Multivariate¶

Table of Contents

Preamble¶

In this preamble, we load the gstlearn library.

We load two data bases:

• a data base dat containing point observations of two variables across Scotland: the elevation (Elevation) and the temperature (January_temp, then renamed as Temperature)
• a data base target containing a grid of points covering Scotland with a selection variable (inshore) selecting the points that are on land, and a variable (Elevation) giving the elevation at every point on land

Exploratory data analysis¶

The target data base a (grid) map of the elevation across Scotland.b

Let us plot its content.

The dat data base contains 236 (point) samples of two variables across Scotland: elevation and temperature.

We can use the dbStatisticsPrint function to compute statistics on variables of a Db. We specify

• the Db of interest (argument db),
• a vector containing the names of the variables of interest (argument names),
• the statistics we want to compute (argument opers). This last argument is set through a EStatOption.fromKeys function, which we call with a vector containing the (abbreviated) names of the statistics (run EStatOption.printAll() for the full list)
• a flag flagIso allowing to specify whether we want to compute the statistics for each variable separately (flagIso=TRUE), or whether we want to compute "filtered" statistics (flagIso=FALSE). In the latter case, we compute the statistics while only using the points where all the variables are defined.

For instance, let us count the number of observations of each variable using the dbStatisticsPrint.

Since the data base dat contains 236 samples, we can conclude that the Elevation variable is defined at every point, but not the Temperature variable. Similarly, we can compute the mean and the extreme values of each variable in the observation data base.

Finally, we compute the mean and the extreme values of the elevation in the target data base.

We can then compute the filtered means for the Elevation and Temperature variables as follows:

As explained above, the first row of the table contains contains the mean, the minimum and the maximum of the observations of the Elevation variable, over all the locations where both the Elevation na Temperature variables are defined (i.e. in our case, all the points where the temperature is defined). Hence, we see that the points where the temperature is observed are located at relatively low altitudes within the Scottish landscape.

To confirm that, we plot the points where the temperature is sampled on top of the elevation map of Scotland.

From observing this last plot, it seems like the bigger points (corresponding to high temperatures) are located where the elevation is smaller: this seems to hint (confirm?) that the temperature is negatively correlated with the elevation. To corroborate this, we plot a correlation plot between the two variables.

Baseline univariate model¶

In this course, we will introduce four methods allowing to deal with multivariate data: Cokriging, Residual modeling, Models with polynomial trends and finally Models with an external drift. As a baseline, we will compare the results obtained with each model with the results obtained with a univariate model.

Model fitting¶

We start by computing an experimental directional variogram vario_raw2dir of the "raw" temperature observations, along two directions ($0^{\circ}\text{C}$ and $90^{\circ}\text{C}$).

We then fit a model on the experimental variogram of raw temperature observations. Since we want to use ordinary kriging, we add a constant drift to the model.

Ordinary kriging¶

We create a "neighborhood" object specifying a unique neighborhood, which we will use throughout the course.

We now perform an ordinary kriging prediction of the temperature on the target grid using the model fitted on the raw observations, and compute statistics on the predicted values.

Cross-validation¶

We perform a cross-validation of the fitted model using Ordinary Kriging, and calculate the Mean Squared cross-validation and standardized errors.

Multivariate Models and Cokriging¶

To create and work with multivariate models, we simply need to work with Db objects containing more than one variable with a z locator. All the variables with a z locator will be considered as part of the multivariate model. Then, the same commands as in the univariate case can be used to create and fit experimental variograms, and to perform (co)kriging predictions.

Let us illustrate our point with our running example. We would like now to consider a bivariate model of the temperature and the elevation. To do so, we simply allocate, in the observation data base dat, a z locator to both variables using the setLocators method.

Fitting a bivariate model¶

To create experimental (directional) variograms and cross-variograms, we use the same commands as in the univariate case: since the data base dat now contains two variables with a z locator, the compute method automatically computes both variograms and cross-variograms for these variables.

We can then plot the experimental variograms and cross-variograms with a simple command: the plot in the i-th row and j-th column corresponds to the cross-variogram between the variables with locators zi and zj (hence the diagonal plots correspond to the variograms of each variable).

To fit a model on the experimental variograms and cross-variograms, we use the same commands as in the univariate case.

Cokriging predictions¶

To compute predictions by (simple) cokriging on the grid, we use the same syntax as in univariate case: a predictor for each variable in the multivariate model is produced. (Note: we revert back to a unique neighborhood to compare with the predictors previously introduced).

We can then represent the cokriging predictor for the temperature.

For this predictor, we get the following statistics:

Finally, we compare the cokriging predictor to the ordinary kriging predictor.

Cross-validation¶

Since cokriging can be time-consuming in Unique Neighborhood, we create a small moving neighborhood for demonstration.

To perform a cross-validation of the bivariate model using co-kriging, we use the same commands as in the univariate case. Then cross-validation errors are computed for each variable of the multivariate model (hence for both the Temperature and the Elevation in our case).

We obtain the following Mean Squared Errors for the temperature.

We obtain the following Mean Squared Errors for the elevation.

Working with residuals¶

In this section, we assume that the variable of interest $Z$ is modeled (at each location $x$) as $$ Z(x) = b+a Y(x) + \varepsilon(x)$$ where $Y$ is an auxiliary variable known at every location, $a$ and $b$ are some (unknown) regression coefficients, and $\varepsilon$ denotes stationary residuals. Our goal will be to model and work directly with the residuals $\varepsilon$ (since they are the one who are assumed to be stationary).

Computing and fitting the residuals¶

In our running example, the variable of interest $Z$ will be the temperature, and the auxiliary variable $Y$ will be the elevation. In the observation data base, we start by assigning the locator z to the Temperature variable (this is our variable of interest), and ensure that it is the only variable with a z locator.

To compute the coefficients $a$ and $b$ of the linear regression between the temperature and the elevation, we can use the regression function. We specify the name of response variable (argument nameResp) and the names of the auxiliary variables (argument names), and set the argument mode=0 to specify that we would like to compute a regression defined from the variables with the specified names. We also set the argument flagCste=TRUE to specify that we are looking for an affine regression model between the variables (i.e. that includes the bias coefficient b).

From these regression coefficients obtained above, we can then compute the residuals explicitly as $ \varepsilon(x)=Z(x) - (b+a Y(x) )$. An alternative method consists in using the dbRegression function: this functions fits a regression model, computes the residuals and add them directly on the data base containing the data. The dbRegression function is called in a similar way as the regression function.

In the next example, we compute the residuals of the linear regression between the temperature and the elevation and add them to the observation data base (with a name starting with "RegRes"and without changing the locators in the data base).

We then compute some statistics about these residuals.

Finally we plot a correlation plot between the residuals and the regressor variable (i.e. the elevation).

Now that the residuals are explicitly computed and available in the observation data base, we can proceed to work with them as with any other variable.

We start by setting their locator to z to specify that they are now our variable of interest within the data base (instead of the raw temperature observations).

Then we can compute an experimental variogram for the residuals and fit a model on them.

Ordinary kriging of the residuals¶

Finally, we can compute an kriging prediction of the residuals and plot the results.

Now that the residuals $\varepsilon^{OK}$ are predicted everywhere on the grid (by ordinary kriging), we can compute a predictor $Z^*$ for the temperature by simply adding the back the linear regression part of the model, i.e. by computing

$$ Z^{\star}(x) = b + a Y(x) + \varepsilon^{OK}(x) $$ We can compute this predictor by directly manipulating the variables of the target data base.

Finally, we compare the predictor obtained by kriging of the residuals to the ordinary kriging predictor.

Models with auxiliary variables¶

Models with polynomial trends (intrinsic model)¶

In this subsection, we move to a model for the data where the temperature $Z$ at each location is considered to have a polynomial trend, i.e. we can write, for any location $x$, $$ Z(x) = P(x) + \varepsilon(x)$$ where $P$ is a polynomial function of the coordinates $x$, and $\varepsilon$ denotes stationary residuals. This model can also be referred to as an intrinsic model.

Fitting a model with a polynomial trend¶

As before, in the observation data base, we start by assigning the locator z to the Temperature variable (this is our variable of interest), and ensure that it is the only variable with a z locator.

It is possible to directly compute the experimental variogram of the residuals $\varepsilon$. To do so, we start by creating a Model object in which we specify, through the setDriftIRF method, the (maximal) degree of the polynomial trend P.

Then, computing the (possibly directional) experimental variogram of the residuals is done in the same way as for an "ordinary" experimental variogram, the only difference being that we now add in the compute method of the Vario class, the Model object specifying the polynomial trend (argument model). Then, the methods takes care of filtering out of the observations a polynomial trend of the specified degree, before computing the experimental variogram on the remaining residuals.

For instance, if we want to compute an experimental variogram of the residuals from a model with a linear trend (polynomial of degree 1), then we can run the following commands.

When plotting the experimental variograms, we notice that in one of the directions, the variogram does not reach a plateau. This suggests that a non-stationary model could be more suited for the analysis of these temperature observations.

Then, we can now fit that a model on the experimental variorgram using the fit method in the usual way.

Finally, note that it is possible to compute the coefficients of the polynomial trend from the intrinsic model defined above, using the function regression. This function allows to perform linear regressions on variables defined in a data base.

For instance, to fit a linear trend on the temperature observations in the dat data base, we specify the name of response variable (argument nameResp) and the drift model (argument model), and set the argument mode=2 to specify that we would like to compute a regression defined from a drift model. Finally we set the argument verbose=TRUE to print the results.

Note: The regression results are stored in an object. We can access the coefficients through the slot coeffs, the initial variance of the data through the slot variance and the variance of the residuals through the slot varres.

Universal Kriging¶

To work with universal kriging, it suffices to call the kriging (to compute predictions) or xvalid (to perform cross-validation) with a Model object that includes polynomial drift functions. This model should be fitted on a experimental variogram computed on residuals where the polynomial trend has been filtered out.

Let's compute an universal kriging prediction of the temperature on the target grid using the model fitted on the residuals.

We compute some statistics on the predicted values.

Let us compare the predictions obtained when considering a polynomial trend in the model, with the ordinary kriging predictor based on the "raw" temperature observations. We plot a correlation plot between the ordinary and universal kriging predictions, compare their respective statistics.

Cross-validation¶

To perform cross-validation using Universal Kriging, we simply call the xvalid function with the model we fitted on the residuals.

The Mean Squared cross-validation and standardized errors of the universal kriging predictor are:

Models with External Drifts¶

In this last section, we investigate how to specify and work with external drifts when modeling data. In a data base, external drifts are identified by allocating a locator f to them in the data bases.

For instance, circling back to our running example, let us assume that we would like to model the temperature with an external drift given by the elevation. Then, in the observation data base, we simply need to allocate a z locator to the temperature variable and a f locator to the elevation variable using the setLocator method. Note: we use the flag cleanSameLocator=TRUE to make sure that only the variable we specify carries the locator.

Then, defining and fitting the models on the one hand, and performing kriging predictions on the other hand, is done using the same approach as the one described earlier for models with a polynomial trend.

Fitting a model with external drifts¶

To create experimental (directional) variograms of the residuals from a model with external drift, we use the same approach as the one described for modeling data with polynomial trends.

First, we create a Model object where we specify that we deal with external drifts. This is once again done through the setDriftIRF function where we specify:

• the number of external drift variables (argument nfex): this is the number of variables with a f locator the we want to work with
• the maximal degree of the polynomial trend in the data (argument order): setting order=0 allows to add a constant drift to the model, and setting order to a value $n$ allows to add all the monomial functions of the coordinates of order up to $n$ as external drifts (on top of the nfex external drift variables)

Circling back to our example, we create a model with a single external drift (the elevation), and add a constant drift (that basically acts like a bias term).

Then, to compute the experimental variogram, we use the same commands as in the case of polynomial trends: we create a Vario object from the data base containing the data, and call the compute method with the model we just created. The experimental variogram is computed on the residuals obtained after "filtering out" the (linear) effect of the drift variables (and possibly of a polynomial trend if specified in the model).

As a reference, we plot the experimental variograms computed on the raw temperature data (dotted lines) and on the residuals from the model with external drift (dashed line).

Finally, we fit our model with external drift using the fit method (which we call on the experimental variogram of residuals).

Kriging with external drifts¶

Similarly to Universal kriging, to perform a cross-validation or predictions using kriging with External Drifts, we simply call the xvalid and kriging functions with models where external drifts are specified.

For instance, to compute kriging predictions with external drift of the temperature on the target.

Finally, we perform an kriging prediction of the temperature on the target.

For this predictor, we get the following statistics:

We can then compare these predictions to the ones obtained by ordinary kriging. We create a correlation plot between the ordinary kriging predictions, and the kriging with external drift (KED) predictions.

Note that negative Estimates are present when using an External Drift.

Cross-validation¶

To perform a cross-validation, we simply call the xvalid function.

The Mean Squared cross-validation and standardized errors of the resulting kriging predictor are:

Comparing the various kriging predictions¶

We compare the Mean Squared cross-validation errors obtained in for the different kriging predictions (UK=Universal kriging, OK=Ordinary kriging, COK= Cokriging, KED= Kriging with external drift, KR=Kriging of residuals).

We then compare various statistics computed for each predictor.

Finally, we compare various statistics computed for the standard-deviation of each predictor.