SC (complexity)

In statistics, first-hitting-time models are a sub-class of survival models. The first hitting time, also called first passage time, of a set $A$ with respect to an instance of a stochastic process is the time until the stochastic process first enters $A$.

Examples

A common example of a first-hitting-time model is a ruin problem, such as Gambler's ruin. In this example, an entity (often described as a gambler or an insurance company) has an amount of money which varies randomly with time, possibly with some drift. The model considers the event that the amount of money reaches 0, representing bankruptcy. The model can answer questions such as the probability that this occurs within finite time, or the mean time until which it occurs.

First-hitting-time models can be applied to expected lifetimes, of patients or mechanical devices. When the process reaches an adverse threshold state for the first time, the patient dies, or the device breaks down.

First passage time of a 1D Brownian Particle

The probability density function (PDF) for a particle in one dimension is found by solving the one-dimensional diffusion equation. (This equation states that the position probability density diffuses out over time. It is analogous to say, cream in a cup of coffee if the cream was all contained within some small location initially. In the long time limit the cream has diffused throughout the entire drink evenly.)

${\frac {\partial p(x,t\mid x_{0})}{\partial t}}=D{\frac {\partial ^{2}p(x,t\mid x_{0})}{\partial x^{2}}},$

given the initial condition $p(x_{0},t={0}\mid x_{0})=\delta (x-x_{0})$; where $x(t)$ is the position of the particle at some given time, $x_{0}$ is the tagged particle's initial position, and $D$ is the diffusion constant with the S.I. units $m^{2}s^{-1}$ (an indirect measure of the particle's speed). The bar in the argument of the instantaneous probability refers to the conditional probability. The diffusion equation states that the speed at which the probability for finding the particle at $x(t)$ is position dependent.

It can be shown that the one-dimensional PDF is

$p(x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {(x-x_{0})^{2}}{4Dt}}\right).$

This states that the probability of finding the particle at $x(t)$ is Gaussian, and the width of the Gaussian is time dependent. More specifically the Full Width at Half Maximum (FWHM) - technically, this is actually the Full Duration at Half Maximum as the independent variable is time - scales like

${\rm {{FWHM}\sim {\sqrt {t}}.}}$

Using the PDF one is able to derive the average of a given function, $L$, at time $t$:

$\langle L(t)\rangle \equiv \int _{-\infty }^{\infty }L(x,t)p(x,t)dx,$

where the average is taken over all space (or any applicable variable). The First Passage Time Density (FPTD) is the probability that a particle has first reached a point $x_{c}$ at time $t$. This probability density is directly calculable from the Survival probability (a more common probability measure in statistics). The Survival probability is calculated from the PDF, namely

$S(t)\equiv \int _{-\infty }^{x_{c}}p(x,t\mid x_{0})dx,$

which is the probability that a particle is still at a position $x<x_{c}$ at time $t$.(The subscript c for the absorption point $x_{c}$ is an abbreviation for cliff used in many texts as an analogy to an absorption point.) The relation between the Survival probability and the FPTD is as follows(the probability that a particle has reached the absorption point between times $t$ and $t+dt$ is $f(t)dt=S(t)-S(t+dt)$. If one uses the first-order Taylor approximation, the definition of the FPTD follows):

$f(t)=-{\frac {\partial S(t)}{\partial t}}.$

By working through the relevant mathematics (using a technique known as the Method of Images - borrowed from Electrodynamics), the explicit FPTD is

$f(t)\equiv {\frac {|x_{c}-x_{0}|}{\sqrt {4\pi Dt^{3}}}}\exp \left(-{\frac {(x_{c}-x_{0})^{2}}{4Dt}}\right).$

For $t\gg {\frac {(x_{c}-x_{0})^{2}}{4D}}$, it follows from above that

$f(t)={\frac {\Delta x}{\sqrt {4\pi Dt^{3}}}}\sim t^{-3/2},$

where $\Delta x\equiv x_{c}-x_{0}$. This equation states that the probability for a Brownian particle achieving a first passage at some long time (defined in the paragraph above) becomes increasingly small, but always finite.

The first moment of the FPTD diverges (as it is a so-called heavy-tailed distribution), therefore one cannot calculate the average FPT, so instead, one can calculate the typical time, the time when the FPTD is at a maximum ($\partial f/\partial t=0$), i.e.

$\tau _{\rm {ty}}={\frac {\Delta x^{2}}{6D}}.$

Latent vs observable

In many real world applications, the process is latent, or unobservable. When first hitting time models are equipped with regression structures, accommodating covariate data, we call such regression structure Threshold regression. The threshold state, parameters of the process, and even time scale may depend on corresponding covariates.

A first-hitting-time (FHT) model has two underlying components: (1) a parent stochastic process $\{X(t)\}\,\,$, and (2) a threshold. The first hitting time is defined as the time when the stochastic process first reaches the threshold. It is very important to distinguish whether the sample path of the parent process is latent (i.e., unobservable) or observable, and such distinction is a characteristic of the FHT model. By far, latent processes are most common. To give an example, we can use a Wiener process $\{X(t),t\geq 0\,\}\,$ as the parent stochastic process. Such Wiener process can be defined with the mean parameter ${\mu }\,\,$, the variance parameter ${\sigma ^{2}}\,\,$, and the initial value $X(0)=x_{0}>0\,$.

See also

• Survival analysis
• Proportional hazards models

References

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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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