Confidence Band

This example demonstrates adding confidence bands to a trend line. It also demonstrates how to display linear regression parameters on the chart.

In linear regression, the data points are assumed to be related by:

 y = m * x + c + err 

where m and c are constants, and err is a random variable. Linear regression estimates m, c and err based on available data.

As m and c are estimated values, the trend line y = m * x + c may be inaccurate. In ChartDirector, the uncertainties can be represented as a confidence band using TrendLayer.addConfidenceBand. For example, the 95% confidence band means there are 95% probability that the "real" trend line is in that band.

To predict a data point (infer y given x), we use y = m * x + c + err. As m, c and err are estimated values, the predicted data point may be inaccurate. In ChartDirector, the uncertainties can be represented as a prediction band using TrendLayer.addPredictionBand.

Note that the prediction band is always wider than the confidence band because of the extra uncertainties contributed by the err term.

In ChartDirector, the linear regression parameters slope, intercept, correlation coefficient and standard error can be obtained by using TrendLayer.getSlope, TrendLayer.getIntercept, TrendLayer.getCorrelation and TrendLayer.getStdError. (Please refer to appropriate statistics text books for explanation of these values.)

In this example, the linear regression parameters are formatted into a table using CDML, which is then added to the chart as custom text box using BaseChart.addText.

[File: phpdemo/confidenceband.php]

<?php
require_once("../lib/phpchartdir.php");

# The XY data of the first data series
$dataX = array(50, 55, 37, 24, 42, 49, 63, 72, 83, 59);
$dataY = array(3.6, 2.8, 2.5, 2.3, 3.8, 3.0, 3.8, 5.0, 6.0, 3.3);

# Create a XYChart object of size 450 x 420 pixels
$c = new XYChart(450, 420);

# Set the plotarea at (55, 65) and of size 350 x 300 pixels, with white background
# and a light grey border (0xc0c0c0). Turn on both horizontal and vertical grid lines
# with light grey color (0xc0c0c0)
$c->setPlotArea(55, 65, 350, 300, 0xffffff, -1, 0xc0c0c0, 0xc0c0c0, -1);

# Add a title to the chart using 18 point Times Bold Itatic font.
$c->addTitle("Server Performance", "timesbi.ttf", 18);

# Add titles to the axes using 12 pts Arial Bold Italic font
$c->yAxis->setTitle("Response Time (sec)", "arialbi.ttf", 12);
$c->xAxis->setTitle("Server Load (TPS)", "arialbi.ttf", 12);

# Set the axes line width to 3 pixels
$c->yAxis->setWidth(3);
$c->xAxis->setWidth(3);

# Add a scatter layer using (dataX, dataY)
$c->addScatterLayer($dataX, $dataY, "", DiamondSymbol, 11, 0x008000);

# Add a trend line layer for (dataX, dataY)
$trendLayer = $c->addTrendLayer2($dataX, $dataY, 0x008000);

# Set the line width to 3 pixels
$trendLayer->setLineWidth(3);

# Add a 95% confidence band for the line
$trendLayer->addConfidenceBand(0.95, 0x806666ff);

# Add a 95% confidence band (prediction band) for the points
$trendLayer->addPredictionBand(0.95, 0x8066ff66);

# Add a legend box at (50, 30) (top of the chart) with horizontal layout. Use 10 pts
# Arial Bold Italic font. Set the background and border color to Transparent.
$legendBox = $c->addLegend(50, 30, false, "arialbi.ttf", 10);
$legendBox->setBackground(Transparent);

# Add entries to the legend box
$legendBox->addKey("95% Line Confidence", 0x806666ff);
$legendBox->addKey("95% Point Confidence", 0x8066ff66);

# Display the trend line parameters as a text table formatted using CDML
$textbox = $c->addText(56, 65, sprintf(
    "<*block*>Slope\nIntercept\nCorrelation\nStd Error<*/*>   <*block*>%.4f ".
    "sec/tps\n%.4f sec\n%.4f\n%.4f sec<*/*>", $trendLayer->getSlope(),
    $trendLayer->getIntercept(), $trendLayer->getCorrelation(),
    $trendLayer->getStdError()), "arialbd.ttf", 8);

# Set the background of the text box to light grey, with a black border, and 1 pixel
# 3D border
$textbox->setBackground(0xc0c0c0, 0, 1);

# Output the chart
header("Content-type: image/png");
print($c->makeChart2(PNG));
?>