No abstract
In the 1940s, Karhunen and Loève proposed a method for processing a one-dimensional numeric time series by converting it into multidimensional by shifts. In fact, a one-dimensional number series was decomposed into several orthogonal time series. This method has many times been independently developed and applied in practice under various names (EOF, SSA, Caterpillar, etc.). Nowadays, the name ‘SSA’ (Singular Spectral Analysis) is the most often used. It turned out that it is universal, applicable to any time series without requiring stationary assumptions, automatically decomposes time series into a trend, cyclic components and noise. By the beginning of the 1980s, Takens had shown that for a dynamical system such a method makes it possible to obtain an attractor from observing only one of these variables, thereby bringing the method to a powerful theoretical basis. In the same years, the practical benefits of phase portraits became clear. In particular, it was used in the analysis and forecast of animal abundance dynamics. In this paper we propose to extend SSA to a one-dimensional sequence of any type of elements, including numbers, symbols, figures, etc., and, as a special case, to a molecular sequence. Technically, the problem is solved using an algorithm like SSA. The sequence is cut by a sliding window into fragments of a given length. Between all fragments, the matrix of Euclidean distances is calculated. This is always possible. For example, the square root of the Hamming distance between fragments is a Euclidean distance. For the resulting matrix, the principal components are calculated by the principal-coordinate method (PCo). Instead of a distance matrix, one can use a matrix of any similarity/dissimilarity indexes and apply methods of multidimensional scaling (MDS). The result will always be PCs in some Euclidean space. We called this method ‘PCA-Seq’. It is certainly an exploratory method, as is its particular case SSA. For any sequence, in cluding molecular, PCA-Seq without any additional assumptions allows presenting its principal components in a numerical form and visualizing them in the form of phase portraits. A long history of SSA application for numerical data gives all reason to believe that PCA-Seq will be not less useful in the analysis of non-numerical data, especially in hypothesizing. PCA-Seq is implemented in the freely distributed Jacobi 4 package (http://jacobi4.ru/).
5 2 Институт систематики и экологии животных СО РАН, Новосибирск, Россия 6 3 Новосибирский государственный университет, Новосибирск, Россия 7 4 Томский государственный университет, Томск, Россия 8 5 Московский физико-технический институт (государственный университет), 9 Москва, Россия 10 * efimov@bionet.nsc.ru 11 12 В 40-х годах прошлого столетия Карунен и Лоев предложили метод обработки 13 одномерного числового временного ряда через его преобразование в 14 многомерный путем сдвига несколько раз подряд и разложения на несколько 15 ортогональных временных рядов методом главных компонент (PCA). Метод 16 много раз независимо возникал и применялся на практике под различными 17 названиями (EOF, SSA, Гусеница и т.д.). Оказалось, что он является 18 универсальным, применим к любому временному ряду, не требуя предположения 19 стационарности автоматически разлагает его на тренд, циклические 20 составляющие и шум. В наши дни чаще всего используется название SSA 21 (Сингулярный Спектральный Анализ). В начале 80-х годов Такенс показал, что 22 для динамической системы сдвиги только одной наблюдаемой переменной 23 позволяют построить аттрактор всей системы и, тем самым, подвел под SSA 24 мощную теоретическую базу. В те же годы выяснилась практическая польза 25 фазовых портретов. В частности, это было использовано при анализе и прогнозе 26 динамики численности животных. 27 В настоящей работе предлагается распространить SSA на одномерную 28 последовательность элементов любого типа, включая числа, символы, фигуры и 29 т.д., и, в качестве частного случая, на молекулярную последовательность. 30 Технически проблема решается практически тем же алгоритмом, что и SSA. 31 Последовательность режется скользящим окном на фрагменты заданной длины. 32 2 Между всеми фрагментами вычисляется матрица евклидовых расстояний. Это 33 всегда возможно. Например, квадратный корень из p-дистанции (дистанции 34 Хэмминга) является евклидовым расстоянием. Для полученной матрицы методом 35 главных координат (PCo) вычисляются главные компоненты. 36 Вместо расстояний можно использовать любые индексы сходства/различия и 37 применить методы многомерного шкалирования (MDS). В итоге все равно 38 получатся главные компоненты в некотором евклидовом пространстве.39 Мы назвали этот метод PCA-Seq. Он, безусловно, является разведочным методом, 40 как и его частный случай SSA. Для любой последовательности, в том числе 41 молекулярной, PCA-Seq без всяких дополнительных предположений позволяет 42 получить ее главные компоненты в числовом виде и визуализировать их в виде 43 графиков и фазовых портретов. Многолетний опыт применения SSA для 44 числовых данных дает все основания полагать, что PCA-Seq окажется не менее 45 полезным при анализе нечисловых данных, особенно при выдвижении гипотез. 46 PCA-Seq реализован в свободно распространяемом пакете Jacobi4 47 62 63In the 40s of the last century, Karhunen and Loève proposed a method for processing of 64 one-dimensional numeric time series by converting it into a multidimensional by shifts. 65In fact, a one-dimensional number se...
In addition to probabilities of monetary gains and losses, personality traits, socio-economic factors, and specific contexts such as emotions and framing influence financial risk taking. Here, we investigated the effects of joyful, neutral, and sad mood states on participants’ risk-taking behaviour in a simple task with safe and risky options. We also analysed the effect of framing on risk taking. In different trials, a safe option was framed in terms of either financial gains or losses. Moreover, we investigated the effects of emotional contagion and sensation-seeking personality traits on risk taking in this task. We did not observe a significant effect of induced moods on risk taking. Sad mood resulted in a slight non-significant trend of risk aversion compared to a neutral mood. Our results partially replicate previous findings regarding the presence of the framing effect. As a novel finding, we observed that participants with a low emotional contagion score demonstrated increased risk aversion during a sad mood and a similar trend at the edge of significance was present in high sensation seekers. Overall, our results highlight the importance of taking into account personality traits of experimental participants in financial risk-taking studies.
When analyzing a 1D time series, it is traditional to represent it as the sum of the trend, cyclical components and noise. The trend is seen as an external influence. However, the impact can be not only additive, but also multiplicative. In this case, not only the level changes, but also the amplitude of the cyclic components. In the PCA-Seq method, a generalization of SSA, it is possible to pre-standardize fragments of a time series to solve this problem. The algorithm is applied to the Anderson series – a sign alternating version of the well-known Wolf series, reflecting the 22-year Hale cycle. The existence of this cycle is not disputed at high solar activity, but there are doubts about the constancy of its period at this time, as well as its existence during the epoch of low solar activity. The processing of the series by the PCA-Seq method revealed clear oscillations fluctuations of almost constant amplitude with an average period of 21.9 years, and it was found that the correlation of these oscillations with the time axis for 300 years does not differ significantly from zero. This confirms the hypothesis of the existence of 22-year oscillations in solar activity even at its minima, like the Maunder minimum.
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