> For the complete documentation index, see [llms.txt](https://zavalab.gitbook.io/statschemebook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://zavalab.gitbook.io/statschemebook/chapter-1-introduction/01-motivation.md).
# Motivation
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**Slides:** [stats\_ch1\_intro.pdf](https://github.com/vzavala/StatsBook/blob/master/Slides/stats_ch1_intro.pdf) · [4-up handout](https://github.com/vzavala/StatsBook/blob/master/Slides/handouts/stats_ch1_intro_4up.pdf) · **Code:** [Python notebooks](https://github.com/vzavala/StatsBook/tree/master/Python) · [MATLAB scripts](https://github.com/vzavala/StatsBook/tree/master/Matlab) · [MATLAB live scripts](https://github.com/vzavala/StatsBook/tree/master/Matlab-Live)
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As engineers, we often use *fundamental laws* to make *decisions*:
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* Discovery of fundamental laws has been the result of extensive collection and analysis of observations (data)
* Fundamental law is often expressed in the form of a mechanistic model
* Mechanistic model provides a concise summary of observations (knowledge) that allows us to predict and generalize
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Can you think of fundamental laws used in chemical engineering?
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Fundamental laws are powerful but only provide limited descriptions of phenomena:
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* Laws are applicable under specific settings (e.g., continuum vs. atomistic)
* Discovering laws and new mechanistic models might be challenging or cost-prohibitive (e.g., climate)
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To account for this, we also often build predictive models based purely on observations (data); such models are known as *empirical models* and also embed knowledge.
Engineering decisions rely on a combination of mechanistic and empirical knowledge.
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Can you think of empirical models used in chemical engineering?
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However:
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* Model predictions will *always* face a certain degree of *uncertainty* (due to limited knowledge/understanding/data or due to *random* phenomena affecting systems).
* Despite these limitations, we still need to be able to *make decisions*. In fact, we (as humans) make predictions and decisions in our daily lives.
* The human brain naturally gathers knowledge (learns) from observations by building empirical models and has the ability to blend such empirical knowledge with mechanistic knowledge.
* The human brain has a natural ability to hedge against uncertainty and to adapt decisions based on new knowledge.
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**Examples of random phenomena:** time to failure of a material, “pop” time of a corn kernel, molecular fluctuations.
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**Examples of brain learning:** riding a bicycle, cooking.
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Decision-making relies on the ability to characterize what is known (predictable) and not known (not predictable).
*Statistics* is the branch of mathematics that offers tools to:
* Collect, analyze, and extract knowledge (models) from data
* Characterize and model the unknown (uncertainty)
* Systematically make decisions in the face of uncertainty
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From an engineering perspective, *statistics* aids the discovery of fundamental laws and the development of mechanistic and empirical models as well as the characterization of random phenomena.
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From a scientific perspective, *statistics* provides a framework for thinking about the world that can help us understand how humans extract knowledge from data and use this to make decisions in the face of uncertainty.
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## Example: Flow Mixer
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* Consider a mixing system with input flows $$F\_1$$, $$F\_2$$ and output flow $$F\_3$$
* Sensors measure flows at time $$\omega$$: $$f\_1(\omega)=101.5$$, $$f\_2(\omega)=50.5$$, $$f\_3(\omega)=151$$ gpm
* Conservation laws tell us that $$f\_3(\omega)=f\_1(\omega)+f\_2(\omega)$$ (but this is not true). Why?
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* What is known and unknown about this system?
* What is the source of uncertainty?
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Collect 100 observations and monitor mismatch $$\epsilon(\omega)=f\_3(\omega)-f\_1(\omega)-f\_2(\omega)$$
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#### Python implementation
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Notebook [`ch1_mixer_example.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch1_mixer_example.ipynb) — open it in Colab and run the cells alongside this section.
MATLAB live script: [`ch1_mixer_example.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch1_mixer_example.mlx)
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Flow mixer example (Victor Z and Ugochukwu Ikegwu, UW-Madison, 2024).
| Slides | Python | Meaning |
| ---------------------------------------------------- | ---------------- | ---------------------------------------- |
| $$\omega$$ | `omega` | observation (time) index |
| $$S$$ | `S` | number of observations |
| $$f\_1(\omega)$$, $$f\_2(\omega)$$, $$f\_3(\omega)$$ | `f1`, `f2`, `f3` | measured flows \[gpm] |
| $$\epsilon(\omega)$$ | `epsilon` | conservation mismatch $$f\_3-f\_1-f\_2$$ |
**Note:** This notebook reproduces Example 1.1 (Section 1.1) of *Statistics for Chemical Engineers*: three flow sensors measure the flows of a mixer, and mass conservation says that f3 = f1 + f2.
```python
import numpy as np
import matplotlib.pyplot as plt
```
Simulate $$S=100$$ sensor readings of each flow (sensor noise of about 1%) and compute the mismatch $$\epsilon(\omega)=f\_3(\omega)-f\_1(\omega)-f\_2(\omega)$$.
```python
np.random.seed(0) # reproducibility
S = 100
omega = np.arange(S)
f1 = np.random.normal(100, 100*0.01/2.5, S)
f2 = np.random.normal(50, 50*0.01/2.5, S)
f3 = np.random.normal(150, 150*0.01/2.5, S)
epsilon = f3 - f1 - f2 # conservation mismatch
```
```python
fig, axs = plt.subplots(1, 3, figsize=(14, 4))
for ax, f_k, nominal, label in zip(axs, (f1, f2, f3), (100, 50, 150), ('f_1', 'f_2', 'f_3')):
ax.scatter(omega, f_k, marker='o', s=20, color='white', edgecolor='black')
ax.plot(omega, nominal * 1.01 * np.ones(S), 'k--')
ax.plot(omega, nominal * 0.99 * np.ones(S), 'k--')
ax.set_xlabel(r'Observation $\omega$')
ax.set_ylabel(rf'${label}(\omega)$ [gpm]')
ax.axis([0, S, nominal * 0.98, nominal * 1.02])
ax.grid(True)
plt.tight_layout()
plt.show()
```
```python
plt.scatter(omega, epsilon, marker='o', s=20, color='white', edgecolor='black')
plt.plot(omega, np.zeros(S), 'k--')
plt.xlabel(r'Observation $\omega$')
plt.ylabel(r'$\epsilon(\omega)$ [gpm]')
plt.grid(True)
plt.show()
```
**Note:** As in the book, the mismatch in mass conservation comes from random sensor errors, not from a flaw in the conservation law. Enough data is needed to see this, and both engineering knowledge (conservation) and data are needed to understand the system.
***
## Example: Gibbs Reactor
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* Consider a reactor in which the reaction $$CO+2H\_2\leftrightarrow CH\_3OH$$ takes place
* Data seem to indicate that the reaction is favored (achieves higher conversion $$\xi$$) at high pressure ($$P$$) and low temperature ($$T$$) but this is blurred by variability.
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* What is known and unknown about this system?
* Note that trends are blurred by uncertainty/noise.
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#### Python implementation
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Notebook [`ch1_gibbs_data.ipynb`](https://colab.research.google.com/github/vzavala/StatsBook/blob/master/Python/ch1_gibbs_data.ipynb) — open it in Colab and run the cells alongside this section.
MATLAB live script: [`ch1_gibbs_data.mlx`](https://github.com/vzavala/StatsBook/blob/master/Matlab-Live/ch1_gibbs_data.mlx)
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Data for the Gibbs reactor (Victor Z and Ugochukwu Ikegwu, UW-Madison, 2024).
| Slides | Python | Meaning |
| ------- | ------ | ---------------- |
| $$P$$ | `P` | pressure \[bar] |
| $$T$$ | `T` | temperature \[K] |
| $$\xi$$ | `xi` | conversion \[-] |
**Note:** This notebook creates the reactor data of Example 1.2 (Section 1.1) of *Statistics for Chemical Engineers*; the Gibbs reactor is used throughout the book.
```python
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
```
Each file has columns $$(T, P, \xi)$$; we stack the high- and low-temperature data and add a little noise to blur the pattern, as in measured data.
```python
data_high = pd.read_csv(path+'Data/gibbs_hightemp_class.dat', sep="\t", header=None)
data_low = pd.read_csv(path+'Data/gibbs_lowtemp_class.dat', sep="\t", header=None)
data = pd.concat([data_high, data_low])
n = len(data)
np.random.seed(1) # reproducibility
T = data.iloc[:, 0].to_numpy() + np.random.randn(n)*0.1 # temperature
P = data.iloc[:, 1].to_numpy() + np.random.randn(n)*0.1 # pressure
xi = data.iloc[:, 2].to_numpy() + np.random.randn(n)*0.1 # conversion
```
```python
fig, axs = plt.subplots(1, 2, figsize=(12, 5))
axs[0].scatter(P, xi, marker='o', s=20, color='white', edgecolor='black')
axs[0].set_xlabel('$P$ [bar]')
axs[0].set_ylabel(r'$\xi$ [-]')
axs[0].grid(True)
axs[1].scatter(T, xi, marker='o', s=20, color='white', edgecolor='black')
axs[1].set_xlabel('$T$ [K]')
axs[1].set_ylabel(r'$\xi$ [-]')
axs[1].grid(True)
plt.tight_layout()
plt.show()
```
**Note:** As discussed in the book, data reveal trends but cannot explain them alone; a model (thermodynamics) helps, and some aspects of the system remain unknown, so new experiments may still be needed.
***
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$$
\begin{aligned}
\mu^{out}*k&=\mu\_k+\gamma\_k\cdot \xi\cdot \mu*{CO},;k\in K\\
\xi&=1-\frac{\mu^{out}*{CO}}{\mu*{CO}}\\
K\_{eq}&=\frac{(\mu\_{CH\_3OH}+\xi\cdot \mu\_{CO})}{(\mu\_{CO}-\xi\mu\_{CO})(\mu\_{H\_2}-2\cdot \xi\cdot \mu\_{CO})^2}\left(\frac{\mu\_{out}}{P}\right)^2\\
\frac{\partial \log K\_{eq} }{\partial T}&=\frac{\Delta H\_0}{R\cdot T^2}
\end{aligned}
$$
* We know fundamental conservation and thermodynamic laws hold and the mechanistic model allows us to predict $$\xi$$ from $$P$$ and $$T$$.
* This knowledge, however, is limited (e.g., makes assumptions, cannot explain random behavior).
* How to predict $$\xi$$ as a function of $$P$$ and $$T$$ without any mechanistic knowledge?
* What are trade-offs between mechanistic and empirical knowledge?
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***
## Motivation
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**Don't forget:**
Inherent limitation of engineering practice: no matter how sophisticated our predictive models and sensing devices are, we will *always* have a certain degree of uncertainty.
Characterizing both known and unknown aspects of a system is key.
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