Correlation
1.In the statistical lexicon, the word correlation is ud to describe a linear statistical
relationship between two random variables. The phra ‘linear statistical’ indicates that the mean of one of the random variables is linearly dependent upon the random component of the other. The stronger the linear relationship, the stronger the correlation. A correlation coefficient of +1 (-1) indicates a pair of variables that vary together precily, one variable being related to the other by means of a positive (negative) scaling factor.
2.While this concept ems to be intuitively simple, it does warrant scrutiny. For
example, consider a satellite instrument that makes radiance obrvations in two different frequency bands. Suppo that the radiometers have been designed in such
a way that instrumental error in one channel is independent of that in the other. This
形容搞笑的成语means that knowledge of the noi in one channel provides no information about that in the other. However, suppo also that the radiometers drift (go out of calibration) together as they age becau both share the same physical environment, share the same power supply and are expod to the sam
e physical abu（机械损伤）.
Reasonable models for the total error as a function of time in the two radiometer channels might be:
e1t =α1(t- t0)+ε1t ;
e2t =α2(t- t0)+ε2t ;
where t0 is the launch time of the satellite and α1 and α2 are fixed constants describing the rates of drift of the two radiometers. The instrumental errors, ε1t and ε2t , are statistically independent of each other, implying that the correlation between the two, ρ(ε1t and ε2t), is zero. Conquently the total errors, e1t and e2t are also statistically independent even though they share a common systematic component. However, simple estimates of correlation between e1t and e2t that do not account for the deterministic drift will suggest that the two quantities are correlated.
3.Correlations manifest themlves in veral different ways in obrved and simulated
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climates. Several adjectives are ud to describe correlations depending upon whether they describe relationships in time (rial correlation, lagged correlation), space (spatial correlation, tele-connection), or between different climate variables (cross-correlation).
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4. A good example of rial correlation is the monthly Southern Oscillation Index
黄芪建中汤的功效与作用
(SOI), which is defined as the anomalous monthly mean pressure difference between Darwin (Australia) and Papeete (Tahiti) (Figure 1.2).
关于端午节的习俗5.The time ries is basically stationary, although variability during the first 30 years
ems to be somewhat weaker than that of late. Despite the noisy nature of the time ries, there is a distinct tendency for the SOI to remain positive or negative for extended periods, some of which ar
e indicated in Figure 1.2. This persistence in the sign of the index reflects the rial correlation of the SOI.
海安特产6. A quantitative measure of the rial correlation is the auto-correlation function, ρSOI (t;
t+Δ), shown in Figure 1.3, which measures the similarity of the SOI at any time difference Δ. The autocorrelation is greater than 0.2 for lags up to about six months and varies smoothly around zero with typical magnitudes between 0.05 and 0.1 for lags greater than about a year. This tendency of estimated auto-correlation functions not to converge to zero at large lags, even though the real auto-correlation is zero at long lags, is a natural conquence of the uncertainty due to finite samples.
7. A good example of a cross-correlation is the relationship that exists between the SOI
有关历史的英文and various alternative indices of the Southern Oscillation. The characteristic low-frequency variations in Figure 1.2 are also prent in area averaged Central Pacific a-surface temperature (Figure 1.4). The correlation between the two time ries displayed in Figure 1.4 is 0.67.
8.Pattern analysis techniques, such as Empirical Orthogonal Function analysis,
人才管理Canonical Correlation Analysis（典型相关分析）and Principal Oscillation Patterns （主振荡型）, rely upon the assumption that the fields under study are spatially correlated. The Southern Oscillation Index (Figure 1.2) is a manifestation of the negative correlation between surface pressure at Papeete and that at Darwin. Variables such as pressure, height, wind, temperature, and specific humidity vary smoothly in the free atmosphere and conquently exhibit strong spatial interdependence（相互依赖）. This correlation is prent in each weather map (Figure 1.5, left). Indeed, without this feature, routine weather forecasts would be all but impossible given the sparness of the global obrving network as it exists even today. Variables derived from moisture, such as cloud cover, rainfall and snow amounts, and variables associated with land surface process tend to have much smaller spatial scales (Figure 1.5, right), and also tend not to have nor
mal distributions. While mean a-level pressure (Figure 1.5, left) will be more or less constant on spatial scales of tens of kilometres, we may often travel in and out of localized rain showers in just a few kilometres. This dichotomy is illustrated in Figure 1.5, where we e a cold front over Ontario (Canada). The left panel, which displays mean a-level pressure, shows the front as a smooth curve. The right panel displays a radar image of precipitation occurring in southern Ontario as the front pass through the region.
