Ttl Mode

[Vinnie Frevert]

Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.

The mode is not necessarily unique in a given discrete distribution since the probability mass function may take the same maximum value at several points x1, x2, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.

A mode of a continuous probability distribution is often considered to be any value x at which its probability density function has a locally maximum value.[2] When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution, so any peak is a mode. Such a continuous distribution is called multimodal (as opposed to unimodal).

In symmetric unimodal distributions, such as the normal distribution, the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode.

The mode of a sample is the element that occurs most often in the collection. For example, the mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6. Given the list of data [1, 1, 2, 4, 4] its mode is not unique. A dataset, in such a case, is said to be bimodal, while a set with more than two modes may be described as multimodal.

For a sample from a continuous distribution, such as [0.935..., 1.211..., 2.430..., 3.668..., 3.874...], the concept is unusable in its raw form, since no two values will be exactly the same, so each value will occur precisely once. In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as for making a histogram, effectively replacing the values by the midpoints of theintervals they are assigned to. The mode is then the value where the histogram reaches its peak. For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval width if chosen too narrow or too wide; typically one should have a sizable fraction of the data concentrated in a relatively small number of intervals (5 to 10), while the fraction of the data falling outside these intervals is also sizable. An alternate approach is kernel density estimation, which essentially blurs point samples to produce a continuous estimate of the probability density function which can provide an estimate of the mode.

The algorithm requires as a first step to sort the sample in ascending order. It then computes the discrete derivative of the sorted list and finds the indices where this derivative is positive. Next it computes the discrete derivative of this set of indices, locating the maximum of this derivative of indices, and finally evaluates the sorted sample at the point where that maximum occurs, which corresponds to the last member of the stretch of repeated values.

Unlike mean and median, the concept of mode also makes sense for "nominal data" (i.e., not consisting of numerical values in the case of mean, or even of ordered values in the case of median). For example, taking a sample of Korean family names, one might find that "Kim" occurs more often than any other name. Then "Kim" would be the mode of the sample. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.

Unlike median, the concept of mode makes sense for any random variable assuming values from a vector space, including the real numbers (a one-dimensional vector space) and the integers (which can be considered embedded in the reals). For example, a distribution of points in the plane will typically have a mean and a mode, but the concept of median does not apply. The median makes sense when there is a linear order on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the geometric median and the centerpoint.

For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined. For a data sample it is the "halfway" value when the list of values is ordered in increasing value, where usually for a list of even length the numerical average is taken of the two values closest to "halfway". Finally, as said before, the mode is not necessarily unique. Certain pathological distributions (for example, the Cantor distribution) have no defined mode at all.[citation needed][4] For a finite data sample, the mode is one (or more) of the values in the sample.

A well-known class of distributions that can be arbitrarily skewed is given by the log-normal distribution. It is obtained by transforming a random variable X having a normal distribution into random variable Y = eX. Then the logarithm of random variable Y is normally distributed, hence the name.

Taking the mean μ of X to be 0, the median of Y will be 1, independent of the standard deviation σ of X. This is so because X has a symmetric distribution, so its median is also 0. The transformation from X to Y is monotonic, and so we find the median e0 = 1 for Y.

Pearson uses the term mode interchangeably with maximum-ordinate. In a footnote he says, "I have found it convenient to use the term mode for the abscissa corresponding to the ordinate of maximum frequency."

Freshly Restored from Original Super 8 Sources, the Compilation of GroundbreakingDepeche Mode Videos, Directed and Filmed by Anton Corbijn, includes 11 Music Videos + 6 Previously Unreleased Outtake "Vignettes"

Sony Music Entertainment will release Depeche Mode's Strange/Strange Too (a much sought-after collection of music videos directed and filmed in Super 8 by Anton Corbijn) on Friday, December 8. Strange/Strange Too will be released through SME worldwide, excluding USA, Canada and Mexico (where the title will be released through Warner Music Group).

Previously available as individual titles in VHS and Laserdisc formats (now out-of-print and highly collectible), Strange (1988) and Strange Too (1990) are compilations of the provocative and visionary short films lensed by master photographer/director Anton Corbijn, in collaboration with Depeche Mode, to create a new visual iconography for the band and their music.

Available for the first-time in DVD and Blu-ray configurations and as a single collection, Strange/Strange Too presents 11 Anton Corbijn/Depeche Mode music films, newly restored from original Super 8mm sources, alongside six previously unseen outtake "vignettes" from the DM archives. When assembling the final edits for Strange and Strange Too, Corbijn created a visual running order where the individual music videos are perceived as one continuous film, with additional interstitial content not seen in the original clips.

According to Depeche Mode: "Anton Corbijn's photography and art direction have played an indispensable part in the evolution of the Depeche Mode aesthetic. Strange and Strange Too are essential titles in both the Depeche Mode and Anton Corbijn catalogues, and are the perfect example of Anton's unique ability to capture the spirit of DM on film."

The physical nature of Super 8 film means that the stock's rough and grainy quality becomes part of the finished film's inherent aesthetic, and Corbijn's mastery of Super 8 is a key element in Strange/Strange Too and the development of Depeche Mode's visual components. The film restoration underwent a rigorous process over the course of several years with the participation of personnel involved in making the original films including Anton Corbijn. Because this Blu-Ray/DVD release was created from the original Super 8 film stock, the final result may seemingly lack the visual clarity that modern viewers associate with contemporary HD 4k reproduction. In some occasions where the original footage had deteriorated too much, the next best source was used.

"To be seen as 'strange' in a creative field is no bad thing as it probably means 'different' which I find is a very positive description of one's struggle to be just that. The idea to make this into a connected series of little films and fake interviews came late into the process of shooting these," wrote Corbijn. "I was shooting all the films myself on black/white Super 8 film. We put all this together on a shoestring budget; those were that kind of days. We have to look at these films in the light of us being young; we were experimenting and I am happy we were given that space at the time by Daniel Miller."

The Delta Machine The 12" Singles box set includes replicas of the original 12" single releases plus three newly compiled discs featuring remixes, dub mixes, instrumental versions, live sessions and radio mixes of the singles as well as the bonus track "Goodbye (Gesaffelstein Remix)."

Each box set in the series contains the singles from an individual Depeche Mode album pressed on audiophile-quality 12" vinyl. The artwork for the exterior of each of the box sets draws on iconography inspired by the original releases, while the vinyl sleeves themselves feature the original single artwork.

Delta Machine The 12" Singles is the chronological 13th volume in SME's Depeche Mode 12" Singles Series with plans to release a Spirit box set in a similar deluxe audio-grade collector's edition.

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