You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]



package jsat.distributions.multivariate;

import java.util.ArrayList;
import java.util.List;
import java.util.Random;
import jsat.linear.CholeskyDecomposition;
import jsat.linear.DenseVector;
import jsat.linear.Matrix;
import jsat.linear.MatrixStatistics;
import jsat.linear.SingularValueDecomposition;
import jsat.linear.Vec;
import static java.lang.Math.*;
import jsat.linear.IndexValue;

/**
 * Class for the multivariate Normal distribution. It is often called the Multivariate Gaussian distribution. 
 * 
 * @author Edward Raff
 */
public class NormalM extends MultivariateDistributionSkeleton
{

    private static final long serialVersionUID = -7043369396743253382L;
    /**
     * When computing the PDF of some x, part of the equation is only dependent on the covariance matrix. This part is
     * <pre>
     *       -k
     *       --          -1
     *        2          --
     * /  __\             2
     * \2 ||/   (|Sigma|)
     * </pre>
     * where k is the dimension, Sigma is the covariance matrix, and || denotes the determinant. <br>
     * Taking the negative log of this gives
     * <pre>
     *         /  __\
     * (-k) log\2 ||/ - log(|Sigma|)
     * -----------------------------
     *               2
     * </pre>
     * 
     * This can then be added to the log of the x dependent part, which, when exponentiated, gives the correct result of dividing by this term. 
     */
    private double logPDFConst;
    /**
     * When we compute the constant {@link #logPDFConst}, we only need the inverse of the covariance matrix. 
     */
    private Matrix invCovariance;
    private Vec invCov_diag;
    private Vec mean;
    /**
     * Lower triangular cholesky decomposition used for sampling such that L * L<sup>T</sup> = Covariance Matrix
     */
    private Matrix L;
    private Vec L_diag;
    /**
     * The determinant of the covariance matrix. 
     */
    private double log_det;

    public NormalM(Vec mean, Matrix covariance)
    {
        setMeanCovariance(mean, covariance);
    }
    
    public NormalM(Vec mean, Vec diag_covariance)
    {
        this.mean = mean.clone();
        setCovariance(diag_covariance);
    }

    public NormalM()
    {
    }
    
    /**
     * Sets the mean and covariance for this distribution. For an <i>n</i> dimensional distribution,
     * <tt>mean</tt> should be of length <i>n</i> and <tt>covariance</tt> should be an <i>n</i> by <i>n</i> matrix.
     * It is also a requirement that the matrix be symmetric positive definite. 
     * @param mean the mean for the distribution. A copy will be used. 
     * @param covariance the covariance for this distribution. A copy will be used. 
     * @throws ArithmeticException if the <tt>mean</tt> and <tt>covariance</tt> do not agree, or the covariance is not 
     * positive definite. An exception may not be throw for all bad matrices. 
     */
    public void setMeanCovariance(Vec mean, Matrix covariance)
    {
        if(!covariance.isSquare())
            throw new ArithmeticException("Covariance matrix must be square");
        else if(mean.length() != covariance.rows())
            throw new ArithmeticException("The mean vector and matrix must have the same dimension," +
                    mean.length() + " does not match [" + covariance.rows() + ", " + covariance.rows() +"]" );
        //Else, we are good!
        this.mean = mean.clone();
        setCovariance(covariance);
    }
    
    /**
     * Sets the covariance matrix for this matrix. 
     * @param covMatrix set the covariance matrix used for this distribution
     * @throws ArithmeticException if the covariance matrix is not square, 
     * does not agree with the mean, or is not positive definite.  An 
     * exception may not be throw for all bad matrices. 
     */
    public void setCovariance(Matrix covMatrix)
    {
        if(!covMatrix.isSquare())
            throw new ArithmeticException("Covariance matrix must be square");
        else if(covMatrix.rows() != this.mean.length())
            throw new ArithmeticException("Covariance matrix does not agree with the mean");
        
        CholeskyDecomposition cd = new CholeskyDecomposition(covMatrix.clone());
        System.out.println();
        L = cd.getLT();
        L.mutableTranspose();
        log_det = cd.getLogDet();
        
        int k = mean.length();
        if(Double.isNaN(log_det) || log_det < log(1e-10))
        {
            //Numerical unstable or sub rank matrix. Use the SVD to work with the more stable pesudo matrix
            SingularValueDecomposition svd = new SingularValueDecomposition(covMatrix.clone());
            //We need the rank deficient PDF and pesude inverse
            this.logPDFConst = 0.5*log(svd.getPseudoDet()) + svd.getRank()*0.5*log(2*PI);
            this.invCovariance = svd.getPseudoInverse();
        }
        else
        {
            this.logPDFConst = (-k*log(2*PI)-log_det)*0.5;
            this.invCovariance = cd.solve(Matrix.eye(k));
        }
        this.invCov_diag = null;
        this.L_diag = null;
    }
    
    public void setCovariance(Vec cov_diag)
    {
        if(cov_diag.length()!= this.mean.length())
            throw new ArithmeticException("Covariance matrix does not agree with the mean");
        
        int k = mean.length();
        
        log_det = 0;
        for(IndexValue iv : cov_diag)
            log_det += Math.log(iv.getValue());
        L_diag = cov_diag.clone();
        L_diag.applyFunction(Math::sqrt);//Cholesky is L*L' = C, sicne just diag, that means sqrt
        invCov_diag = cov_diag.clone();
        this.logPDFConst = (-k*log(2*PI)-log_det)*0.5;
        this.invCov_diag.applyFunction(f->f > 0 ? 1/f : 0.0);
        this.invCovariance = null;
        this.L = null;
    }

    public Vec getMean() 
    {
        return mean;
    }

    @Override
    public double logPdf(Vec x)
    {
        if(mean == null)
            throw new ArithmeticException("No mean or variance set");
        Vec xMinusMean = x.subtract(mean);
        //Compute the part that is depdentent on x
        double xDependent;
        if(invCov_diag != null)
        {
            xDependent = 0;
            for(IndexValue iv : xMinusMean)
                xDependent += iv.getValue()*iv.getValue()*invCov_diag.get(iv.getIndex());
            xDependent *= -0.5;
        }
        else
            xDependent = xMinusMean.dot(invCovariance.multiply(xMinusMean))*-0.5;
        return logPDFConst + xDependent;
    }
    
    @Override
    public double pdf(Vec x)
    {
        double pdf = exp(logPdf(x));
        if(Double.isInfinite(pdf) || Double.isNaN(pdf))//Ugly numerical error has occured
            return 0;
        return pdf;
    }

    @Override
    public <V extends Vec> boolean setUsingData(List<V> dataSet, boolean parallel)
    {
        Vec origMean = this.mean;
        try
        {
            Vec newMean = MatrixStatistics.meanVector(dataSet);
            Matrix covariance = MatrixStatistics.covarianceMatrix(newMean, dataSet);

            this.mean = newMean;
            setCovariance(covariance);
            return true;
        }
        catch(ArithmeticException ex)
        {
            this.mean = origMean;
            return false;
        }
    }

    @Override
    public NormalM clone()
    {
        NormalM clone = new NormalM();
        if(this.invCovariance != null)
            clone.invCovariance = this.invCovariance.clone();
        if(this.mean != null)
            clone.mean = this.mean.clone();
        clone.logPDFConst = this.logPDFConst;
        return clone;
    }
    
    @Override
    public List<Vec> sample(int count, Random rand)
    {
        List<Vec> samples = new ArrayList<>(count);
        Vec Z = new DenseVector(L == null ? L_diag.length() : L.rows());
        
        for(int i = 0; i < count; i++)
        {
            for(int j = 0; j < Z.length(); j++)
                Z.set(j, rand.nextGaussian());
            Vec sample;
            if(L != null)//full diag
                sample = L.multiply(Z);
            else
                sample = L_diag.pairwiseMultiply(Z);
            sample.mutableAdd(mean);
            samples.add(sample);
        }
        
        return samples;
    }
}