How do I characterise the relationship between one variable and many other variables for a complex socio-technical problem? I am doing a PhD on a socio-technical problem – this is a building physics problem, where I am trying to determine the relationship between one variable (x) which affects a number (up to 20 other) of other variables (a, b, c etc).
I have calculated x, which can be characterized as both discrete and continuous. All the other variables (a, b, c etc) are continuous. The relationships are complex, as one key influencer over all variables is (unpredictable) human behaviour, and this behaviour can be seasonal.
Due to the seasonal nature of the behaviour, I will likely subset the data based on the seasons and then start to investigate the relationships.
What are the different available methods and techniques for determining the relationship between one variable (x) and a number of other variables (a,b,c)?
I am aware of the following possible methods, but interested in any comments/observations and if there are any more that i should also consider…

*

*Multiple linear (continuous dependent variable) regression. Must
have linear relation between dependent and independent variables.

*Polynomial regression. Nonlinear.

*Logistic regression. Discrete dependent variable.

*Correlation. However, this is only for one variable vs another?

Thank you for any advice.
Edit 16.08.2021
My project is specifically looking at 'The relationship between ventilation practices, indoor air quality, noise and overheating, and their impact on health.' 3-minute video on my project here.
Variable x is ventilation, which can be expressed as both a discrete variable (ventilation practices, window open for example) and as a continuous variable (air exchange rate).
Other variables are (a,b,c etc): temperature, relative humidity, carbon monoxide, carbon dioxide, PM1, PM2.5, PM10, (total) volatile organic compounds, noise, nitrogen dioxide. These are all continuous (measurements).
A side note, is that I can also express the above variables as a rate of decay (for periods of decay).
The relationships here are complex, as human behaviour is the main driving 'force' behind ventilation, by opening and closing windows for example. But human behaviour is also the main driver for the other variables - cooking for example can produce CO, CO2, NO2, VOC, PM etc. Vacuuming, burning candles, heating home, bathing etc all produce pollutants.
My data comprises of 5 minute time-series data for all the variables above.
But...the hypothesis is that there is a relationship between ventilation and the other variables (or the rate of decay of the other variables). So, I am looking at different ways to investigate this relationship in a number of big datasets (using Python). I am looking for advice on different methods for investigating this.
 A: Correlation is typically what I use to screen for association between variables in a complex system with unknown characteristics.  The correlation matrix is a snapshot which shows the association between all possible pairs of variables.  In addition, I typically use Spearman rank correlation, since it's not-sensitive to outliers and doesn't really require central tendency, i.e., bulk of data in the middle of a histogram.  Pearson correlation, on the other hand, does require central tendency and few outliers in order to be less biased than Spearman.  Pearson correlation, and many hypothesis tests in statistics, assume varying degrees of normality, and therefore a unimodal distribution.  If you have multimodal distributions among your variables, then Pearson is inapplicable, but Spearman may recover some informativeness.  Spearman can also handle triangular distributions quite well.
For the regression models you listed, you'd have to run all three and then compare fits and significance.  Non-linear may be more promising than linear.  Logistic makes no assumptions about the distribution of the predictors (inputs), so that may be helpful.
You might also consider running PCA (principal component analysis) on the variables, and then plot the first two principal component score vectors (n-tuples) against one another using variable labels to see if variables cluster together.
Last, I would consider unsupervised manifold learning to reduce dimensionality and look for patterns among variables (or objects with the data transposed) using e.g. crisp K-means cluster, fuzzy-K-means cluster, diffusion maps, self-organizing maps, localized linear embeddings, Laplacian eigenmaps, locally preserved projections, unsupervised artificial neural networks, stochastic neighbor embedding,  Sammon mapping, non-negative matrix factorization, classic multidimensional scaling, non-metric multidimensional scaling, unsupervised neural gas, Gaussian mixture models, unsupervised random forests, kernel distance-based PCA, kernel Gaussian radial basis function PCA, kernel Tanimoto distance-based PCA, and hierarchical cluster analysis.
The ROI will be greater if you first focus on knowledge discovery through pattern recognition among objects and features.  The key point about non-linear manifold learning methods is that you will always be reducing dimensionality to make 2D and 3D plots of the major 2 or 3 dimensions obtained from the various procedures.  Clustering among objects or features may become apparent with several of the manifold learning approaches.
