Michael Faraday
by Thomas Phillipsoil on canvas, 1841-1842
Tuesday, July 31, 2012
Thursday, July 5, 2012
Higgs Boson Experiment Discovery at the world's largest atom smasher
Physicists at the world's largest atom smasher announced July 4 that they are more than 99 percent sure they've found a new, and heavy, boson particle, that may be the Higgs boson.
Two experiments at the Large Hadron Collider (LHC) in Geneva, Switzerland, show this new particle has a mass of about 125 GeV, with 1 gigaelectron volt about the mass of a proton. The LHC is the most powerful machine on Earth, capable of producing huge explosions of energy that generate new and exotic particles inside the 17-mile (27 kilometer) loop underneath Switzerland and France. [New Particle Is Likely the Higgs Boson
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http://www.foxnews.com/scitech/2012/07/05/elusive-particle-5-implications-finding-higgs-boson/#ixzz1zlfj15Sw
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If the discovery can be confirmed as the Higgs boson, it will have wide wide-reaching implications. Here are five of the biggest.
1. The origin of mass
The Higgs boson has long been thought the key to resolving the mystery of the origin of mass. The Higgs boson is associated with a field, called the Higgs field, theorized to pervade the universe. As other particles travel though this field, they acquire mass much as swimmers moving through a pool get wet, the thinking goes.
is associated with a field, called the Higgs field, theorized to pervade the universe. As other particles travel though this field, they acquire mass much as swimmers moving through a pool get wet, the thinking goes.
"The Higgs mechanism is the thing that allows us to understand how the particles acquire mass," said Joao Guimaraes da Costa, a physicist at Harvard University who is the Standard Model Convener at the LHC's ATLAS experiment. "If there was no such mechanism, then everything would be massless."
If physicists confirm that the detection of the new elementary particle is indeed the Higgs boson, and not an imposter, it would also confirm that the Higgs mechanism for particles to acquire mass is correct. "This discovery bears on the knowledge of how mass comes about at the quantum level, and is the reason we built the LHC. It is an unparalleled achievement," Caltech professor of physics Maria Spiropulu, co-leader of the CMS experiment, said in a statement.
And, it may offer clues to the next mystery down the line, which is why individual particles have the masses that they do. "That could be part of a much larger theory," said Harvard University particle physicist Lisa Randall."Knowing what the Higgs boson is, is the first step of knowing a little more about what that theory could be. It's connected."
And, it may offer clues to the next mystery down the line, which is why individual particles have the masses that they do. "That could be part of a much larger theory," said Harvard University particle physicist Lisa Randall."Knowing what the Higgs boson is, is the first step of knowing a little more about what that theory could be. It's connected."
2. The Standard Model
The Standard Model is the reigning theory of particle physics that describes the universe's very small constituents. Every particle predicted by the Standard Model has been discovered — except one: the Higgs boson.
is the reigning theory of particle physics that describes the universe's very small constituents. Every particle predicted by the Standard Model has been discovered — except one: the Higgs boson.
"It's the missing piece in the Standard Model," said Jonas Strandberg, a researcher at CERN working on the ATLAS experiment. "So it would definitely be a confirmation that the theories we have now are right." If the newly detected particle turns out not to be the Higgs boson, it would mean physicists made some assumptions that are wrong, and they'd have to go back to the drawing board.
While the discovery of the Higgs boson would complete the Standard Model, and fulfill all its current predictions, the Standard Model itself isn't thought to be complete. It doesn't encompass gravity (so don't count on catching that fly ball), for example, and leaves out the dark matter thought to make up 98 percent of all matter in the universe. [6 Weird Facts About Gravity]
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"The Standard Model describes what we have measured, but we know it doesn’t have gravity in it, it doesn't have dark matter," said CERN physicist William Murray, the senior Higgs convener at ATLAS and a physicist at the U.K.'s Science and Technology Facilities Council. "So we're hoping to extend it to include more."
3. The Electroweak Force
A confirmation of the existence of the Higgs boson would also help explain how two of the fundamental forces of the universe — the electromagnetic force that governs interactions between charged particles, and the weak force that's responsible for radioactive decay — can be unified. [9 Unsolved Physics Mysteries]
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Every force in nature is associated with a particle. The particle tied to electromagnetism is the photon, a tiny, massless particle. The weak force is associated with particles called the W and Z bosons, which are very massive.
The Higgs mechanism is thought to be responsible for this.
"If you introduce the Higgs field, the W and Z bosons mix with the field, and through this mixing they acquire mass," Strandberg said. "This explains why the W and Z bosons have mass, and also unifies the electromagnetic and weak forces into the electroweak force."
Though other evidence has helped buffer the union of these two forces, the discovery of the Higgs would seal the deal. "That's already pretty solid," Murray said. "What we're trying to do now is find really the crowning proof."
4. Supersymmetry
Another theory that would be affected by the discovery of the Higgs is called supersymmetry. This idea posits that every known particle has a "superpartner" particle with slightly different characteristics.
. This idea posits that every known particle has a "superpartner" particle with slightly different characteristics.
Supersymmetry is attractive because it could help unify some of the other forces of nature, and even offers a candidate for the particle that makes up dark matter. The newly detected particle is in the low-mass range, at 125.3 or so GeV, something that lends credence to supersymmetry.
. The newly detected particle is in the low-mass range, at 125.3 or so GeV, something that lends credence to supersymmetry.
"If the Higgs boson is found at a low mass, which is the only window still open, this would make supersymmetry a viable theory," Strandberg said."We'd still have to prove supersymmetry exists."
5. Validation of LHC
The Large Hadron Collider is the world's largest particle accelerator. It was built for around $10 billion by the European Organization for Nuclear Research (CERN) to probe higher energies than had ever been reached on Earth. Finding the Higgs boson was touted as one of the machine's biggest goals.
is the world's largest particle accelerator. It was built for around $10 billion by the European Organization for Nuclear Research (CERN) to probe higher energies than had ever been reached on Earth. Finding the Higgs boson was touted as one of the machine's biggest goals.
Finding the Higgs would offer major validation for the LHC and for the scientists who've worked on the search for many years.
"This discovery bears on the knowledge of how mass comes about at the quantum level, and is the reason we built the LHC. It is an unparalleled achievement," Spiropulu said in a statement. "More than a generation of scientists has been waiting for this very moment and particle physicists, engineers, and technicians in universities and laboratories around the globe have been working for many decades to arrive at this crucial fork. This is the pivotal moment for us to pause and reflect on the gravity of the discovery, as well as a moment of tremendous intensity to continue the data collection and analyses."
The discovery of the Higgs would also have major implications for scientist Peter Higgs and his colleagues who first proposed the Higgs mechanism in 1964.
And a Nobel Prize may be another result: "If it is found there are several people who are going to get a Nobel prize," said Vivek Sharma, a physicist at the University of California, San Diego, and the leader of the Higgs search at LHC's CMS experiment.
http://www.foxnews.com/scitech/2012/07/05/elusive-particle-5-implications-finding-higgs-boson/#ixzz1zlfj15Sw
Sunday, June 17, 2012
Box–Jenkins
In time series analysis, the Box–Jenkins methodology, named after the statisticians George Box and Gwilym Jenkins, applies autoregressive moving average ARMA or ARIMA models to find the best fit of a time series to past values of this time series, in order to make forecasts.
Modeling approach
The original model uses an iterative three-stage modeling approach:
For higher-order autoregressive processes, the sample autocorrelation needs to be supplemented with a partial autocorrelation plot. The partial autocorrelation of an AR(p) process becomes zero at lag p + 1 and greater, so we examine the sample partial autocorrelation function to see if there is evidence of a departure from zero. This is usually determined by placing a 95% confidence interval on the sample partial autocorrelation plot (most software programs that generate sample autocorrelation plots will also plot this confidence interval). If the software program does not generate the confidence band, it is approximately
, with N denoting the sample size.
The sample partial autocorrelation function is generally not helpful for identifying the order of the moving average process.
Mixed models difficult to identify
In practice, the sample autocorrelation and partial autocorrelation functions are random variables and will not give the same picture as the theoretical functions. This makes the model identification more difficult. In particular, mixed models can be particularly difficult to identify.
Although experience is helpful, developing good models using these sample plots can involve much trial and error. For this reason, in recent years information-based criteria such as FPE (final prediction error) and AIC (Akaike Information Criterion) and others have been preferred and used. These techniques can help automate the model identification process. These techniques require computer software to use. Fortunately, these techniques are available in many commercial statistical software programs that provide ARIMA modeling capabilities.
For additional information on these techniques, see Brockwell and Davis (1987, 2002).
The main approaches to fitting Box–Jenkins models are non-linear least squares and maximum likelihood estimation. Maximum likelihood estimation is generally the preferred technique. The likelihood equations for the full Box–Jenkins model are complicated and are not included here. See (Brockwell and Davis, 1987,2002) for the mathematical details.
That is, the error term At is assumed to follow the assumptions for a stationary univariate process. The residuals should be white noise (or independent when their distributions are normal) drawings from a fixed distribution with a constant mean and variance. If the Box–Jenkins model is a good model for the data, the residuals should satisfy these assumptions.
If these assumptions are not satisfied, one needs to fit a more appropriate model. That is, go back to the model identification step and try to develop a better model. Hopefully the analysis of the residuals can provide some clues as to a more appropriate model.
One way to assess if the residuals from the Box–Jenkins model follow the assumptions is to generate statistical graphics (including an autocorrelation plot) of the residuals. One could also look at the value of the Box–Ljung statistic.
Modeling approach
The original model uses an iterative three-stage modeling approach:
- Model identification and model selection: making sure that the variables are stationary, identifying seasonality in the dependent series (seasonally differencing it if necessary), and using plots of the autocorrelation and partial autocorrelation functions of the dependent time series to decide which (if any) autoregressive or moving average component should be used in the model.
- Parameter estimation using computation algorithms to arrive at coefficients which best fit the selected ARIMA model. The most common methods use maximum likelihood estimation or non-linear least-squares estimation.
- Model checking by testing whether the estimated model conforms to the specifications of a stationary univariate process. In particular, the residuals should be independent of each other and constant in mean and variance over time. (Plotting the mean and variance of residuals over time and performing a Ljung-Box test or plotting autocorrelation and partial autocorrelation of the residuals are helpful to identify misspecification.) If the estimation is inadequate, we have to return to step one and attempt to build a better model.
[edit] Box-Jenkins model identification
[edit] Stationarity and seasonality
The first step in developing a Box–Jenkins model is to determine if the time series is stationary and if there is any significant seasonality that needs to be modeled.[edit] Detecting stationarity
Stationarity can be assessed from a run sequence plot. The run sequence plot should show constant location and scale. It can also be detected from an autocorrelation plot. Specifically, non-stationarity is often indicated by an autocorrelation plot with very slow decay.[edit] Detecting seasonality
Seasonality (or periodicity) can usually be assessed from an autocorrelation plot, a seasonal subseries plot, or a spectral plot.[edit] Differencing to achieve stationarity
Box and Jenkins recommend the differencing approach to achieve stationarity. However, fitting a curve and subtracting the fitted values from the original data can also be used in the context of Box–Jenkins models.[edit] Seasonal differencing
At the model identification stage, the goal is to detect seasonality, if it exists, and to identify the order for the seasonal autoregressive and seasonal moving average terms. For many series, the period is known and a single seasonality term is sufficient. For example, for monthly data one would typically include either a seasonal AR 12 term or a seasonal MA 12 term. For Box–Jenkins models, one does not explicitly remove seasonality before fitting the model. Instead, one includes the order of the seasonal terms in the model specification to the ARIMA estimation software. However, it may be helpful to apply a seasonal difference to the data and regenerate the autocorrelation and partial autocorrelation plots. This may help in the model identification of the non-seasonal component of the model. In some cases, the seasonal differencing may remove most or all of the seasonality effect.[edit] Identify p and q
Once stationarity and seasonality have been addressed, the next step is to identify the order (i.e., the p and q) of the autoregressive and moving average terms.[edit] Autocorrelation and partial autocorrelation plots
The primary tools for doing this are the autocorrelation plot and the partial autocorrelation plot. The sample autocorrelation plot and the sample partial autocorrelation plot are compared to the theoretical behavior of these plots when the order is known.[edit] Order of autoregressive process (p)
Specifically, for an AR(1) process, the sample autocorrelation function should have an exponentially decreasing appearance. However, higher-order AR processes are often a mixture of exponentially decreasing and damped sinusoidal components.For higher-order autoregressive processes, the sample autocorrelation needs to be supplemented with a partial autocorrelation plot. The partial autocorrelation of an AR(p) process becomes zero at lag p + 1 and greater, so we examine the sample partial autocorrelation function to see if there is evidence of a departure from zero. This is usually determined by placing a 95% confidence interval on the sample partial autocorrelation plot (most software programs that generate sample autocorrelation plots will also plot this confidence interval). If the software program does not generate the confidence band, it is approximately
, with N denoting the sample size.[edit] Order of moving-average process (q)
The autocorrelation function of a MA(q) process becomes zero at lag q + 1 and greater, so we examine the sample autocorrelation function to see where it essentially becomes zero. We do this by placing the 95% confidence interval for the sample autocorrelation function on the sample autocorrelation plot. Most software that can generate the autocorrelation plot can also generate this confidence interval.The sample partial autocorrelation function is generally not helpful for identifying the order of the moving average process.
[edit] Shape of autocorrelation function
The following table summarizes how one can use the sample autocorrelation function for model identification.Mixed models difficult to identify
In practice, the sample autocorrelation and partial autocorrelation functions are random variables and will not give the same picture as the theoretical functions. This makes the model identification more difficult. In particular, mixed models can be particularly difficult to identify.
Although experience is helpful, developing good models using these sample plots can involve much trial and error. For this reason, in recent years information-based criteria such as FPE (final prediction error) and AIC (Akaike Information Criterion) and others have been preferred and used. These techniques can help automate the model identification process. These techniques require computer software to use. Fortunately, these techniques are available in many commercial statistical software programs that provide ARIMA modeling capabilities.
For additional information on these techniques, see Brockwell and Davis (1987, 2002).
[edit] Box–Jenkins model estimation
Estimating the parameters for the Box–Jenkins models is a quite complicated non-linear estimation problem. For this reason, the parameter estimation should be left to a high quality software program that fits Box–Jenkins models. Fortunately, many statistical software programs now fit Box–Jenkins models.The main approaches to fitting Box–Jenkins models are non-linear least squares and maximum likelihood estimation. Maximum likelihood estimation is generally the preferred technique. The likelihood equations for the full Box–Jenkins model are complicated and are not included here. See (Brockwell and Davis, 1987,2002) for the mathematical details.
[edit] Box–Jenkins model diagnostics
[edit] Assumptions for a stable univariate process
Model diagnostics for Box–Jenkins models is similar to model validation for non-linear least squares fitting.That is, the error term At is assumed to follow the assumptions for a stationary univariate process. The residuals should be white noise (or independent when their distributions are normal) drawings from a fixed distribution with a constant mean and variance. If the Box–Jenkins model is a good model for the data, the residuals should satisfy these assumptions.
If these assumptions are not satisfied, one needs to fit a more appropriate model. That is, go back to the model identification step and try to develop a better model. Hopefully the analysis of the residuals can provide some clues as to a more appropriate model.
One way to assess if the residuals from the Box–Jenkins model follow the assumptions is to generate statistical graphics (including an autocorrelation plot) of the residuals. One could also look at the value of the Box–Ljung statistic.
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