Linear multivariate statistical analysis in .NET Drawer data matrix barcodes in .NET Linear multivariate statistical analysis

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Linear multivariate statistical analysis use visual .net barcode data matrix integrated todisplay datamatrix 2d barcode with .net USPS PLANET Barcode number. Note that .net framework barcode data matrix for time series, a j is a function of time while e j is a function of space, hence the names temporal coef cients and spatial patterns describe them well.

However, in many cases, the dataset may not consist of time series. For instance, the dataset could be plankton collected from various oceanographic stations t then becomes the label for a station, while space here could represent the various plankton species, and the data y(t) = [y1 (t), . .

. , ym (t)]T could be the amount of species 1, . .

. , m found in station t. Another example comes from the multi-channel satellite image data, where images of the Earth s surface have been collected at several frequency channels.

Here t becomes the location label for a pixel in an image, and space indicates the various frequency channels. There are two important properties of PCAs. The expansion kj=1 a j (t)e j (x), with k m, explains more of the variance of the data than any other linear combination kj=1 b j (t)f j (x).

Thus PCA provides the most ef cient way to compress data, using k eigenvectors e j and corresponding time series a j . The second important property is that the time series in the set {a j } are uncorrelated. We can write a j (t) = eT (y y) = (y y)T e j .

j For i = j, cov(ai , a j ) = E[eiT (y y)(y y)T e j ] = eiT E[(y y)(y y)T ]e j = eiT Ce j = eiT j e j = j eiT e j = 0, (2.32) implying zero correlation between ai (t) and a j (t). Hence PCA extracts the uncorrelated modes of variability of the data eld.

Note that no correlation between ai (t) and a j (t) only means no linear relation between the two, there may still be nonlinear relations between them, which can be extracted by nonlinear PCA methods ( 10). When applying PCA to gridded data over the globe, one should take into account the decrease in the area of a grid cell with latitude. By scaling the variance from each grid cell by the area of the cell (which is proportional to the cosine of the latitude ), one can avoid having the anomalies in the higher latitudes overweighted in the PCA (North et al.

, 1982). This scaling of the variance can be accomplished simply by multiplying the anomaly yl yl at the lth grid cell by the factor (cos )1/2 for that cell. 2.

1.5 PCA of the tropical Paci c climate variability Let us illustrate the PCA technique with data from the tropical Paci c, a region renowned for the El Ni o phenomenon (Philander, 1990). Every 2 10 years, a sudden warming of the coastal waters occurs off Peru.

As the maximum warming (2.31). 2.1 Principal comp onent analysis (PCA). occurs around Chri datamatrix 2d barcode for .NET stmas, the local shermen called this warming El Ni o (the Child in Spanish), after the Christ child. During normal times, the easterly equatorial winds drive surface waters offshore, forcing the cool, nutrient-rich, sub-surface waters to upwell, thereby replenishing the nutrients in the surface waters, hence the rich biological productivity and sheries in the Peruvian waters.

During an El Ni o, upwelling suddenly stops and the Peruvian sheries crash. A major El Ni o can bring a maximum warming of 5 C or more to the surface waters off Peru. Sometimes the opposite of an El Ni o develops, i.

e. anomalously cool waters appear in the equatorial Paci c, and this has been named the La Ni a (the girl in Spanish) (also called El Viejo , the old man, by some researchers). Unlike El Ni o episodes, which were documented as far back as 1726, La Ni a episodes were not noticed until recent decades, because its cool sea surface temperature (SST) anomalies are located much further offshore than the El Ni o warm anomalies, and La Ni a does not harm the Peruvian sheries.

The SST averaged over some regions (Ni o 3, Ni o 3.4, Ni o 4, etc.) of the equatorial Paci c (Fig.

2.3) are shown in Fig. 2.

4, where El Ni o warm episodes and La Ni a cool episodes can easily be seen. Let us study the monthly tropical Paci c SST from NOAA (Reynolds and Smith, 1994; Smith et al., 1996) for the period January 1950 to August 2000, (where the original 2 by 2 resolution data had been combined into 4 by 4 gridded data; and each grid point had its climatological seasonal cycle removed, and smoothed by a 3 month moving average).

The SST eld has two spatial dimensions, but can easily be rearranged into the form of y(t) for the analysis with PCA. The rst six PCA modes account for 51.8%, 10.

1%, 7.3%, 4.3%, 3.

5% and 3.1%, respectively, of the total SST variance. The spatial patterns (i.

e. the eigenvectors) for the rst three.
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