Soil resistivity

CLs (from Confidence Levels) is a statistical method for setting upper limits on model parameters, a particular form of interval estimation used for parameters that can take only non-negative values. It was first introduced by physicists working at the LEP experiment at CERN and has since been used by many high energy physics experiments. It is a frequentist method in the sense that the properties of the limit are defined by means of error probabilities, however it differs from standard confidence intervals in that the stated confidence level of the interval is not equal to its coverage probability. The reason for this deviation is that standard upper limits based on a most powerful test necessarily produce empty intervals with some fixed probability when the parameter value is zero, and this property is considered undesirable by most physicists and statisticians. Upper limits derived with the CLs method always contain the zero value of the parameter and hence the coverage probability at this point is always 100%. The definition of CLs does not follow from any precise theoretical framework of statistical inference and is therefore described sometimes as ad hoc. It has however close resemblance to concepts of statistical evidence proposed by the statistician Allan Birnbaum.

Definition

Let X be a random sample from a probability distribution with a real non-negative parameter $\theta \in [0,\infty )$. A CLs upper limit for the parameter θ, with confidence level $1-\alpha '$, is a statistic (i.e., observable random variable) $\theta _{up}(X)$ which has the property:

The inequality is used in the definition to account for cases where the distribution of X is discrete and an equality can not be achieved precisely. If the distribution of X is continuous then this should be replaced by an equality. Note that the definition implies that the coverage probability $\mathbb {P} (\theta _{up}(X)\geq \theta |\theta )$ is always larger than $1-\alpha '$.

An equivalent definition can be made by considering a hypothesis test of the null hypothesis $H_{0}:\theta =\theta _{0}$ against the alternative $H_{1}:\theta =0$. Then the numerator in (Template:EquationNote), when evaluated at $\theta _{0}$, correspond to the type-I error probability ($\alpha$) of the test (i.e., $\theta _{0}$ is rejected when $\theta _{up}(X)<\theta _{0}$) and the denominator to the power ($1-\beta$). The criterion for rejecting $H_{0}$ thus requires that the ratio $\alpha /(1-\beta )$ will be smaller than $\alpha '$. This can be interpreted intuitively as saying that $\theta _{0}$ is excluded because it is $\alpha '$ less likely to observe such an extreme outcome as X when $\theta _{0}$ is true than it is when the alternative $\theta =0$ is true.

The calculation of the upper limit is usually done by constructing a test statistic $q_{\theta }(X)$ and finding the value of $\theta$ for which

${\frac {\mathbb {P} (q_{\theta }(X)\geq q_{\theta }^{*}|\theta )}{\mathbb {P} (q_{\theta }(X)\geq q_{\theta }^{*}|0)}}=\alpha '.$

where $q_{\theta }^{*}$ is the observed outcome of the experiment.

Usage in high energy physics

Upper limits based on the CLs method were used in numerous publications of experimental results obtained at particle accelerator experiments such as LEP, the Tevatron and the LHC. Perhaps most notable are the upper limits placed on the production cross section of Higgs bosons.

Origin

The original motivation for CLs was based on a conditional probability calculation suggested by physicist G. Zech for an event counting experiment. Suppose an experiment consists of measuring $n$ events coming from signal and background processes, both described by Poisson distributions with respective rates $s$ and $b$, namely $n\sim {\text{Poiss}}(s+b)$. $b$ is assumed to be known and $s$ is the parameter to be estimated by the experiment. The standard procedure for setting an upper limit on $s$ given an experimental outcome $n^{*}$ consists of excluding values of $s$ for which $\mathbb {P} (n\leq n^{*}|s+b)\leq \alpha$, which guarantees at least $1-\alpha$ coverage. Consider, for example, a case where $b=3$ and $n^{*}=0$ events are observed, then one finds that $s+b\geq 3$ is excluded at 95% confidence level. But this implies that $s\geq 0$ is excluded, namely all possible values of $s$. Such a result is difficult to interpret because the experiment cannot essentially distinguish very small values of $s$ from the background-only hypothesis, and thus declaring that such small values are excluded (in favor of the background-only hypothesis) seems inappropriate. To overcome this difficulty Zech suggested conditioning the probability that $n\leq n^{*}$ on the observation that $n_{b}\leq n^{*}$, where $n_{b}$ is the (unmeasurable) number of background events. The reasoning behind this is that when $n_{b}$ is small the procedure is more likely to produce an error (i.e., an interval that does not cover the true value) than when $n_{b}$ is large, and the distribution of $n_{b}$ itself is independent of $s$. That is, not the over-all error probability should be reported but the conditional probability given the knowledge one has on the number of background events in the sample. This conditional probability is easily seen to be

$\mathbb {P} (n\leq n^{*}|n_{b}\leq n^{*},s+b)={\frac {\mathbb {P} (n\leq n^{*}|s+b)}{\mathbb {P} (n_{b}\leq n^{*}|s+b)}}={\frac {\mathbb {P} (n\leq n^{*}|s+b)}{\mathbb {P} (n\leq n^{*}|b)}}.$

which correspond to the above definition of CLs.

Generalization of the conditional argument

Zech's conditional argument can be formally extended to the general case. Suppose that $q(X)$ is a test statistic from which the confidence interval is derived, and let

$p_{\theta }=\mathbb {P} (q(X)>q^{*}|\theta )$

where $q*$ is the outcome observed by the experiment. Then $p_{\theta }$ can be regarded as an unmeasurable (since $\theta$ is unknown) random variable, whose distribution is uniform between 0 and 1 independent of $\theta$. If the test is unbiased then the outcome $q*$ implies

$p_{\theta }\leq \mathbb {P} (q(X)>q^{*}|0)\equiv p_{0}^{*}$

from which, similarly to conditioning on $n_{b}$ in the previous case, one obtains

$\mathbb {P} (q(X)\geq q^{*}|p_{\theta }\leq p_{0}^{*},\theta )={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{\mathbb {P} (p_{\theta }\leq p_{0}^{*}|\theta )}}={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{p_{0}^{*}}}={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{\mathbb {P} (q(X)>q^{*}|0)}}.$

Relation to foundational principles

The arguments given above can be viewed as following the spirit of the conditionality principle of statistical inference, although they express a more generalized notion of conditionality which do not require the existence of an ancillary statistic . The conditionality principle however, already in its original more restricted version, formally implies the likelihood principle, a result famously shown by Birnbaum . CLs does not obey the likelihood principle, and thus such considerations may only be used to suggest plausibility, but not theoretical completeness from the foundational point of view. (The same however can be said on any frequentist method if the conditionality principle is regarded as necessary).

Interestingly, Birnbaum himself suggested in his 1962 paper that the CLs ratio $\alpha /(1-\beta )$ should be used as a measure of the strength of statistical evidence provided by significance tests, rather than $\alpha$ alone. This followed from a simple application of the likelihood principle: if the outcome of an experiment is to be only reported in a form of a "accept"/"reject" decision, then the overall procedure is equivalent to an experiment that has only two possible outcomes, with probabilities $\alpha$,$(1-\beta )$ and $1-\alpha$,$(\beta )$ under $H_{1},(H_{2})$. The likelihood ratio associated with the outcome "reject $H_{1}$" is therefore $\alpha /(1-\beta )$ and hence should determine the evidential interpretation of this result. (Since, for a test of two simple hypotheses, the likelihood ratio is a compact representation of the likelihood function). On the other hand, if the likelihood principle is to be followed consistently, then the likelihood ratio of the original outcome should be used and not $\alpha /(1-\beta )$, making the basis of such an interpretation questionable. Birnbaum later described this as having "at most heuristic, but not substantial, value for evidential interpretation".

A more direct approach leading to a similar conclusion can be found in Birnbaum's formulation of the Confidence principle, which, unlike the more common version, refers to error probabilities of both kinds. This is stated as follows :

"A concept of statistical evidence is not plausible unless it finds 'strong evidence for $H_{2}$ as against $H_{1}$' with small probability ($\alpha$) when $H_{1}$ is true, and with much larger probability (1 -$\beta$) when

$H_{2}$ is true. "

Such definition of confidence can naturally seem to be satisfied by the definition of CLs. It remains true that both this and the more common (as associated with the Neyman-Pearson theory) versions of the confidence principle are incompatible with the likelihood principle, and therefore no frequentist method can be regarded as a truly complete solution to the problems raised by considering conditional properties of confidence intervals.

Calculation in the large sample limit

If certain regularity conditions are met, then a general likelihood function will become a Gaussian function in the large sample limit. In such case the CLs upper limit at confidence level $1-\alpha '$ (derived from the uniformly most powerful test) is given by

$\theta _{up}={\hat {\theta }}+\sigma \Phi ^{-1}(1-\alpha '\Phi ({\hat {\theta }}/\sigma )),$

where $\Phi$ is the standard normal cumulative distribution, ${\hat {\theta }}$ is the maximum likelihood estimator of $\theta$ and $\sigma$ is its standard deviation. $\sigma$ might be estimated from the inverse of the Fisher information matrix or by using the "Asimov" data set. This result happens to be equivalent to a Bayesian credible interval if a uniform prior for $\theta$ is used.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• 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
• 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
• Template:Cite arXiv

External links

• The Particle Data Group (PDG) review of statistical methods