Here's how you can evaluate performance in statistics using various methods.

Mean Squared Error-MSE measures the average squared difference between the predicted values and the actual values. Root Mean Squared Error- RMSE is the square root of MSE, providing a measure of the typical magnitude of error. Mean Absolute Error-MAE measures the average absolute difference between the predicted values and the actual values. Coefficient of Determination-R-squared indicates the proportion of the variance in the dependent variable that is predictable from the independent variables. ?Values range from 0 to 1, where 1 indicates a perfect fit. ? Accuracy, Precision, Recall, F1 Score-for classification problems ? Confusion Matrix-A table showing the counts of true positives, true negatives, false positives, and false negatives.

Descriptive statistics are not the first step of any data project for no reason: if done appropriately, they might guide your whole analysis and even completely shift the problem. Indeed, they are the business cards of your data and this first glance can let spark questions we did not even think about at first. These metrics are often the easier to compute, to understand and to communicate - so make sure to always start with them.

Descriptive statistic is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics is the process of using and analysing those statistics.

Descriptive statistics are the cornerstone of data analysis, providing a simple yet powerful way to summarize and describe data features. This method includes calculating measures such as mean, median, mode, range, and standard deviation. These statistics are straightforward to compute and can be easily understood by laypeople. For example, the mean provides an average performance measure, the median offers a midpoint of data, and the standard deviation indicates the spread or variability in performance. These are tangible figures that can help clients grasp the basic properties of the data without needing to understand complex statistical theories.

Exploratory Data Analysis (EDA) with a focus on descriptive statistics is vital in data analysis and machine learning. Large differences between the mean and median could be an indication of presence of outliers or non-normal distribution of the data. Standard deviation measures the level of dispersion in the data, and a common outlier removal method is excluding data points >3 standard deviations away from the mean. Another technique, the Interquartile Range (IQR), considers points below Q1 - 1.5*IQR or above Q3 + 1.5*IQR as outliers. Effective outlier management is key to unbiased statistical results. For instance, in a recent Kaggle Machine Learning competition, removing such outliers doubled our model's performance.

There are parallels in applying Hypothesis Testing to jurisprudence & empirical economics. True, many will concede the cost of misjudgement in the latter is neither false imprisonment (Type II error) or a free criminal (Type I). Now, in the British legal system the Null Hypothesis is Innocence. In empirical economics we have to choose what exactly to make the Null. & since we are most comfortable with the status quo, we give it the benefit of the doubt, & so risk confirmation bias leading us into a Type I error. What of Confidence Intervals that Fan Out? Let’s think of a drinker in a bar, & predicting his position in 5min intervals: ever-wider concentric circles. As to his most likely position on his alarm chiming: well, PRECISELY his bed!

While inferential tests allow generalizations about a population based on the sample data, it is important to consider the assumptions involved with each method. These tests are widely used for hypothesis testing but errors are common and often missed even by professionals. Some examples include t-tests, chi-square tests, etc.

Inferential statistics are crucial for making reliable predictions about larger populations based on sample data. For instance, pharmaceutical companies use hypothesis testing in clinical trials to establish if a drug genuinely improves patient outcomes, rather than effects occurring by chance. Market researchers use confidence intervals to determine the probable range of consumer preferences. Environmental scientists apply p-values to evaluate if changes in ecosystems are likely due to pollutants. These statistical tools are essential for drawing valid conclusions and informing decision-making across different fields.

Inferential tests basically are statistical tests used to determine whether the results of a particular research on a given sample can be generalised to the general population. It involves hypothesis testing, which is evaluation of a sample and estimating the value or potential range of values of some characteristics of the population based on that of the sample.

Tests are some of the most used statistical tools out there: their fields of application are multiple, they help guide decision making and are generally quick to put in place with respect to their power. However, what often gets left on the side for the sake of efficiency are hypothesis: as they can make or break your results, be mindful about them - if we set our problem in the world of fairies, of course everything will be possible...

As a fresher to econometrics, regression analysis was the 1st tool I was taught, one illustrated, quite literally, by Francis Galton’s work on predicting a child’s adult height by using the mean height of its parents. Whilst this ‘regression to the mean’ was a powerfully logical introduction to empirics, it came with clear caveats. Cautions best captured, or should that be shaped?, by Anscombe’s Quartet & The Yule-Simpson Paradox. My favourite ‘illustrations’ of why to avoid confusing coincidence with causality came when presented with data on the number of storks on the x-axis & level of child births on the y. ‘Their close correlation’ hardly captured storks delivered babies, rather their common positive integration over time.

Regression analysis is a valuable method for evaluating performance by understanding relationships between variables. It involves identifying whether a dependent variable (e.g., performance outcome) can be predicted from one or more independent variables (e.g., time spent training, investment levels). Linear regression, the simplest form, models relationships in a way that is interpretable for non-statisticians: it shows how much we expect the dependent variable to increase whenever the independent variable increases by one unit. This can be visually represented through a regression line on a scatter plot, making it relatively easy for laypeople to see trends and relationships in the data.

Regression analysis measures the strength and direction of relationships between variables. They are a way of quantifying the model's explanatory power but they can often be misinterpreted and/or lead to the ignorance of other more comprehensive metrics. That said, large or small R-squared values can be an indicator of whether your variable selection is on the right track but optimizing for this value can lead to overfit models quite easily.

What never ceases to amaze is the power of simplicity. Just as arithmetics, regression analysis impact and power are far bigger than the tools needed to put it in place. Indeed, it is maybe the most basic principle of data science, but this does not limit its results in any way - when in doubt, regress!

One of the critical aspects of regression analysis is interpreting the coefficients, which quantifies the relationship between each predictor variable and the outcome. It goes beyond a simple "one unit changein the predictor leads to a certain change in the outcome" narrative. While this interpretation is foundational, the real challenge lies in making sense of these results to derive meaningful insights. Being a skilled analyst involves creativity in interpreting coefficients to extract meaningful insights. It requires thinking beyond the numerical output to understand the story behind the data.

Nonparametric methods in statistics are valuable alternatives to parametric methods when assumptions about data distribution or variance are not met. Unlike parametric tests that assume specific population parameters, nonparametric methods make fewer assumptions and are robust to outliers and non-normal data. These methods include tests like the Mann-Whitney U test for comparing two independent groups, the Wilcoxon signed-rank test for paired samples, and the Kruskal-Wallis test for comparing more than two independent groups. I think people should be more open to exploring nonparametric tests because rarely will primary data be symmetric, and these methods offer reliable analysis without unachievable distributional assumptions.

It is also expected that in cases of data non-linearity then Non-parametric methods should be preferred in analysis over parametric methods.

That we are most comfortable without risk to our health & not least wealth, is testified by the considerable size of the insurance & legal professions. And yet like it or not, we live & work surrounded by uncertainly. Indeed, for those of us involved in finance & economics, risk is not so much an occupational hazard, as the very reason we find employment. Now, our work is made so much easier when we accept we are not in a closed lab, & so employ, or rather borrow from Control Engineering, their disciplines & skills across State-Space Modelling & Kalman Filtering. The plain truth is we have to accept that in practical financial modelling, parameters are not only rarely fixed, but at times as dynamic as the ‘variable’ they are attached to.

Nonparametric methods in climate and sustainability research offer flexibility when the data doesn't assume a specific distribution, which is common with complex environmental datasets. These methods are crucial for analyzing non-linear and non-normal data, such as the irregular growth patterns of vegetation under varying climate conditions or the distribution of pollutants in different ecosystems. For example, using the Mann-Whitney U test, researchers can compare the health of ecosystems in polluted versus unpolluted areas without assuming a normal distribution of data. These methods provide crucial insights for environmental management and policy-making, adapting to the unique, often complex data found in sustainability studies.

Many will have heard the justification for economists as: they exist to make weather forecasters look good; or is it the other way round? Either way, in their keenness to predict meteoritical & market futures, both employ similar ARIMA models. After all, in predicting tomorrows weather or equity index, anchoring at the recent past makes sense. So much has momentum become part of investing, NOT to use auto-regression would put one at a market disadvantage. Indeed, the most common moving average durations are so cemented in trading strategies to have practically become self-fulfilling, re the Keynes beauty contest. With so many others attracted to ARIMA models, so must those who FAR prefer fundamental macro & micro economic & financial data.

This method is great for data indexed in time order but may require more specialized knowledge to interpret and understand, making it less ideal for a lay audience unless the presentation is adjusted for simplicity.

Time series method of forecasting involves analysing historical data collected overtime in order to identify patterns and trends. These data should have been collected at regular intervals (days, months, years). It is important to note that the distribution of an observation at a certain time point conditional on the previous value of the series depends on the outcome of those previous observations. Basically, the outcomes are not independent.

Time series analysis is pivotal in the climate and sustainability sectors for predicting future environmental changes based on historical data. For instance, using ARIMA models, scientists forecast rising sea levels by analyzing decades of ocean data. Similarly, ACF and PACF help in understanding the periodicity of natural phenomena like El Ni?o events, aiding in agricultural planning and disaster readiness. Such analyses enable more precise predictions of climate patterns, assisting policymakers in developing informed environmental strategies and resilience plans.

Signal-reception systems need safeguards. As with all great fiction there are fascinating kernels of fact. In The Count of Monte Cristo the semaphore ruse was inspired by a scam performed in the early 1830s by the Blanc brothers; the MAN IN THE MIDDLE trick. The truth is hacking a ‘signalling system’ began the moment man began to transmit & receive information in search of superior capital gain. Be in no doubt each signal-reception-action strategy has always, & will forever be, open to false-flares, head fakes etc. Those putting their faith & wealth into AI-inspired algos, or any quant-system, promising to exploit ‘advance signal recognition to lucratively act’, should be conscious of the risk that what man codes, some Count will game.

Machine learning revolutionizes the climate change and sustainability sectors with predictive and analytical capabilities. Supervised learning forecasts renewable energy outputs from variables like weather, enhancing grid management. Unsupervised learning identifies ecological zones vulnerable to climate impacts, guiding conservation priorities. Additionally, it optimizes routes for electric vehicle fleets by analyzing traffic and charging data, reducing emissions and promoting cleaner transportation. This underscores machine learning’s critical role in driving sustainability, with its impact echoed across numerous other sectors.

Machine learning is a field of Artificial intelligence which is basically the ability of a machine to imitate human behaviour. It focuses on algorithms that learn from experience and improve their decision-making or predictive performance over time. The types of machine learning are; supervised, semi supervised, unsupervised and reinforcement. Examples are, facial recognition, natural Language Processing and many others

One thing I find helpful is to always do an EDA (Exploratory Data Analysis)...even when the raw data seems basic, or straightforward. The EDA approach identifies general patterns in the data, informing the type(s) of analysis one may use. I find, its never good to exclude obvious outliers or unexpected data features. The EDA includes all data, and the visualizations help me really explore the inferences of the dataset as a whole. It's also nice to investigate what the data has to offer without focus on confirming any statistical hypothesis...just letting the data show whatever is there, and moving forward without biases.

There ARE 'working' in finance those who believe a: Chow Test, a cooking competition; Occam’s Razor, a grooming tool; Simpson Paradox, a cunning chess strategy; Type II error, misdiagnosis of blood-sugar; & Fallacy of Composition, the hidden truth of Area 51. There are, after all, those working in finance who fail themselves & those they advise; considering modular paradigm shifts, as relating to a musical score. As valuable as it is, time series can easily trip us up if not careful where we are looking when stepping back to glean the future. In relation to say inflation, is history before e-commerce & ESG relevant? There are those too in financial markets whose art, is just that; aphotheniacs charting lines absent of any fundamentals.

Goodness-of-fit tests like the Kolomogorov-Smirnov test can assess how well a statistical model fits the observed data. Cross-validation is the splitting of data into training & test sets multiple times to ensure the model's performance is consistent and not overly dependent on a specific data partition. Residual analysis examines the difference between observed and predicted values to check for patterns that might indicate model inadequacies or violations of assumptions. Bayesian methods are becoming more common. They update the probability estimate for a hypothesis as more evidence or information becomes available, incorporating prior knowledge and uncertainty.

Your analysis is only as good as your data: focus on data, mainly how to get it, structure it and integrate it in a larger infrastructure, the rest will come later