knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
(The title here alludes to Andriy Burkov's excellent work, The Hundred-Page Machine Learning Book. Note too my own book, The Art of Machine Learning: Algorithms+Data+R.)
Here we give an overview of the most widely used predictive methods in statistical/machine learning (ML). For each one, we present
background
overview of how it works
function in the R package, qeML (readers without R background or who simply wish to acquire an overview of ML may skip the R code without loss of comprehehnsion of the text)
Advanced methods, e.g. Generative Adversial Networks, and specialized methods for language models and image analysis, are not covered.
For convenience, we'll let Y denote the variable to be predicted, often termed the response variable or outcome, and let X denote the set of predictor variables/features. (ML people tend to use the term features, while In statistics, the term predictors is common. In applied fields, e.g. economics or psychology, some use the term independent variables.)
We develop our prediction rule from available training data, consisting of n data points, denoted by (X_{1}, Y_{1}),.., (X_{n}, Y_{n}). We wish to predict new cases (X_{new},Y_{new}) in the future, in which X_{new} is known but Y_{new} needs to be predicted.
If our data is in the form of an R data frame, then we will have n rows. The number of columns, other than our Y column, is generally denoted by p. Both n and p could be very large in some applications, say in the millions for n and the hundreds for p.
So, we may wish to predict human weight Y from height X, or from height and age in a two-component vector X. Say we have the latter situation, and data on n = 100 people. Then for instance X_{23} would be the vector of height and age for the 23rd person in our training data, and Y_{23} would be that person's weight.
Indicator variables:
The vector X may include indicator variables, which have values only 1 or 0. We may for instance predict weight from height, age and gender, the latter being 1 for female, 0 for male.
If Y represents a binary variable, we represent it as an indicator variable. In the famous Pima Diabetes dataset in the UCI Machine Learning Repository, 1 means diabetic, 0 means not.
The 0,1 form is statistical; often in ML circles the coding -1,1 is used.
If Y itself is categorical, we represent it by several indicator variables, one for each category. In another disease-related UCI dataset, Y is status of a person's vertebrae condition; there are three classes: normal (NO), disk hernia (DH), or spondilolysthesis (SP). Y = (1,0,0) for a normal person, for instance. Thus Y can be a vector too.
If we have data on an indicator variable, note that its mean is the proportion of 1s in the data. On the population level, this becomes the probability of a 1.
The package's built-in dataset mlb consists of data on major league baseball players (courtesy of the UCLA Dept. of Statistics).
Here is a glimpse of the data:
> data(mlb) > head(mlb) Name Team Position Height Weight Age PosCategory 1 Adam_Donachie BAL Catcher 74 180 22.99 Catcher 2 Paul_Bako BAL Catcher 74 215 34.69 Catcher 3 Ramon_Hernandez BAL Catcher 72 210 30.78 Catcher 4 Kevin_Millar BAL First_Baseman 72 210 35.43 Infielder 5 Chris_Gomez BAL First_Baseman 73 188 35.71 Infielder 6 Brian_Roberts BAL Second_Baseman 69 176 29.39 Infielder
Here "qe" stands for "quick and easy," and we really mean that!
The functions have a simple, uniform interface, and most importantly, require no setup. To fit an SVM model, say, one simply calls qeSVM, no preparation calls to define the model etc.
The call form is
model fit <- qe<model name>(<data name>,<Y name>)
Just a one-liner!
Let's predict weight from height and age, using two methods, k-Nearest Neighbor and random forests. (Details of the methods will be explained shortly; for now, let's just see how to invoke them.)
mlb1 <- mlb[,4:6] # columns for height, weight and age knnout <- qeKNN(mlb1,'Weight') # fit k-Nearest Neighbor model rfout <- qeRF(mlb1,'Weight') # fit random forests model
It's that easy. As noted, no prior calls are needed to define the model, etc.
Prediction of new cases is equally easy, in the form
predict(<model fit>, <new X value>)
E.g. to predict the weight of a new player of height 70 and age 28, using our random forests model created above, run
> predict(rfout,c(70,28)) 2 184.1626
Such a player would be predicted to weigh about 184 pounds.
The return objects also give an indication of overall predictive power of our model.
> rfout$testAcc # mean absolute prediction error [1] 15.16911
So, on average, our predictions are off by about 15 pounds. What about the k-NN model?
> knnout$testAcc [1] 13.20277
Seems to be better, though we should try other values of the hyperparameters, and we should run the model multiple times. (See below.)
Most methods have hyperparameters, values that can be used to tweak the analysis in various ways. In k-NN, for instance, the number of neighbors k is a hyperparameter for the algorithm.
In qeML, default values of hyperparameters, e.g. k = 25 for k-NN, are set but can and should be overridden.
By default, the data is partitioned by the qe- functions into a training set and a holdout set, with the model being fit on the former and then tested on the latter. The point is that, in assessing the predictive accuracy of our fit, we should do so on "fresh, new" data. This is known as cross-validation (there are many variants). The size of the holdout set size can be set differently from the default, or suppressed entirely.
The typical strategy regarding selection of the values of hyperparameters is to perform a grid search, trying all possible combinations. This computation can be quite voluminous.
Prediction applications in which Y is a continuous variable, or at least ordinal, are called regression settings. Applications in which Y is categorical, i.e. Y is a factor variable in R, are classification settings. In the mlb dataset, for instance, prediction of player weight is a regression application, while predicting a player's position would be a classification application.
Somewhat confusingly, both settings make use of the regression function, m(t) = E(Y | X = t), the mean value of Y in the subpopulation defined by X = t. If say we are predicting weight from height and age in the mlb data, then for instance m(71,23) would denote the mean weight among all players of height 71 inches and 23 years old. To predict the weight of a new player, say height 77 and age 19, we use m(77,19).
Readers who have seen **linear regression** models should note that the term **regression** is much more general than that linear case. In the general context, the regression function is simply the mean function, expressing the mean of one variable in terms of the values of other variables.
In classification problems, Y is converted to a set of indicator variables. For the position 'pitcher' in the mlb data, we would have Y = 1 or 0, depending on whether the player is a pitcher or not. Then E(Y | X = t) reduces to P(Y = 1 | X = t), the probability that the player is a pitcher given the player's height, weight and age, say. We would typically find probabilities for each position, then guess the one with the highest probability.
In other words, the regression function m(t) is central to both regression and classification settings. The statistical/machine learning methods presented here amount to ways to estimate m(t). The methods are presented below in an order that shows connection between them.
Even though each ML method has its own special tuning parameters or hyperparameters, used to fine-tune performance, they all center around the regression function m(t).
The **qe** series function sense whether the user is specifying a regression setting or a classification setting, by noting whether the Y variable is numeric or an R factor.
We now present the "30,000 foot" view of the major statistical/machine learning methods. But first, a point on the role of prediction.
In most ML applications, our goal is prediction of an outcome variable Y, in contrast to statistics, in which most applications center on estimated effects of the features on Y. Say we have data on diabetes. In ML, we will likely be interested in how to diagnose the disease, based on features such blood glucose, body mass index and family history. In statistics, we would be typically interested in estimating, for instance, the effect of BMI on the likelihood of developing diabetes.
The significance of this distinction is that in predictive applications, classical issues of model validity, say homogeneous Y variance in the linear model setting, are not of much interest. If our model predicts well, we are happy. In an effect measurement setting, the assumptions may matter a lot, in terms of validity of confidence intervals and hypothesis tests.
This method was originally developed by statisticians, starting in the 1950s and 60s.
It's very intuitive. To predict, say, the weight of a new player of height 72 and age 25, we find the k closest players in our training data to (72,25), and average their weights. This is our estimate of m(72,25), and we use it as our prediction. The default value of k in qeKNN is 25.
The qeKNN function wraps kNN in regtools. The main hyperparameter is the number of neighbors k. As with any hyperparameter, the user aims to set a "Goldilocks" level--not too big, not too small. Setting k too small will result in our taking the average of just a few Y values, too small a sample. On the other hand, too large a value for k will mean that some distant data points may be used that are not representative. To choose k, we could try various values, run qeKNN on each one, then use the value that produced the best (i.e. smallest) testAcc.
Again, if Y is an indicator variable, its average works out to be the proportion of 1s. This will then be the estimated probability that Y = 1 for a new case. If that is greater than 0.5 in the neighborhood of the new case, we guess Y = 1 for the new case, otherwise 0.
If Y is categorical, i.e. an R factor as in the case of predicting player position in the mlb dataset, we then find a probability for each category in this manner, and guess Y to be whichever category has the highest probability.
The hyperparameter k as default value 25. Let's see if, say, 10 is better:
replicMeans(50,"qeKNN(mlb1,'Weight')$testAcc") [1] 13.61954 attr(,"stderr") [1] 0.1346821 replicMeans(50,"qeKNN(mlb1,'Weight',k=10)$testAcc") [1] 14.25737 attr(,"stderr") [1] 0.1298224
Since the holdout set is randomly generated, we did 50 runs in each case. The average test accuracy over 50 runs is printed out, along with a standard error for the figure. (1.96 times the standard error will be the radius of an approximate 95% confidence interval.) Changing k to 10 reduced accuracy.
One potential problem is bias at the edge of our dataset Say we are predicting weight from height, and have a new case involving a very tall person. Most data points in the neighborhood of this particular height value will be for people who are shorter than the new case. Those neighbors are thus likely to be lighter than the new case, making our predicted value for that case biased downward.
Rare for k-NN implementations, qeKNN has a remedy for this bias. Setting the argument smoothingFtn = loclin removes a linear trend within the neighborhood, and may improve predictive accuracy for new cases that are located near the edge of the training set.
See also the function **qeLinKNN**.
This method stems from decision trees, which were developed mainly by statisticians in the 1980s, and which were extended to random forests in 1990s. Note too the related (but unfortunately seldom recognized) random subspaces work of Tin Kam Ho, who did her PhD in computer science.
This is a natural extension of k-NN, in that it too creates neighborhoods and averages Y values within the neighborhoods. However, it does so in a different way, by creating tree structures.
Say in some dataset we are predicting blood pressure from height, weight and age. We could, say, first ask whether the height is above or below a certain threshold. After that, we ask whether weight is above or below a certain (different) threshold. This partitions height-weight space into 4 sectors. We then might subdivide each sector according to whether age is above or below a threshold, now creating 8 sectors of height-weight-age space. Each sector is now a "neighborhood." To predict a new case, we see which neighborhood it belongs to, then take our prediction to be the average Y value among training set points in that neighborhood.
This produces a tree structure, as shown below. Each "neighborhood" is termed a node, and terminal nodes are called leaves.
Example:
Here is an illustration using the vertebrae dataset. Again, there are 6 predictor variables, named V1 through V6, consisting of various bone measurements. The variable to be predicted is disease status, a categorical variable with levels DH, NO, SL and VC. NO designates a normal patient, no disease.
This picture was generated using the **rpart.plot** package, via **qeRpart()**.
At the root, if the variable V6 in our new case to be predicted is < 16, we go left, otherwise right. Say we go left. Then if V4 < 28 in the new case, we go left again, getting to a leaf, in which we guess DH. The 0.74 etc. mean that for the training data that happen to fall into that leaf, 74% of them are in class DH, 26% are NO and 0% are SL.
Constructing the tree:
In generating the tree, the algorithm decides whether to split the current node, according to the current feature. (There are various methods used for this, so that choice of method becomes a hyperparameter.) Thus the the process may stop early, if the current subdivision is judged by the algorithm to be fine enough to produce good accuracy.
In the above example, for instance, if V6 is at least 16, we do not check other variables, and the next node, the green one oApparently n the far right, becomes a leaf. Just knowing that V6 ≥ 16 is enough to reasonably guess that this case is in the SL class, with probability 98%.
We also might check a feature more than once, as seen above for V4.
Prediction:
When presented with a new case X_{new}, for which we wish to predict Y_{new}, we check to see which leaf X_{new} belongs to. For instance, say this new case has V6 = 12 and V4 = 8. Then following the tree links, we see that this case falls into the leftmost leaf, and we predict Y_{new} to be DH, with there being an estimated 74% chance that this new case is of type DH.
Another hyperparameter:
Remember, in some settings we may have dozens, or even hundreds of predictor variables, so our decision tree could have many levels. One of the hyperparameters common in tree software indirectly controls the number of levels, as follows.
One generally wants to avoid having leaves that don't have many data points. This is like setting k to too small a value in k-NN. We often control this via a hyperparameter in the form of a minimum number of data points per node. If a split would violate that rule, then don't split. A small minimum leaf size is analogous to a small k, and the same is true for large minimum leaf size and large k. Again we need to try to find a "Goldilocks" level.
In the above discussion, the user decides the input order of predictor variables used in constructing the tree. But of course, some orders may be better than others.
In random forests (RFs), many trees are generated at random, using random orders of entry of the features. The number of trees is another hyperparameter. Each tree gives us a prediction for the unknown Y. In a regression setting, those predictions are averaged to get our final prediction. In a classification setting, we see which class was predicted most often among all those trees.
The qeRF function wraps the function of the same name in the randomForests package.
The qeML package also offers other implementations of RF, notably qeRFgrf, as follows.
The leaves in any tree method, such as random forests and boosting, are essentially neighborhoods, different in structure from those of k-NN, but with similar properties. In particular, they have an "edge bias" problem similar to that described for k-NN above. In the case of random forests, the qeRFgrf function (which wraps the grf package) deals with the same bias problem via removal of a linear trend.
This method has been developed both by CS and statistics people. The latter have been involved mainly in gradient boosting, the technique used here.
The basic idea is to iteratively build up a sequence of trees, each of which is an update of the last, hopefully an improved one. At the end, all the trees are combined, with more weight given to the more recent trees.
The qeGBoost wraps gbm in the package of the same name. It is gradient tree-based, with hyperparameters similar to the random forests case, plus a learning rate. The latter controls the size of iteration steps; more on this below.
The package also offers other implementations of boosting, including qeXGBoost, based on the popular XGBoost algorithm that incorporates regularization into the tree-building process. (Discussed below.)
This of course is the classical linear regression model, invented 200 years ago (!) and developed by statisticians. It is mainly for regression settings.
For motivation, below is a graph of mean weight vs. height for the mlb data. (There are many players for each height level, and we find and plot the mean weight at each height.)
The means seem to lie near a straight line. That suggests modeling m(t) is a linear function.
The points don't **exactly** lie on a straight line. It's important to keep in mind: (a) We are merely setting up an approximate model, one that is close enough to reality to be useful (see "A note on prediction" above); and (b) these are sample means, subject to sampling variation.
For example, a linear model for mean weight, given height and age, would be
m(height,age) = β_{0} + β_{1} height + β_{2} age
for unknown population constants β_{i}, which are estimated from our training data using the classic least-squares approach. Our estimates of the β_{i}, denoted b_{i}, are calculated by minimizing
Σ_{i} [ weight_{i} - (b_{0}+b_{1}height_{i}+b_{2}age_{i}) ] ^{2}
This is a simple calculus problem. We find the partial derivatives of the sum of squares with respect to the b_{i}, and set them to 0. This gives us 3 equations in 3 unknowns, and since these equations are linear, it is easy to solve them for the b_{i}.
There are no hyperparameters in the linear model.
The function qeLin wraps the ordinary lm. It mainly just calls the latter, but does some little fixes.
This is a generalization of the linear model (hence a generalized linear model), developed by statisticians and economists.
This model is only for classification settings. As noted, since Y is
now 1 or 0, its mean becomes the probability of 1. Since m(t) is now a
probability, we need it to have values in the interval [0,1]. This is
achieved by feeding a linear model into the logistic function,
l(u) = (1 + exp(-u))^{-1}, which does take values in (0,1).
Here u will be a linear form in the features.
So for instance, to predict whether a player is a catcher (Y = 1 if yes, Y = 0 if no), we fit the model
probability of catcher given height, weight, age =
m(height,weight,age) =
1 / [1 + exp{-(β_{0} + β_{1} height + β_{2} weight + β_{3} age)}]
The β_{i} are estimated from the sample data, using a technique called iteratively reweighted least squares.
The function qeLogit wraps the ordinary R function glm, but adds an important feature: glm only handles the 2-class setting, e.g. catcher vs. non-catcher. The qeLogit handles the c-class situation via the One vs. All method (applicable to any ML algorithm): It calls glm one class at a time, generating c glm outputs. When a new case is to be predicted, it is fed into each of the c glm outputs, yielding c probabilities. It then predicts the new case as whichever class has the highest probability.
Here is an example using the UCI vertebrae data;
> library(fdm2id) > data(spine) > str(spine) 'data.frame': 310 obs. of 8 variables: $ V1 : num 39.1 53.4 43.8 31.2 48.9 ... $ V2 : num 10.1 15.9 13.5 17.7 20 ... $ V3 : num 25 37.2 42.7 15.5 40.3 ... $ V4 : num 29 37.6 30.3 13.5 28.9 ... $ V5 : num 114 121 125 120 119 ... $ V6 : num 4.56 5.99 13.29 0.5 8.03 ... $ Classif2: Factor w/ 2 levels "AB","NO": 1 1 1 1 1 1 1 1 1 1 ... $ Classif3: Factor w/ 3 levels "DH","NO","SL": 1 1 1 1 1 1 1 1 1 1 ... > spine <- spine[,-7] # skip 2-class example > u <- qeLogit(spine,'Classif3') > u$testAcc [1] 0.1935484 > u$baseAcc [1] 0.5053763
So, we would have a misclassification rate of about 19%, whereas the error rate would be about 50% if we didn't use the features. Where does this latter figure come from? Look at this tabulation:
> table(spine$Classif3) DH NO SL 60 100 150
If we were to not use the body measurements V1 etc., we would always guess SL, the most prevalent class. We would be wrong
(60+100)/(60+100+150) = 0.51629
of the time, similar to the 0.5053763 value found above. (The discrepancy is due to the latter figure being based on a holdout set.)
By making use of the measurement data, we can reduce the misclassification rate to about 19%.
Let's try a prediction. Consider someone like the first patient in the dataset but with V6 = 6.22. What would our prediction be?
> newx <- spine[1,-7] # omit "Y" > newx$V6 <- 6.22 > predict(u,newx) $predClasses [1] "DH" $probs DH NO SL [1,] 0.7432193 0.2420913 0.01468937
We would predict the DH class, as our estimated probability for the class is 0.74, the largest among the three classes.
Some presentations describe the logistic model as "linearly modeling the logarithm of the odds of Y = 1 vs. Y = 0." While this is correct, it is less informative, in our opinion. Why would we care about a logarithm being linear? The central issue is that the logistic function models a probability, just what we need.
Some people tend to shrink when they become older. Thus we may wish to model a tendency for people to gain weight in middle age but then lose weight as seniors, a nonlinear relation. We could try a quadratic model:
m(height,age) = β_{0} + β_{1} height + β_{2} age + β_{3} age^{2}
where presumably β_{3} < 0.
We may even include a height x age product term, allowing for interactions. Polynomials of degree 3 and so on could also be considered. The choice of degree is a hyperparameter; in qePolyLin it is named deg.
Such model for mean weight for given height and age would at first seem nonlinear, but that would be true only in the sense of being nonlinear in age. It is still linear in the β_{i} -- e.g. if we double each β_{i} in the above expression, the value of the expression is doubled -- so qeLin could in principle be used, or qeLogit for classification settings.
But forming the polynomial terms by hand would be tedious, especially since we would also have to do this for predicting new cases. Instead, we use qePolyLin (regression setting) and qePolyLog (classification), which do all that work for us. They make use of the package polyreg.
Polynomial models can in many applications hold their own with the fancy ML methods. One must be careful, though, about overfitting, just as with any ML method. In particular, polynomial functions tend to grow rapidly near the edges of one's dataset, causing both bias and variance problems.
Some deep mathematical theory due to James and Stein implies that in linear models it may be advantageous to shrink the estimated b_{i}. Ridge regression and the LASSO do this in a mathematically rigorous manner. Each of them minimizes the usual sum of squared prediction errors, subject to a limit being placed on the size of the b vector; for ridge, the size is defined as the sum of the b_{i}^{2}, while for the LASSO it's the sum of |b_{i}|.
Limiting the size of some statistical estimator in general is termed regularization. Shrinkage methods are often also applied to other ML algorithms, such as we previously mentioned for XGBoost.
The function qeLASSO wraps cvglmnet in the glmnet package. The main hyperparameter alpha specifies ridge (0) or the LASSO (1, the default). There are various other shrinkage methods, such as Minimax Convex Penalty (MCP). This and some others are available via qeNCVregCV, which wraps the ncvreg package.
Why regularize?
As mentioned, mathematically theory suggests it.
Ridge regression was originally proposed as a remedy for multicollinearity, a situation in which high intercorrelations among the features can make predictions numerically and statistically unstable.
The LASSO's appealing property is that it can be used for feature selection; it tends to produce solutions in which some of the b_{i} are 0.
Here is an example using pef, a dataset from the US Census, included with qeML. The features consist of age, education, occupation and gender. Those last three are categorical variables, which are
The function **regtools::factorsToDummies** can be used to do this conversion manually.
automatically converted by most qe-series functions to indicator variables.
Let's predict wage income.
> w <- qeLASSO(pef,'wageinc') > w$coefs 11 x 1 sparse Matrix of class "dgCMatrix" s1 (Intercept) -8983.986 age 254.475 educ.14 9031.883 educ.16 9182.592 occ.100 -2707.388 occ.101 -1029.592 occ.102 3795.166 occ.106 . occ.140 . sex.1 4472.672 wkswrkd 1178.889
There are six occupations (this dataset is just for programmers and engineers),
> levels(pef$occ) [1] "100" "101" "102" "106" "140" "141"
thus five indicator variables. By inspection, we see no indicator variable for occupation 141, so it is the base, e.g. it is estimated that on average, holding other feature values constant, occupation 102 pays about $3795 more than than does occupation 141.
The LASSO gave coefficients of 0 for occupations 106 and 140, so we might decide not to use them, even if we utimately use some other ML algorithm.
As mentioned, ridge and the LASSO shrink the estimated coefficients vector b by placing a limit on the size of that vector; the smaller the limit, the greater the amount of shrinkage. So, what is the best value for that amount? The glmnet packages handles that via cross-validation.
These were developed originally in the AI community, and later attracted interest among statisticians. They are used mainly in classification settings.
Say in the baseball data we are predicting catcher vs. non-catcher, based on height and weight. We might plot a scatter diagram, with height on the horizontal axis and weight on the vertical, using red dots for the catchers and blue dots for the non-catchers. We might draw a line that best separates the red and blue dots, then predict new cases by observing which side of the line they fall on. This is what SVM does.
With more predictors, the line become a plane or hyperplane.
Here is an example, using the Iris dataset built in to R:
There are 3 classes, but we are just predicting setosa species (shown by + symbols) vs. non-setosa (shown by boxes) here. Below the solid line, we predict setosa, otherwise non-setosa.
SVM philosophy is that we'd like a wide buffer separating the classes, called the margin, denoted by the dashed lines. Data points lying on the edge of the margin are termed support vectors, so called because if any other data point were to change, the margin would not change, so these points on the edge of margin "support" the margin..
In most cases, the two classes are not linearly separable. So we allow curved boundaries, implemented through polynomial (or similar) transformations to the data. The degree of the polynomial is a hyperparameter.
Another hyperparameter is cost: Here we allow some data points to be within the margin. The cost variable is roughly saying how many exceptions we are willing to accept.
The qeSVM function wraps svm in the e1071 package. The package also offers various other implementations.
These were developed almost exclusively in the AI community.
An NN consists of layers, each of which consists of a number of neurons (also called units or nodes). Say for concreteness we have 10 neurons per layer. The values of the features for a given data point are fed into the first layer, the output of which will be 10 different linear combinations of those values. This is essentially 10 different linear regression models. Those outputs will be fed into the second layer, yielding 10 "linear combinations of linear combinations," and so on.
In regression settings, the outputs of the last layer will be averaged together to produce our estimated m(t). In the classification case with c classes, our final layer will have c outputs; whichever is largest will be our predicted class.
"Yes," you say, "but linear combinations of linear combinations are still linear combinations. We might as well just use linear regression in the first place." True, which is why there is more to the story: activation functions. Each output of a layer is fed into a function A(t) before being passed on to the next layer; this is for the purpose of allowing nonlinear effects in our model of m(t).
For instance, say we take A(t) = t^2 (not a common choice in practice, but a simple one to explain the issues). The output of the first layer will be quadratic functions of our features. Since we square again at the outputs of the second layer, the result will be 4th-degree polynomials in the features. Then 8th-degree polynomials come out of the third layer, and so on. So, NNs are doing something akin to polynomial linear regression. And the hyperparameter, Number of Layers, is analogous to the degree of the polynomial.
Another hyperparameter is the number of neurons per layer.
One common choice for A(t) is the logistic function l(u) we saw earlier. Another popular choice is ReLU, r(t) = max(0,t). No matter what we choose for A(t), the point is that we have set up a nonlinear model for m(t).
At the training stage, the coefficients ("weights") of the linear combinations are computed by least squares. Due to the activation function, this is now a nonlinear optimization problem, so the computation is iterative.
Then to predict Y_{new} for a new case X_{new}, we run the latter through the layers, taking linear combinations of inputs at each layer, using the coefficients found in the training stage.
The qeNeural function allows specifying the numbers of layers and neurons per layer, and the number of iterations. It wraps krsFit from the regtools package, which in turn wraps the R keras package (and there are further wraps involved after that).
Here's an example, again with the vertebrae data: The predictor variables V1, V2 etc. for a new case to be predicted enter on the left, and the predictions come out the right; whichever of the 3 outputs is largest, that will be our predicted class. Weights are shown on the edges.
Let n_{w} denote the number of weights. This can be quite large, even in the millions. Moreover, the n_{w} equations we get by setting the partial derivatives to 0 are not linear.
Though far more complex than in the linear case, we are still in the calculus realm. We compute the partial derivatives of the sum of squares with respect to the n_{w} weights, and set the results to 0s. So, we are finding roots of a very complicated function in n_{w} dimensions, and we need to do so iteratively.
Typical NN methods also use back propogation, in which the accuracy found in one iteration is used to aid in making the next one better.
Many ML algorithms, e.g. XGBoost and NNs, do some kind of iterative optimization, making use of a hyperparameter known as the learning rate. A simplified version of the role of this quantity in the iteration process is as follows. Consider the function f graphed below:
There is an overall minimum at approximately x = 2.2. This is termed the global minimum. But there is also a local minimum, at about x = 0.4; that term means that this is the minimum value of the function only for points near -- "local to" -- 0.4. Let's give the name x_{0} to the value of x at the global minimum. Denote our guess for x_{0} at iteration i by g_{i}. Say our initial guess g_{0} = 1.1.
The tangent line is pointing upward to the right, i.e. has positive slope, so it tells us that by going to the left we will go to smaller values of the function. We do want smaller values, but in this case, the tangent is misleading us. We should be going to the right, towards 2.2, where the global minimum is.
If our current guess were near 2.2, the tangent line would guide us in the right direction. But we see above that it can send us in the wrong direction. One remedy (among several typically used in concert) is to not move very far in the direction the tangent line sends us. The idea behind this is, if we are going to move in the wrong direction, let's limit our loss. The amount we move is called the step size in general math, but the preferred ML term is the learning rate. And, this is yet another hyperparameter.
So, to try to avoid settling on a local minimum, we should try different values of the learning rate, as well as different initial values for the weights.
Up to a point, the more complex a model is, the greater its predictive power. "More complex" can mean adding more predictor variables, using a higher-degree polynomial, adding more layers etc.
As we add more and more complexity, the model bias will decrease, meaning that our models become closer--in the sense of expected value over all possible samples--to the actual m(t), in principle. But the problem is that at the same time, the variance of a predicted Y_{new} is increasing.
Let M(X_{new}) denote our estimate of the true m(X_{new}), obtained from one of the methods covered here. M(X_{new}) has a distribution over all possible samples. For fixed n, the greater the complexity, the closer this distribution will be centered on m(X_{new})--i.e. smaller bias--BUT the larger will be the variance of that distribution.
Say again we are predicting human weight from height, with polynomial models. With a linear model, we use our training data to estimate two coefficients, b_{0} and b_{1}. With a quadratic model, we estimate three, and estimate four in the case of a cubic model and so on. But it's same training data in each case, and intuitively we are "spreading the data thinner" with each more complex model. As a result, the standard error of our predicted Y_{new} (estimated standard deviation) increases.
Hence the famous Bias-Variance Tradeoff. If we use too complex a model, the increased variance overwhelms the reduction in bias, and our predictive ability suffers. We say we have overfit.
So there is a "Goldilocks" level of complexity for any given n: an optimal polynomial degree, optimal number of nearest neighbors, optimal NN network architecture (configuration of layers and neurons), and so on. How do we find it?
Alas, there are no magic answers here. We will definitely try fitting our model using cross-validation on various degrees of complexity, but we must keep in mind that the resulting values are themselves random.
The output of qeFT, the package's grid search function, includes standard errors, indicating the accuracy of our cross-validation. A cautious policy would be that if two combinations of hyperparameter values give similar predictive power, we choose the one with lesser complexity.
See the 'Overfitting' vignette in this package for more on this topic, including the intriguing phenonemon of double descent.
The usual answer given for this question is, "The best method is very dependent on which dataset you have." True, but are there some general principles?
Let's first consider the views of core ML specialists (CMLSs). In our view they tend to be biased by their focus on image and language settings (see below), and by an assumption that the user can and will do voluminous hyperparameter optimization, but their views are of course still worth considering.
CMLSs tend to distinguish between "tabular" and "nontabular" data. A disease diagnosis application may be considered tabular--rows of patients, and columns of measurements such as blood glucose, body temperature and so on. On the other hand, CMLSs view a facial recognition application as nontabular, which is confusing, since it too has the form of a table--one table row per row of the picture, and one column per pixel position within a picture row.
The strategies currently in vogue among CMLSs are:
Use XGBoost for tabular applications.
Use deep NNs or other advanced, application-specific methods in image and language settings.
SVM is definitely not in vogue among CMLSs.
CMLSs view linear and generalized linear models as valuable, especially as initial analyses of a given dataset or in settings in which elaborate hyperparameter optimization will not be done. They would view k-NN and RFs similarly.
Many analysts find that SVM works well for them, in spite of disdain by the CMLSs.
XGBoost is nice but in a typical application the default values for the hyperparameters will not give impressive results. Extensive, application-specific hyperparameter tuning will be needed.
A key question is whether there is a general monotonic relationship between Y and the features. In predicting diabetes, say, the higher the glucose level, the more likely the patient is diabetic. Though monotonic relations may exist in image classification, they maybe more localized.
"Monotonic" applications may be best served by ML methods that have either linear components or low-degree polynomials, such as linear or generalized linear models, SVM and low-degree polynomial versions of these. For other applications, one might try k-NN, RFs, NNs etc.
As the saying goes, "Your mileage may vary," depending on your application and data. Our recommended strategy for tabular data (image and language applications require extensive experience in those realms) would be to first try a linear or logistic model, possibly with a quadratic term. If you then wish to go further, first try k-NN (in regression settings, also the "best of both worlds" algorithm, qeLinKNN), then heavy grid search with qeXGBoost or qeNCVregCV. For feature selection, qeLASSO and qeFOCI.
For grid search the qeFT function can alleviate you of a lot of the work.
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