PLS regression relies on iterative algorithms (e.g., NIPALS, SIMPLS). Your description of the main ideas is correct: we seek one (PLS1, one response variable/multiple predictors) or two (PLS2, with different modes, multiple response variables/multiple predictors) vector(s) of weights, $u$ (and $v$), say, to form linear combination(s) of the original variable(s) such that the covariance between Xu and Y (Yv, for PLS2) is maximal. Let us focus on extracting the first pair of weights associated to the first component. Formally, the criterion to optimize reads
$$\max\text{cov}(Xu, Yv).\qquad (1)$$
In your case, $Y$ is univariate, so it amounts to maximize
$$\text{cov}(Xu, y)\equiv \text{Var}(Xu)^{1/2}\times\text{cor}(Xu, y)\times\text{Var}(y)^{1/2},\quad st. \|u\|=1.$$
Since $\text{Var}(y)$ does not depend on $u$, we have to maximise $\text{Var}(Xu)^{1/2}\times\text{cor}(Xu, y)$. Let's consider X=[x_1;x_2], where data are individually standardized (I initially made the mistake of scaling your linear combination instead of $x_1$ and $x_2$ separately!), so that $\text{Var}(x_1)=\text{Var}(x_2)=1$; however, $\text{Var}(Xu)\neq 1$ and depends on $u$. In conclusion, maximizing the correlation between the latent component and the response variable will not yield the same results.
I should thank Arthur Tenenhaus who pointed me in the right direction.
Using unit weight vectors is not restrictive and some packages (pls. regression in plsgenomics, based on code from Wehrens's earlier package pls.pcr) will return unstandardized weight vectors (but with latent components still of norm 1), if requested. But most of PLS packages will return standardized $u$, including the one you used, notably those implementing the SIMPLS or NIPALS algorithm; I found a good overview of both approaches in Barry M. Wise's presentation, Properties of Partial Least Squares (PLS) Regression, and differences between Algorithms, but the chemometrics vignette offers a good discussion too (pp. 26-29). Of particular importance as well is the fact that most PLS routines (at least the one I know in R) assume that you provide unstandardized variables because centering and/or scaling is handled internally (this is particularly important when doing cross-validation, for example).
Given the constraint $u'u=1$, the vector $u$ is found to be $$u=\frac{X'y}{\|X'y\|}.$$
Using a little simulation, it can be obtained as follows:
set.seed(101)
X <- replicate(2, rnorm(100))
y <- 0.6*X[,1] + 0.7*X[,2] + rnorm(100)
X <- apply(X, 2, scale)
y <- scale(y)
# NIPALS (PLS1)
u <- crossprod(X, y)
u <- u/drop(sqrt(crossprod(u))) # X weights
t <- X%*%u
p <- crossprod(X, t)/drop(crossprod(t)) # X loadings
You can compare the above results (u=[0.5792043;0.8151824], in particular) with what R packages would give. E.g., using NIPALS from the chemometrics package (another implementation that I know is available in the mixOmics package), we would obtain:
library(chemometrics)
pls1_nipals(X, y, 1)$W # X weights [0.5792043;0.8151824]
pls1_nipals(X, y, 1)$P # X loadings
Similar results would be obtained with plsr and its default kernel PLS algorithm:
> library(pls)
> as.numeric(loading.weights(plsr(y ~ X, ncomp=1)))
[1] 0.5792043 0.8151824
In all cases, we can check that $u$ is of length 1.
Provided you change your function to optimize to one that reads
f <- function(u) cov(y, X%*%(u/sqrt(crossprod(u))))
and normalize u afterwards (u <- u/sqrt(crossprod(u))), you should be closer to the above solution.
Sidenote: As criterion (1) is equivalent to
$$\max u'X'Yv,$$
$u$ can be found as the left singular vector from the SVD of $X'Y$ corresponding to the largest eigenvalue:
svd(crossprod(X, y))$u
In the more general case (PLS2), a way to summarize the above is to say that the first PLS canonical vectors are the best approximation of the covariance matrix of X and Y in both directions.
References
- Tenenhaus, M (1999). L'approche PLS. Revue de Statistique Appliquée, 47(2), 5-40.
- ter Braak, CJF and de Jong, S (1993). The objective function of partial least squares regression. Journal of Chemometrics, 12, 41–54.
- Abdi, H (2010). Partial least squares regression and projection on latent structure regression (PLS Regression). Wiley Interdisciplinary Reviews: Computational Statistics, 2, 97-106.
- Boulesteix, A-L and Strimmer, K (2007). Partial least squares: a versatile tool for the analysis of high-dimensional genomic data. Briefings in Bioinformatics, 8(1), 32-44.
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:
A short primer on PLS regression with univariate $y$ (aka PLS1, aka SIMPLS)
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:
- If all PLS1 components are used, then PLS will be equivalent to OLS. So the number of components serves as a regularization parameter: the lower the number, the stronger the regularization.
- If the predictors $\mathbf X$ are uncorrelated and all have the same variance (i.e. $\mathbf X$ has been whitened), then there is only one PLS1 component and it is equivalent to OLS.
- Weight vectors $\beta_i$ and $\beta_j$ for $i\ne j$ are not going to be orthogonal, but will yield uncorrelated components $\mathbf z_i=\mathbf X \beta_i$ and $\mathbf z_j=\mathbf X \beta_j$.
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.
Best Answer
Standardize the log transformed data, if you are going to tranform, not the original dataset. The goal of standardizing (aka auto-scaling) is to have mean zero and variance 1 data input into PLSR, so as to force a y=0 intercept in model, and to have PLSR initially weigh all variables equally, respectively. Auto-scaling can introduce noise, however, so it is not always a good option. There are other types of scaling that are compromises between auto-scaling and just centering, such as pareto scaling, where you divide by the square root of standard deviation before centering. If you do not scale, larger magnitude variables will be weighed more heavily in PLSR than lower magnitude variables.
Log transformations are often used when your x variables are e.g. chemical concentrations, that are known to be lognormally distributed in the environment. But in other applications, (e.g. spectroscopy), log transformation may not make the most sense. There are other transformation options, such as the rank-based inverse normal transformation, Box-Cox, or Tukey that you could consider, that will almost absolutely force a normal result. What matters is that when you run PLSR, the relationship between the x-scores and y-scores are linear (t vs u plots). If you see a lot of curvature in these plots, you could probably obtain better results if you were to transform something.