Module 1:
In general, models should reflect a quantitative relationship between variables.
y=a+b x
o The formula above shows a linear relationship between the dependent and
independent variable.
o x: independent variable. Something we can control in time (pH, time,
temperature).
o y: dependent variable. Something related to x, that we can observe or
measure (concentration, enzyme activity, texture).
o a, b: parameters. They are what makes the model fit a certain data set. They
form the core of a model and summarize the information in the data.
In linear regression, you estimate the intercept and slope of the reaction. These
parameters describe the relation between the variables.
Sometimes, a theory may predict a certain mathematical model. However, almost
always, mathematical models must be based on experimental observations. Models
based on experimental observations are called empirical models.
o To select an empirical model, you first analyze the structure of the data (make
a plot!) to determine how the data changes with changing x-value linear,
non-linear, hyperbolic, parabolic, etc.
o Based on the trend of your data, you select a mathematical model.
In food science, we usually have mathematical models in the form of:
o Algebraic equations.
−a ( ρ p− ρf ) d 2
v=
18 η p
μ
ln ( N ) =ln ( N 0 ) + Aexp( ∞ ( λ−t ) +1)
A
When you know the parameters and independent variables, you can
directly calculate the independent variables.
o Ordinary differential equations.
−dc
=kc
dt
These models give the change in something (e.g., concentration), with
respect to time.
o Partial differential equations.
2
dc =D d 2c
dt d x
These models give the change in something (e.g., concentration), with
respect to time and location. There are thus 2 independent variables.
Typically used for diffusion phenomena.
Parameter values in a model are used to calculate an outcome (dependent variable).
These parameters are estimated from experiments. Those experiments have variation
in their outcome, so the parameter estimates also contain some unexplained variation.
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, When it comes to deterministic models, they produce 1 output value.
Stochastic models/probabilistic models, on the other hand, produce a whole range of
outcomes, based on the uncertainty of the parameters.
o An error term (ε ) is added to make an
equation a stochastic equation. The
error refers to uncertainties.
Variability and uncertainty are 2 sources that contribute to the total uncertainty.
- Variability: inherent variation in the system under study. Cannot be reduced for a
given system.
o E.g., biological variability in raw materials for foods.
Uncertainty: reflects our state/lack of knowledge about the system under study. Can
be reduced by better and more measurements.
o E.g., lack of knowledge on the average vitamin C concentration of apples in a
batch, because only 1 apple was measured.
Separation of variability and uncertainty is of importance!
Systematic errors are indicated by the term
‘accuracy’. These errors cannot be
characterized or corrected by statistics; the
researcher has the responsibility to prevent
these (e.g., by calibration). These errors are
avoidable.
Random errors are indicated by the term
‘precision’. They are characterized by mean ±
standard deviation, or CV, or CIs. They can be
reduced upon taking more samples. These
errors are unavoidable.
There are different ways of characterizing uncertainty quantitively. For instance, by
using descriptive statistics.
o Mean: x=
∑x
n
n
o Variance: v=∑ ¿ 1 ¿ ¿ ¿
i
o Standard deviation: s= √ v
Will tell you something about the variation/variability in the data. Will
not change upon having more samples.
s
o Standard error of the mean (SEM): SEM =
√n
Will tell you something about the certainty of the average value in the
whole batch. Upon taking more samples, certainty increases.
s
o Variation coefficient: ∗100 %
x
o Confidence interval: x ± t α ,df ∗SEM
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