Load Forecasting using Bagged Regression Trees

This example demonstrates an alternate model for building relationships between historical weather and load data to build and test a short term load forecasting. The model used is a set of aggregated Regression Trees.


Import Data

The data set used is a table of historical hourly loads and temperature observations from the New England ISO for the years 2004 to 2008. The weather information includes the dry bulb temperature and the dew point. This data set is imported from an Access database using the auto-generated function fetchDBLoadData. This function was generated by the Visual Query Builder in Database Toolbox and then modified.

If using the MATLAB Central File Exchange script, load the MAT-files from the Data folder and skip to line 72.

data = fetchDBLoadData('2004-01-01', '2008-12-31');
addpath ..\Util

Import list of holidays

A list of New England holidays that span the historical date range is imported from an Excel spreadsheet

[num, text] = xlsread('..\Data\Holidays.xls');
holidays = text(2:end,1);

Generate Predictor Matrix

The function genPredictors generates the predictor variables used as inputs for the model. For short-term forecasting these include

If the goal is medium-term or long-term load forecasting, only the inputs hour of day, day of week, time of year and holidays can be used deterministically. The weather/load information would need to be specified as an average or a distribution

% Select forecast horizon
term = 'short';

[X, dates, labels] = genPredictors(data, term, holidays);

Split the dataset to create a Training and Test set

The dataset is divided into two sets, a training set which includes data from 2004 to 2007 and a test set with data from 2008. The training set is used for building the model (estimating its parameters). The test set is used only for forecasting to test the performance of the model on out-of-sample data.

% Create training set
trainInd = data.NumDate < datenum('2008-01-01');
trainX = X(trainInd,:);
trainY = data.SYSLoad(trainInd);

% Create test set and save for later
testInd = data.NumDate >= datenum('2008-01-01');
testX = X(testInd,:);
testY = data.SYSLoad(testInd);
testDates = dates(testInd);

save Data\testSet testDates testX testY
clear X data trainInd testInd term holidays dates ans

Build the Load Forecasting Model

The next few cells builds a Bagged Regression Trees model for day-ahead load forecasting given the training data. This model is then used on the test data to validate its accuracy.

Build the Bootstrap Aggregated Regression Trees

The function TreeBagger is used to build the model, ie. a set of regression trees each with a different set of rules for performing the non-linear regression. We build an aggregate of 20 such trees, with a minimum leaf size of 20. The larger the leaf size the smaller the tree. This provides a control for overfitting and performance.

model = TreeBagger(20, trainX, trainY, 'method', 'regression', 'minleaf', 20)

simpleTree = prune(model.Trees{1}, 500);
simpleTree = prune(simpleTree, simpleTree.prunelist(1)-10);
view(simpleTree, 'names', labels);
model = 

Ensemble with 20 bagged decision trees:
               Training X:            [35064x8]
               Training Y:            [35064x1]
                   Method:           regression
                    Nvars:                    8
             NVarToSample:                    3
                  MinLeaf:                   20
                    FBoot:                    1
    SampleWithReplacement:                    1
     ComputeOOBPrediction:                    0
         ComputeOOBVarImp:                    0
                Proximity:                   []

Save Trained Model

We can compact the model (to remove any stored training data) and save for later reuse

model = compact(model);
save Models\TreeModel model

Forecast Using Tree Model

Once the model is built, perform a forecast on the independent test set.

load Data\testSet
forecastLoad = predict(model, testX);

Compare Forecast Load and Actual Load

Create a plot to compare the actual load and the predicted load as well as compute the forecast error. In addition to the visualization, quantify the performance of the forecaster using metrics such as mean absolute error (MAE), mean absolute percent error (MAPE) and daily peak forecast error.

err = testY-forecastLoad;
fitPlot(testDates, [testY forecastLoad], err);

errpct = abs(err)./testY*100;

fL = reshape(forecastLoad, 24, length(forecastLoad)/24)';
tY = reshape(testY, 24, length(testY)/24)';
peakerrpct = abs(max(tY,[],2) - max(fL,[],2))./max(tY,[],2) * 100;

MAE = mean(abs(err));
MAPE = mean(errpct(~isinf(errpct)));

fprintf('Mean Absolute Percent Error (MAPE): %0.2f%% \nMean Absolute Error (MAE): %0.2f MWh\nDaily Peak MAPE: %0.2f%%\n',...
    MAPE, MAE, mean(peakerrpct))
Mean Absolute Percent Error (MAPE): 2.24% 
Mean Absolute Error (MAE): 338.76 MWh
Daily Peak MAPE: 2.27%

Examine Distribution of Errors

In addition to reporting scalar error metrics such as MAE and MAPE, the plot of the distribution of the error and absolute error can help build intuition around the performance of the forecaster

subplot(3,1,1); hist(err,100); title('Error distribution');
subplot(3,1,2); hist(abs(err),100); title('Absolute error distribution');
line([MAE MAE], ylim); legend('Errors', 'MAE');
subplot(3,1,3); hist(errpct,100); title('Absolute percent error distribution');
line([MAPE MAPE], ylim); legend('Errors', 'MAPE');

Group Analysis of Errors

To get further insight into the performance of the forecaster, we can visualize the percent forecast errors by hour of day, day of week and month of the year

[yr, mo, da, hr] = datevec(testDates);

% By Hour
boxplot(errpct, hr+1);
xlabel('Hour'); ylabel('Percent Error Statistics');
title('Breakdown of forecast error statistics by hour');

% By Weekday
boxplot(errpct, weekday(floor(testDates)), 'labels', {'Sun','Mon','Tue','Wed','Thu','Fri','Sat'});
ylabel('Percent Error Statistics');
title('Breakdown of forecast error statistics by weekday');

% By Month
boxplot(errpct, datestr(testDates,'mmm'));
ylabel('Percent Error Statistics');
title('Breakdown of forecast error statistics by month');

Generate Weekly Charts

Create a comparison of forecast and actual load for every week in the test set.

generateCharts = true;
if generateCharts
    step = 168*2;
    for i = 0:step:length(testDates)-step
        fitPlot(testDates(i+1:i+step), [testY(i+1:i+step) forecastLoad(i+1:i+step)], err(i+1:i+step));
        title(sprintf('MAPE: %0.2f%%', mean(errpct(i+1:i+step))));