AshayPanchal_Homework2
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Northeastern University *
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Course
5100
Subject
Industrial Engineering
Date
Jan 9, 2024
Type
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10
Uploaded by SargentElectron11417
Machine Learning DS. 6140
Ashay Panchal
Homework 2
Q1.
1.
The choice of a Gaussian distribution in a regression model for predicting soil moisture based
on voltage readings is rooted in several key intuitions and principles. Let's explore why the
mean and variance of the Gaussian distribution are pivotal in shaping the effectiveness and
insights of the regression model in this context:
1. Central Tendency (Mean):
-
The mean of the Gaussian distribution represents the expected or average value of the
soil moisture given a particular voltage reading. In regression, this mean serves as the
predicted moisture value associated with a given input (voltage reading).
-
By using the mean as the prediction, the model assumes that on average, the moisture
content is centered around this value for a specific voltage reading. This is important
because it provides a point estimate that minimizes the squared differences (errors)
between the predicted and actual moisture content.
2. Uncertainty and Variability (Variance):
-
The variance of the Gaussian distribution indicates the spread or uncertainty in the
predicted moisture values for a given voltage reading. A larger variance implies higher
uncertainty, while a smaller variance means lower uncertainty.
-
In the context of soil moisture prediction, the variance reflects how much the actual
moisture content may deviate from the mean prediction for a specific voltage reading.
High variance suggests that moisture levels can vary widely for that reading, while low
variance implies more confidence in the prediction.
-
Understanding the variance is crucial because it provides insights into the reliability of the
model's predictions. It informs users about the range within which the true moisture
percentage is likely to fall, and it can guide decision-making in agriculture. For instance, if
the variance is high, it may be prudent to consider risk mitigation strategies.
3. Model Interpretation:
-
The mean and variance of the Gaussian distribution can also influence the interpretation
of the relationship between voltage readings and soil moisture. If the mean is consistently
off from the actual moisture values, it suggests a systematic bias in the model, which may
need correction.
-
If the variance is large, it implies that the sensor readings have limited predictive power,
and there might be other unaccounted factors influencing soil moisture. This can guide
further research or data collection efforts to improve the model's accuracy.
4. Model Evaluation:
-
When evaluating the performance of the regression model, examining the residuals (the
differences between the actual and predicted values) and their distribution can provide
insights into whether the Gaussian assumption holds. Ideally, the residuals should follow a
normal distribution with a mean close to zero and a consistent variance. Deviations from
this pattern might indicate model inadequacies.
2.
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3.
4.
Regarding how this prediction aligns with the known data and assumptions that might impact
its accuracy:
Model Fit:
The accuracy of the prediction depends on how well the quadratic regression model
fits the data. If the model adequately captures the underlying relationship between voltage and
soil moisture, the prediction is likely to be accurate. However, if the relationship is not truly
quadratic, the prediction may deviate from the actual data.
Assumptions:
The model assumes that the errors follow a Gaussian distribution with constant
variance. This means it assumes that the variability in soil moisture for a given voltage reading is
the same across the entire range of voltages. If this assumption is not met (e.g., if the variance
varies with voltage), the model's predictions may not be accurate, especially for values far from
the observed data points.
Extrapolation:
Predicting outside the range of observed data (e.g., at 7.3 V) is a form of
extrapolation. The accuracy of predictions in extrapolation depends on the validity of the
assumed model beyond the observed range. It's possible that the model may not perform well
in regions where there are no data points to inform the prediction.
Model Error:
There may be unmodeled factors or sources of error that are not accounted for in
the model. These can include environmental conditions, sensor calibration, or other variables
that impact soil moisture and are not considered in the model.
Q2.
1.Assumptions of the Model:
- Independence and Identically Distributed (i.i.d) Data: In this context, we assume that
brainwave readings and their corresponding cognitive states are independent of each other and
identically distributed across the entire dataset. Independence is essential because it ensures
that the predictions for one individual's cognitive state do not influence the predictions for
others. Identically distributed data is crucial for the reliability and interpretability of the model
because it means that the model's performance is consistent across different observations,
allowing for generalization to new data points.
- Linearity of the Logistic Regression Model: We assume that the log-odds of an individual
being in a "focused" state is a linear combination of the brainwave features. This linearity is
fundamental to the logistic regression model.
- Poisson Error Distribution: We assume that the errors in our logistic regression model follow
a Poisson distribution. This assumption is made to model the count nature of the binary
outcomes (0 for "distracted" and 1 for "focused"). Poisson distribution is appropriate when
dealing with count data, where the outcome can be viewed as a count of events in a fixed
interval. In this case, the "event" corresponds to an individual being in a "focused" state.
2. Intuition about the Poisson Distribution:
- Poisson Distribution Suitability: The Poisson distribution is suitable for modeling the error in
this classification context because it is often used for modeling count data, which is the case
here, where we have binary labels (0 and 1). Poisson distribution describes the number of
events (in this case, instances of being "focused") occurring in a fixed interval of time or space.
It is appropriate when events are rare, and the outcomes are non-negative integers.
- Implications of Poisson Distribution: When using the Poisson distribution to model errors in
logistic regression, it suggests that the likelihood of observing a certain number of "focused"
states given brainwave features follows this distribution. The Poisson distribution has only one
parameter, λ (lambda), which represents the mean and variance of the distribution. In the
context of logistic regression, λ is related to the expected number of individuals in a "focused"
state given the brainwave features.
3. Formulate the Likelihood Function and Discuss MLE:
- The likelihood for the logistic regression model with Poisson errors can be expressed as
follows:
L(θ; x, y) = Π [e^(−e^(-θ^Tx_i)) * (e^(-θ^Tx_i))^y_i / y_i!]
Where:
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- θ represents the logistic regression model parameters.
- x_i is the brainwave feature vector for the i-th observation.
- y_i is the binary label (0 or 1) for the i-th observation.
- e is the base of the natural logarithm.
To maximize this likelihood and estimate the parameters θ, we would typically take the
natural logarithm of the likelihood (log-likelihood) and then differentiate it with respect to θ to
find the maximum likelihood estimate (MLE).
When we differentiate the log-likelihood with respect to θ, we'll find that the sample mean of
the brainwave feature vectors, i.e., (∑(x_i) / n), emerges as a part of the equation. This term
signifies the relationship between sample statistics and population parameters in the context of
maximum likelihood estimation. In the case of logistic regression, it implies that the MLE for the
parameters θ is closely related to the observed sample means of the features.
Essentially, it means that the model's parameter estimates (θ) will be influenced by the
relationships between your brainwave features and the observed outcomes. The sample mean
of the features is a measure of central tendency in your data and affects how the logistic
regression model maps brainwave patterns to cognitive states. This demonstrates how the MLE
leverages sample statistics to estimate the parameters that best fit the data and maximize the
likelihood of the observed outcomes.
Q3.
In the context of Linear Discriminant Analysis (LDA), prior probabilities are essential because
they play a critical role in the classification process. LDA is a supervised machine learning
algorithm that is used for classification tasks, assuming that the data follows a multivariate
normal distribution. The primary goal of LDA is to find the decision boundary that maximizes
the separation between different classes, in this case, the species Alpha and Beta.
Here's why prior probabilities are important in LDA:
1.
Class Separation:
LDA determines the decision boundary by estimating the parameters of multivariate
normal distributions for each class (Alpha and Beta). These parameters include the
means and covariance matrices. The prior probabilities, denoted as P(Alpha) and
P(Beta), represent the likelihood of encountering each class in the dataset. These priors
help in weighing the contributions of each class to the decision boundary. In other
words, they help LDA account for the unequal prevalence of Alpha and Beta in the
dataset.
2.
Posterior Probability Estimation:
LDA not only classifies data points into one of the classes but also provides the posterior
probabilities for each class. These posterior probabilities represent the probability that a
data point belongs to a particular class given its features. These probabilities are
calculated using Bayes' theorem, and the prior probabilities are an essential part of this
calculation.
3.
Regularization:
In cases where the dataset is imbalanced, and one class is much more prevalent than
the other, prior probabilities can help to regularize the LDA model. Without the priors,
LDA may be biased towards the more prevalent class, leading to suboptimal
performance in classifying the minority class.
In this specific scenario, where we know that the likelihood of encountering an Alpha is twice
that of encountering a Beta in the given region, we can determine the prior probabilities as
follows:
- P(Alpha) = 2/3
- P(Beta) = 1/3
By incorporating these prior probabilities into your LDA model, we ensure that the model takes
into account the differences in class prevalence, which is critical for accurate classification when
working with imbalanced datasets.
2.
To formulate the discriminant functions for Linear Discriminant Analysis (LDA), we need to
calculate the means and variances for each species and then use them to derive the
discriminant functions. LDA assumes that the data for each class follows a multivariate normal
distribution with distinct means and covariances.
For Alpha:
- Genetic Marker Values: 5.1, 5.8, 5.4
- Mean (μ_alpha) = (5.1 + 5.8 + 5.4) / 3 = 5.4333
- Variance (σ_alpha^2) = [(5.1 - μ_alpha)^2 + (5.8 - μ_alpha)^2 + (5.4 - μ_alpha)^2] / 2 ≈ 0.0689
(assuming unbiased variance estimator)
For Beta:
- Genetic Marker Values: 6.9, 7.2, 6.5
- Mean (μ_beta) = (6.9 + 7.2 + 6.5) / 3 = 6.8667
- Variance (σ_beta^2) = [(6.9 - μ_beta)^2 + (7.2 - μ_beta)^2 + (6.5 - μ_beta)^2] / 2 ≈ 0.0844
(assuming unbiased variance estimator)
Now that we have the means and variances for both species, we can proceed to formulate the
discriminant functions.
LDA assumes that the discriminant functions are linear combinations of the features (genetic
marker values) that maximize class separation. The discriminant function for LDA can be
expressed as follows:
For a two-class problem (Alpha and Beta):
Discriminant function for Alpha: D_alpha(x) = x * (μ_alpha) - 0.5 * (μ_alpha^2) + ln(P(Alpha))
Discriminant function for Beta: D_beta(x) = x * (μ_beta) - 0.5 * (μ_beta^2) + ln(P(Beta))
In these formulas, "x" represents the genetic marker value for an observation, μ_alpha and
μ_beta are the means of the genetic marker values for Alpha and Beta, and ln(P(Alpha)) and
ln(P(Beta)) are the natural logarithms of the prior probabilities for Alpha and Beta, which you
can calculate using the information you have (P(Alpha) = 2/3, P(Beta) = 1/3).
plug in the values:
For Alpha:
D_alpha(x) = x * 5.4333 - 0.5 * (5.4333^2) + ln(2/3)
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For Beta:
D_beta(x) = x * 6.8667 - 0.5 * (6.8667^2) + ln(1/3)
These discriminant functions will help classify new genetic marker values into either the Alpha
or Beta species based on their likelihood under the assumed normal distributions. The class
with the higher discriminant function value will be the predicted class for a given observation.
3.
let's classify the new plant with a genetic marker value of 6.2 using the LDA model:
Calculate the discriminant functions for Alpha and Beta:
D_alpha(x) = x * μ_alpha - 0.5 * μ_alpha^2 + ln(P(Alpha))
D_beta(x) = x * μ_beta - 0.5 * μ_beta^2 + ln(P(Beta))
Plug in the values:
D_alpha(6.2) = 6.2 * 5.4333 - 0.5 * (5.4333^2) + ln(2/3)
D_beta(6.2) = 6.2 * 6.8667 - 0.5 * (6.8667^2) + ln(1/3)
Calculate the discriminant function values for both Alpha and Beta:
D_alpha(6.2) ≈ 32.5681
D_beta(6.2) ≈ 31.6719
The classification will be based on the criteria that the plant is assigned to the species with the
higher discriminant function value. In this case, the plant with a genetic marker value of 6.2 is
classified as Alpha because D_alpha(6.2) > D_beta(6.2).