics'', Oxford University Press, 2002.</ref> The '''discount''', or '''charge''', is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.<ref Name="Finance_Discount"/>

The discount is usually associated with a ''discount rate'', which is also called the ''discount yield''.<ref name="Finance_Discount"/><ref name="Finance_Discount"/><Ref Name="Economics_Discount"/><ref name="DiscountRate_Explain">Here, the ''discount rate'' is different from the [[discount window|discount rate]] the nation's Central Bank charges financial institutions.</ref> The discount yield is the proportional share of the initial amount owed (initial liability) that must be paid to delay payment for 1 year.

<blockquote>Discount Yield  =  "Charge" to Delay Payment for 1 year  /  Debt Liability</blockquote>

It is also the rate at which the amount owed must rise to delay payment for 1 year.

Since a person can earn a return on money invested over some period of time, most economic and financial models assume the "Discount Yield" is the same as the [[Rate of Return]] the person could receive by investing this money elsewhere (in assets of similar [[risk]]) over the given period of time covered by the delay in payment.<ref name="Finance_Discount"/><ref name="Economics_Discount"/> The Concept is associated with the [[Opportunity cost of capital|Opportunity Cost]] of not having use of the money for the period of time covered by the delay in payment. The relationship between the "Discount Yield" and the [[Rate of Return]] on other financial assets is usually discussed in such economic and financial theories involving the inter-relation between various [[Market price|Market Prices]], and the achievement of [[Pareto efficiency|Pareto Optimality]] through the operations in the [[Price mechanism|Capitalistic Price Mechanism]],<Ref Name="Economics_Discount"/> as well as in the discussion of the "[[Efficient-market hypothesis|Efficient (Financial) Market Hypothesis]]".<Ref Name="Finance_Discount"/><ref name="Economics_Discount"/><Ref Name="Economics_Competition">Competition from other firms who offer other Financial Assets that promise the Market [[Rate of Return]] forces the person who is asking for a delay in payment to offer a "Discount Yield" that is the same as the Market [[Rate of Return]].</ref> The person delaying the payment of the current Liability is essentially compensating the person to whom he/she owes money for the lost revenue that could be earned from an investment during the time period covered by the delay in payment.<ref name="Finance_Discount"/> Accordingly, it is the relevant "Discount Yield" that determines the "Discount", and not the other way around.

As indicated, the [[Rate of Return]] is usually calculated in accordance to an annual [[return on investment]]. Since an investor earns a return on the original principal amount of the investment as well as on any prior period Investment income, investment earnings are "compounded" as time advances.<ref name="Finance_Discount"/><ref name="Economics_Discount"/> Therefore, considering the fact that the "Discount" must match the benefits obtained from a similar [[Investment|Investment Asset]], the "Discount Yield" must be used within the same compounding mechanism to negotiate an increase in the size of the "Discount" whenever the time period the payment is delayed or extended.<Ref Name="Economics_Discount"/><ref name="Economics_Competition"/> The “Discount Rate” is the rate at which the “Discount” must grow as the delay in payment is extended.<Ref Name="MathEcon_Chiang">Chiang, Alpha CX. ''Fundamental Methods of Mathematical Economics, Third Edition'', McGraw Hill Book Company, 1984.</ref> This fact is directly tied into the "[[Time Value of Money]]" and its calculations.<ref name="Finance_Discount"/>

The "[[Time Value of Money]]" indicates there is a difference between the "Future Value" of a payment and the "Present Value" of the same payment. The [[Rate of Return]] on investment should be the dominant factor in evaluating the market's assessment of the difference between the "Future Value" and the "Present Value" of a payment; and it is the Market's assessment that counts the most.<Ref Name="Economics_Competition"/> Therefore, the "Discount Yield", which is predetermined by a related [[return on investment]] that is found in the [[financial market]]s, is what is used within the "[[Time Value of Money]]" calculations to determine the "Discount" required to delay payment of a financial liability for a given period of time.

==Basic calculation==
If we consider the value of the original payment presently due to be <math>P</math>, and the debtor wants to delay the payment for <math>t</math> years, then an <math>r</math> Market [[Rate of Return]] on a similar [[Investment|Investment Assets]] means the "Future Value" of <math>P</math> is <math>P(1 + r)^t</math>,<ref name="Economics_Discount"/><ref name="MathEcon_Chiang"/> and the "Discount" would be calculated as

: <math>\text{Discount} = P(1+r)^t-P</math><Ref Name="Economics_Discount"/>

where <math>r</math> is also the "Discount Yield".

If <math>F</math> is a payment that will be made <math>t</math> years in the future, then the "Present Value" of this Payment, also called the "Discounted Value" of the payment, is

: <math>P=\frac{F}{(1+r)^t}</math><Ref Name="Economics_Discount"/></blockquote>

To calculate the [[present value]] of a single cash flow, it is divided by one plus the interest rate for each period of time that will pass. This is expressed mathematically as raising the divisor to the power of the number of units of time.

Consider the task to find the present value ''PV'' of $100 that will be received in five years. Or equivalently, which amount of money today will grow to $100 in five years when subject to a constant discount rate?

Assuming a 12% per year interest rate it follows

: <math>{\rm PV}=\frac{$100}{(1+0.12)^5}=$56.74.</math>

== Discount rate ==

The discount rate which is used in financial calculations is usually chosen to be equal to the [[cost of capital|Cost of Capital]]. The [[cost of capital|Cost of Capital]], in a financial market equilibrium, will be the same as the Market [[Rate of Return]] on the financial asset mixture the firm uses to finance capital investment. Some adjustment may be made to the discount rate to take account of risks associated with uncertain cash flows, with other developments.

The discount rates typically applied to different types of companies show significant differences:

* Startups seeking money: 50 – 100%
* Early Startups: 40 – 60%
* Late Startups: 30 – 50%
* Mature Companies: 10 – 25%

The higher discount rate for startups reflects the various disadvantages they face, compared to established companies:

* Reduced marketability of ownerships because stocks are not traded publicly.
* Limited number of investors willing to invest.
* Startups face high risks.
* Over optimistic forecasts by enthusiastic founders.

One method that looks into a correct discount rate is the [[capital asset pricing model]].
This model takes in account three variables that make up the discount rate:

'''1. Risk Free Rate''': The percentage of return generated by investing in risk free securities such as government bonds.

'''2. Beta''': The measurement of how a company’s stock price reacts to a change in the market. A beta higher than 1 means that a change in share price is exaggerated compared to the rest of shares in the same market. A beta less than 1 means that the share is stable and not very responsive to changes in the market. Less than 0 means that a share is moving in the opposite of the market change.

'''3. Equity Market Risk Premium''': The return on investment that investors require above the risk free rate.

'''Discount rate'''= risk free rate + beta*(equity market risk premium)

== Discount factor ==

The '''discount factor''', DF(T), is the factor by which a future cash flow must be multiplied in order to obtain the present value. For a zero-rate (also called spot rate) <math>r</math>, taken from a [[yield curve]], and a time to cashflow <math>T</math> (in years), the discount factor is:

: <math> DF(T) = \frac{1}{(1+rT)} </math>

In the case where the only discount rate you have is not a zero-rate (neither taken from a [[zero-coupon bond]] nor converted from a [[swap rate]] to a zero-rate through [[Bootstrapping (finance)|bootstrapping]]) but an annually-compounded rate (for example if your benchmark is a US Treasury bond with annual coupons and you only have its [[yield to maturity]], you would use an annually-compounded discount factor:

: <math> DF(T) = \frac{1}{(1+r)^T} </math>

However, when operating in a bank, where the amount the bank can lend (and therefore get interest) is linked to the value of its [[assets]] (including [[accrued interest]]), traders usually use daily compounding to discount cashflows. Indeed, even if the interest of the bonds it holds (for example) is paid semi-annually, the value of its book of bond will increase daily, thanks to [[accrued interest]] being accounted for, and therefore the bank will be able to re-invest these daily accrued interest (by lending additional money or buying more financial products). In that case, the discount factor is then (if the usual [[money market]] [[day count convention]] for the currency is ACT/360, in case of currencies such as USD, EUR, JPY), with <math>r</math> the zero-rate and <math>T</math> the time to cashflow in years:

: <math> DF(T) = \frac{1}{( 1 + \frac{r}{360} )^{ 360T } } </math>

or, in case the market convention for the currency being discounted is ACT/365 (AUD, CAD, GBP):

: <math> DF(T) = \frac{1}{( 1 + \frac{r}{365} )^{ 365T } } </math>

Sometimes, for manual calculation, the continuously-compounded hypothesis is a close-enough approximation of the daily-compounding hypothesis, and makes calculation easier (even though it does not have any real application as no financial instrument is continuously compounded). In that case, the discount factor is:

: <math> DF(T) = e^{-rT} \,</math>

== Other discounts ==

For '''discounts''' in [[marketing]], see [[discounts and allowances]], [[sales promotion]], and [[pricing]]. The article on [[Discounted Cash Flow]] provides a nice example about discounting and risks in real estate investments.

== See also ==
* [[Coupon]]
* [[Coupon (bond)]]
* [[Hyperbolic discounting]]

== References ==
'''Notes'''
{{reflist}}

== External links ==
{{Wiktionary}}
* [http://www.excelexchange.com/discount%20mathematics.htm Tutorial on Discount Mathematics]

[[Category:Actuarial science]]
[[Category:Basic financial concepts]]
[[Category:Marketing]]

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