Exercise 7-4 Decomposition using Pr

Exercise 7-4
Decomposition using Principal Components


Earth observation imagery typically shows a great deal of variability over time. Thus it is common to want to decompose that variability into its underlying constituents. One of the most popular ways of doing this is through Principal Compo- nents Analysis (PCA -- also known as Empirical Orthogonal Function (EOF) Analysis).

a)

If you have not done so already, read the Principal Components section of the Earth Trends Modeler chapter in
the IDRISI Manual. Then open the PCA panel on the Analysis tab. Select the SST data set as the input series. The defaults are set for their typical use in time series analysis so you can immediately click the Run button. When it has finished, ETM will automatically switch to the Explore PCA/EOT/Fourier PCA/Wavelet panel of the Explore tab. The first component will be displayed.


Note: A Clarification About Terminology. Please note the Climatology/Atmospheric Science communities use a different terminology from that used in the Geography and Remote Sensing communities. This goes beyond the issue of calling it EOF rather than PCA. The starting point for a standardized PCA/EOF is a correlation matrix (or a variance/ covariance matrix if it is unstandardized). In Geography/Remote Sensing applications (as it is in ETM), this correlation matrix is between images over time. Thus, if you have 300 images over time, this is a 300 x 300 matrix of correlations. In the climatological community, the correlations are between pixels over space. Thus if you have an image series with 100 columns and 100 rows, the correlation matrix will be a 10,000 by 10,000 matrix. Both procedures produce a series of images and a corresponding set of graphs, which are identical. In other words, there is only one solution regardless of how you construct the correlation matrix. This is because the solution is orthogonal over both space and time. However, the terminology is different. In the implementation here, the images are called components and the graphs are called loadings. If your correlations are between pixels, the graphs are the components and the images are the loadings. Note also that some climatologists refer to each component/loading pair as a mode.

b)








c)

Look at the first loading graph. This shows time on the X axis and correlation on the Y axis. Notice that the val-
ues are all very high. What this tells us is that every image has this pattern present within it. Thus, this is essentially the pattern of the long term average sea surface temperature. Note that in interpreting the components, you should focus on the pattern over space and not the absolute values of the component scores (the values in the image). Because it is a standardized analysis and successive components are based on residuals from previous components, it becomes increasingly hard to relate these values back to the original imagery. However, we can see in the title of the loading graph that this first component accounts for 98.22% of the variability in sea surface temperature over space and time. All remaining variability is contained within the remaining 1.78%.

Now in the Explore PCA panel, select Component 2 and click the Display (map) icon to its right. The compo- nent image will display. Notice that the loadings follow an annual cycle that is symmetric about the 0 correlation position. The loadings are positive during the northern hemisphere late summer/early autumn and negative in the early spring. Then notice that the component image also has positive and negative values. This is a case where it is best that the contrast stretch be symmetric about 0 so that it is unambiguous as to where there are negative values and where there are positive values. Therefore, make sure that the PCA layer is highlighted in Composer (it might not be if you have an automatic vector overlay), and click the middle STRETCH button at the bottom of Composer to create a symmetric stretch about zero.

Notice the hemispheric (north/south) differences in the component scores (the image values). Also notice in the Atlantic how the division between the hemispheres falls in the same position as the Atlantic Equatorial Counter Current noted earlier. Clearly this is the annual seasonal cycle. Notice also that while the component explains only a little over 1.5% of the total variance in SST over space and time, this represents over 85% of the variance




Exercise 7-4 Decomposition using Principal Components 303

remaining after the effects of Component 1 are removed.

Looking at the loadings graph and the component image as a pair, the loadings say that geographically the pattern looks most like this during the boreal late summer/early autumn (August/September - i.e., when the load- ings are high) and the opposite of this during the boreal early spring months (February/March, when the loadings are highly negative). The nearly perfect sinusoidal pattern of the loadings supports the interpretation of this as the annual cycle, but there is evidently a lag in its maximum impact.

d) Now display the loading graph and component image for Component 3. Also use the STRETCH button on
Composer to stretch the image symmetrically. This is also an annual cycle, but notice that it is aligned more with the early winter (December) and early summer (June) and that it is much smaller in its accounting of variance (only about 4% of the variance explained by Component 2).

1 Compare the areas that have the strongest seasonality in Components 2 and 3. Given the timing of loadings, what does
this suggest about the relationship between components over space and time? We know that components are independent of each other. Are they independent of each other in time, space or over both?


e)







f)




g)










h)


Now display and examine the loading graphs for Components 4, 5 and 6. Stretch each of the component images symmetrically using the middle STRETCH option in Composer. Component 4 is also clearly a seasonal cycle; however it is semiannual. Component 5 is clearly an interannual cycle (we will have more to say about this shortly), while Component 6 appears to be a mix between a seasonal cycle (again, semi-annual) and an interan- nual oscillation. This highlights an interesting issue regarding PCA/EOF. Although the components can repre- sent true underlying sources of variability, they can also represent mixtures. We will explore this further in subsequent exercises.

Often it is these interannual oscillations that are a key interest in image time series analysis. If this is the case, then it is usually advisable to run the PCA on deseasoned data. Therefore, let's go back to the Analysis tab and run PCA again, but this time use the anomalies in SST you created in an earlier exercise. Use all the same parameters that you did the first time (i.e., the defaults).

Now look at Component 1 from this new analysis and compare it to Component 5 from your previous one. Clearly they are the same thing (although the loading for Component 1 of the anomalies in SST is more coherent over time), but the patterns are inverted in the component images and the loading graphs. Since they are both inverted, they therefore represent the same thing. It's like taking the negative of a negative number which yields
a positive. This leads to an important issue. It is mathematically permissible to invert the loadings graph (by multiplying by -1) if you also invert the component image. The end result is identical mathematically, but in some cases may be easier to explain. Don't hesitate to do this. For the graph, export the data to a spreadsheet (right- click on empty space in the graph and choose the clipboard text option to paste into your spreadsheet, and then subsequently multiply by -1); for the component image, use the SCALAR module or Image Calculator to multiply by -1.

If you have not yet stretched Component 1 from your anomalies analysis, do so now (with the symmetric option). This is the El Niño / La Niña phenomenon (also known as the El Niño / Southern Oscillation, abbre- viated as ENSO). ENSO is an irregular oscillation typically in the 2.5-7 year range. El Niño events are associated with a weakening (or even a reversal) of the prevailing easterlies (trade winds) along the equator. Normally, the frictional effect of these easterlies on the sea surface causes a movement of warm surface waters to the Asian side of the Pacific. In fact, normally, the Asian side is actually higher (by about 40 cm) than the South American side. When the trade winds weaken, this warm pool of water flows back to the South American side under the force of gravity. After a period of about 6-12 months of warming, the trade winds resume and the pattern reverses. In fact, El Niño events are characteristically followed by an abnormal strengthening of the trades, producing the opposite effect known as a La Niña.





Exercise 7-4 Decomposition using Principal Components 304

2 Looking at your loading graph, the big peaks and big valleys represent El Niño and La Niña events, respectively. Tabulate the periods when you think El Niño conditions existed, when the La Niña pattern was prevalent and when neither was present (some call this "La Nada"). What do you think is the typical length of a complete El Niño event? What about the typical length of a La Niña? How normal are La Nada conditions?


i)










j)


ENSO is known as a climate teleconnection because it leads to correlated climate conditions over widely dispersed areas of the globe. A teleconnection can also be defined as a characteristic pattern of variability. There is great interest in the study of teleconnections because of their utility in seasonal forecasting. By monitoring SST in the central Pacific, we now have good warning about the development of ENSO conditions, which has facilitated seasonal forecasting around

Kết quả (Latinh) 1:

Exercise 7-4
Decomposition Principal Components usura simulacri typice terra observatione, ostendit multum tempus variabilitatem. Et sic manifestum est quod commune est dissolutum, ut ex subiecta in quibus ipsa constituitur variabilitatem. Unum ex maxime popularis modis, ex compo- nents examinatione maiori (PCA - quoque notus ut empirica Orthogona Muneris (EOF) Analysis). a) Si autem non feci ita iam legere lacinia principalibus orbis terrarum Trends Modeler in capite de IDRISI Manual. Analysis aperiesque PCA panel in tab. Lego series SST data set sicut input. De defaltis sunt pro eorum amet tempus usus seriem in analysi sic vos can click Run puga pyga immediate. Quam, cum consummatum fuerit, generat ETM mos automatically switch ad Explore PCA / EOT / Fourier PCA / Wavelet panel de Explore tab. Prima pars demonstrabuntur. nota A ferendo circa terminos. Please note Climatology / Atmospheric Science communitates utor a diversus ex terminis, qui in Sensu Remote, geographiam et communitates. Et vocabant illam, et hoc quod superexcedit eventum potius quam PCA EOF. Quod a principio ad standardized PCA / EOF matrix est reciproci (or do variacione sue / covariance matrix si unstandardized). In Geographia / remotus Sensus pertinens (sicut in ETM) Aequa relatio inter imagines matrix tempus. Ita si statuas CCC subinde est CCC X matricem correlationes CCC. In climatological communitate, in correlationibus sunt super spatium inter pixels. Sic si vos have an image C series et ordines columns cum C, erit habitudo matrix 10,000 10,000 a matrice. Et par est ratio et imaginum series producatur a paro of graphs, quae idem sunt. Praeterea, quantumcumque sit unica ratione tibi construere matrice. Tum quia haec est solutio orthogonalis super continuum et tempus. Sed aliud est locutio. In exsecutionem hic, praesto sunt imagines dicuntur partes et graphs loadings vocantur. Si correlations sunt inter elementa, et ea quae sunt ad graphs loadings imagines sunt. Item nota, quod quaedam referuntur ad climatologists singulis component / loading par ut modus. b), c) primum loading Aspice Aliquam lacinia purus. Inde constare potest tempus in X Y axis et reciproci in axis. Animadverto ut val- ues sunt valde. Quid est hoc quod dicit, quod imago exemplaris omnis inest. Sic, quod est mare superficiem exemplar longa mediocris temperatus. Nam interpretes componentibus, ut focus in spatio, non autem formam absolutis super componentia turpis (valores ad imaginem). Quia ita est, et successive a normatum analysis fundatur in partes sunt a priore residuals componentibus, quibus valoribus fit magis magisque difficile dictu retro ad originale simulacri. Potest tamen in hoc titulo prima pars oneratum graph rationem 98,22% de superficie temperies variabilitatem in mare et in praemissis. Reliqua omnia quae in medio, quod reliquum est in pituitam abire, 1,78%. In Explore PCA panel, Component II elige quod click Display (map) icon ad eius dexteram. Ut ostendam tibi quoniam imago et compo- nent. Animadverto ut loadings an annuum cyclum sequi, quae aequaliter ad 0 ratione situm. Quod certo in hemisphaerio septentrionali loadings aestate / autumnus in vere et negative. Et nota quod habet quandam imaginem positivum sive negativum. Sed haec ita, quod est optimum, ut anceps tractus erit aequaliter 0, unde et ubi sunt valorem negativum. Ergo, quod planto certus ut accumsan est highlighted in PCA Composer (ut si non sit tibi habere automatic vector inaurabis), quod click in puga pyga in fundo de medio SPATIUM Composer creare uiarum circa tractum zero. Animadverto hemisphærico (aquilonem / meridiem) component in in turpis differentia (imago values). Et tamen et in modum divisionis Atlantic inter hemisphaeria cadit in eodem loco sunt cum Atlanticum Equatorial Counter Current supra dictum est. , Manifestum est quod annui temporis cyclum. Et nota quod non tantum est component, cum paulo plus 1.5% of totalis super SST contentiones in spatio et in tempore, hoc repraesentatur per LXXXV% of the contentiones Exercise 7-4 Decomposition usura Principal Components CCCIII remanere post effectus Component I removentur . et ipse elevatis oculis in loadings graph imago component ut par est, loadings geographicam descriptionem, quod spectat sicut hoc maxime in borealibus aestate / mane autumnus (December / - ie, cum in ho load- alta sunt) et vere primo in contrarium boreales mensibus (June / Martiis, quo loadings valde negative). The sinusoidal fere perfectum exemplar sustentat loadings interpretatio ut annuum in cyclum, sed manifestum est, maximum labefactum in TARDO. d) autem et quandam imaginem ostentare loading Aliquam lacinia purus pro Component 3. utor SPATIUM puga pyga Composer ad imaginem extendam aequalibus. Quod etiam annuos, sed animadverto ut est aligned cum prima hieme (December) et prima aestate (June) et multo minus sit in ratione discordant (circa IV% variationem modo explicari cursus II ). Compare the areas quæ habent I et II 3. Praeterea, posita in infirmitate seasonality in lacinia leo loadings, quid suadere hanc ad necessitudinem pertinent inter partes supra spatium et tempus? Scimus non dependent ab invicem membra. Sunt non dependens a se invicem in tempore, aut in utrisque? e) f) g) h) Sed et examinare propono loading graphs quia Components IV, V et aetate simulacra 6. extendas singulis component usura bene in medio SPATIUM Composer. Patet etiam temporis cursus cursus IV; tamen est semiannual. V Component plane interannual cycli (ut paulo plura dicamus hac de), dum Component VI videtur esse commistum inter a seasonal cyclum (etiam semi-annuis) et interan- tate oscillationis. Hoc dubium de lumine an interesting PCA / EOF. Licet ea quae possunt repraesentandum verum subjectam exitus mutabilitas, non possunt, etiam mixtures. Nos explorare hoc in sequentibus ulterius exercitia. Saepe his qui interannual oscillationum sunt interest in a key series imaginem analysis tempus. Hoc si ita est, ex hoc genere est visa ratio ad currendam PCA deseasoned elit. Igitur redeamus ad tab Analysis PCA currunt iterum, sed in hac vice SST creasti argumentis utendum in priore exercitation. Uti omnibus idem parametri primum fecistis (scilicet defectus). Ecce enim ex hoc I cursus et analysi cursus comparant V praemissae a. Nempe illi sunt idem (quamvis oneratum in I de anomalias in SST cursus tempus aptius est), sed imagines et exempla inversi oneratum graphs componentes. Cum utrumque traditur, idem ipsi repraesentetur. Est assumpta numerus negativus cedentem negativum positivum. Haec ratio momenti exitus. Est licitum mathematice invertant loadings graph (a -1, ducendo) component si etiam imago invertunt. Finis fit idem mathematice, sed in quibusdam ut facilius explicabo. Non temere facere. Nam lacinia purus elit ut patefacio spreadsheet (Bl iustitiam click in vacuo spatio optio elige clipboard est igitur in textu Spreadsheet et multiplicamini postea a -1); singulares enim res imago, utor modulus scalari Image Computus multiplicandi per -1. Si vos have nondum extentis Component analysis I de anomalias, ita nunc (cum quaecunque bene). El hoc est Niño / La Niña phaenomenon (quoque notus ut El Niño / Southern oscillationis, quod abbre- viated ENSO). ENSO oscillationis impar plerumque 2.5-7 anno in range. El eventus coniungitur cum a Nino deminutum (vel etiam transuerso) praevalentem easterlies (trade ventis) per equator. Northmanni, per huiusmodi frictional easterlies superficies aquae calidae, motum a superficie ad mare partem Asiae urna. In facto, plerumque in actu superioris Asiae parte est (circiter XL cm), quam ad meridiem American latus. Cum ventum commercium infirmat quidem in America meridionali parte regurgitat ad stagna aquarum vi gravitatis. 6-12 post menses fere tepidus, et omnium vasorum in quaestu adversis ventis resumere. In facto, accidere El Niño indolem deinde an abnormal roborandam trades, ad producendum oppositum effectus cognoscitur tamquam Niña La. Exercise 7-4 Decomposition Principal Components CCCIV usura II loading Vultus procul vestri Aliquam lacinia purus, et electas abietes illius, et magnus magnum vallibus La Niña El Nino et rerum caput est. El Nino Tabulate putatis ea tempora fuisse, neque cum pluribus et de Niña exemplar fuit presens (hoc dicitur "La Nada"). Quid putas, totam longitudinem sit amet El Nino coniunx? Quid de typicam longitudinem a La Niña? Quid sunt communis condicionibus La Nada? i) j) ENSO est notus ut a climate teleconnection connectuntur, quia est ad climate areas of conditionibus super orbem latius dissipatur. A exemplar proprium definiri potest teleconnection variabilitatem. Est autem quaestus magnus interest quia de eorum utilitate, in tantum studiis teleconnections praevidens temporis. By vigilantia SST in central Pacifici sumus, nunc habere bonum monet ENSO progressionem conditiones, quae mihi hunc temporis circa praevidens