Forecasting and Statistical Modeling Practicum

A series of Jupyter notebooks on regression diagnostics and classical time-series modeling. The work moves from least-squares model construction and statistical significance tests to trend/seasonality analysis, residual diagnostics, autoregressive and moving-average processes.

Python · NumPy · pandas · SciPy · statsmodels · scikit-learn · Matplotlib

What is covered

The notebooks include:

• construction of regression design matrices;
• ordinary least squares implemented from matrix formulas;
• comparison of polynomial and nonlinear model forms;
• coefficient significance testing;
• confidence intervals for coefficients and predictions;
• Fisher and Student statistical tests;
• residual diagnostics and model-adequacy checks;
• trend removal and differencing;
• stationarity diagnostics with the Augmented Dickey–Fuller test;
• seasonal-pattern estimation with Fourier terms, time-slot indicators and seasonal differencing;
• autocorrelation analysis with ACF/PACF;
• stationarity and invertibility conditions;
• autoregressive AR models;
• moving-average MA models;
• classical ARIMA tooling from statsmodels.

Notebook progression

1.ipynb — model construction and least squares

The first notebook works with different functional model forms, builds the experiment/design matrix explicitly and implements least-squares estimation. It compares fitted relationships using numerical and visualization tools rather than relying only on a ready-made estimator.

2.ipynb — significance and confidence intervals

The second notebook focuses on statistical inference around a fitted regression model:

• Student tests for coefficient significance;
• Fisher comparison of nested models;
• confidence intervals for coefficients;
• confidence and prediction intervals;
• a model containing polynomial and Fourier components.

3.ipynb — model adequacy and residual diagnostics

The third notebook checks whether the regression assumptions are reasonable. The analysis includes residual behavior, tests related to constant expectation/variance and autocorrelation diagnostics using statsmodels.

4.ipynb — trend, stationarity and seasonality

The fourth notebook studies two time series from TimeSeries.xls and separates deterministic trend from periodic behavior.

For the pressure series it covers:

• visual trend and ACF analysis;
• least-squares estimation of a linear trend;
• detrending and first differences;
• Augmented Dickey–Fuller tests under several deterministic specifications;
• comparison of the original and differenced series from a stationarity perspective.

For the flow series it identifies a daily seasonal period of 12 observations at a two-hour sampling interval and compares several ways of removing that periodic component:

• a non-singular Fourier basis for period 12;
• time-of-day indicator variables;
• direct per-slot seasonal means;
• seasonal differencing with lag 12.

The notebook also compares residual ACF and variance across the seasonal-adjustment approaches. The indicator and per-slot-mean formulations are shown as equivalent representations of the same 12-slot seasonal profile.

5.ipynb — autoregressive models

The AR section studies autoregressive processes analytically and numerically. Examples include:

• stationarity through characteristic roots;
• ACF recurrence using Yule–Walker relationships;
• ACF/PACF visualization;
• fitting/working with autoregressive models through statsmodels.

6.ipynb — moving-average models

The final notebook covers MA processes, including:

• stationarity and invertibility;
• characteristic roots;
• analytical autocorrelation structure;
• ACF/PACF diagnostics;
• use of ARIMA tooling for classical time-series modeling.

Data

The repository contains the original .xls datasets used by the notebooks:

TimeSeries.xls
ls_variants.xls

The notebooks expect these files to be available in the repository root when run locally.

Running locally

Install the dependencies:

pip install -r requirements.txt

Then start Jupyter and run the notebooks from the repository root so the relative paths to the .xls files remain valid.

Why this repository is in the portfolio

This is primarily a statistical-modeling repository rather than an end-to-end production forecasting service. Its value is that it shows the mathematical side behind common modeling tools: least squares, statistical tests, uncertainty intervals, trend and seasonality, residual diagnostics, ACF/PACF and AR/MA model assumptions.

That foundation is useful when a forecasting or regression problem requires more than calling a library model and comparing a single metric.