Math & Data Science

From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array indexes start at 1.

Percentile is a statistical measure used to describe the position of a particular value in a dataset relative to the remainder of the values. In education, percentiles are commonly used to rank students’ performance. For instance, if a student is in the 90th percentile, it implies the score of the student is higher than 90% of the other students. To calculate percentile of a student, use the following formula:

For instance, if the scores of five students are given as an array {30, 60, 80, 72, 15}, then the percentile array for each student will be {25, 50, 100, 75, 0}. This implies that the number of students with scores less than 30 is 1; thus, the percentile for the first student is (1 ÷ (5 - 1) × 100) = 25, the number of students with scores less than 60 is 2, and the percentile for the second student is (2 ÷ (5 - 1) × 100) = 50. The same applies thereafter.

The function calculatePercentile receives the scores of each student in the argument arr and calculates and returns the percentile of each student. Assume that the argument arr contains two or more numbers between 0 and 100.

From the answer group below, select the correct combination of answers to be inserted into A and B in the description.

Bag of Words is a representation of text that describes the occurrence of words in a document. It is a feature extraction method in text mining. From a document collection (or corpus), a feature vector for each document can be created by considering the count of each word in that document.

The following are two documents that construct a text corpus:

Here, the word sequence, defined by the unique words (ignoring case and punctuation) in the documents, is as follows:

Now, feature vectors for document-I and document-II can be created using the unique words.

Again, two new text documents and the corresponding word sequence are given below:

Here, the word sequence for those two documents is {He, is, a, an, good, boy, honest, the}. Then, the Feature Vectors for document-III and document-IV will be A and B, respectively.

From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array index starts at 1.

Inverse document frequency (IDF) is a metric used to evaluate the importance of a word in a document relative to a document collection. The function calcIDF receives a string term and an array of strings corpus as arguments, and calculates and returns the IDF score for the term based on the given corpus. The corpus of documents corpus is given as a string array. An example of corpus that indicates 3 documents is as follows:

The table shows the description of the functions used in the program.

Traditionally, IDF is computed using the following formula:

Where N is the total number of documents, t refers to a specific term or word in a document, and df(t) is the number of documents containing the term t.

For example, the return value of calcIDF("ITPEC", corpus) is log(3 ÷ (1 + 2)) = 0 when array corpus is assigned as shown in the above example. Because the total number of documents is 3, and the total number of documents that include the word “ITPEC” is 2.

From the answer group below, select the correct combination of answers to be inserted into blank in the program. Here, the array index starts at 0.

The Jaccard similarity index, J, measures the similarity between two sets by computing the ratio of the number of elements in their intersection to the number of elements in their union, i.e., J(A, B) = |A∩B| / |A∪B|, where A and B represent sets, ∩ and ∪ represent the intersection and the union set operations, respectively, and |…| is the operator to calculate the number of elements in a set. The intersection operation builds a new set taking members that are common to both sets; while the union operation builds a set taking all members of both sets, where each member will appear only once. For instance, A = {3, 4, 6, 7} and B = {2, 4, 5}, then A∩B = {4}, A ∪ B = {2, 3, 4, 5, 6, 7}, |A| = 4, | B| = 3, |A∩B| = 1, and |A∪B| = 6.

The function jaccardSimilarity calculates the Jaccard similarity stated above and returns it. The definitions of variables used in the function are stated in the below table.

From the answer group below, select the correct combination of answers to be inserted into A and B in the description. Here, the array index starts at 1.

The calcDistance returns a value based on array p1, p2 and positive integer n. Assuming that the number of elements in arrays p1 and p2 are the same. The function abs(x) returns the absolute value of x, and pow(a, b) returns a raised to the power of b.
In the case of p1 is {3, 1, 5, 2} and p2 is {4, 6, 2, 3}, the calcDistance(p1, p2, 1) returns A, whereas calcDistance(p1, p2, 2) returns B. As the value of n increases, those of calcDistance(p1, p2, n) converges to 5.

From the answer group below, select the correct combination of answers to be inserted into A through C in the program.

Craps is a casino dice game in which players bet on the outcomes of the roll of a pair dice. In the rules of the game, a player rolls two dice and finds their sum. There possible outcomes could lead to a win or loss.

The following program approximates the chance of winning a game of Craps using Monte Carlo method and outputs it. The Monte Carlo method is a technique used to estimate the probability of certain outcomes of an experiment by running multiple trial runs, using random numbers. In the program, the playing Craps is illustrated simply by generating random numbers rather than actually rolling a pair of dice. The function random_int(1, 6) generates a random integer number between 1 and 6, and returns it.
The program also calculates and outputs the relative error of the measured approximate probability of winning Craps. In general, the relative error Er is calculated using the following formula:

where Pm denotes the measured approximate probability and Pt denotes the theoretical probability. The theoretical probability of winning Craps is known to be 244/495.

From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array indexes start at 1.

Bicycle ridership in a city is studied. Examination of several years of data revealed that 30% of the people who regularly ride bicycles in a given year do not regularly ride bicycles in the subsequent year. Additionally, 2% of the people who do not regularly ride bicycles in that year begin to ride bicycles regularly in the subsequent year. If 5,000 people ride bicycles and 100,000 people do not ride bicycles in a given year, then the following program calculates the number of cyclists in the subsequent year (cycling) and that of people who do not ride bicycles in the subsequent year (noncycling):

From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array index starts at 1.

Function calcGeoMean receives the array dataArray (the number of elements ≥ 1) as an argument and returns the geometric mean of values in dataArray as the return value. For the numbers a₁, a₂, …, a_n, the geometric mean is ⁿ√(a₁ × a₂ × ⋯ × a_n). Here, the pow(x, y) function returns a real value by raising x to the power of y. In the program, division is performed in data type real.

From the answer group below, select the correct answer to be inserted into blank in the description. Here, the array indexes start at 1.

The function summarize receives the array sortedData sorted in ascending order and returns five values that characterize the array. The array sortedData must have at least one element. The summarize calls the findRank with two arguments, sortedData and q.
When the function summarize is called as summarize({0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}), the return value is blank.

From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array indexes start at 1.

Suppose the weather in a city, from day to day, follows a Markov process. This implies the next day’s weather depends only on today’s weather and not on those of the previous days. If it rains today in the city, a probability 0.3 exists that it will rain again tomorrow. Similarly, the probability that tomorrow’s weather will be sunny is 0.4 and that it will be cloudy is 0.3. If today’s weather is sunny, the next day will be rainy, sunny, or cloudy with probabilities 0.2, 0.7, and 0.1, respectively. Additionally, if today’s weather is cloudy, the next day will be rainy, sunny, or cloudy with probabilities 0.25, 0.5, and 0.25, respectively.
Let us say that the weather is in state 1 if it is rainy, in state 2 if it is sunny, and in state 3 if it is cloudy. Then, the above is a three-state Markov chain whose transition probabilities are given by the following:

Using this transition matrix, the probability distribution of the weather after several days can be estimated.
The program calculates the probability distribution of the weather of the city after three days from the present day and outputs a 3 × 3 square matrix that shows this probability distribution of the weather. For instance, if the weather on the present day is sunny (state 2), the probability that the weather three days after will be rainy (state 1) is the value in the second row and the first column of the outputted matrix.

From the answer group below, select the correct answer to be inserted into blank in the description. Here, the array indexes start at 1.

The function calcSim takes two vectors (arrays) as input, calculates their similarity, and returns a value that characterizes the similarity. A large similarity indicates that the vectors are similar, and a small similarity indicates that the vectors are dissimilar. For example, to quantify the similarity between two documents, you can create a vector of occurrences of some words for each of the two documents and input them into calcSim. When the function calcSim({2, 2, 1, 0, 4}, {3, 1, 1, 1, 2}) is called, the return value rounded to the first decimal place is blank.

From the answer group below, select the correct answer to be inserted into blank in the description. Here, the array index starts at 1.

This is a program that, based on an array that has one or more elements and is sorted in ascending order, returns five values that characterize the array.
When the function summarize is called as summarize({0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}), the return value is blank.

From the answer group below, select the correct combination of answers to be inserted into A through C in the description. Here, the array index starts at 1.

A matrix in which most of the elements are 0 is called a sparse matrix. The following program converts a matrix stored in a two-dimensional array into a data structure suitable for storing a sparse matrix, in order to reduce the amount of data of the matrix.
The function transformSparseMatrix converts the matrix given as a two-dimensional array in the argument matrix into an array of integer-type arrays, and returns it. When the function transformSparseMatrix is called as transformSparseMatrix({{3, 0, 0, 0, 0}, {0, 2, 2, 0, 0}, {0, 0, 0, 1, 3}, {0, 0, 0, 2, 0}, {0, 0, 0, 0, 1}}), the return value is {{A}, {B}, {C}}.

From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array index starts at 1.

In this question, a representation by an integer-type array in which only one element is 1 and the other elements are 0 is called a One-Hot representation. As shown in the figure, the function oneHotEncoding receives as its argument a string-type array of one or more elements in which names of colors are stored, converts each element into a One-Hot representation based on the names of colors contained in the array (three kinds in the example in the figure), and stores them in an array of integer-type arrays and returns it. An example of conversion by the function oneHotEncoding is shown in the figure.

From the answer group below, select the correct combination of answers to be inserted into A and B in the description. Here, the array index starts at 1.

Based on the tabulated results of data collected to investigate the preventive effect of a vaccination against disease X, the number of persons is calculated on the assumption that whether a person contracts disease X is not affected by whether or not the person received the vaccination. This number of persons is called the theoretical frequency. Table 1 shows an example of the tabulated results, and Table 2 shows the theoretical frequencies calculated based on Table 1.
The function f returns the theoretical frequencies calculated based on the tabulated results received in the argument data. The argument and the return value are two-dimensional arrays, whose rows correspond to the rows of the table and whose columns correspond to the columns of the table.

From the answer group below, select the correct combination of answers to be inserted into A through C in the program. Here, the array index starts at 1.

There is order data that records, for each order, the list of the items purchased in one order. The table shows an example of the order data.

From the order data, the degree of association L_xy, which shows the tendency of item x and item y to be purchased in the same order, is calculated by the following formula.

Here, M_xy represents the number of orders in which item x and item y were purchased in the same order, K_x represents the number of orders in which item x was purchased, and K_y represents the number of orders in which item y was purchased. In the example in the table, M_AB is 2, the total number of orders is 6, K_A is 4, and K_B is 3, so the degree of association L_AB between item A and item B is (2 × 6) / (4 × 3) = 1.0.
The procedure putRelatedItem outputs, based on the order data stored in the global variable orders, one of the items that have the largest degree of association with the item given as the argument, and that degree of association. In the program, an item is represented by a string, an order by an array of the purchased items, and the order data by an array of orders. The order data is assumed to contain two or more kinds of items. In addition, no item other than the items in the order data is given as the argument.

From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array index starts at 1.

Cosine similarity is a measure of the similarity of the directions of two vectors. The function calcCosineSimilarity receives the real-type arrays vector1 and vector2, each of which has n (n ≥ 1) elements, and returns the cosine similarity of the two arrays. When the array vector1 is {a₁, a₂, …, a_n} and the array vector2 is {b₁, b₂, …, b_n}, the cosine similarity is calculated by the following formula. Here, it is assumed that neither the array vector1 nor the array vector2 has 0 stored in all of its elements.