Spaceman Spiff. Best Answer: The coefficients "a" and "b" for least squares regression may be found as follows: For the equation y = ax + b. Yahoo! Answers is currently one of the most popular question answering systems . .. A survey, a taxonomy and an analysis of social information retrieval approaches and platforms, Information Systems, Publication Date, (yyyy-mm-dd) . It has been widely applied to solve L1-regularized logistic regression. Practice: Calculating and interpreting residuals · Calculating the equation of a regression line · Practice: Calculating the equation of the least-squares line.

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Smoothed Returns On a day-to-day basis, market prices are extremely noisy and for the most part completely random. It is often useful to remove some of this noise by smoothing the returns, which in turn develops a more stable signal which can be consumed by a systematic investment process.

### Regression analysis using Python

A 0 day half-life is interpreted as no smoothing, so this simply yields the raw data. The smoothed signal is computed as follows: Return Distribution Many of the statistical techniques in modern finance are predicated on the assumption that asset returns are normally distributed.

However, it is often the case that returns are not normal, which is why during the GFC many funds experienced greater than -6 sigma events one day after the next.

While returns are not normal, the statistical techniques that are commonly used are still a reasonable engineering approximation when markets are not under stress. When they are under stress, all bets are off no pun intendedwhich can lead to bad outcomes for investors.

In addition, the mean and standard deviation of these daily returns are calculated. They are then used to compute a normal probability density function which is scaled appropriately and then overlaid on the chart. Daily Return Frequency Distribution" ; chart.

The standard deviation of SPY daily returns over the past 20 years comes out to be 1. The fitted distribution predicts that a This is not inconsistent with the actual frequency distribution as shown in the second plot below, which is zoomed into the left side of the distribution inorder to magnify this part of the plot. As mentioned earlier, many funds experienced -6 sigma events or greater during the global financial crisis, day after day.

## Find the slope, b of the least squares regression line?

The plot below zooms into the left tail of the distribution demonstrating that extreme events occur far more frequently than a normal distribution would suggest. In the above example we leverage a function called normal to generate a scaled normal distribution that we can superimpose on the plot to get a sense of how well the return histogram fits such a model. The code below simply generates a normal curve given the mean, standard deviation of the daily returns, with an appropriate scale factor to fit the histogram.

De-noising the returns does not yield a better fit, and the positive skew in the distribution becomes more apparent.

In addition, while there is a positive skew, it is also clear that smoothing the returns highlights far more negative extreme events than positive events. In this example, we request the most recent data for several tickers, and we are presented with a DataFrame containing all available fields as columns.

Discrete values are difficult to work with because they are non differentiable so gradient-based optimization techniques don't apply. It works by automatic selecting statistically significant independent variables to include in the regression analysis. This decision tree can be used to help determine the right components for a model.

The least squares method minimizes the sum of the errors squared, where the errors are the residuals between the fitted curve and the set of data points. The residual can be calculated using perpendicular distances or vertical distances. The errors are squared so that the residuals form a continuous differentiable quantity. In the case of vertical offsets the error is equal to the difference between the value from the data-set and the computed value from the regression line where and are additional explanatory variables in a multiple regression.

In the case of perpendicular offsets the error is the sum of the distance,between the data points,and the point along the regression curve,perpendicular to that. For a straight line, this can be calculated by solving a quadratic equation.

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Simple linear regression analysis This optimization problem for straight lines was further simplified by Kenney and Keeping who introduced the concept of the center of mass of the dataset,and related this to the y intercept of the fitted line. This optimization problem is mathematically modelled as, Sum of x squares These two measurements can be combined to calculate the overall sum of squares, Overall sum of squares Using just these three variables,and the center of mass it is possible to construct the straight line linear regression of the form,which minimizes the sum squared error,of the residuals between the line and the datapoints.

These parameters are all that is needed to draw the linear regression analysis which fits a set of observed data points. Lastly, the overall quality of the regression analysis is measured using the correlation coefficient, Iterative methods - for harder problems fun For more complex functions iterative methods need to be applied.

An iterative procedure is one which generates a sequence of improving approximate solutions to a particular problem. This is also known as a search or optimization algorithm. There are two classes of optimization algorithms, exhaustive or heuristic.

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Heuristic methods on the other hand use knowledge about the optimization problem to locate good solutions. One widely used heuristic is the gradient of a function because when this is equal to zero, that point in the function is either a local minima or maxima. Gradient methods such as gradient descentthe Gauss Newton methodand the Levenberg Marquardt algorithm adjust the solution such that the derivate is either minimized or maximized.

Another widely used heuristic is line of sight a. These methods don't use gradients but instead generate points within the search space and "look for" the optima.