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<title>GrADS Function: tregr</title>
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<h2><b>tregr()</b></h2>
<p>
<code>tregr (<i>expr1, expr2, tdim1, tdim2</i>)</code>
<p>
This function calculates the least-squares regression between two
time-dependent variables.
<p>
<ul>
<code><i>expr1</i> </code>-
a valid GrADS expression that varies in time<br>
<code><i>expr2</i> </code>-
a valid GrADS expression that varies in time and may also vary in X and Y<br>
<code><i>tdim1</i> </code>-
starting time dimension expression <br>
<code><i>tdim2</i> </code>-
ending time dimension expression <br>
</ul>
<p>
The result is a grid that matches the X and Y dimensions of
<code><i>expr2</i></code> where each grid point is the temporal
regression of <code><i>expr2</i></code> (the dependent variable) onto
<code><i>expr1</i></code> (the independent variable).
<code>tregr</code> gives the expected value of
the <code><i>expr2</i></code> departure given a 1 unit departure in
<code><i>expr1</i></code>.
<p>
<h3>Usage Notes</h3>
<ol>
<li>If both <code><i>expr1</i></code> and <code><i>expr2</i></code>
vary only in time, the output is a single value.
<p>
<li>The regression is sensitive to the units of the input expressions.
In the examples below, the variable SLP is in units of mb and
the variable elnino is in units of K, so the regression coefficient
of SLP upon elnino is in units of mb per K.
<p>
<li>The result of the least squares regression of Y on X is often
expressed as a linear equation:
<p>
<ul><code>Y = slope * X + intercept</code></ul>
<p>
where X is the independent variable, Y is the dependent variable, and
the slope and intercept are calculated using complicated algebraic
formulas. The calculation is simplified if the time means are
removed. If we define x and y to be the departures from the time
averages of X and Y:
<p>
<ul><code>x = X - Xave</code><br><code>y = Y - Yave</code></ul>
<p>
then the regression equation becomes:
<p>
<ul><code>y = <i>coefficient</i> * x</code></ul>
<p>
Where
<p>
<ul><code><i>coefficient</i> = (sum of x*y over time)/(sum of x*x over time)</code></ul>
<p>
This <code><i>coefficient</i></code> is the output from the <code>tregr</code> function.
The second example below shows how to construct the regression estimate of Y based on X.
<p>
<li>Use the <a href="gradfuncsregr.html"><code>sregr</code></a>
function to do regression over the spatial domain.
<p>
</ol>
<p>
<h3>Example</h3>
<ol>
<li>This example calculates the expected departure from the
slp mean given a unit departure in the defined variable elnino.
<p>
<pre>
set y 1
set z 1
set t 1 100
define elnino = aave(ts, lon=-160, lon=-80, lat=-20, lat=10)
set lon 0 360
set lat -90 90
set z 1
set t 1
d tregr(elnino, slp, t=1, t=100)
</pre>
<p>
<li>This example builds on the previous example by
calculating the regression estimate of slp based on the
defined variable elnino.
<p>
<pre>
define coeff = tregr(elnino, slp, t=1, t=100)
define slpave = ave(slp, t=1, t=100)
define ninoave = ave(elnino, t=1, t=100)
d coeff * (elnino - ninoave) + slpave
</pre>
</ol>
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