每日彙整: 2017-02-12

樹莓派、樹莓派之學習、樹莓派之教育

時間序列︰粉紅 = 白色 + 紅色？

何謂『隨機性』？維基百科中文詞條這麼講︰

隨機性（英語：Randomness）這個詞是用來表達目的、動機、規則或一些非科學用法的可預測性的缺失。

一個隨機的過程是一個不定因子不斷產生的重複過程，但它可能遵循某個機率分布。

術語隨機經常用於統計學中，表示一些定義清晰的、徹底的統計學屬性，例如缺失偏差或者相關。隨機與任意不同，因為「一個變量是隨機的」表示這個變量遵循機率分布。而任意在另一方面又暗示了變量沒有遵循可限定機率分布。

隨機在自然科學和哲學上有著重要的地位。

 

因其不清不楚，特此引用英文詞條作為補充︰

Randomness is the lack of pattern or predictability in events.[1] A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or “trials”) is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.

The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.

Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science.[2] By analogy, quasi-Monte Carlo methods use quasirandom number generators.

Random selection is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random.

 

切莫認為『隨機』是個簡單概念！你將如何『判定』某個時間序列不是

隨機數生成器（Random number generator）是通過一些算法、物理訊號、環境噪音等來產生看起來似乎沒有關聯性的數列的方法或裝置。丟硬幣、丟骰子、洗牌就是生活上常見的隨機數產生方式。

大部分計算機上的偽隨機數，並不是真正的隨機數，只是重複的周期比較大的數列，是按一定的算法和種子值生成的。

 

造成的呢？更別說先‧後驗『機率分佈』恐有爭論乎！！？？

近代西方傳統中用『經驗』的『先後』區分了兩類『知識』，一種是『先於經驗』，無需經驗就有的『先驗知識』；另一種是『後於經驗』，源自某種經驗才有的『後驗知識』。 理性主意者通常相信先驗知識的存在，而經驗主義者認為即使存在著先驗知識，相對於眾多的後驗知識來講它也並不重要。在機率的世界理也用著這樣的術語。舉個 例說，一個骰子我們推論它是『公正的』，所以每個面的『先驗機率』是 \frac{1}{6}；一個公平的硬幣，正反兩面的機率相等是 \frac{1}{2}。而『後驗機率』是一種將相關的『證據』或者『背景』考慮後才給定的『條件機率』。根據條件機率的定義，在事件 C 發生的條件下事件 A 發生的機率是︰

P(A|C)=\frac{P(A \cap C)}{P(C)}。

比方說擲兩顆骰子，在點數和為 6 的條件下，其中有一顆骰子是 2 點的機率為 \frac{\frac{2}{36}}{\frac{5}{36}} = \frac{2}{5}。

一九零零年英國倫敦大學的 Arnold Zuboff 教授發表了一篇寫於一九八六年的『One Self: The Logic of Experience』的論文，提出了『睡美人的問題』。

280px-Brewtnall_-_Sleeping_Beauty

220px-Dornröschen

睡美人被詳細告知細節，自願參加下面的實驗︰

周日她將被安排入睡，實驗過程中會被喚醒一次或者兩次，然後用一種失憶的藥，她將不會記得自己曾經被喚醒過。這個實驗中會擲一個公平的硬幣來決定它將採取的程序︰

如果硬幣的結果是『頭』，她只會在『禮拜一』被喚醒與訪談。
如果硬幣的結果是『尾』，她將會在『禮拜一』和『禮拜二』都被喚醒與訪談。

無論是上面哪種情況，她終會在『禮拜三』被喚醒，而且沒有訪談就結束了實驗。每次她被喚醒與訪談時，她將被問到︰你現在對『硬幣的結果是頭』的『相信度』是什麼？

這個問題至今爭論不休，『三分之一者』 Thirder 認為是 \frac{1}{3}，『二分之一者』 Halfer 認為是 \frac{1}{2}。睡美人真的能有一個『正確答案』嗎？一個只擲一次頭尾兩種結果的硬幣，帶出可能一天或兩天的訪談，將要如何思考『機率』的先驗或後驗說法的呢？一般機率論是用『各種可能出現之狀況』 ── 樣本空間 ── 的『相對發生頻率』來作測度；如果不能測度時，或許用著『無差別』或說『無法區分』去假設它們相對發生頻率都『一樣』。這樣『樣本空間』與『測度假設』就是爭論的緣由的了。假使我們用硬幣結果集合 {頭，尾} 與訪談時間集合 {禮拜一，禮拜二}，從公平硬幣角度來看這個問題中的事件機率︰

機率【頭，禮拜一】= \frac{1}{2}
機率【頭，禮拜二】= 0
機率【尾，禮拜一】= \frac{1}{4}
機率【尾，禮拜二】= \frac{1}{4}

這個『機率【頭，禮拜二】= 0』就是引發爭論的主焦點，因為它是一個『不可能』發生的事件。從機率的經驗事件取樣之觀點來看，也許在考慮『樣本空間』時根本該將之去除，然而這樣的一個『觀察者』又為什麼不該假設『所有可能發生事件』的『機率』不是相同的 \frac{1}{3} 呢？？

─── 摘自《改不改？？變不變！！》

 

所以科學家喜歡與大自然的『雜訊』── 真的『隨機』── 打交道 ，並且醉心的賦予了它『顏色』耶？？！！

雜訊的顏色

雜訊雖作為一個隨機訊號，仍然具有統計學上的特徵屬性。功率譜密度（功率的頻譜分布）即是雜訊的特徵之一，從而人們可以通過它來區分不同類型的雜訊。在一些雜訊扮演著重要角色的研究領域中（例如聲學、電子工程和物理），這種雜訊分類方法通常會給予不同的功率譜密度一個不同的「色彩」稱謂，也就是說不同種類的雜訊會被命名為不同的顏色。但是在不同的專業領域間，或許會有不同的術語稱謂。

冪律雜訊

在雜訊的顏色分類中，很多定義都假設了雜訊訊號在全頻域都有分布，並且在單位頻域內的譜密度正比於  1/f^{{\beta }}\,，因此它們都屬於冪律雜訊（Power-law noise）。例如白雜訊的譜密度函數是平坦的，它具有  \beta =0\,，而閃爍雜訊或粉紅雜訊  \beta =1\,，紅雜訊  \beta =2\,。

白雜訊

白雜訊的名稱來自白光，表示在全頻域內單位頻域下都分布有相同的能量密度[1][2]，在線性空間內它具有平坦的頻譜。

換句話說，一定頻域內的白雜訊在其中任意給定的帶寬內都具有相等的功率（功率譜密度的定義）。例如在40赫茲至60赫茲頻域內的白雜訊具有和4000赫茲至4020赫茲頻域內相同的功率。

需要注意的是，具有無限長帶寬的白雜訊只是一個理論上的概念，因為在任意頻率上都存在相等的功率會導致最終的雜訊總功率為無窮大。在實際應用中的白雜訊是指在某一特定頻域內的譜密度函數是平坦的雜訊。

白雜訊功率譜

White noise

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density.[1] The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustic engineering, telecommunications, statistical forecasting, and many more. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal.

 A “white noise” image

In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance; a single realization of white noise is a random shock. Depending on the context, one may also require that the samples be independent and have identical probability distribution (in other words i.i.d. is the simplest representative of the white noise).[2] In particular, if each sample has a normal distribution with zero mean, the signal is said to be Gaussian white noise.[3]

The samples of a white noise signal may be sequential in time, or arranged along one or more spatial dimensions. In digital image processing, the pixels of a white noise image are typically arranged in a rectangular grid, and are assumed to be independent random variables with uniform probability distribution over some interval. The concept can be defined also for signals spread over more complicated domains, such as a sphere or a torus.

An infinite-bandwidth white noise signal is a purely theoretical construction. The bandwidth of white noise is limited in practice by the mechanism of noise generation, by the transmission medium and by finite observation capabilities. Thus, a random signal is considered “white noise” if it is observed to have a flat spectrum over the range of frequencies that is relevant to the context. For an audio signal, for example, the relevant range is the band of audible sound frequencies, between 20 and 20,000 Hz. Such a signal is heard as a hissing sound, resembling the /sh/ sound in “ash”. In music and acoustics, the term “white noise” may be used for any signal that has a similar hissing sound.

White noise draws its name from white light,[4] although light that appears white generally does not have a flat spectral power density over the visible band.

The term white noise is sometimes used in the context of phylogenetically based statistical methods to refer to a lack of phylogenetic pattern in comparative data.[5] It is sometimes used in non technical contexts, in the metaphoric sense of “random talk without meaningful contents”.[6][7]

Statistical properties

Being uncorrelated in time does not restrict the values a signal can take. Any distribution of values is possible (although it must have zero DC component). Even a binary signal which can only take on the values 1 or –1 will be white if the sequence is statistically uncorrelated. Noise having a continuous distribution, such as a normal distribution, can of course be white.

It is often incorrectly assumed that Gaussian noise (i.e., noise with a Gaussian amplitude distribution – see normal distribution) necessarily refers to white noise, yet neither property implies the other. Gaussianity refers to the probability distribution with respect to the value, in this context the probability of the signal falling within any particular range of amplitudes, while the term ‘white’ refers to the way the signal power is distributed (i.e., independently) over time or among frequencies.

We can therefore find Gaussian white noise, but also Poisson, Cauchy, etc. white noises. Thus, the two words “Gaussian” and “white” are often both specified in mathematical models of systems. Gaussian white noise is a good approximation of many real-world situations and generates mathematically tractable models. These models are used so frequently that the term additive white Gaussian noise has a standard abbreviation: AWGN.

White noise is the generalized mean-square derivative of the Wiener process or Brownian motion.

A generalization to random elements on infinite dimensional spaces, such as random fields, is the white noise measure.

 

粉紅雜訊

粉紅雜訊又稱作1/f雜訊，它的頻譜在對數空間內是平坦的，也就是說在等比例寬度的頻帶內具有相等的功率[3][2]。 例如在40赫茲至60赫茲的區間內，粉紅雜訊具有和它在4千赫茲至6千赫茲頻帶內相等的功率。由於人類對聲音的聽覺與聲波頻率的比例有關：在成比例的頻率 區間內人類聽力所感受到的能量是一樣的，而與頻率的絕對高低無關（在距離和持續時間相同的情形下，40-60赫茲與4000-6000赫茲對人類聽覺來說 沒有差別。）；如此在所有雙倍的頻率區間內人類聽覺都感受到相同的能量，從而在電聲工程中粉紅雜訊經常被用作一種參考訊號，這樣人類的聽覺系統在所有的頻率上所接收到的聲音振幅都是近似相等的。粉紅雜訊和白雜訊在頻譜上的區別是，頻率提高為2倍時，它的譜密度都會降低3dB。基於這個原因，粉紅雜訊的譜密度是隨頻率增加而呈1/f衰減的，因而經常被稱作1/f雜訊。

由於在對數座標下的頻帶在頻譜的低頻端（直流） 和高頻端都可以有無限多個，任何具有有限能量的頻譜在低頻段和高頻端所具有的能量都不能高於粉紅雜訊。粉紅雜訊是僅此一種具有這種性質的冪律雜訊，因為比 它更陡的冪律雜訊在低頻端經過積分後功率將變為無窮大，而比它更平坦的冪律雜訊在高頻端經過積分後功率也將變為無窮大。

粉紅雜訊的頻譜

Pink noise

Pink noise or  1⁄f noise is a signal or process with a frequency spectrum such that the power spectral density (energy or power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave (halving/doubling in frequency) carries an equal amount of noise energy. The name arises from the pink appearance of visible light with this power spectrum.[1]

Within the scientific literature the term pink noise is sometimes used a little more loosely to refer to any noise with a power spectral density of the form

  {\displaystyle S(f)\propto {\frac {1}{f^{\alpha }}},}

where f is frequency, and 0 < α < 2, with exponent α usually close to 1. These pink-like noises occur widely in nature and are a source of considerable interest in many fields. The distinction between the noises with α near 1 and those with a broad range of α approximately corresponds to a much more basic distinction. The former (narrow sense) generally come from condensed-matter systems in quasi-equilibrium, as discussed below.[2] The latter (broader sense) generally correspond to a wide range of non-equilibrium driven dynamical systems.

The term flicker noise is sometimes used to refer to pink noise, although this is more properly applied only to its occurrence in electronic devices. Mandelbrot and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to emphasize that the exponent of the power spectrum could take non-integer values and be closely related to fractional Brownian motion, but the term is very rarely used.

Description

There is equal energy in all octaves (or similar log bundles) of frequency. In terms of power at a constant bandwidth, pink noise falls off at 3 dB per octave. At high enough frequencies pink noise is never dominant. (White noise has equal energy per frequency interval.)

The human auditory system, which processes frequencies in a roughly logarithmic fashion approximated by the Bark scale, does not perceive different frequencies with equal sensitivity; signals around 1–4 kHz sound loudest for a given intensity. However, humans still differentiate between white noise and pink noise with ease.

Graphic equalizers also divide signals into bands logarithmically and report power by octaves; audio engineers put pink noise through a system to test whether it has a flat frequency response in the spectrum of interest. Systems that do not have a flat response can be equalized by creating an inverse filter using a graphic equalizer. Because pink noise has a tendency to occur in natural physical systems, it is often useful in audio production. Pink noise can be processed, filtered, and/or effects can be added to produce desired sounds. Pink-noise generators are commercially available.

One parameter of noise, the peak versus average energy contents, or crest factor, is important for testing purposes, such as for audio power amplifier and loudspeaker capabilities because the signal power is a direct function of the crest factor. Various crest factors of pink noise can be used in simulations of various levels of dynamic range compression in music signals. On some digital pink-noise generators the crest factor can be specified.

Origin

There are many theories of the origin of pink noise. Some theories attempt to be universal, while others are applicable to only a certain type of material, such as semiconductors. Universal theories of pink noise remain a matter of current research interest.

A hypothesis (referred to as the Tweedie hypothesis) has been proposed to explain the genesis of pink noise on the basis of a mathematical convergence theorem related to the central limit theorem of statistics.[24] The Tweedie convergence theorem[25] describes the convergence of certain statistical processes towards a family of statistical models known as the Tweedie distributions. These distributions are characterized by a variance to mean power law, that have been variously identified in the ecological literature as Taylor’s law[26] and in the physics literature as fluctuation scaling.[27] When this variance to mean power law is demonstrated by the method of expanding enumerative bins this implies the presence of pink noise, and vice versa.[24] Both of these effects can be shown to be the consequence of mathematical convergence such as how certain kinds of data will converge towards the normal distribution under the central limit theorem. This hypothesis also provides for an alternative paradigm to explain power law manifestations that have been attributed to self-organized criticality.[28]

 

紅雜訊

紅雜訊又稱作布朗雜訊，它與粉紅雜訊類似，但當頻率提高為2倍時，它的譜密度都會降低6dB，也就是說紅雜訊的譜密度是隨頻率增加而呈  1/f^{2}\,衰減的[2]。需要注意這種雜訊的頻域不能包括直流（即頻率為零），否則經過積分後得到的功率將為無窮大。這種雜訊也可以通過對布朗運動進行算法後得到，因此它在英文中雖然有時被稱作Brown noise，在這裡Brown是布朗運動（Brownian motion）的簡稱，而不應理解為「棕」這種顏色。用顏色表示時它被稱作紅雜訊，這是因為  1/f^{2}\,和 1/f^{0}\,（平坦）之間，而粉紅則介於紅色與白色之間。它有時也被稱作「隨機行走」雜訊或「醉漢行走」雜訊。

紅雜訊的頻譜

Brownian noise

In science, Brownian noise (About this sound 

      1. Sample
 (help·
      2. info
)), also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random-walk noise. The term “Brown noise” comes not from the color, but after Robert Brown, the discoverer of Brownian motion. The term “red noise” comes from the “white noise”/”white light” analogy; red noise is strong in longer wavelengths, similar to the red end of the visible spectrum.

Explanation

The graphic representation of the sound signal mimics a Brownian pattern. Its spectral density is inversely proportional to f 2, meaning it has more energy at lower frequencies, even more so than pink noise. It decreases in power by 6 dB per octave (20 dB per decade) and, when heard, has a “damped” or “soft” quality compared to white and pink noise. The sound is a low roar resembling a waterfall or heavy rainfall. See also violet noise, which is a 6 dB increase per octave.

Strictly, Brownian motion has a Gaussian probability distribution, but “red noise” could apply to any signal with the 1/f 2 frequency spectrum.