Do The Exercises Below Questions Based On Chapter 2

Do The Exercises Belowquestions Based On Chapter 2 Exercises1 Base

Do the exercises below. questions based on chapter 2 exercises: (1) based on exercise 22 A) Describe how to map values from the interval [-1,1] to the interval [0,1]. B) Consider two time series as two sets on numbers: e.g. S1 = {3, 5, 7, 8, ..., 4, 7} and S2 = {5, 5, 8 , ..., 9, 7}. You need to come up with 2 sets, each having 100 elements. You can make them random if you like.

1) Normalize each set in range [0,1]. 2) Find the probabilities of S2 in the ranges [0,0.1], (0.1,0.2], (0.2,0.3], ..., (0.9,1] when S1 are in the same ranges [0,0.1], (0.1,0.2], (0.2,0.3], ..., (0.9,1] respectively. Calculate the probability of the value of S2 falling into each range, given that the value of S1 is in the same range, respectively.

Paper For Above instruction

The process of mapping values from the interval [-1,1] to [0,1] is essential in numerous data normalization contexts, especially in machine learning and data preprocessing. This transformation ensures that data resides within a bounded interval, facilitating standardized analysis. One common method involves linear transformation: for any value x in [-1,1], the mapped value y in [0,1] can be obtained using y = (x + 1) / 2. This mapping shifts the original range from [-1,1] to [0,2], then scales it down to [0,1], preserving the relative position of each data point within the interval. This linear normalization is straightforward, computationally efficient, and maintains the proportions between data points, which is beneficial in many algorithms.

In the context of analyzing two time series, S1 and S2, each comprising numeric data points, it is often useful to normalize these data sets to a common scale. To do this, each set should be normalized within the [0,1] range. The normalization process involves identifying the minimum and maximum values in each set and applying the formula: (x - min) / (max - min). This rescales each data point to a value between 0 and 1, facilitating comparison and statistical analysis across the two series.

Once normalized, the distribution of the data within specified ranges can be analyzed to determine probabilities. Specifically, for each series, we can categorize data points into deciles or equally-sized intervals, such as [0,0.1], (0.1,0.2], and so forth, up to (0.9,1]. These ranges partition the normalized data into ten equal parts. The key analysis involves calculating the probability that, for the second series (S2), a random data point falls into each interval, conditioned on the corresponding data point in series S1 lying in the same interval. This conditional probability offers insights into the relationship and dependence between the two series, potentially revealing patterns or correlations inherent in the data.

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