- In the next couple of weeks
we're going to be talking about empirical models,
mathematical modeling, and regression models.
And so I'm just gonna kind of give you an introduction
to empirical models in this screencast.
Natural phenomena is described by some process,
we have an input this is also known as an
independent variable or a factor or a predictor variable.
This goes into some sort of process
whether that's a natural process or a
synthetic or a chemical process
some sort of chemical reaction or a mechanical process
and we have an output.
This is known as the dependent variable
it depends upon the independent variable.
Oftentimes we will refer to this as the response
and our job as engineers and scientists
is to try to model this process.
We try to put together some sort of
mathematical model where we take
into account the independent variable,
there could be multiple independent variables going
into this by the way, and we try to come up
with a prediction.
This hat above the Y represents the prediction or
the estimate, and ideally, we would create this model such
that the only difference between the model prediction
and the actual output is just random error denoted
by this epsilon.
We would like to determine the form of a mathematical model
and the best values of the parameters
in that model so that the errors are minimized
in some sense and only represent purely
random behavior that we are unable to predict.
Models can take various forms.
Some incorporate physical and chemical principles
with adjustable parameters,
these are known as fundamental models.
Often it is expedient to use models that are
simple mathematically like linear or polynomial equations
and these are empirical models.
Occasionally models are a mix of the two
and these are known as semi empirical models.
When choosing the form of purely empirical model
it is important to make a selection that has
a good potential of fitting the behavior of the data.
The first step in choosing an empirical model is to look
at the behavior of the process via collected data sets
and depicted graphically.
The simplest of models is a straight line, and many
of you have seen the equation for this before.
Our model prediction is some function of an intercept
beta knot and a slope beta one, and the expansion
of this model into a polynomial model to capture curvature
in the data is most common, and here we have
subsequent terms that incorporate
polynomial terms and we have the subsequent beta terms.
For example, beta two is a coefficient for the
X squared term.
There are many other empirical models available such as
an exponential model, a power model, a logarithmic model,
there are also some less commonly used models.
We have an X Reciprocal, Y equals beta not
plus beta one over x.
We have a Y reciprocal one over Y hat is beta not
plus beta one x.
We have an X Y ratio, an X exponential.
So there's lots of different candidate models
that you can look into.
The appropriate way to know whether one
of these models is a candidate is to observe its behavior
for different parameter values.
Here we have an X exponential model and depending upon
the different parameters, B one and B two
you can see that it can change quite dramatically.
The remaining issue is that more
than one model forms may be candidates for the data.
So how do you decide which is best?
You guys are gonna learn
in this class how to choose which model is the best
for the data among several candidates.
The assumption we make in specifying a model is
that the main also known as the
expected value of the output variable.
So that's mu Y, is a function of the input variable.
The main of the output at any value of X
is simply beta, not plus beta one X.
That doesn't mean every time you put
in the same value of X, you're gonna get the same output.
But if you were to look at the probability density function
it's gonna be normally distributed
about some value that depends upon the value
of X that you put into this model.
Since Y equals beta, not plus beta one x plus sum
error term epsilon
the expected value of epsilon is gonna be zero.
Epsilon is going to be a normally distributed
variable centered about zero.
X is sometimes called the regressor variable.
The expected value
of epsilon squared is the variance is sigma squared
which describes the variance of Y at any given value of X.
So in the coming weeks we're gonna talk all
about empirical models, how to create them
and how to analyze them.