Synonyms containing punctured interval
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In statistics, the coverage probability of a technique for calculating a confidence interval is the proportion of the time that the interval contains the true value of interest. For example, suppose our interest is in the mean number of months that people with a particular type of cancer remain in remission following successful treatment with chemotherapy. The confidence interval aims to contain the unknown mean remission duration with a given probability. This is the "confidence level" or "confidence coefficient" of the constructed interval which is effectively the "nominal coverage probability" of the procedure for constructing confidence intervals. The "nominal coverage probability" is often set at 0.95. The coverage probability is the actual probability that the interval contains the true mean remission duration in this example. If all assumptions used in deriving a confidence interval are met, the nominal coverage probability will equal the coverage probability (termed "true" or "actual" coverage probability for emphasis). If any assumptions are not met, the actual coverage probability could either be less than or greater than the nominal coverage probability. When the actual coverage probability is greater than the nominal coverage probability, the interval is termed "conservative", if it is less than the nominal coverage probability, the interval is termed "anti-conservative", or "permissive." A discrepancy between the coverage probability and the nominal coverage probability frequently occurs when approximating a discrete distribution with a continuous one. The construction of binomial confidence intervals is a classic example where coverage probabilities rarely equal nominal levels. For the binomial case, several techniques for constructing intervals have been created. The Wilson or Score confidence interval is one well known construction based on the normal distribution. Other constructions include the Wald, exact, Agresti-Coull, and likelihood intervals. While the Wilson interval may not be the most conservative estimate, it produces average coverage probabilities that are equal to nominal levels while still producing a comparatively narrow confidence interval. The "probability" in coverage probability is interpreted with respect to a set of hypothetical repetitions of the entire data collection and analysis procedure. In these hypothetical repetitions, independent data sets following the same probability distribution as the actual data are considered, and a confidence interval is computed from each of these data sets; see Neyman construction. The coverage probability is the fraction of these computed confidence intervals that include the desired but unobservable parameter value.
In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached. In other words, interval cycles "unfold a single recurrent interval in a series that closes with a return to the initial pitch class". See: wikt:cycle. Interval cycles are notated by George Perle using the letter "C", with an interval class integer to distinguish the interval. Thus the diminished seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0–11 to indicate the lowest pitch class in the cycle. "These interval cycles play a fundamental role in the harmonic organization of post-diatonic music and can easily be identified by naming the cycle.". Here are interval cycles C1, C2, C3, C4 and C6: Interval cycles assume the use of equal temperament and may not work in other systems such as just intonation. For example, if the C4 interval cycle used justly-tuned major thirds it would fall flat of an octave return by an interval known as the diesis. Put another way, a major third above G♯ is B♯, which is only enharmonically the same as C in systems such as equal temperament, in which the diesis has been tempered out.
In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale. Often, especially in the context of the common practice period, most or all of the melody and harmony of a musical work is built using the notes of a single scale, which can be conveniently represented on a staff with a standard key signature.Due to the principle of octave equivalence, scales are generally considered to span a single octave, with higher or lower octaves simply repeating the pattern. A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance (or interval) between two successive notes of the scale. However, there is no need for scale steps to be equal within any scale and, particularly as demonstrated by microtonal music, there is no limit to how many notes can be injected within any given musical interval. A measure of the width of each scale step provides a method to classify scales. For instance, in a chromatic scale each scale step represents a semitone interval, while a major scale is defined by the interval pattern W–W–H–W–W–W–H, where W stands for whole step (an interval spanning two semitones: From C to D), and H stands for half-step (From C to C#). Based on their interval patterns, scales are put into categories including diatonic, chromatic, major, minor, and others. A specific scale is defined by its characteristic interval pattern and by a special note, known as its first degree (or tonic). The tonic of a scale is the note selected as the beginning of the octave, and therefore as the beginning of the adopted interval pattern. Typically, the name of the scale specifies both its tonic and its interval pattern. For example, C major indicates a major scale with a C tonic.
In statistics, a confidence interval is a type of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval, in principle different from sample to sample, that frequently includes the parameter of interest if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient. More specifically, the meaning of the term "confidence level" is that, if confidence intervals are constructed across many separate data analyses of repeated experiments, the proportion of such intervals that contain the true value of the parameter will match the confidence level; this is guaranteed by the reasoning underlying the construction of confidence intervals. Whereas two-sided confidence limits form a confidence interval, their one-sided counterparts are referred to as lower or upper confidence bounds. Confidence intervals consist of a range of values that act as good estimates of the unknown population parameter. However, in infrequent cases, none of these values may cover the value of the parameter. The level of confidence of the confidence interval would indicate the probability that the confidence range captures this true population parameter given a distribution of samples. It does not describe any single sample. This value is represented by a percentage, so when we say, "we are 99% confident that the true value of the parameter is in our confidence interval", we express that 99% of the observed confidence intervals will hold the true value of the parameter. After a sample is taken, the population parameter is either in the interval made or not, there is no chance. The desired level of confidence is set by the researcher. If a corresponding hypothesis test is performed, the confidence level is the complement of respective level of significance, i.e. a 95% confidence interval reflects a significance level of 0.05. The confidence interval contains the parameter values that, when tested, should not be rejected with the same sample. Greater levels of variance yield larger confidence intervals, and hence less precise estimates of the parameter. Confidence intervals of difference parameters not containing 0 imply that there is a statistically significant difference between the populations.
In music a mixed-interval chord is a chord not characterized by one consistent interval. Chords characterized by one consistent interval, or primarily but with alterations, are equal-interval chords. Mixed interval chords "lend themselves particularly" to atonal music since they tend to be dissonant. Equal-interval chords are often of indeterminate root and mixed-interval chords are also often best characterized by their interval content. "Equal-interval chords are often altered to make them 'impure' as in the case of quartal and quintal chords with tritones, chords based on seconds with varying intervals between the seconds."
In music a time point or timepoint (point in time) is "an instant, analogous to a geometrical point in space". Because it has no duration, it literally cannot be heard, but it may be used to represent "the point of initiation of a single pitch, the repetition of a pitch, or a pitch simultaneity", therefore the beginning of a sound, rather than its duration. It may also designate the release of a note or the point within a note at which something changes (such as dynamic level). Other terms often used in music theory and analysis are attack point and starting point. Milton Babbitt calls the distance from one time point, attack, or starting point to the next a time-point interval, independent of the durations of the sounding notes which may be either shorter than the time-point interval (resulting in a silence before the next time point), or longer (resulting in overlapping notes). Charles Wuorinen shortens this expression to just time interval. Other writers use the terms attack interval, or (translating the German "Einsatzabstand"), interval of entry, interval of entrance, or starting interval.
In musical set theory, an interval class, also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod 6, is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 . See modular arithmetic for more on modulo 12. The largest interval class is 6 since any greater interval n may be reduced to 12 − n.
In modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C♭ or B♯ and C. The two notes may more often be described as Enharmonic equivalents. More specifically, in other tunings and repertoires from Western culture, a diminished second is the minute pitch interval produced by narrowing a minor second, or diatonic semitone, by a chromatic semitone. It is therefore the difference between the diatonic and chromatic semitones. For instance, the interval from B to C is a diatonic semitone, the interval from B to B♯ is a chromatic semitone, and their difference, the interval from B♯ to C is a diminished second. Being diminished, it is considered a dissonant interval. The diminished second can be also viewed as a comma, the minute interval between two enharmonically equivalent notes tuned in a slightly different way. This makes it a highly variable quantity between tuning systems. Hence for example C♯ is narrower than D♭ by a diminished second interval, however large or small that may happen to be.
The serial interval, in the epidemiology of communicable diseases, refers to the time between successive cases in a chain of transmission. The serial interval is generally estimated from the interval between clinical onsets, in which case it is the 'clinical onset serial interval' when these quantities are observable. It could, in principle, be estimated by the time interval between infection and subsequent transmission. If it is assumed that infections occur at random during the infectious period, then the average serial interval is the sum of the average latent period and half the average infectious period. Serial intervals can vary widely, and may be lifelong for some diseases. The serial interval for SARs was 7 days. Related but distinct quantities include : the 'average transmission interval' sum of average latent and infectious period; the 'incubation period' between infection and disease onset; the 'latent period' between infection and infectiousness.
In classical music from Western culture, an augmented sixth is an interval produced by widening a major sixth by a chromatic semitone. For instance, the interval from C to A is a major sixth, nine semitones wide, and both the intervals from C♭ to A, and from C to A♯ are augmented sixths, spanning ten semitones. Being augmented, it is considered a dissonant interval. Its inversion is the diminished third, and its enharmonic equivalent is the minor seventh. In septimal meantone temperament, it is specifically equivalent to the harmonic seventh. In the tuning system known as equal temperament the augmented sixth is equal to ten semitones and is a dissonant interval. The augmented sixth is relatively rare. Its most common occurrence is built on the lowered submediant of the prevailing key, in which position the interval assumes a natural tendency to resolve by expanding to an octave built on the dominant tonal degree. In its most common and expected resolution, the lower note of the interval moves downwards by a minor second to the dominant while the upper note, being chromatically inflected, is heard as the leading note of the dominant key, rising naturally by a minor second. It is the strong tendency to resolve in this way that properly identifies this interval as being an augmented sixth rather than its more common enharmonic equivalent: the minor seventh, which has a tendency to resolve inwardly.
In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members. In the diatonic collection the generic interval is one less than the corresponding diatonic interval: ⁕Adjacent intervals, seconds, are 1 ⁕Thirds = 2 ⁕Fourths = 3 ⁕Fifths = 4 ⁕Sixths = 5 ⁕Sevenths = 6 The largest generic interval in the diatonic scale being 7-1 = 6. Myhill's property is the quality of musical scales or collections with exactly two specific intervals for every generic interval. In other words, each generic interval can be made from one of two possible different specific intervals.
In statistical inference, specifically predictive inference, a prediction interval is an estimate of an interval in which a future observation will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis. Prediction intervals are used in both frequentist statistics and Bayesian statistics: a prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter: prediction intervals predict the distribution of individual future points, whereas confidence intervals and credible intervals of parameters predict the distribution of estimates of the true population mean or other quantity of interest that cannot be observed.
In statistics, interval estimation is the use of sample data to calculate an interval of possible values of an unknown population parameter; this is in contrast to point estimation, which gives a single value. Jerzy Neyman (1937) identified interval estimation ("estimation by interval") as distinct from point estimation ("estimation by unique estimate"). In doing so, he recognized that then-recent work quoting results in the form of an estimate plus-or-minus a standard deviation indicated that interval estimation was actually the problem statisticians really had in mind.
In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I. In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take:, and. However, the notation I is most commonly reserved for the closed interval [0,1].
The lunitidal interval, measures the time lag from the moon passing overhead, to the next high or low tide. It is also called the high water interval Tides are known to be mainly caused by the moon's gravity. Theoretically, peak tidal forces at a given location occur when the moon is at the meridian, but there is usually a delay before high tide that depends largely on the shape of the coastline, and the sea floor, therefore, the lunitidal interval varies from place to place. The lunitidal interval further varies within about +/- 30 minutes according to the lunar phase. The approximate lunitidal interval can be calculated if the moon-rise, moon-set and high tide times are known for a location. In the northern hemisphere, the moon is at its highest point when it is southernmost in the sky. Lunar data are available from printed tables and online. Tide tables tell the time of the next high water. The difference between these two times is the lunitidal interval. This value can be used to calibrate certain clocks and wristwatches to allow for simple but crude tidal predictions.