Machine Learning Notation - From Flight Logs to Symbols

The Flight Log Comes First

An RC plane club has been keeping a small flight log.

Some flights land cleanly. Some do not.

The question is simple:

Can we look at a few flight conditions and predict whether the landing will succeed?

This is where machine learning begins.

Not with symbols.

With records.

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Rows and Columns: RC

Our dataset is a table of notes from RC flight experiments.

Each row records one flight: what we observed before landing, and whether the landing succeeded.

Each column tells us one thing about the flight: Wind, Battery, Time, or Landing.

We want the model to look at the first three columns and predict the last column.

Rows move down the table.

Columns move across the table.

The plane gives us the memory hook:

R = Row
C = Column

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Features and Label

Now we can split the columns into two groups.

Wind, Battery, and Time are features — the model’s input.

Flight ID is not a feature.

It is just the row name.

The model learns from Wind, Battery, and Time.

Land is the label — the output we want the model to predict.

In machine learning notation, we use x for features and y for the label.

Because models work better with numbers, we turn Landing into a simple code:

$1$ means the landing succeeded.
$0$ means the landing failed.

For flight F001, the model sees wind 8, battery 92, and a 14-minute flight:

$$x = [8, 92, 14]$$ $$y = 1$$

For flight F002, the model sees wind 22, battery 35, and a 9-minute flight:

$$x = [22, 35, 9]$$ $$y = 0$$

That is the first notation move:

x is the input.
y is the output.

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The Shape of the Data

Now we can describe the shape of the dataset.

$n$ is the number of rows.

These are our training examples.

We logged 4 flights:

$$n = 4$$

$m$ is the number of features.

These are our input columns.

We tracked Wind, Battery, and Time:

$$m = 3$$

Notice what we left out.

We did not count the label column: Land.

Land is the answer key, never a feature.

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Pointing Inside the Table

Now that we know the shape, we need a way to point inside the table.

$i$ points to a row.

Since there are $n$ rows, $i$ can go from $1$ to $n$:

$$1 \le i \le n$$

$j$ points to a feature column.

Since there are $m$ features, $j$ can go from $1$ to $m$:

$$1 \le j \le m$$

Back to the RC hook:

Rows use $n$ and $i$.

Columns use $m$ and $j$.

So:

$i$ steps down. $j$ steps across.

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One Row, One Feature, One Answer

Sometimes we do not want the whole input row.

We want one specific input value from one specific flight.

For example: the Wind value from flight F001.

That sentence has three pieces:

input value, row, and feature column

Now we give each piece a symbol:

input value → $x$
row → $i$
feature column → $j$

So the notation becomes:

$$\text{input}_{\text{Column}}^{(\text{Row})} \rightarrow x_{\text{Wind}}^{(\text{F001})}$$

For flight F001, Wind is the first feature column, and the landing succeeded:

$$x_1^{(1)} = 8 \quad\text{and}\quad y^{(1)} = 1$$

For flight F002, Wind is also the first feature column, but the landing failed:

$$x_1^{(2)} = 22 \quad\text{and}\quad y^{(2)} = 0$$

Read it as:

row $i$, feature column $j$

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Bringing It All Together

Now play with the table below.

Move $i$ down the rows.

Move $j$ across the feature columns.

Watch how $x_j^{(i)}$, $x^{(i)}$, and $y^{(i)}$ change together.

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Flight Log Field Notes

The RC Flight Log

The dataset. A notebook of what we observed.

The RC Flight Log. The dataset. A notebook of what we observed.

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