Time series analysis is a statistical technique that deals with time series data, or trend analysis. Time series data means that data is in a series of particular time periods or intervals. The data is considered in three types: Time series data: A set of observations on the values that a variable takes at different times. Cross-sectional data: Data of one or more variables, collected at the same point in time. Pooled data: A combination of time series data and cross-sectional data. Terms and concepts: Dependence: Dependence refers to the association of two observations with the same variable, at prior time points. Stationarity: Shows the mean value of the series that remains constant over a time period; if past effects accumulate and the values increase toward infinity, then stationarity is not met. Differencing: Used to make the series stationary, to De-trend, and to control the auto-correlations; however, some time series analyses do not require differencing and over-differenced series can produce inaccurate estimates. Specification: May involve the testing of the linear or non-linear relationships of dependent variables by using models such as ARIMA, ARCH, GARCH, VAR, Co-integration, etc. Exponential smoothing in time series analysis: This method predicts the one next period value based on the past and current value. It involves averaging of data such that the nonsystematic components of each individual case or observation cancel out each other The exponential smoothing method is used to predict the short term predication. Alpha, Gamma, Phi, and Delta are the parameters that estimate the effect of the time series data. Alpha is used when seasonality is not present in data. Gamma is used when a series has a trend in data. Delta is used when seasonality cycles are present in data. A model is applied according to the pattern of the data. Curve fitting in time series analysis: Curve fitting regression is used when data is in a non-linear relationship. The following equation shows the non-linear behavior:

Time Series Analysis

Climate science involves various functional relations and needs to investigate the dominance or relative importance of the variables in their relation. In our previous studies, we examined several problems in which causal relations are established, showing how the dependent quantity is affected by two or more independent variables. With linear fitting, the relative contributions of the variables to the variation of the quantity are compared. In this study, we examine constraint relation, which is a simultaneous multivariate relation, with all variables in the relation being equal in position. The relation can generally be nonlinear. To be convenient for examining the dominance, plane equation fitting can be used to linearize the relation. The equation of state for ideal dry air is investigated as a simple case of the relation. For this special case, a linearized relation can be obtained from both the fitting and the derivation. The scale analysis tool used in dynamic meteorology is applied here for the dominance analysis. Through comparing the scales of the terms, we can simplify the equation. The simplified relations correspond, respectively, to Charle’s law, Boyle’s law, and Gay-Lussac’s law. The geographical preferences of the different dominance patterns are exhibited. In addition, when considering the change of the variable that is smallest in scale, we can identify which factor is the dominator. The ideal gas law is intentionally chosen as the example, since the relation is simple in form, and the results of dominance can be deduced analytically. A comparison demonstrates that the methods used here for the dominance analysis are reliable.

/pdf/dominance-patterns-specified-by-the-ideal-gas-equation-65lkp9nd.pdf

Dominance Patterns Specified by the Ideal Gas Equation: Example of Examining Simultaneous Multivariate Relation with Scale Analysis Approach

Eighty subjects estimated the correlation coeffi cient, r, for each of 13 computer-printed scatter plots. Making judgments were 46 students in a graduate-level statistics course and 34 faculty and graduate students in a department of psychology. The actual correlation values ranged from .010 to .995, with 200 observations in each scatterplot and with the order of scatterplot presentation ran domized. As predicted, subjects underestimated the degree of actual correlation. Also as predicted, but with substantial moderation by a method-of-presen tation factor, this underestimation was most pro nounced in the middle of the correlational range—between the 0 and 1 extremes. Though per ception of correlation was shown not to be veridical (i.e., in terms of r), little support was given one al ternative view—its being in terms of r 2.

/pdf/underestimating-correlation-from-scatterplots-47bu4bjbun.pdf

Underestimating Correlation from Scatterplots

The misleading character of the correlation coefficient was investigated in two studies of intuitive statistical behaviour: subjective estimation of partial correlation and subjective estimation of the minimum possible correlation between two variables given their equal correlation with a third. Fourteen Psychology Department faculty and graduate students provided estimates of partial correlations, r[SUBxy.z'] for varying positive values of rxy and rxy = ryz. Mean estimated rxy.z values were uniformly lower than theoretical (actual) values. Twenty-four statistics students and sixteen Statistics Department faculty and graduate students furnished estimates of minimum possible correlations, min rxy' for varying positive values of rxz = ryz. For both samples mean estimated min rxy was higher than the theoretical (actual) min rxy. Thus, in both studies, subjective overestimated the degree of influence on the correlation between two variables, X and Y, of a third, constraining variable, Z. The Hazards of intuitive statistical inference in the interpretation of correlational data are are discussed.

On The Nonintuitive Nature Of The Correlation Coefficient: Subjective Estimation Of Three-Variable Relations.

The tendency for variables to change values in the same direction from one sampling occasion to the next is termed directional correlation. A multiple di- rectional coefficient, D, is described that is a generalization to k variables of the bivariate directional coefficient, d. The similarities of D and Kendall's coefficient of agreement, u, are discussed and their time-series differences illustrated.

A generalized directional coefficient for multiple time-series analysis

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On The Nonintuitive Nature Of The Correlation Coefficient: Subjective Estimation Of Three-Variable Relations.

A generalized directional coefficient for multiple time-series analysis