linear algebrapartial least squaresterminology
I have read in The Elements of Statistical Learning book and particularly in the Partial Least Squares (PLS) section:
Orthogonalize each $x_j^{(m−1)}$ with respect to $z_m$.
I would like to know what "orthogonalize" means in this statement or general.
1- what orthogonal means?
2- how to orthogonalise?
Section 3.5.2 in The Elements of Statistical Learning is useful because it puts PLS regression in the right context (of other regularization methods), but is indeed very brief, and leaves some important statements as exercises. In addition, it only considers a case of a univariate dependent variable $\mathbf y$.
The literature on PLS is vast, but can be quite confusing because there are many different "flavours" of PLS: univariate versions with a single DV $\mathbf y$ (PLS1) and multivariate versions with several DVs $\mathbf Y$ (PLS2), symmetric versions treating $\mathbf X$ and $\mathbf Y$ equally and asymmetric versions ("PLS regression") treating $\mathbf X$ as independent and $\mathbf Y$ as dependent variables, versions that allow a global solution via SVD and versions that require iterative deflations to produce every next pair of PLS directions, etc. etc.
All of this has been developed in the field of chemometrics and stays somewhat disconnected from the "mainstream" statistical or machine learning literature.
The overview paper that I find most useful (and that contains many further references) is:
For a more theoretical discussion I can further recommend:
The goal of regression is to estimate $\beta$ in a linear model $y=X\beta + \epsilon$. The OLS solution $\beta=(\mathbf X^\top \mathbf X)^{-1}\mathbf X^\top \mathbf y$ enjoys many optimality properties but can suffer from overfitting. Indeed, OLS looks for $\beta$ that yields the highest possible correlation of $\mathbf X \beta$ with $\mathbf y$. If there is a lot of predictors, then it is always possible to find some linear combination that happens to have a high correlation with $\mathbf y$. This will be a spurious correlation, and such $\beta$ will usually point in a direction explaining very little variance in $\mathbf X$. Directions explaining very little variance are often very "noisy" directions. If so, then even though on training data OLS solution performs great, on testing data it will perform much worse.
In order to prevent overfitting, one uses regularization methods that essentially force $\beta$ to point into directions of high variance in $\mathbf X$ (this is also called "shrinkage" of $\beta$; see Why does shrinkage work?). One such method is principal component regression (PCR) that simply discards all low-variance directions. Another (better) method is ridge regression that smoothly penalizes low-variance directions. Yet another method is PLS1.
PLS1 replaces the OLS goal of finding $\beta$ that maximizes correlation $\operatorname{corr}(\mathbf X \beta, \mathbf y)$ with an alternative goal of finding $\beta$ with length $\|\beta\|=1$ maximizing covariance $$\operatorname{cov}(\mathbf X \beta, \mathbf y)\sim\operatorname{corr}(\mathbf X \beta, \mathbf y)\cdot\sqrt{\operatorname{var}(\mathbf X \beta)},$$ which again effectively penalizes directions of low variance.
Finding such $\beta$ (let's call it $\beta_1$) yields the first PLS component $\mathbf z_1 = \mathbf X \beta_1$. One can further look for the second (and then third, etc.) PLS component that has the highest possible covariance with $\mathbf y$ under the constraint of being uncorrelated with all the previous components. This has to be solved iteratively, as there is no closed-form solution for all components (the direction of the first component $\beta_1$ is simply given by $\mathbf X^\top \mathbf y$ normalized to unit length). When the desired number of components is extracted, PLS regression discards the original predictors and uses PLS components as new predictors; this yields some linear combination of them $\beta_z$ that can be combined with all $\beta_i$ to form the final $\beta_\mathrm{PLS}$.
Note that:
All that being said, I am not aware of any practical advantages of PLS1 regression over ridge regression (while the latter does have lots of advantages: it is continuous and not discrete, has analytical solution, is much more standard, allows kernel extensions and analytical formulas for leave-one-out cross-validation errors, etc. etc.).
Quoting from Frank & Friedman:
RR, PCR, and PLS are seen in Section 3 to operate in a similar fashion. Their principal goal is to shrink the solution coefficient vector away from the OLS solution toward directions in the predictor-variable space of larger sample spread. PCR and PLS are seen to shrink more heavily away from the low spread directions than RR, which provides the optimal shrinkage (among linear estimators) for an equidirection prior. Thus PCR and PLS make the assumption that the truth is likely to have particular preferential alignments with the high spread directions of the predictor-variable (sample) distribution. A somewhat surprising result is that PLS (in addition) places increased probability mass on the true coefficient vector aligning with the $K$th principal component direction, where $K$ is the number of PLS components used, in fact expanding the OLS solution in that direction.
They also conduct an extensive simulation study and conclude (emphasis mine):
For the situations covered by this simulation study, one can conclude that all of the biased methods (RR, PCR, PLS, and VSS) provide substantial improvement over OLS. [...] In all situations, RR dominated all of the other methods studied. PLS usually did almost as well as RR and usually outperformed PCR, but not by very much.
Update: In the comments @cbeleites (who works in chemometrics) suggests two possible advantages of PLS over RR:
An analyst can have an a priori guess as to how many latent components should be present in the data; this will effectively allow to set a regularization strength without doing cross-validation (and there might not be enough data to do a reliable CV). Such an a priori choice of $\lambda$ might be more problematic in RR.
RR yields one single linear combination $\beta_\mathrm{RR}$ as an optimal solution. In contrast PLS with e.g. five components yields five linear combinations $\beta_i$ that are then combined to predict $y$. Original variables that are strongly inter-correlated are likely to be combined into a single PLS component (because combining them together will increase the explained variance term). So it might be possible to interpret the individual PLS components as some real latent factors driving $y$. The claim is that it is easier to interpret $\beta_1, \beta_2,$ etc., as opposed to the joint $\beta_\mathrm{PLS}$. Compare this with PCR where one can also see as an advantage that individual principal components can potentially be interpreted and assigned some qualitative meaning.
Scale invariance means that rescaling any or all of the columns will not change the results - that is, multiplying or dividing all the values from any variable will not affect the model predictions (ref). As @ericperkeson mentioned, rescaling in this manner is known as dilation (ref). Scale invariance for metrics about contingency tables refers to rescaling rows as well as columns, though I don't believe it applies here (see the scaling property section here).
As to why PLSR is not scale invariant, I'm not completely certain, but I'll leave notes on what I've learned and possibly a better mathematician can clarify. Generally, regression with no regularisation (e.g. OLS) is scale invariant, and regularised regression (e.g. ridge regression) is not scale invariant, because the minimisers of the function change (ref).
Now, I can't see an explicit penalty term in PLSR, but I it's constrained in a similar way to PCA. PCA chooses the axes of maximal variance - so if you rescale a variable, the variance relative to other variables can change (ref). PLSR tries to find the ' multidimensional direction in the X space that explains the maximum multidimensional variance direction in the Y space', hence rescaling an input can change the direction of maximum variance (ref).
Best Answer
I believe the quote refers to this algorithm, where the relevant line reads:
$x_j^m=x_j^{m-1}-\frac{\langle z_m,x_j^{m-1}\rangle}{\langle z_m,z_m\rangle}z_m$
Here the authors are using the angle-brackets to denote an inner product, which is essentially the standard vector dot product from Physics 101.
The second term is the orthogonal projection of $x_j^{m-1}$ onto $z_m$. By subtracting this from $x_j^{m-1}$, the result $x_j^m$ is made orthogonal to $z_m$.