[JKQTPlotterBasicJKQTPDatastoreStatisticsGroupedStat]: @ref JKQTPlotterBasicJKQTPDatastoreStatisticsGroupedStat "1-Dimensional Group Statistics with JKQTPDatastore"
This tutorial project (see `./examples/datastore_statistics/`) explains several advanced functions of JKQTPDatastore in combination with the [[statisticslibrary]] conatined in JKQTPlotter.
The source code of the main application can be found in [`datastore_regression.cpp`](https://github.com/jkriege2/JKQtPlotter/tree/master/examples/datastore_statistics/datastore_regression.cpp).
Now we can caluate the regression line (i.e. the two regression coefficients a and b of the function \c f(x)=a+b*x) using the function `jkqtpstatLinearRegression()` from the [statisticslibrary]:
Sometimes data contains outliers that can render the results of a regression analysis inaccurate. For such cases the [statisticslibrary] offers the function `jkqtpstatRobustIRLSLinearRegression()`, which is a drop-in replacement for `jkqtpstatLinearRegression()` and solves the optimization problem a) in the Lp-norm (which is more robust to outliers) and b) uses the [iteratively reweighted least-squares algorithm (IRLS)](https://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares), which performs a series of regressions, where in each instance the data-points are weighted differently. The method assigns a lower weight to those points that are far from the current best-fit (typically the outliers) and thus slowly comes nearer to an estimate that is not distorted by the outliers.
To demonstrate this method, we use the same dataset as above, but add a few outliers:
```.cpp
std::random_device rd; // random number generators:
The following screenshot shows the result of the IRLS regression analysis and for comparison the normal linear regression for the same dataset (plotted using `jkqtpstatAddLinearRegression(graphD);`):
Another option to react to measurement errors is to take these into account when calculating the regression. To do so, you can use weighted regression that uses the measurement errors as inverse weights. This algorithm is implemented in the function `jkqtpstatLinearWeightedRegression()`.
First we generate again a set of datapoints (x,y), which scatter randomly around a linear function. In addition we calculate an "error" `err` for each datapoint:
```.cpp
std::random_device rd; // random number generators:
We use distribution `de` to draw deviations from the ideal linear function from the range 0.5...1.5. then - for good measure - we use a second distribution `ddecide` (dice tossing) to select a few datapoints to have a 4-fold increased error.
Finally we visualize this data with a simple scatter graph with error indicators:
Now we can caluate the regression line (i.e. the two regression coefficients a and b of the function \c f(x)=a+b*x) using the function `jkqtpstatLinearWeightedRegression()` from the [statisticslibrary]:
***Note*** that in addition to the three data-columns we also provided a C++ functor `jkqtp_inversePropSaveDefault()`, which calculates 1/error. This is done, because the function `jkqtpstatLinearWeightedRegression()` uses the data from the range `datastore1->begin(colWLinE)` ... `datastore1->end(colWLinE)` directly as weights, but we calculated errors, which are inversely proportional to the weight of each data point when solving the least squares problem, as data points with larger errors should be weighted less than thos with smaller errors (outliers).
Again these two steps can be simplified using an "adaptor":
Here the x- and y-columns from the `JKQTPXYGraph`-based graph `graphE` (see above) and the weights from the error column of `graphE` are used as datasources for the plot. This function implicitly uses the function `jkqtp_inversePropSaveDefault()` to convert plot errors to weights, as it is already clear that we are dealing with errors rather than direct weights.
The plot resulting from any of the variants above looks like this:
which performs a simple non-weighted regression. The difference between the two resulting linear functions (blue: simple regression, green: weighted regression) demonstrates the influence of the weighting.
# Linearizable Regression Models
In addition to the simple linear regression model `f(x)=a+b*x`, it is also possible to fit a few non-linear models by transforming the data:
The available models are defined in the enum `JKQTPStatRegressionModelType`. And there exists a function `jkqtpStatGenerateRegressionModel()`, which returns a C++-functor representing the function.
To demonstrate these fitting options, we first generate data from an exponential and a power-law model. Note that we also add normally distributed errors, but in order to ensure that we do not obtain y-values <0,weuseloopsthatdrawnormallydistributedrandomnumbers,untilthisconditionismet:
```.cpp
std::random_device rd; // random number generators:
gFunc->setPlotFunctionFunctor(jkqtpStatGenerateRegressionModel(JKQTPStatRegressionModelType::PowerLaw, cA, cB));
gFunc->setTitle(QString("regression: $%1$").arg(jkqtpstatRegressionModel2Latex(JKQTPStatRegressionModelType::PowerLaw, cA, cB)));
```
The regression models can be plotted using a `JKQTPXFunctionLineGraph`. the fucntion to plot is again generated by calling `jkqtpStatGenerateRegressionModel()`, but now with the parameters determined above the respective lines. Note that `jkqtpstatRegressionModel2Latex()` outputs the model as LaTeX string, which can be used as plot label.
Also note that we used the function `jkqtpstatRegression()` above, which performs a linear regression (internally uses `jkqtpstatLinearRegression()`). But there also exist variants for robust IRLS regression adn weighted regression:
Finally the [statisticslibrary] also supports one option for non-linear model fitting, namely fitting of polynomial models. This is implemented in the function `jkqtpstatPolyFit()`.
To demonstrate this function we first generate data from a poylnomial model (with gaussian noise):
```.cpp
std::random_device rd; // random number generators:
Each model is also ploted using a `JKQTPXFunctionLineGraph`. The plot function assigned to these `JKQTPXFunctionLineGraph` is generated by calling `jkqtp_generatePolynomialModel()`, which returns a C++-functor for a polynomial.
The output of the full test program [`datastore_regression.cpp`](https://github.com/jkriege2/JKQtPlotter/tree/master/examples/datastore_statistics/datastore_regression.cpp) looks like this:
