Standard axes often miss the primary directions of data spread. PCA rotates the coordinate system to align new axes with maximum variation.
- The left panel plots original points on x and y, showing a clear diagonal pattern in the colored dots.
- The middle panel shows the axes rotated so PC1 runs along the main trend and PC2 crosses it at right angles. - The right panel projects the points onto a single line using only PC1, preserving their order and most variation.
It is used to simplify high-dimensional sensor data in quality control for manufacturing processes.
🧭 Basis of a Vector Space: The Building Blocks of Linear Algebra
A basis is the smallest set of vectors needed to describe every vector in a vector space. It provides a unique coordinate system and reveals the true dimension of the space.
Confidence Interval (σ known)
When the population standard deviation (σ) is known, we use the Z-distribution to construct a confidence interval for the population mean. It gives a range of values within which the true mean is expected to lie with a certain level of confidence.