Estimation of covariance matrices

In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R"p"×"p"; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has normal distribution, the sample covariance matrix has Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data require deeper considerations. Another issue is the robustness to outliers, to which sample covariance matrices are highly sensitive.
Statistical analyses of multivariate data often involve exploratory studies of the way in which the variables change in relation to one another and this may be followed up by explicit statistical models involving the covariance matrix of the variables. Thus the estimation of covariance matrices directly from observational data plays two roles:
Estimates of covariance matrices are required at the initial stages of principal component analysis and factor analysis, and are also involved in versions of regression analysis that treat the dependent variables in a data-set, jointly with the independent variable as the outcome of a random sample.
Estimation in a general context.
Given a sample consisting of "n" independent observations "x"1..., "x""n" of a "p"-dimensional random vector "X" ∈ R"p"×1 (a "p"×1 column-vector), an unbiased estimator of the ("p"×"p") covariance matrix
is the sample covariance matrix
where formula_3 is the "i"-th observation of the "p"-dimensional random vector, and the vector
is the sample mean.
This is true regardless of the distribution of the random variable "X", provided of course that the theoretical means and covariances exist. The reason for the factor "n" − 1 rather than "n" is essentially the same as the reason for the same factor appearing in unbiased estimates of sample variances and sample covariances, which relates to the fact that the mean is not known and is replaced by the sample mean (see Bessel's correction).
In cases where the distribution of the random variable "X" is known to be within a certain family of distributions, other estimates may be derived on the basis of that assumption. A well-known instance is when the random variable "X" is normally distributed: in this case the maximum likelihood estimator of the covariance matrix is slightly different from the unbiased estimate, and is given by
A derivation of this result is given below. Clearly, the difference between the unbiased estimator and the maximum likelihood estimator diminishes for large "n".
In the general case, the unbiased estimate of the covariance matrix provides an acceptable estimate when the data vectors in the observed data set are all complete: that is they contain no missing elements. One approach to estimating the covariance matrix is to treat the estimation of each variance or pairwise covariance separately, and to use all the observations for which both variables have valid values. Assuming the missing data are missing at random this results in an estimate for the covariance matrix which is unbiased. However, for many applications this estimate may not be acceptable because the estimated covariance matrix is not guaranteed to be positive semi-definite. This could lead to estimated correlations having absolute values which are greater than one, and/or a non-invertible covariance matrix.
When estimating the cross-covariance of a pair of signals that are wide-sense stationary, missing samples do "not" need be random (e.g., sub-sampling by an arbitrary factor is valid).
Maximum-likelihood estimation for the multivariate normal distribution.
A random vector "X" ∈ R"p" (a "p"×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ R"p" × "p" is a positive-definite matrix and the probability density function of "X" is
where "μ" ∈ R"p"×1 is the expected value of "X". The covariance matrix Σ is the multidimensional analog of what in one dimension would be the variance, and
normalizes the density formula_8 so that it integrates to 1.
Suppose now that "X"1, ..., "X""n" are independent and identically distributed samples from the distribution above. Based on the observed values "x"1, ..., "x""n" of this sample, we wish to estimate Σ.
First steps.
The likelihood function is:
It is fairly readily shown that the maximum-likelihood estimate of the mean vector "μ" is the "sample mean" vector:
See the section on estimation in the article on the normal distribution for details; the process here is similar.
Since the estimate formula_11 does not depend on Σ, we can just substitute it for "μ" in the likelihood function, getting
and then seek the value of Σ that maximizes the likelihood of the data (in practice it is easier to work with log formula_13).
The trace of a 1 × 1 matrix.
Now we come to the first surprising step: regard the scalar formula_14 as the trace of a 1×1 matrix. This makes it possible to use the identity tr("AB") = tr("BA") whenever "A" and "B" are matrices so shaped that both products exist. We get
where
formula_17 is sometimes called the scatter matrix, and is positive definite if there exists a subset of the data consisting of formula_18 affinely independent observations (which we will assume).
Using the spectral theorem.
It follows from the spectral theorem of linear algebra that a positive-definite symmetric matrix "S" has a unique positive-definite symmetric square root "S"1/2. We can again use the "cyclic property" of the trace to write
Let "B" = "S"1/2 Σ −1 "S"1/2. Then the expression above becomes
The positive-definite matrix "B" can be diagonalized, and then the problem of finding the value of "B" that maximizes
Since the trace of a square matrix equals the sum of eigen-values ("trace and eigenvalues"), the equation reduces to the problem of finding the eigen values λ1, ..., λ"p" that maximize
This is just a calculus problem and we get λ"i" = "n" for all "i." Thus, assume "Q" is the matrix of eigen vectors, then
i.e., "n" times the "p"×"p" identity matrix.
Concluding steps.
Finally we get
i.e., the "p"×"p" "sample covariance matrix"
is the maximum-likelihood estimator of the "population covariance matrix" Σ. At this point we are using a capital "X" rather than a lower-case "x" because we are thinking of it "as an estimator rather than as an estimate", i.e., as something random whose probability distribution we could profit by knowing. The random matrix "S" can be shown to have a Wishart distribution with "n" − 1 degrees of freedom. That is:
Alternative derivation.
An alternative derivation of the maximum likelihood estimator can be performed via matrix calculus formulae (see also differential of a determinant and differential of the inverse matrix). It also verifies the aforementioned fact about the maximum likelihood estimate of the mean. Re-write the likelihood in the log form using the trace trick:
The differential of this log-likelihood is
It naturally breaks down into the part related to the estimation of the mean, and to the part related to the estimation of the variance. The first order condition for maximum, formula_29, is satisfied when the terms multiplying formula_30 and formula_31 are identically zero. Assuming (the maximum likelihood estimate of) formula_32 is non-singular, the first order condition for the estimate of the mean vector is
which leads to the maximum likelihood estimator
This lets us simplify
as defined above. Then the terms involving formula_31 in formula_37 can be combined as
The first order condition formula_29 will hold when the term in the square bracket is (matrix-valued) zero. Pre-multiplying the latter by formula_32 and dividing by formula_41 gives
which of course coincides with the canonical derivation given earlier.
Dwyer points out that decomposition into two terms such as appears above is "unnecessary" and derives the estimator in two lines of working. Note that it may be not trivial to show that such derived estimator is the unique global maximizer for likelihood function.
Intrinsic covariance matrix estimation.
Intrinsic expectation.
Given a sample of "n" independent observations "x"1..., "x""n" of a "p"-dimensional zero-mean Gaussian random variable "X" with covariance R, the maximum likelihood estimator of R is given by
The parameter R belongs to the set of positive-definite matrices, which is a Riemannian manifold, not a vector space, hence the usual vector-space notions of expectation, i.e. "E[R^]", and estimator bias must be generalized to manifolds to make sense of the problem of covariance matrix estimation. This can be done by defining the expectation of an manifold-valued estimator R^ with respect to the manifold-valued point R as
where
are the exponential map and inverse exponential map, respectively, "exp" and "log" denote the ordinary matrix exponential and matrix logarithm, and E[·] is the ordinary expectation operator defined on a vector space, in this case the tangent space of the manifold.
Bias of the sample covariance matrix.
The intrinsic bias vector field of the SCM estimator formula_47 is defined to be
The intrinsic estimator bias is then given by formula_49.
For complex Gaussian random variables, this bias vector field can be shown to equal
where
and ψ(·) is the digamma function. The intrinsic bias of the sample covariance matrix equals
and the SCM is asymptotically unbiased as "n" → ∞.
Similarly, the intrinsic inefficiency of the sample covariance matrix depends upon the Riemannian curvature of the space of positive-definite matrices.
Shrinkage estimation.
If the sample size "n" is small and the number of considered variables "p" is large, the above empirical estimators of covariance and correlation are very unstable. Specifically, it is possible to furnish estimators that improve considerably upon the maximum likelihood estimate in terms of mean squared error. Moreover, for "n" < "p" (the number of observations is less than the number of random variables) the empirical estimate of the covariance matrix becomes singular, i.e. it cannot be inverted to compute the precision matrix.
As an alternative, many methods have been suggested to improve the estimation of the covariance matrix. All of these approaches rely on the concept of shrinkage. This is implicit in Bayesian methods and in penalized maximum likelihood methods and explicit in the Stein-type shrinkage approach.
A simple version of a shrinkage estimator of the covariance matrix is represented by the Ledoit-Wolf shrinkage estimator. One considers a convex combination of the empirical estimator (formula_53) with some suitable chosen target (formula_54), e.g., the diagonal matrix. Subsequently, the mixing parameter (formula_55) is selected to maximize the expected accuracy of the shrunken estimator. This can be done by cross-validation, or by using an analytic estimate of the shrinkage intensity. The resulting regularized estimator (formula_56) can be shown to outperform the maximum likelihood estimator for small samples. For large samples, the shrinkage intensity will reduce to zero, hence in this case the shrinkage estimator will be identical to the empirical estimator. Apart from increased efficiency the shrinkage estimate has the additional advantage that it is always positive definite and well conditioned.
Various shrinkage targets have been proposed:
The shrinkage estimator can be generalized to a multi-target shrinkage estimator that utilizes several targets simultaneously. Software for computing a covariance shrinkage estimator is available in R (packages corpcor and ShrinkCovMat), in Python (library scikit-learn), and in MATLAB.


Magnetic level gauge

A magnetic level gauge is a type of level sensor, i.e., a device used to measure the level of fluids. A magnetic level gauge includes a “floatable” device that can float both in high and low density fluids. Magnetic level gauges may also be designed to accommodate severe environmental conditions up to 210 bars at 370 °C.
Magnetic level sensors.
Magnetic float level sensors involve the use of a permanent magnet sealed inside a float whose rise and fall causes the opening or closing of a mechanical switch, either through direct contact or in proximity of a reed switch. With mechanically actuated floats, the float is directly connected to a micro switch. For both magnetic and mechanical float level sensors, chemical compatibility, temperature, specific gravity (density), buoyancy, and viscosity affect the selection of the stem and the float. For example, larger floats may be used with liquids with specific gravities as low as 0.5 while still maintaining buoyancy. The choice of float material is also influenced by temperature-induced changes in specific gravity and viscosity – changes that directly affect buoyancy.
Explanation.
Exploring the physics and engineering behind this design, requires looking at basic magnetism. A standard bar magnet has two magnetic poles: north and south. (The north will read positive on a gauss meter and the south will read negative.) Magnetic field are mapped using magnetic flux lines. These lines are a graphical representation of the magnetic field density. They show the direction of flow for the magnetic field and represent relative field strength – the closer together the lines are, the stronger the magnetic field. Flux lines will always travel from the north pole to the nearest south pole and always leave and enter surfaces at 90°, or perpendicular to the surface. They can only travel in straight lines or curved paths, which means they can never make a sudden, abrupt change in direction. Flux lines will also always follow a path of least magnetic resistance. Most importantly, they can never cross one another.
Considerations.
When selecting a magnetic level gauge it is important to take into account the strength of the magnetic field. The magnetic field is the heart of the magnetic level gauge – the stronger the field, the more reliable the instrument will function. Some manufacturers rely on a single magnet for their magnetic level gauges which causes the strength of the north field to be identical to, and as weak as, the south field. It is apparent that at the location of the indicators, switches and transmitters, the field would not be as intense. Some manufacturers use a single annular ring magnet, others use a series of single bar magnets in a circular array in their float design. In this design the relative field strength of the north and south poles will be equal to one another and less than that of a dual magnet design. Moreover, the field strength as you travel around the circumference will have high and low spots as you pass between the individual bar magnets.


Mathematical principles of reinforcement

The mathematical principles of reinforcement (MPR) constitute of a set of mathematical equations set forth by Peter Killeen and his colleagues attempting to describe and predict the most fundamental aspects of behavior (Killeen & Sitomer, 2003).
The three key principles of MPR, arousal, constraint, and coupling, describe how incentives motivate responding, how time constrains it, and how reinforcers become associated with specific responses, respectively. Mathematical models are provided for these basic principles in order to articulate the necessary detail of actual data.
First principle: arousal.
The first basic principle of MPR is arousal. Arousal refers to the activation of behavior by the presentation of incentives. An increase in activity level following repeated presentations of incentives is a fundamental aspect of conditioning. Killeen, Hanson, and Osborne (1978) proposed that adjunctive (or schedule induced) behaviors are normally occurring parts of an organism's repertoire. Delivery of incentives increases the rate of adjunctive behaviors by generating a heightened level of general activity, or arousal, in organisms.
Killeen & Hanson (1978) exposed pigeons to a single daily presentation of food in the experimental chamber and measured general activity for 15 minutes after a feeding. They showed that activity level increased slightly directly following a feeding and then decreased slowly over time. The rate of decay can be described by the following function:
The time course of the entire theoretical model of general activity is modeled by the following equation:
To better conceptualize this model, imagine how rate of responding would appear with each of these processes individually. In the absence of temporal inhibition or competing responses, arousal level would remain high and response rate would be depicted as an almost horizontal line with a very small negative slope. Directly following food presentation, temporal inhibition is at its maximum level. It decreases quickly as time elapses, and response rate would be expected to increase up to the level of arousal in a short time. Competing behaviors such as goal tracking or hopper inspection are at a minimum directly after food presentation. These behaviors increase as the interval elapses, so the measure of general activity would slowly decrease. Subtracting these two curves results in the predicted level of general activity.
Killeen et al. (1978) then increased the frequency of feeding from daily to every fixed-time seconds. They showed that general activity level increased substantially from the level of daily presentation. Response rate asymptotes were highest for the highest rates of reinforcement. These experiments indicate that arousal level is proportional to rate of incitement, and the asymptotic level increases with repeated presentations of incentives. The increase in activity level with repeated presentation of incentives is called cumulation of arousal. The first principle of MPR states that arousal level is proportional to rate of reinforcement, formula_4, where:
= arousal level
= specific activation
= rate of reinforcement
(Killeen & Sitomer, 2003).
Second principle: constraint.
An obvious but often overlooked factor when analyzing response distributions is that responses are not instantaneous, but take some amount of time to emit (Killeen, 1994). These ceilings on response rate are often accounted for by competition from other responses, but less often for the fact that responses cannot always be emitted at the same rate at which they are elicited (Killeen & Sitomer, 2003). This limiting factor must be taken into account in order to correctly characterize what responding could be theoretically, and what it will be empirically.
An organism may receive impulses to respond at a certain rate. At low rates of reinforcement, the elicited rate and emitted rate will approximate each other. At high rates of reinforcement, however, this elicited rate is subdued by the amount of time it takes to emit a response. Response rate, formula_5, is typically measured as the number of responses occurring in an epoch divided by the duration of an epoch. The reciprocal of formula_5 gives the typical measure of the inter response (IRT), the average time from the start of one response to the start of another (Killeen & Sitomer, 2003). This is actually the cycle time rather than the time between responses. According to Killeen & Sitomer (2003), the IRT consists of two subintervals, the time required to emit a response, formula_7 plus the time between responses, formula_8. Therefore, response rate can be measured either by dividing the number of responses by the cycle time:
or as the number of responses divided by the actual time between responses:
This instantaneous rate, formula_11 may be the best measure to use, as the nature of the operandum may change arbitrarily within an experiment (Killeen & Sitomer, 2003).
Killeen, Hall, Reilly, and Kettle (2002) showed that if instantaneous rate of responding is proportional to rate of reinforcement, formula_12, then a fundamental equation for MPR results. Killeen & Sitomer (2003) showed that:
if formula_13
then formula_14,
and rearranging gives:
formula_15
While responses may be elicited at a rate proportional to formula_16, they can only be emitted at rate formula_5 due to constraint. The second principle of MPR states that the time required to emit a response constrains response rate (Killeen & Sitomer, 2003).
Third principle: coupling.
Coupling is the final concept of MPR that ties all of the processes together and allows for specific predictions of behavior with different schedules of reinforcement. Coupling refers to the association between responses and reinforcers. The target response is the response of interest to the experimenter, but any response can become associated with a reinforcer. Contingencies of reinforcement refer to how a reinforcer is scheduled with respect to the target response (Killeen & Sitomer, 2003), and the specific schedules of reinforcement in effect determine how responses are coupled to the reinforcer. The third principle of MPR states that the degree of coupling between a response and reinforcer decreases with the distance between them (Killeen & Sitomer, 2003). Coupling coefficients, designated as formula_18, are given for the different schedules of reinforcement. When the coupling coefficients are inserted into the activation-constraint model, complete models of conditioning are derived:
This is the fundamental equation of MPR. The dot after the formula_20 is a placeholder for the specific contingencies of reinforcement under study (Killeen & Sitomer, 2003).
Fixed-ratio reinforcement schedules.
The rate of reinforcement for fixed-ratio schedules is easy to calculate, as reinforcement rate is directly proportional to response rate and inversely proportional to ratio requirement (Killeen, 1994). The schedule feedback function is therefore:
Substituting this function into the complete model gives the equation of motion for ratio schedules (Killeen & Sitomer, 2003). Killeen (1994, 2003) showed that the most recent response in a sequence of responses is weighted most heavily and given a weight of formula_22, leaving formula_23 for the remaining responses. The penultimate response receives formula_24, the third back receives formula_25. The formula_26th response back is given a weight of formula_27
The sum of this series is the coupling coefficient for fixed-ratio schedules:
The continuous approximation of this is:
where formula_30 is the intrinsic rate of memory decay. Inserting the reinforcement rate and coupling coefficient into the activation-constraint model gives the predicted response rates for FR schedules:
This equation predicts low response rates at low ratio requirements due to the displacement of memory by consummatory behavior. However, these low rates are not always found. Coupling of responses may extend back beyond the preceding reinforcer, and an extra parameter, formula_32 is added to account for this. Killeen & Sitomer (2003) showed that the coupling coefficient for FR schedules then becomes:
formula_32 is the number of responses preceding the prior reinforcer that contribute to response strength. formula_35 which ranges from 0 to 1 is then the degree of erasure of the target response from memory with the delivery of a reinforcer. (formula_36) If formula_37, erasure is complete and the simpler FR equation can be used.
Variable-ratio reinforcement schedules.
According to Killeen & Sitomer (2003), the duration of a response can affect the rate of memory decay. When response durations vary, either within or between organisms, then a more complete model is needed, and formula_22 is replaced with formula_39 yielding:
Idealized variable-ratio schedules with a mean response requirement of formula_26 have a constant probability of formula_42 of a response ending in reinforcement (Bizo, Kettle, & Killeen, 2001). The last response ending in reinforcement must always occur and receives strengthening of formula_22. The penultimate response occurs with probability formula_44 and receives a strengthening of formula_24 . The sum of this process up to infinity is (Killeen 2001, Appendix):
The coupling coefficient for VR schedules ends up being:
formula_47
Multiplying by degree of erasure of memory gives:
formula_48
The coupling coefficient can then be inserted into the activation-constraint model just as the coupling coefficient for FR schedules to yield predicted response rates under VR schedules:
formula_49
In interval schedules, the schedule feedback function is
formula_50
where formula_51 is the minimum average time between reinforcers (Killeen, 1994). Coupling in interval schedules is weaker than ratio schedules, as interval schedules equally strengthen all responses preceding the target rather than just the target response. Only some proportion formula_52 of memory is strengthened. With a response requirement, the final, target response must receive strength of formula_22. All preceding responses, target or non-target, receive a strengthening of formula_23.
Fixed-time schedules are the simplest time dependent schedules in which organisms must simply wait t seconds for an incentive. Killeen (1994) reinterpreted temporal requirements as response requirements and integrated the contents of memory from one incentive to the next. This gives the contents of memory to be:
N
MN= lò e-lndn
0
This is the degree of saturation in memory of all responses, both target and non-target, elicited in the context (Killeen, 1994). Solving this equation gives the coupling coefficient for fixed-time schedules:
c=r(1-e-lbt)
where formula_52 is the proportion of target responses in the response trajectory. Expanding into a power series gives the following approximation:
c» rlbt
 1+lbt
This equation predicts serious instability for non-contingent schedules of reinforcement.
Fixed-interval schedules are guaranteed a strengthening of a target response, b=w1, as reinforcement is contingent on this final, contiguous response (Killeen, 1994). This coupling is equivalent to the coupling on FR 1 schedules
w1=b=1-e-l.
The remainder of coupling is due to the memory of preceding behavior. The coupling coefficient for FI schedules is:
c= b +r(1- b -e-lbt).
Variable-time schedules are similar to random ratio schedules in that there is a constant probability of reinforcement, but these reinforcers are set up in time rather than responses. The probability of no reinforcement occurring before some time t’ is an exponential function of that time with the time constant t being the average IRI of the schedule (Killeen, 1994). To derive the coupling coefficient, the probability of the schedule not having ended, weighted by the contents of memory, must be integrated.
M= lò e-n’t/te-ln’ dn’
 0
In this equation, t’=n’t, where t is a small unit of time. Killeen (1994) explains that the first exponential term is the reinforcement distribution, whereas the second term is the weighting of this distribution in memory. Solving this integral and multiplying by the coupling constant r, gives the extent to which memory is filled on VT schedules:
c=rlbt
 1+lbt
This is the same coupling coefficient as an FT schedule, except it is an exact solution for VT schedules rather than an approximation. Once again, the feedback function on these non-contingent schedules predicts serious instability in responding.
As with FI schedules, variable-interval schedules are guaranteed a target response coupling of b. Simply adding b to the VT equation gives:
M= b+ lò e-n’t/te-ln’ dn’
 1
Solving the integral and multiplying by r gives the coupling coefficient for VI schedules:
c= b+(1-b) rlbt
 1+lbt
The coupling coefficients for all of the schedules are inserted into the activation-constraint model to yield the predicted, overall response rate. The third principle of MPR states that the coupling between a response and a reinforcer decreases with increased time between them (Killeen & Sitomer, 2003).
Mathematical principles of reinforcement describe how incentives fuel behavior, how time constrains it, and how contingencies direct it. It is a general theory of reinforcement that combines both contiguity and correlation as explanatory processes of behavior. Many responses preceding reinforcement may become correlated with the reinforcer, but the final response receives the greatest weight in memory. Specific models are provided for the three basic principles to articulate predicted response patterns in many different situations and under different schedules of reinforcement. Coupling coefficients for each reinforcement schedule are derived and inserted into the fundamental equation to yield overall predicted response rates.


2396 Kochi


Mean integrated squared error

In statistics, the mean integrated squared error (MISE) is used in density estimation. The MISE of an estimate of an unknown probability density is given by
where "ƒ" is the unknown density, "ƒ""n" is its estimate based on a sample of "n" independent and identically distributed random variables.
Here, E denotes the expected value with respect to that sample.
The MISE is also known as "L"2 risk function.


Tachymeter (watch)

A tachymeter (pronounced ) is a scale sometimes inscribed around the rim of an analog watch with a chronograph. It can be used to conveniently compute the frequency in hours of an event of a known second-defined period, such as speed (distance over hours) based on travel time (seconds over distance), or measure distance based on speed. The spacings between the marks on the tachymeter dial are therefore proportional to , where "t" is the elapsed time.
The function performed by a tachymeter is independent of the unit of distance (e.g. statute miles, nautical miles, kilometres, metres etc.) as long as the same unit of length is used for all calculations. It can also be used to measure the frequency of any regular event in occurrences per hour, such as the units output by an industrial process. A tachymeter is simply a means of converting "elapsed time" (in seconds per unit) to "rate" (in units per hour).
Measuring speed.
To use a tachymeter-equipped watch for measuring speed, the chronograph is started at a starting marker of a known distance. At the next marker, the point on the scale adjacent to the second hand indicates the speed (in distance between markers per hour) of travel between the two. The typical tachymeter scale on a watch converts between the number of seconds it takes for an event to happen and the number of times that event will occur in one hour. The formula used to create this type of tachymeter scale is
where T is the tachymeter scale value; t is the time in seconds that it takes for the event to occur; and 3600 is the number of seconds in an hour.
As a sample calculation, if it takes 35 seconds to travel one mile, then the average speed is 103 miles/hour. On the watch, 35 seconds gives scale value 103. Similarly, if one kilometre takes 35 seconds then the average speed would be 103 km/hour.
Note that the tachymeter scale only calculates the average speed.
As a second example, if it takes 20 seconds to travel one unit of distance, then the average speed on the watch used for the purpose of the picture only is 180 units of distance per hour (examine the picture of the watch which is here to simplify the idea, actual tachymeters may vary slightly).
For events that happen either very quickly or slowly, one can adjust the sixty-second tachymeter scale commonly found on watches. Smaller fractional units can be used for slower objects, but the same X/hour function remains constant. The scale on a watch is only valid for things that happen in 60 seconds or fewer, and the scale is also difficult to resolve for events that take fewer than 10 seconds or so to occur. As an example, if it takes 100 seconds to eat an apple, cutting that number in half allows one to say that it takes 50 seconds to eat half an apple. Using the tachymeter scale one can calculate that 72 half apples (36 whole apples) could be eaten in one hour. Some watches, not common, have "wraparound" or "scroll" scales, which extend the readings to lower speeds, typically 45 units.
A counter-intuitive method of measuring speeds slower than can be directly indicated was published by Michael Bilow on the web site TimeZone.com: "Because the tachymeter scale is calibrated quite simply as divisors of 3600, it is easy to use the 0-60 segment as a reference scale for the reciprocals. That is, since something which traverses the unit distance in 45 seconds is going 80 units per hour, then it follows mathematically that something which traverses the unit distance in 80 seconds is going 45 units per hour. Since something which traverses the unit distance in 30 seconds is going 120 units per hour, then it follows that something which traverses the unit distance in 120 seconds is going 30 units per hour... Using this method, it is fairly simple to read by inspection nearly any slowly moving object down to the range of 8-12 units per hour." The method was later explained in detail: "It is not intuitive to use the tachymeter scale in this way because there will, in all likelihood, not be any hands pointing to the important places on it. Unlike the conventional use of the tachymeter scale to measure things which take 60 seconds or less to traverse the unit distance, you must mentally move from the number of seconds measured on the chronograph to the same number on the tachymeter scale, and only then read the number opposite on the dial. The ability of the tachymeter scale to be used 'backwards' like this, freely interchanging time and rate, is a direct consequence of the fact that it is created as a table of reciprocals in the first place."
Measuring distance.
A tachymeter-equipped watch can be used to measure distance by timing the travel over the distance while the speed is held constant. The tachymeter scale is rotated to align with the second hand at the start of the length to be measured. When the second hand reaches the point on the scale where the speed indicated equals the speed of the vehicle, one unit of distance (miles if speed is miles per hour, kilometres if kilometres per hour, etc.) has been covered. For example, if you travel at a constant 80 mph (or at 80 km/h), then the distance travelled while the second hand sweeps to "80" (45 seconds) will be exactly 1 mile (or 1 kilometre at 80 km/h).
Rotating scale.
Some tachymeter scales are on a rotating, indexed bezel. This allows two additional modes of use: The tachymeter bezel can be aligned with a free running second hand, and, more subtly, can be used to find the average speed over longer times/distances. Set the rotary bezel index to the position of the minute hand, note the current mileage/distance. Glance at the position of the minute hand on the tachymeter scale 60 units of distance later, and "average" speed will be indicated. A little mental math allows interim averages, easiest at 1/4 (15 unit) and other integer values. Alternatively, instead of using minute hand, align index bezel to the second hand and observe passing one unit of distance when position of the second hand will then indicate average speed.


222 Lucia

Lucia (minor planet designation: 222 Lucia) is a large Themistian asteroid. It was discovered by Johann Palisa on 9 February 1882 in Vienna and named after Lucia, daughter of Austro-Hungarian explorer Graf Wilczek.
This object is spectral C-type and is probably composed of primitive carbonaceous material. Based upon analysis of infrared spectra, it has a diameter of 59.8 ± 0.8 km. This object belongs to the Themis family, which was formed by the break-up of a larger parent body about a billion years ago.


Hook gauge evaporimeter

A Hook gauge evaporimeter is a precision instrument used to measure changes in water levels due to evaporation. It is used
to precisely measure the level of a free water surface as an evaporation pan or a tank. The main users are meteorologists and water engineers, especially in hot, arid countries where water conservation is of vital importance.
Description.
The device consists of a sharp hook suspended from a micrometer cylinder, with the body of the device having arms which rest on the rim of a still well inside the pan. The still well serves to isolate the device from any ripples that might be present in the sample being measured, while allowing the water level to equalize. The measurement is taken by turning the knob to lower the hook through the surface of the water until capillary action causes a small depression to form around the tip of the hook. The knob is then turned slowly until the depression "pops," with the measurement showing on the micrometer scale.
Use.
Evaporation rate is determined by a sequence of measurements over a set time interval, usually every 24 hours.


Web server benchmarking

Web server benchmarking is the process of estimating a web server performance in order to find if the server can serve sufficiently high workload.
Key parameters.
The performance is usually measured in terms of:
The measurements must be performed under a varying load of clients and requests per client.
Tools for benchmarking.
Load testing (stress/performance testing) a web server can be performed using automation/analysis tools such as:
Web application benchmarks.
Web application benchmarks measure the performance of application servers and database servers used to host web applications. TPC-W was a common benchmark emulating an online bookstore with synthetic workload generation.


Benjamin Richard Parker

Benjamin Richard Parker (often called Benjy by his sister) is a fictional character appearing in American comic books published by Marvel Comics. The character is depicted in the alternate future MC2 universe. He is the younger brother of May Parker / Spider-Girl, and the son of Mary Jane and Peter Parker / Spider-Man.
Fictional character biography.
Benjamin is born after a complicated pregnancy. Because his father's genetic code was altered when he received his powers, Ben is at a high risk of being born with some kind of genetic abnormality: deformity, disability, or perhaps even mutant powers. Due to the risk to Mary Jane's health, her obstetrician advises her to consider abortion. However, remembering that she had faced similar risks when pregnant with her daughter May, Mary Jane decides to proceed with the pregnancy.
Ben is born while his sister is fighting Seth, and to his family's relief, is apparently a perfectly healthy, normal little boy. His first name Benjamin is in honor of his uncle and great uncle, while his second name Richard is in honor of his grandfather.
Benjamin seems to display some superhuman abilities, namely balancing a toy block on one finger while spinning his arm at a fast speed. He was also seen dangling the block from his finger on a web-like string. May discovers Ben crawling on the ceiling of their home.
He was once possessed by a miniature version of the Carnage symbiote. After his sister freed him from the symbiote by using the ultrasonic weaponry of the villain Reverb, his father notices that the baby's ears were bleeding, and realized that Ben has lost his hearing, likely because Ben's ears were far too underdeveloped to withstand the sonic waves. The doctors in the hospital were trying to determine whether Ben's hearing loss is temporary or permanent. May is deeply upset over this and blames herself. Nevertheless, Ben still seems to be his usual, happy self. Since discovering her baby brother crawling on the ceiling she fears that due to his exposure to the symbiote, his abilities have somehow been jump started far too early (as hers only came about in her teens).
Normie Osborn agreed to fund an operation to restore Benjamin's hearing. The operation was a success, restoring most, if not all of Ben's hearing. When Benjy was going through it, Normie closely examined his body and found out that he was already developing his powers, but the exposure to the symbiote accelerated the process. Peter begins to fear that Benjy may be more powerful than him and his sister. This is strongly implied to be true when Ben spins organic webs to save both himself and Mary Jane after being thrown off a bridge by the Green Goblin, something neither Peter nor May can do. Despite being a baby, he was also strong enough for Mary Jane to hold on to without hurting him.
It is also known that Peter is the only one who can get him to burp "in the morning" (as Mary Jane says it) by feeding him chili.
Spider-Verse.
In the 2014/2015 crossover event "Spider-Verse", Benjy's family come under attack from Daemos, a relation of the 616 Spider-Man's former nemesis Morlun. During the attack, Mary Jane, Mayday's boyfriend Wes, and Peter are apparently all killed and their home destroyed. Mayday flees with Ben and is rescued by visiting Spider-Men from other dimensions who are trying to save as many Spiders as possible from similar attacks by Morlon and Daemos' family, who call themselves The Inheritors. Mayday and Ben are taken to a safe zone where the Spiders plan their next course of action. The safe zone is eventually compromised and Ben is captured by the Inheritors. It is revealed that the infant Ben is a vital part of a prophecy that will help bring about the downfall of the Inheritors and involves "The Other" (Kaine), "The Bride" (Silk) and "The Scion" (Ben himself). However, conversely if the three specific totems were sacrificed together, their deaths will ensure not only the Inheritors remain in power forever but also will stop future spider-people from appearing, and thus preventing the prophecy. Benjy is eventually saved by Ben Parker- his great-uncle and possible namesake- and Spider-Ham in the final fight, Uncle Ben taking Benjy to safety and Spider-Ham taking Benjy's place to catch the Inheritors off-guard. Afterwards, it is revealed Benjamin's mother and Wes survived the Inheritor attacked, but unfortunately his father did not.
Web Warriors.
Benjamin makes a few cameo appearances in 2015's Web Warriors series, looked after by Mayday, Mary Jane, Uncle Ben and is often visited by Anya Corzon.
Spider-Geddon.
Benjamin is referenced several times by his sister Mayday in the event Spider-Geddon. After the latest battle with The Inheritors has concluded, Mayday comments that her brother is very likely still the Scion of the Spider-Scroll Prophecy, her alternate world sister Annie May Parker, Spiderling, informs her that The Other is still in play too and is closer than she knows. On Mayday and Benjamin's Earth, it is revealed that The Other has resurrected their father.
Other versions.
Universe X.
On Earth X, when Spiders Man, an African American villain mutated by the Terrigen Mist, attacked a middle aged Peter using his web illusions and trapping Peter on a world of suffer. However, Peter took control of the illusion and fabricated a world where he had his ideal life, including his son Benjamin who had grown up and took the mantle of Spider-Man. Then Venom, Peter's real life daughter May who had bonded to the Venom symbiote, defeated Spiders Man and broke her out of that fabricated world and convinced Peter to accept his real life.
Arachnamorphosis.
A version of Benjamin appears in "What If?... Starring Spider-Man: Arachnamorphosis", in which he is the son of Peter Parker and Gwen Stacy. In this reality Spider-Man's powers began to mutate him into a monster along with Ben who inherited traits from the spider powers.
Peter and MJ happily ever after.
In this universe, Peter had overcome every hardship and he with his wife Mary Jane lived a good life, growing old along with their children, May and Ben.
Last Avenger.
In another universe, Ben was featured, where Spider-Man refused to join the Avengers.
Spider-Dreams.
Benjamin along with his family, May, Mary Jane and Peter, with the help of S.H.I.E.L.D. got new identities and moved away after Peter gave up being Spider-Man. He along with his family lived a full life and died of old age, while Peter, who gained longevity, outlived his own family.
Spider-Girl: The End.
In the alternate reality where April had accidentally killed her sister, April started impersonating her dead sister, but Benji saw through her disguise and revealed it MJ.
Adventures in Spidey-Baby-Sitting.
In this alternate universe, the son of Peter and MJ was born a spider-like human mutant, but despite of this, his parents still love him.
Spider-Man: Last Stand.
In the universe of "Spider-Man: Last Stand", Peter had a son named Ben.
House of M.
In the reality of House of M, Peter married Gwen and had a son named Richie.
Fantastic Four: The End.
In the reality of , the son of Peter wore his father's suit and under the alias of Spider-Kid joined the Insect Squad.


Geographical mile

The geographical mile is a unit of length determined by 1 minute of arc along the Earth's equator. For the international ellipsoid 1924 this equalled 1855.4 metres. "The American Practical Navigator" 2017 defines the geographical mile as . Greater precision depends more on choice of ellipsoid than on more careful measurement: the length of the equator in the World Geodetic System WGS-84 is which makes the geographical mile 1855.3248 m, while the "IERS Conventions (2010)" takes the equator to be making the geographical mile 1855.3250 m, 0.2 millimetres longer. In any ellipsoid, the length of a degree of longitude at the equator is thus exactly 60 geographical miles.
The shape of the Earth is a slightly flattened sphere, which results in the Earth's circumference being 0.168% larger when measured around the equator as compared to through the poles. The geographical mile is slightly larger than the nautical mile (which was historically linked to the circumference measured through both poles); one geographic mile is equivalent to approximately .
Related units.
It was closely related to the nautical mile, which was originally determined as 1 minute of arc along a great circle of the Earth, but is nowadays defined as exactly 1852 metres. The US National Institute of Standards and Technology (NIST) notes that: "The international nautical mile of 1 852 meters (6 076.115 49...feet) was adopted effective July 1, 1954, for use in the United States. The value formerly used in the United States was 6 080.20 feet = 1 nautical (geographical or sea) mile." (The deprecated value, 6080.2 feet, is n) A separate reference identifies the geographic mile as being identical to the international nautical mile of 1852 metres (and slightly shorter than the British nautical mile of ). The unit is not used much, but is cited in some United States laws (e.g., Section 1301(a) of the Submerged Lands Act, which defines state seaward boundaries in terms of geographic miles). While debating what became the Land Ordinance of 1785, Thomas Jefferson's committee wanted to divide the public lands in the west into "hundreds of ten geographical miles square, each mile containing 6086 and 4-10ths of a foot" and "sub-divided into lots of one mile square each, or 850 and 4-10ths of an acre".
The Danish and German geographical mile ("geografisk mil" and "geographische Meile" or "geographische Landmeile", respectively) is 4 minutes of arc, and was defined as approximately 7421.5 metres by the astronomer Ole Rømer of Denmark. In Norway and Sweden, this 4-minute geographical mile was mainly used at sea ("sjømil"), up to the beginning of the 20th century.


List of popular music songs featuring Andalusian cadences

Following is a list of popular music songs which feature a chord progression commonly known as Andalusian cadences.
Items in the list are sorted alphabetically by the band or artist's name. Songs which are familiar to listeners through more than one version (by different artists) are mentioned by the earliest version known to contain Andalusian cadences (which is most frequently the original version). Songs whose composers are unknown are at the bottom of the list, sorted alphabetically in their turn by title.


WebVR

WebVR was an experimental JavaScript application programming interface (API) that enabled applications to interact with virtual reality devices, such as the HTC Vive, Oculus Rift, Google Cardboard or Open Source Virtual Reality (OSVR) in a web browser. It was implemented in Firefox and Chromium-based browsers before being deprecated and removed. The WebVR API was superseded by a more powerful WebXR API that is capable of representing virtual reality and augmented reality devices simultaneously.
This API is designed with the following goals in mind:
History.
WebVR API was first conceived in spring 2014 by Vladimir Vukićević from Mozilla. The API's contributors include Brandon Jones, Boris Smus and others from the Mozilla team. On March 1, 2016, the Mozilla VR team and the Google Chrome team announced the version 1.0 release of the WebVR API proposal. The resulting API refactoring brought many improvements to WebVR.
The last tagged version is 1.1, which was last edited on April 5, 2017. The editors of the document include members from Mozilla and Google teams. However some Microsoft members have joined, and are actively collaborating in the drafting process of version 2.0 for the WebVR API.
Design.
The WebVR API exposes a few new interfaces (such as VR Display, VR pose) that allow web applications to present content in virtual reality, by using WebGL with the necessary camera settings and device interactions (such as controllers or point of view). The API has been designed to follow a certain path, which is very similar to other intrusive Web API like the Geolocation API. The necessary steps are:
Support.
WebVR 1.0 is currently supported in the release version of Firefox 55+ for Windows (64 bit version only) and was supported by Chrome for Android until version 80 as an Origin Trial Experiment, which means that developers can request a token to add to their website which will seamlessly enable WebVR. WebVR 1.1 is supported in Microsoft Edge from build 15002+, and in Samsung Internet, Chromium, Servo and Oculus Carmel.
Chrome for Windows supports WebVR 1.1 in a special build. Firefox 55+ on macOS also supports WebVR if a setting is enabled by the user.
Similar technologies.
Although WebVR is unique as an API, there are native applications on most hardware allowing for networked experiences and access to web content. Several key tools, such as Unity and Blender, are also able to export for the web and provide users a way to use their content without installing a dedicated application.
WebXR.
In 2018, the WebXR API superseded WebVR, being designed for both augmented reality and virtual reality devices.


Farad

The farad (symbol: F) is the SI derived unit of electrical capacitance, the ability of a body to store an electrical charge. It is named after the English physicist Michael Faraday. In SI base units 1F = 1s4⋅A2⋅m−2⋅kg−1.
Definition.
One farad is defined as the capacitance across which, when charged with one coulomb, there is a potential difference of one volt. Equally, one farad can be described as the capacitance which stores a one-coulomb charge across a potential difference of one volt.
The relationship between capacitance, charge, and potential difference is linear. For example, if the potential difference across a capacitor is halved, the quantity of charge stored by that capacitor will also be halved.
For most applications, the farad is an impractically large unit of capacitance. Most electrical and electronic applications are covered by the following SI prefixes:
Equalities.
A farad is a derived unit based on four of the seven base units of the International System of Units: kilogram (kg), metre (m), second (s), and ampere (A).
Expressed in combinations of SI units, the farad is:
where , , , , , , , , , , , , Hz = hertz, .
History.
The term "farad" was originally coined by Latimer Clark and Charles Bright in 1861, in honor of Michael Faraday, for a unit of quantity of charge, but by 1873, the farad had become a unit of capacitance. In 1881 at the International Congress of Electricians in Paris, the name farad was officially used for the unit of electrical capacitance.
Explanation.
A capacitor generally consists of two conducting surfaces, frequently referred to as plates, separated by an insulating layer usually referred to as a dielectric. The original capacitor was the Leyden jar developed in the 18th century. It is the accumulation of electric charge on the plates that results in capacitance. Modern capacitors are constructed using a range of manufacturing techniques and materials to provide the extraordinarily wide range of capacitance values used in electronics applications from femtofarads to farads, with maximum-voltage ratings ranging from a few volts to several kilovolts.
Values of capacitors are usually specified in farads (F), microfarads (μF), nanofarads (nF) and picofarads (pF). The millifarad is rarely used in practice (a capacitance of 4.7 mF (0.0047 F), for example, is instead written as ), while the nanofarad is uncommon in North America. The size of commercially available capacitors ranges from around 0.1 pF to (5 kF) supercapacitors. Parasitic capacitance in high-performance integrated circuits can be measured in femtofarads (1 fF = 0.001 pF =  F), while high-performance test equipment can detect changes in capacitance on the order of tens of attofarads (1 aF =  F).
A value of 0.1 pF is about the smallest available in capacitors for general use in electronic design, since smaller ones would be dominated by the parasitic capacitances of other components, wiring or printed circuit boards. Capacitance values of 1 pF or lower can be achieved by twisting two short lengths of insulated wire together.
The capacitance of the Earth's ionosphere with respect to the ground is calculated to be about 1 F.
Informal and deprecated terminology.
The picofarad (pF) is sometimes colloquially pronounced as "puff" or "pic", as in "a ten-puff capacitor". Similarly, "mic" (pronounced "mike") is sometimes used informally to signify microfarads.
Nonstandard abbreviations were and are often used. Farad has been abbreviated "f", "fd", and "Fd". For the prefix "micro-", when the Greek small letter "μ" or the legacy micro sign "μ" is not available (as on typewriters) or inconvenient to enter, it is often substituted with the similar-appearing "u" or "U", with little risk of confusion. It was also substituted with the similar-sounding "M" or "m", which can be confusing because M officially stands for 1,000,000 (or 1,000), and m preferably stands for 1/1000. In texts prior to 1960, and on capacitor packages until more recently, "microfarad(s)" was abbreviated "mf" or "MFD" rather than the modern "μF". A 1940 Radio Shack catalog listed every capacitor's rating in "Mfd.", from 0.000005 Mfd. (5 pF) to 50 Mfd. (50 μF).
"Micromicrofarad" or "micro-microfarad" is an obsolete unit found in some older texts and labels, contains a nonstandard metric double prefix. It is exactly equivalent to a picofarad (pF). It is abbreviated μμF, uuF, or (confusingly) "mmf", "MMF", or "MMFD".
Summary of obsolete capacitance units: (upper/lower case variations are not shown)
Related concepts.
The reciprocal of capacitance is called electrical elastance, the (non-standard, non-SI) unit of which is the daraf.
CGS units.
The abfarad (abbreviated abF) is an obsolete CGS unit of capacitance equal to farads (1 gigafarad, GF).
The statfarad (abbreviated statF) is a rarely used CGS unit equivalent to the capacitance of a capacitor with a charge of 1 statcoulomb across a potential difference of 1 statvolt. It is 1/(10−5"c"2) farad, approximately 1.1126 picofarads.


Edexcel

Edexcel (also known since 2013 as Pearson Edexcel) is a British multinational education and examination body formed in 1996 and wholly owned by Pearson plc since 2005. It is the only privately owned examination board in the United Kingdom. Its name is a portmanteau term combining the words education" and excellence".
Edexcel regulates school examinations under the British Curriculum and offers qualifications for schools on the international and regional scale. It is the UK's largest awarding organisation offering academic and vocational qualifications in schools, colleges and work places in the UK and abroad. It is also recognised internationally. In 2019, Edexcel was the focus of significant controversy following a leak of an A-level examination.
History.
Edexcel was formed in 1996 by the merger of two bodies, the BTEC (Business & Technology Education Council) and ULEAC (University of London Examinations and Assessment Council). In 2003, the Edexcel Foundation (the charity that managed the board) formed a partnership with Pearson plc to set up a new company called London Qualifications Ltd, which was 75% owned by Pearson and 25% by the Edexcel Foundation. London Qualifications Limited changed its name to Edexcel Limited in November 2004 and is now known colloquially as Edexcel and formally as Edexcel Pearson - London Examinations.
In 2005, Edexcel became the only large examination board to be held in private hands, when Pearson plc took complete control. Edexcel subsequently received investment from their new parent company.
Edexcel also offers IAL, known as International Advanced Levels. It is offered only to schools outside the UK. It is considered by the UK NARIC (the National Recognition Information Centre for the United Kingdom) to be of a comparable standard as the GCE Advanced Level. In addition, Edexcel provides the Edexcel International Diploma (ID) which involves the study of 4 A-Levels (3 full A-Levels and 1 AS-Level in either General Studies or Global Development). The British curriculum offered by the Edexcel International board is among the most taught at International Schools along with the curricula offered by Cambridge International Examinations and, the International Baccalaureate Organization.
On 3 April 2013, Pearson rebranded Edexcel and all of its exams 'Pearson Edexcel', in line with a name change that added the prefix 'Pearson' to all of its brands.
Controversies.
As Edexcel is the only privately owned examination board in the UK, questions have been raised on whether the examination board is acting in the best interest of students, or solely as a profit making business, due to the wide range of Edexcel-endorsed text books published by Pearson, the international multi-billion company which owns the board.
2007.
In 2007 it was reported that teachers using Edexcel Music examinations were allowing students to listen to confidential listening paper CDs several days before the examination, by abusing the trust given by the exam board to only check for technical issues. Other exam boards do not allow the practice of checking discs, with AQA specifically instructing teachers not to open the packages containing the CDs before exams.
2014.
In 2014 the loss of an A-Level Core Mathematics C3 Mathematics exam being delivered to an international school in Amsterdam led to a replacement paper being published for the Summer examination series; however, 60 students in the UK took the original paper due to it mistakenly being handed out in two UK and two overseas centres, while the replacement paper was taken by 34,000 students. The replacement paper was criticised for including questions that were not present on the syllabus, and that the students taking the original paper would be unfairly marked.
2015.
Difficulty.
In June 2015, students across the United Kingdom who had taken an Edexcel GCSE Maths paper expressed anger and confusion over questions that "did not make sense" and were "ridiculous", mocking the exam on Twitter. On a Sky News segment, presenter Adam Boulton answered one of the paper's 'hardest' questions with a former maths teacher. As a result, GCSE students across the UK signed a petition made by a candidate requesting that the exam board lowers the grade boundaries as the examination was too hard.
Gender representation.
An additional petition was started by a 17-year-old student requesting to "Ensure the representation of women on the A-Level Music syllabus." The petition asks that music composed by women be added to the Edexcel A-Level Music syllabus which "has a total of 63 different set works from a variety of musical genres and eras." After six days, the petition had over 1,800 signatures and was featured in "The Guardian".
2019.
Question duplication.
In June, Paper 3 of the Mathematics GCSE (Higher Tier, 1MA1/03) appeared to contain an exam question which was published in an AQA (another British exam board) Further Mathematics textbook. The exam question had the same diagram, values and answer as the question in the textbook. Pearson Edexcel said that they were investigating how this might have happened.
Exam difficulty concerns from students.
After students sat the Paper 2 of the new specification Mathematics A-Level (9MA0), many students complained online expressing that the difficulty of the exam was too high and unlike anything seen in past papers. One online petition to lower grade boundaries was signed by thousands of users. Before results day, Pearson Edexcel released a video addressing students about the difficulty concerns and mentioned that independent experts did confirm that the exam was "fair and valid". They acknowledged the first two questions of the exam were more difficult than anticipated, and also claimed that they would make "small adjustments" to "improve the experience" of the exam for future students.
Leak.
Further controversy ensued after the Paper 3 of the new specification Mathematics A-Level (9MA0/3H – sat 14 June), appeared to have been accessed by someone who, late in the night before the exam sitting, posted on Twitter that they would sell a copy of the exam paper for £70. Pearson Edexcel launched an investigation after becoming aware that some scribbled out images of the exam paper were circulated online prior the exam sitting, and found one centre (out of thirty-eight possible suspect centres in one geographic location), who were visited at a very short notice, to have committed a serious and "possibly criminally motivated" security breach of the A-Level maths paper.
Students expressed frustration about the incident; BBC News reported that students believed the incident could have the potential to affect their results. Pearson Edexcel stated that they have "well-established processes to ensure fair and accurate results" and that "grade boundaries will not be affected". On 17 June Edexcel stated that the compromised questions could be removed from the overall assessment, as well as undertaking additional statistical analysis to identify irregular result patterns for particular centres or students.
About 60,000 students were affected in this incident. Two men aged 29 and 32 were arrested on 24 June on suspicion of theft. The men were released under investigation while inquiries continued. Ahead of results day, it was revealed that 78 students who were suspected of being at an advantage from malpractice would have their results withheld. This is the largest number of candidates to be withheld results from a single malpractice incident.
Extremely low A-level mathematics grade boundaries.
After a leak of the A-Level Grade Boundaries ahead of results day, it emerged that in order to pass the new specification Mathematics A-Level (9MA0), candidates needed to score 14% (43 marks out of 300) to pass. OCR (another British exam board) also had similarly low grade boundaries. After these extremely low grade boundaries added flavour to many news headlines, Ofqual said that they were confident the grade boundaries this year were "sound", so shifted their focus onto the previous year's grade boundaries for the new Mathematics A-Level for the 2,000 students who sat it after studying it for one year. Ofqual said "We want to understand why the grade boundaries were so different between the two years.", and had called the significant shifts in boundaries "unusual".
None of the exam boards decided to re-open the 2018 award, after being asked to look at them again by the regulator. This could potentially result in students who sat the new specification Mathematics A-Level (9MA0) in 2018 having their results changed.


Neyman construction

Neyman construction is a frequentist method to construct an interval at a confidence level formula_1 such that if we repeat the experiment many times the interval will contain the true value of some parameter a fraction formula_2 of the time. It is named after Jerzy Neyman.
Theory.
Assume formula_3 are random variables with joint pdf formula_4, which depends on k unknown parameters. For convenience, let formula_5 be the sample space defined by the n random variables and subsequentially define a sample point in the sample space as formula_6
Neyman originally proposed defining two functions formula_7 and formula_8 such that for any sample point,formula_9,
Given an observation, formula_12, the probability that formula_13 lies between formula_14 and formula_15 is defined as formula_16 with probability of formula_17 or formula_18. These calculated probabilities fail to draw meaningful inference about formula_13 since the probability is simply zero or unity. Furthermore, under the frequentist construct the model parameters are unknown constants and not permitted to be random variables.
For example if formula_20, then formula_21. Likewise, if formula_22, then formula_23
As Neyman describes in his 1937 paper, suppose that we consider all points in the sample space, that is, formula_11, which are a system of random variables defined by the joint pdf described above. Since formula_25 and formula_26 are functions of formula_9 they too are random variables and one can examine the meaning of the following probability statement:
That is,formula_41 where formula_42 where formula_31 and formula_32 the upper and lower confidence limits for formula_13
Coverage probability.
The coverage probability, formula_46, for Neyman construction is the frequency of experiments that the confidence interval contains the actual value of interest. Generally, the coverage probability is set to a formula_47 confidence. For Neyman construction, the coverage probability is set to some value formula_46 where formula_49. This value formula_46 tells how confidently that the true value is contained in the interval.
Implementation.
A Neyman construction can be carried out by performing multiple experiments that construct data sets corresponding to a given value of the parameter. The experiments are fitted with conventional methods, and the space of fitted parameter values constitutes the band which the confidence interval can be selected from.
Classic Example.
Suppose formula_9~formula_52, where formula_53 and formula_54 are unknown constants where we wish to estimate formula_53. We can define (2) single value functions, formula_25 and formula_26, defined by the process above such that given a pre-specified confidence level ,formula_46, and random sample formula_59=(formula_60)
Another Example.
formula_68 are iid random variables, and let formula_69. Suppose formula_70. Now to construct a confidence interval with formula_71 level of confidence. We know formula_72 is sufficient for formula_73. So,
This produces a formula_77 confidence interval for formula_73 where,


VAX Unit of Performance

The VAX Unit of Performance, or VUP for short, is an obsolete measurement of computer performance used by Digital Equipment Corporation (DEC). 1 VUP was equivalent to the performance of a VAX 11/780 completing a given task.
Other VAX machines and later workstation designs were compared in performance terms by defining system speed in VUPs, for instance, the VAXft Model 310 ran at 3.8 VUPs, meaning it ran roughly 3.8 times as fast as the 11/780.
The term was used largely within the DEC and its community, and fell from use as more standard ratings like SPEC became more widely used. This was especially true with the introduction of DEC workstations running Unix, in which case the VUP was of little use comparing the platforms to competition systems.


School-based assessment

The Hong Kong Examinations and Assessment Authority (HKEAA) has moved from norm-referenced to standards-referenced assessment, including the incorporation of a substantial school-based summative oral assessment component(SBA) into the compulsory English language subject in the Hong Kong Certificate of Education Examination (HKCEE), a high-stakes examination for all Form 4–5 students (Davison, 2007).
Background.
School-based Assessment (SBA) was first proposed in the Reform Proposals for the Education System in Hong Kong, published in September 2000. According to the proposal, the modes, content and the assessment methods of public examinations should be reviewed (p. 44). A "core-competence" approach, similar to criterion-referencing, will be adopted. In this approach, basic skills and knowledge required by a Form 5 graduate will be indicated and students do not have to compare with the others. Under the education reforms in which a new culture of learning and teaching is to be cultivated, schools can use different modes of broad-based assessments, including observation of students' performance in classroom and participation in project work to promote learning in a more flexible manner. Another benefit is advocating students' all-round development which gives a more comprehensive picture of individual students' learning needs, as well as fosters the positive washback effects of public examinations. It also helps to address the limitations of judging students on their performance in one single examination.
Rationales.
The rationales laid down by the SBA Consultancy Team (2005) for implementing SBA are as follows:
Content.
The SBA component, worth 15% of the total HKCEE English mark, involves four assessments of English oral language skills based on texts—one from each category: print fiction, print non-fiction, non-print fiction, non-print non-fiction—drawn from a programme of independent extensive reading and viewing (SBA Consultancy Team, 2005). At the time the SBA was introduced, students were required to choose at least four texts to read or view over the course of 2 years; keep brief notes in a logbook; and undertake a number of activities in and out of class to develop their independent reading, speaking, and generic skills. Due to teachers' concerns over excessive workload, the requirements have been modified to "at least three texts over the course of 2 years" (Davison, 2007).
The assessment format and requirements as originally specified in the introduction to the SBA in September 2005 are summarised in Table 1.
 Source: (SBA Consultancy Team, 2005, p. 5).
Assessment criteria.
For assessment students are required to partake in interactions with classmates on a particular aspect of the text they have read or viewed, leading up to making a more formal group interaction or an individual presentation on a specific text and responding to questions from their audience (HKEAA, 2008). Students are assessed according to a set of assessment criteria, consisting of a set of descriptors at each of six levels across four domains (HKEAA, 2007). The domains are briefly described as follows:
Future development.
The current SBA for HKCEE English Language will be extended to the Hong Kong Diploma of Secondary Education (HKDSE) English Language Examination from 2012 onwards(HKEAA, 2007). The SBA component will also be extended to other humanities and science subjects in HKDSE (HKEAA, 2009).
Controversies.
Using Brindley’s (1998) framework, the SBA component is controversial in terms of political issues, to do with the purposes and intended use of the assessment; technical issues, primarily to do with validity and reliability; and practical issues, to do with the means by which the assessment was put into practice. These issues were gathered from key stakeholder groups, including teachers, students and education bureau officials.
Politically, in Davison’s (2007) words, “In fact, the SBA guidelines asked teachers to set their own time limits according to the needs of the students, but this was interpreted through the prism of teachers’ existing experience—many schools used buzzers and stopwatches to allocate an identical period of time to each students, with the result that in some schools students’ stress levels were high and their “performance” very contrived and/or rushed. As an outcome-oriented standards-referenced system, SBA is a significant cultural and attitudinal change, not only for teachers but for the whole school community, including students and parents. Hence, it is not surprising that fairness was a deep-seated sociocultural, not just political concern."
Technically, many teachers and students are concerned over validity, reliability and fairness of the SBA component. While teacher-educators and researchers view that SBA will enhance the validity and reliability of the HKCEE theoretically, some frontline teachers and students are sceptical on this.
As for practical issues, teachers raised such concerns as:


Susan J. Devlin

Susan J. Devlin is an American statistician who has contributed to highly-cited research on robust statistics and local regression.
Education and career.
Devlin earned a bachelor's degree in mathematics from William Smith College in 1968, and has a master's degree in statistics from Rutgers University.
After completing her bachelor's degree, she began working for Bell Labs, and completed her master's degree on a part-time basis while working there. After the 1984 breakup of the Bell system, she moved to Bellcore, and in 1987 her responsibilities at Bellcore shifted from research in statistics to its application in modeling client satisfaction with Bellcore's services. In 1997, she retired from Bellcore and became a founding principal of The Artemis Group, a New Jersey-based marketing consulting firm.
After moving to Thomaston, Maine she became president of the Thomaston Historical Society.
Service.
Devlin chaired the Committee on Women in Statistics (CoWiS) of the American Statistical Association for 1984–1985. She was chair of the Statistical Consulting Section of the American Statistical Association for 2005.
Recognition.
Devlin became a Fellow of the American Statistical Association in 2005.


Bang on a Can

Bang on a Can is a multi-faceted contemporary classical music organization based in New York City. It was founded in 1987 by three American composers who remain its artistic directors: Julia Wolfe, David Lang, and Michael Gordon. Called "the country's most important vehicle for contemporary music" by the "San Francisco Chronicle", the organization focuses on the presentation of new concert music, and has presented hundreds of musical events worldwide.
Notable performances.
Bang on a Can is perhaps best known for its Marathon Concerts, during which an eclectic mix of pieces are performed in succession over the course of many hours while audience members, who are encouraged to maintain a "jeans-and-tee-shirt informality," are welcome to come and go as they please. For the twentieth anniversary of their Marathon Concerts, Bang on a Can presented twenty-six hours of uninterrupted music at the World Financial Center Winter Garden Atrium in New York City.
Among Bang on a Can's early events were performances by John Cage, premieres of Glenn Branca’s epic symphonies for massed electric guitars, and fully staged operas by Harry Partch, featuring the composer's original instruments.
Programs.
Bang on a Can Summer Festival at MASS MoCA.
In 2002, Bang on a Can began the yearly Summer Institute of Music, a program at the Massachusetts Museum of Contemporary Art (MASS MoCA) for young composers and performers. This program is sometimes referred to by the nickname "Banglewood" in reference to the nearby but far more traditional Tanglewood Music Festival.
Staged Works.
The three artistic directors occasionally collaborate by jointly composing large staged works, often without revealing which sections each contributed. Examples include:
People's Commissioning Fund (PCF).
Bang on a Can has commissioned and premiered pieces by composers including Steve Reich, Terry Riley, Michael Nyman, John Adams, Somei Satoh, Iva Bittová, Roberto Carnevale, Ornette Coleman, Donnacha Dennehy and Bun-Ching Lam. In 1998 the organization began the People's Commissioning Fund, which supports the creation of new musical compositions by pooling contributions from numerous member-commissioners whose donations range from $5 to $5,000.
Bang on a Can All-Stars.
The Bang on a Can All-Stars is an amplified sextet formed by its parent organization in 1992. The All-Stars tour internationally and have received awards and public recognition for their work in the contemporary classical music field.
The instrumentation of the Bang on a Can All-Stars is clarinet, cello, electric guitar, piano/keyboard, percussion, double bass. Current members include Ashley Bathgate, Robert Black, Vicky Chow, David Cossin, Arlen Husklo, Mark Stewart and Ken Thomson.
Asphalt Orchestra.
Asphalt Orchestra is Bang on a Can's 12-piece marching band. The ensemble's premiere performance was in 2009 at the Lincoln Center Out of Doors festival, and featured new commissioned works by Tyondai Braxton (of experimental rock group Battles), Goran Bregovic, and Stew and Heidi Rodewald, alongside arrangements of songs by Björk, Meshuggah, Mingus, Nancarrow, and Zappa.
"The New York Times" has called Asphalt Orchestra's members "12 top-notch brass and percussion players", and praised their performance as "coolly brilliant and infectious."
Found Sound Nation.
An independent project founded in 2009 and produced by Bang on a Can, Found Sound Nation (FSN) engages at-risk youth and underrepresented communities producing original audio and video projects across the globe in economically disparate settings. The work of FSN emphasizes a mobile, accessible, collaborative way of recording and producing professional quality music, a technique developed by combining the art music traditions of Bang on a Can with traditions of musique concrète, hip hop, and contemporary composition.
FSN has led site-specific projects in New York City, India, Zimbabwe, Mexico, Italy, Switzerland and Haiti.
In 2012, the project received an award from the U.S. Department of State to produce OneBeat, an international music exchange which will bring together innovative musicians from around the world to compose, produce and perform original music.
Found Sound Nation was co-founded by Christopher Marianetti and Jeremy Thal. The project is directed by Marianetti, Thal and Elena Moon Park.
Recordings.
In the past, Bang on a Can released recordings on Composers Recordings Inc. (CRI), Sony Classical, Point Music (Universal), and Nonesuch, but now the majority of its recordings are found on its own record label, Cantaloupe Music. In addition to releasing works by Gordon, Wolfe, and Lang, the label releases CDs of music by composers and musical groups affiliated with the organization, including Evan Ziporyn, Phil Kline, Alarm Will Sound, Icebreaker, Ethel, Gutbucket, R. Luke DuBois, and Don Byron among many, many others.
Below is a partial discography of released works performed by Bang on a Can:


9996 ANS


White light scanner

A white light scanner (WLS) is a device for performing surface height measurements of an object using coherence scanning interferometry (CSI) with spectrally-broadband, "white light" illumination. Different configurations of scanning interferometer may be used to measure macroscopic objects with surface profiles measuring in the centimeter range, to microscopic objects with surface profiles measuring in the micrometer range. For large-scale non-interferometric measurement systems, see structured-light 3D scanner.
Description.
Vertical scanning interferometry is an example of low-coherence interferometry, which exploits the low coherence of white light. Interference will only be achieved when the path length delays of the interferometer are matched within the coherence time of the light source. VSI monitors the fringe contrast rather than the shape of the fringes.
Fig. 2 illustrates a Twyman–Green interferometer set up for white light scanning of a macroscopic object. Light from the test specimen is mixed with light reflected from the reference mirror to form an interference pattern. Fringes appear in the CCD image only where the optical path lengths differ by less than half the coherence length of the light source, which is generally on the order of micrometers. The interference signal (correlogram) is recorded and analyzed as either the specimen or reference mirror is scanned. The focus position of any particular point on the surface of the specimen corresponds to the point of maximum fringe contrast (i.e. where the modulation of the correlogram is greatest).
Fig. 3 illustrates a white light interferometric microscope using a Mirau interferometer in the objective. Other forms of interferometer used with white light include the Michelson interferometer (for low magnification objectives, where the reference mirror in a Mirau objective would interrupt too much of the aperture) and the Linnik interferometer (for high magnification objectives with limited working distance). The objective (or alternatively, the sample) is moved vertically over the full height range of the sample, and the position of maximum fringe contrast is found for each pixel.
The chief benefit of low-coherence interferometry is that systems can be designed that do not suffer from the 2 pi ambiguity of coherent interferometry, and as seen in Fig. 1, which scans a 180 μm × 140 μm × 10 μm volume, it is well suited to profiling steps and rough surfaces. The axial resolution of the system is determined by the coherence length of the light source and is typically in the micrometer range. Industrial applications include in-process surface metrology, roughness measurement, 3D surface metrology in hard-to-reach spaces and in hostile environments, profilometry of surfaces with high aspect ratio features (grooves, channels, holes), and film thickness measurement (semi-conductor and optical industries, etc.).
Technical.
White-light interferometry scanning (WLS) systems capture intensity data at a series of positions along the vertical axis, determining where the surface is located by using the shape of the white-light interferogram, the localized phase of the interferogram, or a combination of both shape and phase. The white light interferogram actually consists of the superposition of fringes generated by multiple wavelengths, obtaining peak fringe contrast as a function of scan position, that is, the red portion of the object beam interferes with the red portion of the reference beam, the blue interferes with the blue, and so forth.
In a WLS system, an imaging interferometer is vertically scanned to vary the optical path difference. During this process, a series of interference patterns are formed at each pixel in the instrument field of view. This results in an interference function, with interference varying as a function of optical path difference. The data are stored digitally and processed in a variety of ways depending on the system manufacturer, including being Fourier-transformed into frequency space, subject to cross-correlation methods, or analysis in the spatial domain.
If a Fourier transform is used, the original intensity data are expressed in terms of interference phase as a function of wavenumber. Wavenumber k is a representation of wavelength in the spatial frequency domain, defined by k = 2π/λ. If phase is plotted versus wavenumber, the slope of the function corresponds to the relative change in group-velocity optical path difference DG by Dh = DG/2nG where nG is group-velocity index of refraction. If this calculation is performed for each pixel, a three-dimensional surface height map emerges from the data.
In the actual measuring process, the optical path difference is steadily increased by scanning the objective vertically using a precision mechanical stage or piezoelectric positioner. Interference data are captured at each step in the scan. In effect, an interferogram is captured as a function of vertical position for each pixel in the detector array. To sift through the large amount of data acquired over long scans, many different techniques can be employed. Most methods allow the instrument to reject raw data that do not exhibit sufficient signal-to-noise. The intensity data as a function of the optical path difference are processed and converted to height information of the sample.


709 Fringilla

709 Fringilla is a minor planet orbiting the Sun.


Orders of magnitude (force)

The following list shows different orders of magnitude of force.
Since weight under gravity is a force, several of these examples refer to the weight of various objects. Unless otherwise stated, these are weights under average Earth gravity at sea level.


Policy Aptitude Test

The Policy Aptitude Test (PAT) is an offline written test held in India. This test scores a person on the bases of "General Knowledge", "Numerical Problem Solving", "Policy Aptitude", "Logical Reasoning", "English Language" and "Policy Analysis Ability". The National Law School of India University, Bengaluru (NLSIU) started this exam and use the test for selecting students for its Public Policy Programme. The test is conducted every year.
Exam format.
Policy Aptitude Test (PAT) have two parts. First part is multiple choice section with negative marking. Second part is descriptive analytical writing about a policy issue. Only those who obtain minimum of 40% of marks in the first part will be evaluated for the second part.
First part of Policy Aptitude Test have five components: General Knowledge, Numerical Problem Solving, Policy Aptitude, Logical Reasoning and English Language. Each component have 20 questions. Each question carries 1 mark, and negative marking is -0.25. All components have equal weight.
Second part of Policy Aptitude Test is for 50 marks and it concerns descriptive writing about an issue that is provided. The second part examine candidate's policy analysis ability as well as English comprehension. Both parts together, Policy Aptitude Test carries 150 marks. Total duration of Policy Aptitude Test is of two hours.


Romanian Baccalaureate

The Bacalaureat (or "bac" for short) is an exam held in Romania and Moldova when one graduates high school ().
Romania.
History.
The Romanian Baccalaureate evolved over time.
Present.
Unlike the French Baccalaureate, the Romanian one has a single degree. The subjects (except subject A) depend on the profile studied (): mathematics and computer science (), philology (), natural sciences (), social sciences (), or various other vocational tracks (), and the candidate's choice.
The exam covers the whole high school curriculum and the marking scale is between 1 and 10. In order to pass, students must obtain at least 5.00 in every subject with a minimum of 6.00 overall.
Although a student obtained 5.00 every year in high school at a subject, they can obtain 1.00 as well as 10.00 at the National Baccalaureate Exam at that same subject. The grades obtained in high school do not affect or interact in any way with the grade written on the Romanian "Diploma de Bacalaureat".
In order to overcome a national problem of discrimination, the final grade written on the Romanian "Diploma de Bacalaureat" is equally obtained by every student, regardless of the difference of study intensity between schools and profiles. All the graduates in the country take the exam at the same time, having to solve the same subjects. The subjects vary from profile to profile.
For example, in high school, a computer-science student will obtain harder a high grade at Mathematics than a natural sciences student, as the difficulty varies.
In addition to this, all the high schools are ranked by the Ministry of Education, so a low grade obtained in the top schools does not mean that one student is less qualified than another who obtained a higher grade in a lower ranked school.
Moldova.
Moldavian Bacalaureat consists of four exams. Pupils from schools with a language of instruction other than Romanian must take one supplementary exam in their native language and literature.
Exams usually take place in June. All of them are written and each is graded separately.
Students must also choose one more subject taught in school. The majority take Geography or Chemistry.
Subjects.
The baccalaureate has up to 5 modules, each one graded separately. Module E, the written examination, is the one considered for admission to higher education.
Competence exams.
A, B and D are marked with: Beginner, Average, Advanced and Experienced.
Written exams.
Each exam takes three hours.


Lancashire Evening Post

The Lancashire Evening Post is a daily newspaper based in Fulwood, a suburb of the city of Preston, Lancashire, England. According to the British Library, its first edition was published on 18 October 1886. It is known locally as the "LEP".


Nipperkin

The nipperkin is a unit of measurement of volume, equal to one-half of a quarter-gill, one-eighth of a gill, or one thirty-second of an English pint. In other estimations, one nip (an abbreviation that originated in 1796) is either one-third of a pint, or any amount less than or equal to half a pint. A nipperkin is also one-eighth of a pint of beer or any other liquor.


D-flat minor

D-flat minor is a theoretical key based on D, consisting of the pitches D, E, F, G, A, B, and C. Its key signature has six flats and one double flat. Its relative major is F-flat major (usually replaced by E major), and its parallel major is D-flat major. Its direct enharmonic equivalent, C-sharp minor, is normally used.
The D-flat natural minor scale is:
\override Score.TimeSignature #'stencil = ##f
\relative c' {
 \clef treble \key des \minor \time 7/4 des4^\markup { Natural minor scale } es fes ges aes beses ces des ces beses aes ges fes es des2
 \clef bass \key des \minor
</score>
Changes needed for the melodic and harmonic versions of the scale are written in with accidentals as necessary. The D-flat harmonic minor and melodic minor scales are:
\override Score.TimeSignature #'stencil = ##f
\relative c' {
 \clef treble \key des \minor \time 7/4 des4^\markup { Harmonic minor scale } es fes ges aes beses c des c beses aes ges fes es des2
</score>
\override Score.TimeSignature #'stencil = ##f
\relative c' {
 \clef treble \key des \minor \time 7/4 des4^\markup { Melodic minor scale (ascending and descending) } es fes ges aes bes c des ces! beses! aes ges fes es des2
</score>
D-flat minor is usually notated as the enharmonic key of C-sharp minor, as in the second and third measures of Amy Beach's "Canticle of the Sun". However, unusually, two of Verdi's most well-known operas, "La traviata" and "Rigoletto", both end in D-flat minor (although written with the five-flat key signature of the parallel major). Mahler's thematic motif "der kleine Appell" ("call to order") from his Fourth and Fifth Symphonies uses both notations: in his Symphony No. 4 (first movement) it is in D-flat minor, but in his Symphony No. 5 it is in C-sharp minor. In the Adagio of his Symphony No. 9 a solo bassoon interpolation following the main theme appears first in D-flat minor, returning twice more notated in C-sharp minor. Likewise, in the Adagio of Bruckner's Symphony No. 8, phrases that are tonally in D-flat minor are notated as C-sharp minor.


Ancient mode