text:
/**
 * To change this template, choose Tools | Templates and open the template in
 * the editor.
 */
package tools.sqltool.lov;

import java.io.File;
import java.io.FileInputStream;
import java.io.FileOutputStream;
import java.io.FileReader;
import java.io.FilenameFilter;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.OutputStreamWriter;
import java.util.HashMap;
import java.util.Map;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import tools.sqltool.utils.Assert;
import tools.sqltool.utils.JaxbUtils;

/**
 *
 * @author yyi
 */
public class LovStore {

    public static final String LOV_FOLDER = "lov";
    private static Logger logger = LoggerFactory.getLogger(LovStore.class);
    private String storeFolder;
    private JaxbUtils jaxbUtils = new JaxbUtils("tools.sqltool.lov");
    private Map<String, LovTable> lovTables = new HashMap<String, LovTable>();

    public String getMapperFolder() {
        return storeFolder;
    }

    public Map<String, LovTable> getMappers() {
        return lovTables;
    }

    public LovStore(String storeFolder) {
        Assert.notNull(storeFolder);
        this.storeFolder = storeFolder;
        this.storeFolder = this.storeFolder.replace('\\', '/');
        if (this.storeFolder.endsWith("/") == false) {
            this.storeFolder += "/";
        }
        File folder = new File(this.storeFolder);
        if (folder.exists() == false) {
            folder.mkdirs();
        }

    }

    public void save() {
        for (LovTable lovTable : lovTables.values()) {
            saveLovTable(lovTable);
        }
    }

    public void load() {
        lovTables.clear();
        File folder = new File(storeFolder);
        File[] listFiles = folder.listFiles(new FilenameFilter() {
            @Override
            public boolean accept(File dir, String name) {
                if (name.endsWith(".xml")) {
                    return true;
                }
                return false;
            }
        });
        for (File file : listFiles) {
            LovTable mapper = loadLovTable(file);
            if (mapper != null) {
                lovTables.put(mapper.getTableName(), mapper);
            }
        }

    }

    public void saveLovTable(LovTable lovTable) {

        lovTables.put(lovTable.getTableName(), lovTable);

        String fullname = getFullFileName(lovTable.getTableName());
        OutputStreamWriter writer = null;
        try {
            writer = new OutputStreamWriter(new FileOutputStream(fullname, false), "UTF-8");
            jaxbUtils.marshal(lovTable, writer);
        } catch (IOException ex) {
            throw new RuntimeException("Can not save mapper " + lovTable.toString(), ex);
        } finally {
            try {
                writer.close();
            } catch (IOException ex) {
            }
        }

    }

    private String getFullFileName(String tableName) {
        return storeFolder + tableName + ".xml";
    }

    public LovTable loadLovTable(File file) {

        InputStreamReader reader = null;
        try {
            reader = new InputStreamReader(new FileInputStream(file), "UTF-8");
            LovTable mapper = (LovTable) jaxbUtils.unmarshal(reader);
            return mapper;
        } catch (IOException ex) {
            logger.warn("Failed to load mapper " + file.getName(), ex);
            return null;
        } finally {
            try {
                reader.close();
            } catch (IOException ex) {
            }
        }

    }

    public LovTable loadLovTable(String tablename) {
        String fullname = storeFolder + tablename + ".xml";
        if (new File(fullname).exists() == false) {
            return null;
        }
        InputStreamReader reader = null;
        try {
            reader = new FileReader(fullname);
            LovTable mapper = (LovTable) jaxbUtils.unmarshal(reader);
            return mapper;
        } catch (IOException ex) {
            throw new RuntimeException("Failed to load mapper " + tablename, ex);
        } finally {
            try {
                reader.close();
            } catch (IOException ex) {
            }
        }
    }

    public LovTable getLovTable(String lovTableName) {
        return this.lovTables.get(lovTableName);
    }

    public boolean deleteLovTable(String lovTableName) {
        LovTable lovTable = this.lovTables.remove(lovTableName);
        if (lovTable != null) {

            String fullname = getFullFileName(lovTable.getTableName());
            File file = new File(fullname);
            if (file.exists()) {
                return file.delete();
            }
        }
        return true;
    }

    public String genAllMergeIntoSql() {
        StringBuilder sb = new StringBuilder();
        for (LovTable table : lovTables.values()) {
            sb.append(table.genMergeIntoSql());
        }
        return sb.toString();
    }

    public String genAllInsertSql() {
        StringBuilder sb = new StringBuilder();
        for (LovTable table : lovTables.values()) {
            sb.append(table.genInsertSql());
        }
        return sb.toString();
    }

    public void genJavaClass(String folder) {
        File file = new File(folder);
        if (file.exists() == false) {
            if (file.mkdirs() == false) {
                throw new RuntimeException("The folder " + folder + " can not be created.");
            }
        }

        for (LovTable table : lovTables.values()) {
            if (table.isGenEnumClass()) {
                JavaClass javaClass = table.genJavaClass();
                OutputStreamWriter writer = null;
                try {

                    File javaFile = new File(file, javaClass.getClassName() + ".java");
                    writer = new OutputStreamWriter(new FileOutputStream(javaFile, false), "UTF-8");
                    writer.write(javaClass.getCode());
                } catch (IOException ex) {
                    throw new RuntimeException(javaClass.getClassName() + " can not be generated in folder " + folder, ex);
                } finally {
                    try {
                        writer.close();
                    } catch (IOException ex) {
                    }
                }
            }
        }

    }
}


"When Jesus was awakened to meet the storm, He was in perfect peace. There was no trace of fear in word or look, for no fear was in His heart. But He rested not in the possession of almighty power. It was not as the "Master of earth and sea and sky" that he reposed in quiet. That power He had laid down, and He says, "I can of Mine own self do nothing." John 5:30. He trusted in the Father's might. It was faith—faith in God's love and care—that Jesus rested, and the power of that word which stilled the storm was the power of God" (The Desire of Ages, p. 336).
So, we see that the "work" of stilling the storm was by the faith of Jesus. This work sprung up and out of His faith in His Fathers power; His works are always the product of His faith.
Paul instructs us further in Galatians 5.6 teaching us that it is faith that works through love (agape). It is never faith + works, but always a faith that works by love.
The year following the 1888 General Conference, E. J. Waggoner wrote, "The great characteristic of faith is that it works. We do not mean that works are attached to it, but works come from it. "As the body without the spirit is dead, so faith without works is dead also." James 2:26. Faith 'works by love,' says Paul. There may be that which is called faith, but if no works proceed from it, it is not faith.
"A just man is one who acts in harmony with the law of God. Therefore a just man is an obedient man. So we may understand ... that an obedient man shall live (a life of obedience) by his faith. The obedience springs from his faith, and therefore there is no chance for him to be lifted up, since the act of obedience is not his personal action, but is the action of his faith, and credited to him as his own. The man, whose soul is lifted up in him, is the man who thinks that he can of himself do all that is required, ... thus, 'pride goeth before destruction'" (Signs of the Times, Feb. 4, 1889).
God has given to the world His only begotten Son (John 3:16) and His Son comes with His own faith. This is the faith that is given to us in Romans 12:3. He comes with His faith so that we may live the same just life that Jesus lived, for the just shall live by His faith. Faith comes by hearing and hearing by the word of God (Rom. 10:17).
The Father chose us in Christ from before the foundation of the world (Eph. 1:3, 4), therefore when Christ went to the cross all mankind went with him. When Jesus died, we all died in Him (2 Cor. 5:14, 15). By this righteous act, Christ gave all men justification of life (Rom. 5:18). In the believing of this good news, the justification of life becomes justification by faith. Paul nails this truth for all to see: "I have been crucified with Christ; it is no longer I who live, but Christ lives in me; and the life which I now live in the flesh I live by the faith [of] the Son of God, who loved me and gave Himself for me" (Gal. 2:20).
Hence by the faith of Jesus we are a new creation. "Therefore, if anyone is in Christ, he is a new creation; old things have passed away; behold, all things have become new" (2 Cor. 5:17). We now walk in the Spirit and not after the flesh. Even though we still retain our flesh, the Spirit controls us and we are safe because God has condemned sin in our flesh that His entire law might be perfectly fulfilled in us by the faith of Jesus! (Rom. 8:3, 4).
We have not to worry about "helping" God to fulfill His promises in us. Our efforts to contribute to salvation are a boastful demonstration of unbelief. Our fingerprints need to be off of everything! "For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand that we should walk in them" (Eph. 2:10). These works are the fruit of the Spirit—"The fruit of the Spirit is love, joy, peace long suffering, kindness, goodness, faithfulness, gentleness, self-control. ... If we live in the Spirit, let us also walk in the Spirit (Gal. 5.22, 23, 25). This fruit grows naturally and is demonstrated evidence of the new creation.
The mystery of God, which was hidden from the ages and generations, is now made known among us: this mystery is Christ in you, the hope of glory (Col. 1.26, 27). Revelation 10:5-7 speaks of this present mystery: " ... in the days of the sounding of the seventh angel, when he is about to sound, the mystery of God would be finished, as He declared to His servants the prophets." God will have a people ready before He comes again! Christ in you cleanses the sanctuary by the blotting out of sin and the reproduction of the character of Christ in His people. Seen at last are the saints who have the faith of Jesus that comes with His obedience to all the commandments of God. Like Jesus they can of themselves do nothing; like Him they rest in the love and care of the Father.
"It is finished" - the cross is the wonder of His work! We, the object of His love and the perfection of His workmanship have been given His life to excercise His faith, in doing His works that He has prepared for us from the foundation of the world. (Eph. 2.10). In accepting His finished works, we enter into His rest.
Today, if we will hear His voice, let us not harden our hearts like we are accustomed to doing, but let us believe and enter His rest.


Anna Susanna Kegel 1783– – webtrees Nos Trois Branches - Le G.A.C.
Anna Susanna Kegel was born about 1783. She married Daniel Gottlieb Rauhut in 1800 (VIII) in Zaborowo, Lissa (Lezno), Posen.
Anna Susanna Rauhut, daughter of Daniel Gottlieb Rauhut and Anna Susanna Kegel, was born on March 12, 1805 (Ventose 21, XIII) in Lissa (Lezno), Lissa (Lezno), Posen.
Benjamin Friedrich Rauhut, son of Daniel Gottlieb Rauhut and Anna Susanna Kegel, was born on May 9, 1807 in Zaborowo, Lissa (Lezno), Posen. He married Anna Dorothea Fiebig, daughter of Johann Samuel Fiebig and Hedwig Kegel, in 1836 in Zaborowo, Lissa (Lezno), Posen. She was born about 1819 in Lissa (Lezno), Lissa (Lezno), Posen.
Daniel Gottlieb Rauhut, son of Daniel Gottlieb Rauhut and Anna Susanna Kegel, was born on June 15, 1812 in Zaborowo, Lissa (Lezno), Posen.
Carl Samuel Rauhut, son of Daniel Gottlieb Rauhut and Anna Susanna Kegel, was born on May 21, 1814 in Zaborowo, Lissa (Lezno), Posen.
Johanna Susanna Rauhut, daughter of Daniel Gottlieb Rauhut and Anna Susanna Kegel, was born on July 7, 1816 in Zaborowo, Lissa (Lezno), Posen. She married Johann Gottlieb Walter, son of Friedrich Walter and Anna Dorothea Walter, in 1839 in Zaborowo, Lissa (Lezno), Posen.
Daniel Fabian Rauhut, son of Daniel Gottlieb Rauhut and Anna Susanna Kegel, was born on January 19, 1819 in Zaborowo, Lissa (Lezno), Posen. He married Johanna Elisabeth Schoen, daughter of Johann Gottfried Schoen and Therese Bäsche, in 1842 in Zaborowo, Lissa (Lezno), Posen. She was born about 1828 in Lissa (Lezno), Lissa (Lezno), Posen.
Adelheid Friederike Rauhut, daughter of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on December 16, 1840 in Zaborowo, Lissa (Lezno), Posen.
Benjamin Heinrich Fabian Rauhut, son of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on January 19, 1843 in Zaborowo, Lissa (Lezno), Posen. He married Bianka Klara Wilhelmine Fiebig, daughter of Samuel Gottlieb Fiebig and Albertine Weigt, on November 30, 1880 in Schmiegel, Schmiegel (Śmigiel), Posen. She was born on February 10, 1841 in Schmiegel (Śmigiel), Posen, PREUSSEN.
Samuel Gottlieb Rudolph Rauhut, son of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on October 11, 1847 in Zaborowo, Lissa (Lezno), Posen.
Carl Friedrich Conrad Rauhut, son of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on January 28, 1849 in Zaborowo, Lissa (Lezno), Posen.
Alexander Otto Rauhut, son of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on March 18, 1852 in Zaborowo, Lissa (Lezno), Posen.
Ludwig Theodor Rauhut, son of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on August 12, 1853 in Zaborowo, Lissa (Lezno), Posen.
Daniel Gottlieb Reinhold Rauhut, son of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on January 13, 1855 in Zaborowo, Lissa (Lezno), Posen and died on March 4, 1855 at the age of 50 days.
Samuel Benjamin Wilhelm Rauhut, son of Benjamin Friedrich Rauhut and Anna Dorothea Fiebig, was born on August 27, 1861 in Zaborowo, Lissa (Lezno), Posen and died on March 29, 1865 at the age of 3.
Carl Gottfried Hermann Rauhut, son of Daniel Fabian Rauhut and Johanna Elisabeth Schoen, was born on March 12, 1846 in Zaborowo, Lissa (Lezno), Posen. He married 2 times. The first time he married Auguste Amalie Ernestine Bruschwitz, daughter of Christian Bruschwitz, in 1878 in Lissa (Lezno), Lissa (Lezno), Posen. She was born calculated 1842. The second time he married Anna Maria Harder, daughter of Ferdinand Harder and Anna Walter, on May 8, 1900 in Lissa (Lezno), Posen, PREUSSEN. She was born on October 13, 1856 in Bärsdorf (Bojanowo), Rawitsch (Rawicz), Posen.
Johanne Renate Rauhut, daughter of Daniel Fabian Rauhut and Johanna Elisabeth Schoen, was born on February 25, 1849 in Preussen, Zaborowo, Lissa (Lezno).
Paul Daniel Robert Rauhut, son of Daniel Fabian Rauhut and Johanna Elisabeth Schoen, was born in September 1851 in Zaborowo, Lissa (Lezno), Posen and died on December 27, 1924 in Zaborowo, Lissa (Lezno), Posen at the age of 73. He married 2 times. The first time he married Rosina Marie Hirschel, daughter of Johann Gottfried Hirschel, in 1876 in Zaborowo, Lissa (Lezno), Posen. She was born on August 2, 1853 in Zaborowo, Lissa (Lezno), Posen and died on November 27, 1886 in Zaborowo, Lissa (Lezno), Posen at the age of 33. The second time he married Marie Ottilie Artin, daughter of August Artin, in 1897 in Zaborowo, Lissa (Lezno), Posen. She was born calculated 1874.
Friedrich Albin Rauhut, son of Daniel Fabian Rauhut and Johanna Elisabeth Schoen, was born on March 1, 1853 in Preussen, Zaborowo, Lissa (Lezno). He married Johanna Deck in 1879 in Lissa (Lezno), Posen, PREUSSEN.


text:
Q: Access dynamically added Component in Unity3d C# After adding a Component to a GameObject via the GameObject.AddComponent method, how can I access this Component from another script?

here is myScript code (not attach to a GameObject):
using UnityEngine;
using System.Collections;

public class MyScript : MonoBehaviour {
    public string       myName = "myName";
    public Vector3      pos;
    public bool         visible;
}

and here is the main code attached to a gameobject in the scene:
using UnityEngine;
using System.Collections;

public class Main : MonoBehaviour {
    public GameObject cube;

    void Update() {
        if(Input.GetKeyDown(KeyCode.Space)) {
            cube = GameObject.CreatePrimitive(PrimitiveType.Cube);
            cube.AddComponent<MyScript>();
        }

        if(Input.GetKeyDown(KeyCode.LeftShift)) {
            var a = GameObject.Find("Cube");

            print("cube name/visible: " +
            /* here is the problem, how do i access the MyScript variables? */
            a.myName + "/" + a.visible);
        }
    }
}

A: To access a Component from a Unity3d GameObject, you should use the GameObject.GetComponent<T>() method. There is more than one way to do this.
MyScript m = gameObject.GetComponent<MyScript>();
MyScript m = gameObject.GetComponent("MyScript") as MyScript;
MyScript m = (MyScript) gameObject.GetComponent(typeof(MyScript));

From your example code, there are a few things that should really be addressed:

*

*If you want to create components in your scene dynamically, it is better practice to create a prefab, then add that into your scene

*If you are planning to add large amounts of objects into your scene at runtime, you should use pooling, rather than expensive creating and destroying objects

*GameObject.Find("Cube"); will only find a single object named "Cube" in the heirarchy. You should use FindObjectsOfType<MyScript>() if you want to find all GameObjects that have the MyScript component (FindObjectOfType if you only want to find 1)

Based on that, you can get to the variables of MyScript like this:
MyScript a = FindObjectOfType<MyScript>();
print(a.myName + "/" + a.visible);

or if you want to check the status of all MyScript active components:
MyScript[] myScripts = FindObjectsOfType<MyScript>();
foreach (MyScript myScript in myScripts) {
    print(myScript.myName + "/" + myScript.visible);
}

redpajama_set_name: RedPajamaStackExchange


Recently, friend of Firestorm Jeremy Gilbert of Dixon Hughes Goodman (DHG) discussed protecting personal identity with Charleston, SC Live5 News. DHG is one of the top 20 accounting firms in the nation.
Jeremy, a Master Analyst in Financial Forensics serves as Manager of the Digital Forensics lab in DHG's Charleston, South Carolina office.
The article explores the warning of a lesser-known way cyber thieves can steal from you – a used airline boarding pass.
But Jeremy observed that Crosby's boarding pass was not the only one in the trash; four others were found and given to Jeremy for review.
Jeremy goes on to demonstrate with a bar code scanner that the bar-code on a boarding pass reveals a frequent flier number. Airlines and the TSA use the bar-codes, and according to the International Air Transport Association, they can be accessed from anywhere, even a mobile phone. Bottom line, Gilbert said, it's important to take steps to protect yourself.
Learn more at the Dixon Hughes Goodman website, and through the video below. You can follow DHG on Twitter.


Diesel needs no explanation. This Italian jeans brand is so well known ... Everyone has undoubtedly a pair of Diesel jeans hanging in their closet, or at least has tried a pair on or seen them on somebody else. Obviously, because the jeans (and other clothing from Diesel) are always a perfect fit.
Find 843 Diesel products from 11 shops on SALE at Fashiola.com.
Get up to 80% off on Diesel products!


New inventions and latest technology is making human life a lot easier than before. A new-fangled pieces of cutting edge technologies are radically altering the use of it in today's world. One such technology is IoT, which is connecting the world. Here comes SENSORO, with a great vision to help the human being towards the advancement with the help of cutting age, long-range low-power IoT technology…..


Colorado AG "Smells" a Loan? Colorado AG "Smells" a Loan?
Colorado AG Smells a Loan!
Colorado Attorney General John Suthers recently filed a lawsuit against a couple of our competitors regarding the availability of pre-settlement financing in Colorado. Attorney General Suthers wants to portray lawsuit pre-settlement funding as a "loan" so that it will fall under the ambit of the various consumer protection laws which are purportedly designed to protect consumers.
It's comically ironic that the "Consumer Protection Code," which by definition is designed to protect "consumers," ends up helping the big insurance companies. Who gets hurt if pre-settlement lawsuit funding is declared a "loan" in Colorado and therefore unlawful if over the pre-determined satisfactory loan interest rates? The big insurance companies will continue the practice of dragging out settlements as long as possible. They know most of America lives paycheck to paycheck. They know that Joe Consumer can't afford to go another month without funds to make his truck payment. (And, of course, the insurance payment on his truck as well.) Ultimately who gets hurt is the poor consumer, who will lose that truck and his house as a result of someone else's negligence for which the insurance company is ultimately responsible. I don't hear Mr. Suthers, or any other of the "consumer protectionists" screaming about the age old insurance company practice of "starving the plaintiff" over time into a settlement. When Mr. Suthers fighs to declare pre-settlement funding a "lawsuit loan," he should ask himself who is going to pay his family's grocery bill when he gets hurt?


Doors open at 8:30 a.m., program begins at 9:30 a.m.
ANSWER Coalition leaders and other movement organizers will be joining an exciting conference to be held in Chicago this Saturday, Nov.5, and Sunday, Nov. 6. We urge everyone who can to attend this important event.
The Nov. 5 and 6 event will feature two days of speakers, panels, organizing sessions and discussion at the University of Illinois at Chicago. Hosted by the Party for Socialism and Liberation, hundreds of workers, students, organizers, anti-war activists and progressive people will come together from across the United States.
Doors will open at 8:30 a.m. and the program will begin at 9:30 a.m. on both days. We encourage everyone to attend the entire conference.
Mark Clements: Anti-death penalty activist, leader in struggle to save Troy Davis. Clements was tortured by Chicago Police and unjustly incarcerated for nearly three decades.
Michael Prysner: Iraq war veteran; Co-founder, March Forward!
Organizations listed for identification purposes only.
See the complete schedule for the Midwest Socialism Conference this Saturday, Nov. 5 and Sunday, Nov. 6.
We urge you register for the Midwest Socialism Conference today. Be there and bring your friends, family, co-workers, co-students and everyone you know.
Pay the registration fee or make a donation by clicking here.


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HomeGrownFun opens to smaller, but enthusiastic crowd of fairgoers
Unicyclist Mark Wilder entertains fair goers by jumping rope on his unicycle on the midway during opening day for HomeGrownFun, a scaled-down version of the San Diego County Fair which will take place June 11 through July 4 at the Del Mar Fairgrounds.
(Nancee E. Lewis/The San Diego Union-Tribune)
Scaled-down version of the San Diego County Fair still a draw for the "regulars" who love the food, shopping and piglet races
By Pam Kragen
DEL MAR —
In a normal year, as many as 1,000 people are usually lined up at the main gate for the opening morning of the San Diego County Fair. But this hasn't been a normal year.
When the gates opened at 11 a.m. Friday, June 11, for the debut of HomeGrownFun — a sharply scaled-down version of the fair at the Del Mar Fairgrounds — there were only a couple dozen people waiting to get in. Among them was Patti Hyss of Poway, who is always among the first to arrive each year. Hyss said she was very upset the fair was canceled last year because of the pandemic, so even though HomeGrownFun is much smaller than the usual event, she wouldn't think of missing it.
"I love the fair so much. I've been coming for too many years to count," Hyss said. "I always get a season pass so I can come back to enjoy all the people, the music, the food and the exhibits. Today I'm going to go see the flowers, eat some cinnamon rolls and have a Dole Whip."
Swifty Swine Racing Pigs are a center stage attraction at this year's HomeGrownFun, a scaled-down version of the San Diego County Fair. Included in this race were Nancy Piglosi, Hamala Harris, Donald Trumproast and Dr. Fauci.
HomeGrownFun, which runs through July 4, is selling online tickets for up to 13,000 visitors a day, compared to the fair's usual 60,000 or so daily visitors. There are no midway games or fun zone, no full-size garden show, no livestock or pet exhibitions, no grandstand concerts, no beer garden and no art, woodworking or collectibles shows.
But there are more than 280 vendors selling their wares in the exhibit halls, nearly 40 food booths, a carousel and Ferris wheel, pony rides, a model train and garden walk, a family-friendly "Agri-Land" exhibit with cow-milking demonstrations and child-size toy tractors, a variety of local entertainment and a closing-day fireworks show on July 4. Also, beginning June 16 — the day after the state officially reopens — 12 children's rides will open in the Midway area.
Christina Negron, of Imperial Beach, who hasn't missed a fair in over 20 years, dives into an order of fried frog legs on opening day for HomeGrownFun, a scaled-down version of the San Diego County Fair which will take place June 11 through July 4 at the Del Mar Fairgrounds.
Also at the front of the line Friday morning, June 11, were Olivia and Joseph Valencia of Lakeside, who've been coming to the fair's opening day every year since 1999. Their checklist for the day included eating at their two favorite food booths, Australian Battered Potatoes and Golden West BBQ, and shopping for some new housewares in Bing Crosby Hall.
"We know it's not the same as before, but we're so excited to be here," Olivia Valencia said.
At the fair's two entry gates, workers made sure everyone entering had on a face covering. But inside the fairgrounds Friday, June 11, about 25 percent of fair visitors were either wearing their masks below their chins or not at all. A volunteer at a fair information booth said masks are indeed required on property for the length of the fair, but that guests will need to flag down security officers to make a complaint about unmasked individuals.
One of the first big attractions of the day Friday, June 11, was the Swifty Swine Racing Pigs, a crowd-pleasing show that always draws capacity crowds. The first race of the morning, featuring 5-week-old market piglets, was won by Britney Spare Rib, while the second race went to the fleet-footed pot-bellied pig Hamala Harris, who beat out Donald Trumproast for the victor's prize: an Oreo cookie.
Matrix, an Australian Shepard mix, catches a flying disc in the Extreme Dogs show on opening day of HomeGrownFun, a scaled-down version of the San Diego County Fair, which runs through July 4 at the Del Mar Fairgrounds.
Swifty Swine owners Jack and Shannon Johnson of Van, Texas, have been touring the fair and festival circuit with their Swifty Swine piglets for 23 years. Their daughter, Hannah, helps run the races, cares for the piglets and sells $2 souvenirs like strap-on rubber pig noses. Hannah said her family usually spends nine months of each year on the road, so the pandemic was devastating for their family business and their morale.
"Last year was really hard," she said. "I grew up in this business and I love doing it. It's super fun. So being back out again feels good."
Inside Bing Crosby Hall, a smaller-than-usual number of vendors were selling clip-on hair pieces, hot sauce, eyebrow services, toe rings, water purifiers, mattresses, kitchen knives, cowboy hats and wellness products. At the bustling Hardware Mercantile booth, the hottest-selling gadget was a $5.50 plastic funnel for emptying one inverted bottle of shampoo or chocolate sauce into another.
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Mauricio Castro of TCM International in Los Angeles is running two side-by-side booths, selling the Spiralizer vegetable slicer and a line of fast-drying, odor-free kitchen sponges. Since 1996, Castro has been traveling to fairs, conventions and other events 12 months of the year. The San Diego County Fair is traditionally the first fair in the nation to open each year, so for Castro and many others in the hall, Friday was the first day back for many vendors since March 2020.
"We're all a little nervous about the small size of this fair, but people are coming to me and saying that people are spending money," Castro said. "Everyone is bored from being inside and they got their stimulus money and they're ready to buy things again. The world is still upside down, but I think by this time next year everything will be back to normal."
Owen Jana, 5, of Del Mar holds on tight to an oversized serving of curly fries on opening day for HomeGrownFun on Friday.
With fewer people to serve at midday Friday, June 11, virtually all of the food booths had short or no lines. One exception was Chicken Charlie's, where an employee said the top-selling items during the opening hours were the Maui chicken in a pineapple bowl and the new ice cream nachos.
One of the booth's first customers was Christina Negron of Imperial Beach, who came with her partner, Vanessa Farias, and her daughter, Elisa Hossfeld. Every year their first stop is Chicken Charlie's where they try some new deep-fried food items. For Negron this year, it was fried frog legs. For Hossfeld it was batter-fried brownie bites topped with spicy buffalo sauce and ice cream.
"It's pretty weird," Hossfeld said of the sweet and spicy combo.
Three generations of the Welsh family of Valley Center arrived at 11 a.m. Friday for a day that they hoped would include visits to Tom Thumb Donuts, the pig races, the Extreme Dogs show and a few exhibits. Although the Welshes weren't crazy with the face mask requirement, they were looking forward to exploring everything without the usual crowds.
"We're just going to take our time and enjoy being out with everyone again," Kathy Welsh said.
HomeGrownFun
Hours: 11 a.m. to 9 p.m. Wednesdays-Sundays, through July 4
Where: Del Mar Fairgrounds, 2260 Jimmy Durante Blvd., Del Mar
Tickets: $10; children 5 and under are free (tickets are sold for reserved times)
Parking: Prepaid parking is $12 at the Solana and Stable gates.
Online ticket sales: sdfair.com
— Pam Kragen is a reporter for The San Diego Union-Tribune
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Pam Kragen
Pam Kragen is the arts and entertainment editor, as well as the chief theater, dining and opera critic, for the San Diego Union-Tribune. She joined the Union-Tribune staff in October 2012 after 27 years at the North County Times, where she served as the Arts & Features Editor, as well as the paper's longtime arts writer and theater and opera critic. She is the co-founder of the San Diego Theatre Critics Circle. She holds a bachelor of arts degree in journalism from San Diego State University and completed fellowships in theater criticism at the University of Southern California and opera-classical music criticism at Columbia University's Graduate School of Journalism. And in 2022, she was chosen as one of the fellows in the prestigious National Critics Institute, run by the Eugene O'Neill Theater Center in New London, Conn.
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Encinitas Advocate E-Edition

redpajama_set_name: RedPajamaCommonCrawl


Log Cabin Republicans spring intern Ashley Rooney is a gay female Republican of faith. In this Townhall op-ed she asserts the importance of this underestimated voting bloc. You can read the full op-ed here.


Kitchen is one of the places that is used many times in our homes. The kitchen is the one that is used to cook meals. Some kitchens that have enough space can be used by families to take snacks together. A lot of people decide to remodel their kitchen to make it look beautiful because of the many functional offers it provides to them. It is costly to remodel the kitchen, and that's why many people do not do that.
You will enjoy many benefits when you choose to remodel your kitchen. Like in other rooms that your home has, kitchen remodelling is not something that you can do at once. You will remodel your kitchen when you want or when you have enough budget. It is cheap to replace the faucets or light fixtures because it is a project that you can complete within a day. Painting of your kitchen walls and cabinets or changing of cabinet hardware can be done when you want.
The ways that you will use in making the essential changes are less expensive when you decide to remodel your kitchen. For instance, installation of another countertop right over the old granite counter top is possible. You will have saved your money because you will not have to buy a new granite counter when you do that. You can choose to sand and paint your old countertop if you would like to save more money.
When you remodel your kitchen, you make it more functional. Those who purchase homes that are already built live with another person's idea. If you choose to remodel the kitchen, you will make it functional according to your thoughts. What might have been functional to others may not work on you, and that's why it is vital to remodel your kitchen according to your way. The value of homes will be raised when people choose to renovate your home.
If you are about to sell your home, you should remodel the kitchen because it counts a lot. Your home will look friendlier and welcoming when you make your kitchen look beautiful and functional. Buyers will even get the value of their money when the kitchen of the home they are about to buy is remodelled. Remodeling of kitchens makes them look gorgeous which on the other hand make people enjoy the time they will spend there. Preparing meals and cooking will be done pleasantly if you will remodel your kitchen also. Spending the time in a kitchen that is not remodeled will make someone feel bored because it will be unfunctional and also dark. A lot of people like bright kitchens.


text:
Q: Why labeling these nodes with an equation doesn't work? \begin{tikzpicture}
\node[state] (x11) {$X_1^1$};
\node[state, right of=x11] (x12) {$X^2_1$};
\node[state, right of=x12] (x13) {$X^3_1$};
\node[state, right of=x13] (x14) {$X^4_1$};
\node[state, right of=x14] (x15) {$X^5_1$};

\node[state, below of=x11,label=left:$s\leq 3$] (x21) {$X^1_2$};
\node[state, below of=x12] (x22) {$X^2_2$};
\node[state, below of=x13] (x23) {$X^3_2$};
\node[state, below of=x14] (x24) {$X^4_2$};
\node[state, below of=x15] (x25) {$X^5_2$};

\node[state, below of=x21] (x31) {$X^1_3$};
\node[state, below of=x22] (x32) {$X^2_3$};
\node[state, below of=x23] (x33) {$X^3_3$};
\node[state, below of=x24] (x34) {$X^4_3$};
\node[state, below of=x25] (x35) {$X^5_3$};

\draw

(x11) edge[above] node{} (x21)
(x11) edge[above] node{} (x22)
(x13) edge[above] node{} (x24)
(x13) edge[above] node{} (x25)
(x15) edge[above] node{} (x23)
(x22) edge[above] node{} (x33)
(x22) edge[above] node{} (x31)
(x23) edge[above] node{} (x32)
(x25) edge[above] node{} (x34)
(x25) edge[above] node{} (x35)
;

\end{tikzpicture}

How come if I change label=left:$s\leq 3$ to label=left:$s=3$ I can no longer compile the code...
I get ! Extra }, or forgotten $.

A: Solution of your problem is given @ferahfeza comment. Note, sign = is used for positioning of node labels. If you have two, parser is lost. Therefore = in equation had to be "hidden" , i.e. had to be enclosed in curly braces.  Good practice is enclosed complete equation into curly braces, since with this spacings in equation are retained.
So, I suggest some off-topic improvements using tikz libraries chains, positioning and quotes:
\documentclass[tikz, margin=3mm]{standalone}
\usetikzlibrary{automata,
                chains,
                positioning,
                quotes}

\begin{document}
    \begin{tikzpicture}[
      node distance = 4mm and 4mm,
        start chain = going right,
state/.append style = {inner sep=0pt, on chain}
                    ]
\node[state] (x11) {$X_1^1$};
\node[state] (x12) {$X^2_1$};
\node[state] (x13) {$X^3_1$};
\node[state] (x14) {$X^4_1$};
\node[state] (x15) {$X^5_1$};
%
\node[state, below= of x11,
      label=left:{$s=3$}]    % <---
             (x21) {$X^1_2$};
\node[state] (x22) {$X^2_2$};
\node[state] (x23) {$X^3_2$};
\node[state] (x24) {$X^4_2$};
\node[state] (x25) {$X^5_2$};
%
\node[state, below= of x21] (x31) {$X^1_3$};
\node[state] (x32) {$X^2_3$};
\node[state] (x33) {$X^3_3$};
\node[state] (x34) {$X^4_3$};
\node[state] (x35) {$X^5_3$};

\draw   (x11) edge ["?"]    (x21)
        (x11) edge          (x22)
        (x13) edge          (x24)
        (x13) edge          (x25)
        (x15) edge          (x23)
        (x22) edge          (x33)
        (x22) edge          (x31)
        (x23) edge          (x32)
        (x25) edge          (x34)
        (x25) edge          (x35)
        ;
    \end{tikzpicture}
\end{document}


Sophie and Lucy are sisters. They work in the creative industries and share everything with each other. When Sophie did Alpha, their lives began to look very different. However, the impact was far beyond what they could have imagined.
We weren't born into a Christian family, but when mum did Alpha she became a completely different person. She could be quite stressed and anxious, but after Alpha she became loving and patient – it was amazing.
In my third year of uni, I was trying to recover from several years of bulimia. I wasn't fully cured at that point and I was very unhappy, even though I didn't realise it at the time.
I met a girl, Amanda, who was always so kind and lovely, and I was drawn to her. When I found out she was a Christian, I mentioned that I was interested in Alpha and she invited me to her church.
It was challenging at first. I found the small talk difficult, and had never enjoyed talking about myself. But after a few weeks, I realised that I was learning something amazing. I'd never felt so fulfilled in an evening.
"I'd never felt so fulfilled in an evening."
The next year was a turbulent one. I was in a bad relationship at the time. My boyfriend hated that fact that I had become a Christian. He was very insecure and jealous because he wasn't my God anymore. I worked at Vogue and my life looked amazing, but I was still bulimic and run down. I wasn't content but I was so used to feeling that way I almost thought it was normal.
I remember crying to my friend Katie in a café, telling her I was ready to give up. I didn't feel I was good enough to become a Christian. I was too tired to pray and I couldn't read my Bible.
She said, 'Sophie, you don't have to do anything. You're already saved. It's finished. You can't earn it. It's love.' That just changed it all for me. All of a sudden I realised that I had experienced his love, and that I didn't need to struggle any longer.
I resigned from my job and set up my own jewellery company with a business partner from church. We made a lot of money supplying shops around the world. We thought we were going to take over the world.
But the recession hit us really hard and I'm not working at the moment. I don't see it as a punishment, though. I think God's scaled it all back because up until now, my work has been my identity. When I read about all the people in the Bible who were broken down and built back up, I feel like I'm in the middle of this major restoration programme. It's tough, but I'm learning to be content every day, even if I'm not doing a really glitzy or exciting job.
Fashion is an industry that can provide so much for the poor, but it's a place where God isn't considered. I would love to start some kind of social enterprise. He has made us all so unique, and given us incredible gifts, so it's great to be able to use them in whatever capacity we can.
Our family are close, passionate and very loud. When mum did Alpha, it had a powerful impact on life at home.
Sophie and I are very different. When she became Christian, I would see her and Amanda, the friend who took her to Alpha, and think: 'They're just the lovely, pretty ones who like to do girly things.' Christianity was this perfect, flowery thing that had no place for me.
I didn't get it and would bombard her with questions. One day, she said, 'You have so many questions, why don't you do Alpha?' I agreed to go, taking my strongest atheist friend as an ally. I knew he would have the answers that I couldn't articulate, though by the end of the course, they fell short. His answers didn't bring any hope or satisfaction.
One Sunday, I went to church with a few of Sophie's friends and one of the girls asked if she could pray for me. I just cringed. I didn't want to let her but I had already started to feel God speak to me. Over many months, he'd been revealing how much he loved me.
When she prayed, I felt myself saying, 'I want to know you. I want to know who you are.' The Holy Spirit swept in and filled me. For someone normally quite self-conscious, I didn't care what everyone else was doing. I was totally wrapped up in God.
"I was totally wrapped up in God."
When I met some of Sophie's other Christian friends – ones that were both 'wild' and Christian, I realised that God didn't want me to give up my personality. He'd made me to be feisty and loud.
God changed the way I saw my work too. After I gave my life to Jesus, I panicked, thinking that my career as an art director in the fashion industry was entirely superficial. I emailed African orphanages to organise some sort of mission trip. None of them got back to me; it seemed that God wanted me to stay put.
I carried on with my job, working on different shoots and sets. Whenever I was asked about my weekend, people were always so surprised that I would talk about church. I am so in love with Jesus that I naturally talk about what he has done and is doing in my life. That's when I realised that God had put me exactly where he wants me to be, to share his love and light in an industry where there is a lot of darkness.
It's hard to see the people around you struggling and carrying really heavy burdens. You just want to say, 'You don't know how amazing you are, you don't have to live like this.' It was great when two of our family friends did Alpha. It encouraged our Dad to go along as well. Even though he was sceptical, he felt a lot lighter and less burdened during that time. Sophie and I found God through Alpha, and it's having a roll-on effect in our family. It's been incredible.


This branch looks beautiful in just one color, but for some, it's a lot of fun to try it in many different colors. If you've never done this before, check out this Tutorial for multiple color stenciling.
SIZE: Branch measures 21 x 43 inches.


Senior politicians and NHS Executives accept that healthcare technology can assist with transformational improvements for the NHS. The improvements will deliver higher productivity, safer care and substantial cost savings. To enable technology to deliver transformational changes requires the government to not only inject capital, but also to have an approach that is inclusive of NHS stakeholders (Clinicians, nurses, scientists, engineers, and managers, etc). These stakeholders hold the knowledge required to jointly deliver a transformational technological solution but unfortunately, although there is momentum, we must ensure that the processes and policies that guide the transformation are properly thought through. Although senior politicians and NHS executives are focussing on technology to deliver transformational improvements, there is a concern that portable and fixed assets (that are used by stakeholders) have not been properly considered as part of this technological transformation.
The NHS spends approximately £4.6billion of its budget on equipment and consumables. (NHS Protect, 2015). For 2015/16, the overall NHS budget was £116.4 billion. This equates to a 3.9% spend on healthcare technology and consumables. Although technology spend is relatively small (in comparison with overall NHS spend) there is still pressure from NHS executives to avoid investment in new medical equipment thus attempting to reduce the £4.6 billion that is currently being spent. This goes against the rhetoric coming from senior politicians about using technology to trigger process change.
When considering the connections between stakeholders, processes and technology, we must analyse any processes that might need to change, and what technologies will help us make that change. Also, people's roles often change as a result of the new technology, so it is imperative to consider the impact on stakeholders, and involve them in making the changes. Ultimately, how are patients and staff likely to feel about the changes and the technology - and what needs to happen to support them? This is especially important when looking at introducing step-down care options such as hospital hotels or acute home warding.
The NHS is currently being challenged by Lord Carter to develop operating models for step down care. Innovative models are being proposed such as hospital hotels.
The hospital hotel concept is not new but also not yet well established or developed within the UK. This concept was first developed as Patient Hotels in Sweden in 1988 to meet increasing demand but within a restricted budget. The concept in Sweden was based on a radical re-think of the healthcare delivery model and the introduction of a graduated care model where appropriate sub-acute patients were transferred to the hotel. The concept was based on a philosophy of re-enablement and engaging the family where this was possible. The model has been successfully duplicated within other locations in Scandinavia, but has not been developed in the UK, potentially because of concerns of patient deterioration or re-admission. These concerns can now be addressed using technology that is Bluetooth and wireless enabled.
Devices are becoming much simpler to use, to the point where patients can now take their own observations such as: Oxygen saturation, pulse rate, blood pressure, spirometry, respiration rate, blood sugar, weight, temperature, etc. using wifi or Bluetooth devices that automatically 'push' the data into the patient record allowing doctors and nurses to view the patient data from their PC, tablet, or phone.
Doctors and nurses can follow the condition of their patients, even though they are in the hospital hotel or at home. This connectivity allows them the comfort to step them down into the hospital hotel, or even to their homes (subject to them being able to care for themselves or have support from family or friends). In summary the hospital hotel model offers the NHS an opportunity to meet the challenges of improving patient experience and the quality of care whilst reducing costs at the same time.
When implementing new healthcare technology processes, it is important to consider how new technology fits with existing systems. Opportunities are wasted when implementing new IT systems if they do not communicate with other systems and devices. Therefore, NHS executives must ensure adequate planning has gone into specifying, setting up, and testing that different technologies do actually 'talk' to each other thus leading to improved productivity. As with any healthcare technology, user training is imperative if technology is to be used safely and effectively. Finally, data security issues need to be addressed ensuring patient information is secure.
By using modern software, providing 24/7 helpdesk support for users, and linking it to the medical devices (using Artificial Intelligence software systems such as Visensia), we can access the data, analyse the data, and put 24/7 high level nursing care at every bed thereby reducing pressures on nurses and doctors. Using Artificial Intelligence software to monitor the patient and respond early to any signs of deterioration means that, particularly in single room environments such as the Hospital Hotel, lower cost models of care be used to monitor patients with the assurance of 24/7 high quality support.
In summary, healthcare technology is changing current working practices, but the transformation needs to be managed carefully if the systems are to deliver real improvements in productivity.
These new technology solutions can improve NHS best practice and feed into other areas such as producing statistics for reporting and planning future improvements. These technologies will ultimately save time and resources assisting the NHS to deliver better care at a lower cost.
NHS Commissioning Assembly, 2015. Technology Enabled Care Services, s.l.: Quality and Outcomes Working Group - NHS Commissioning Assembly.
NHS Institute for Innovation and Improvement, 2007. Technology to Improve Service, Coventry: NHS Institute for Innovation and Improvement.
NHS Protect, 2015. Guidance for NHS health bodies on the security and management of NHS assets, London: NHS.


text:
\section{Introduction}

We consider the linear regression model
\[
y_{i}=\mathbf{x}_{i}^{\prime}\boldsymbol{\beta}+\varepsilon_{i},\text{
\ \ \ }i=1,...,n;
\]
where the error $\varepsilon_{i}$ is assumed independent from $\mathbf{x}%
_{i}\in \mathbb{R}^{d}$, with $E(  \varepsilon_{i})  =0$ and $Var(
\varepsilon_{i})  =\sigma^{2}.$ \ We focus on the estimation of
linear functions, $\mathbf{a}^{\prime}\boldsymbol{\beta}$, of the regression
coefficients. For example, $\mathbf{a}^{\prime}\boldsymbol{\beta}=\beta_{1}$,
when the interest resides in a particular effect, or $\mathbf{a}^{\prime
}\boldsymbol{\beta}=\mathbf{x}_{0}^{\prime}\boldsymbol{\beta}$, when the
interest resides in the prediction of a new outcome $y_{0}=$
$\mathbf{x}_{0}^{\prime}\boldsymbol{\beta}+\varepsilon_{0}$. We aim for
estimators $\widehat{\boldsymbol{\theta}}$ of $\boldsymbol{\beta}$ such that
the mean squared error, $E(\mathbf{a}^{\prime}%
\widehat{\boldsymbol{\theta}}-\mathbf{a}^{\prime}\boldsymbol{\beta})
^{2}$, is minimized. It holds that
\begin{equation}
\MSE(\mathbf{a}^{\prime}\widehat{\boldsymbol{\theta}})  =
E(\mathbf{a}^{\prime}\widehat{\boldsymbol{\theta}}-\mathbf{a}^{\prime}\boldsymbol{\beta})  ^{2}=
(\mathbf{a}^{\prime}(E(\widehat{\boldsymbol{\theta}})  - \boldsymbol{\beta}) )^{2} +
\mathbf{a}^{\prime}\Cov(  \widehat{\boldsymbol{\theta}})\mathbf{a}, \label{MSE1}%
\end{equation}
where the two terms on the right hand side are the squared bias and the
variance, respectively.
The standard estimator for $\boldsymbol{\beta}$ is the
ordinary least squares (LS) estimator $\widehat{\boldsymbol{\beta}}=(
X^{\prime}X)  ^{-1}X^{\prime}\mathbf{y,}$ where $X$ \ is the $n \times d$ design
matrix and $\mathbf{y} \in \mathbb{R}^n$ is the response vector. Since
$\widehat{\boldsymbol{\beta}}$ is unbiased, we obtain that $\MSE(\mathbf{a}%
^{\prime}\widehat{\boldsymbol{\beta}})  =\mathbf{a}^{\prime}\Cov(
\widehat{\boldsymbol{\beta}})  \mathbf{a},$ with
$\Cov(\widehat{\boldsymbol{\beta}})  =\sigma^{2}(X^{\prime}X)^{-1}$.
Moreover, by the Gauss-Markov theorem we know that
$\MSE(\mathbf{a}^{\prime}\widehat{\boldsymbol{\beta}})  \leq
\MSE(\mathbf{a}^{\prime}\widehat{\boldsymbol{\theta}})$ for any linear
unbiased estimator $\widehat{\boldsymbol{\theta}}.$ Therefore, one focuses on
biased estimators when one aims to reduce $\MSE(\mathbf{a}^{\prime}\widehat{\boldsymbol{\theta}})$.

A common approach in this regard is regularization by shrinkage ~\citep{Tikhonov63,Hoerl62}. The most
popular family of shrinkage estimators is ridge regression~\citep{Hoerl70},
given by
\[
\widehat{\boldsymbol{\beta}}_{\lambda}=(  X^{\prime}X+\lambda I)
^{-1}X^{\prime}\mathbf{y,}\text{ \ \ \ }\lambda\geq0.
\]
It can be shown that
$\Cov(\widehat{\boldsymbol{\beta}}_{\lambda})  \preceq \Cov(\widehat{\boldsymbol{\beta}})$,
for any given $\lambda\geq0,$ where $\preceq$ represents inequality in the
Loewner partial order for symmetric matrices [i.e. $A\succeq B$ $\ $if $A-B$
is non-negative definite, see e.g.~\citet{Horn12}]. Indeed, taking $\sigma
^{2}=1$ (w.l.g.) we have that $\Cov(\widehat{\boldsymbol{\beta}%
}_{\lambda})^{-1}=X^{\prime}X+\lambda(2I+\lambda(
X^{\prime}X)^{-1})  \succeq X^{\prime}X$ $=\Cov(\widehat{\boldsymbol{\beta}})  ^{-1}.$

Another well-known shrinkage family is the non-negative
garrote~\citep{Breiman95}. A simplified version of this method (which we use
in this paper) is defined as
\[
\widehat{\boldsymbol{\beta}}_{\boldsymbol{\omega}}=D\widehat{\boldsymbol{\beta
}},\text{ \ \ \ }D=\diag(  \boldsymbol{\omega})  ,\text{\ }%
\]
with $0\leq\omega_{j}\leq1,$ $\ j=1,...,d.$ \ In Section 3 we show that both
the total variance and the generalized variance of $\widehat{\boldsymbol{\beta}%
}_{\boldsymbol{\omega}}$ \ are smaller than those of
$\widehat{\boldsymbol{\beta}}$. There is an extensive literature on shrinkage
in linear regression which mainly focuses on automatic variable selection with the Lasso and the elastic net as the most prominent examples,
see e.g.~\citet{Hastie15}.

In Section 2 we consider a new family of estimators, which we call split
regression estimators (SPLIT).
In Section 3, we show that the total variance and generalized variance of the
simplified garrote and SPLIT are smaller than those of LS. We illustrate the potential usefulness of
SPLIT in Section 4 through results of some numerical comparisons with
ridge regression, garrote and Lasso. We simulate the theoretical mean squared prediction errors (MSPEs) curves for the methods considered in Section 5, illustrating the benefits in higher dimensions. Finally, Section 6 concludes.

\section{\bigskip Split Estimators}

Consider a partitioning of the design matrix $X=(X_{1}\text{ }X_{2})$ with $X_{1}$ a $n\times d_1$ matrix and $X_{2}$ a $n\times (d-d_1)$ matrix.
According to this partition, consider the matrices
\[
A=X^{\prime}X=
\begin{pmatrix}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{pmatrix}
 \text{ \ and \ }
 \overline{A}=
 \begin{pmatrix}
\ \ A_{11} & -A_{12}\\
-A_{21} & \ \ A_{22}
\end{pmatrix}.
\]
Then, we define the corresponding split LS estimator by
\[
\ \widetilde{\boldsymbol{\beta}}=\diag(  A_{11}^{-1},A_{22}^{-1})
X^{\prime}\mathbf{y,}%
\]
which has mean%
\[
E(\widetilde{\boldsymbol{\beta}})  =\diag(A_{11}^{-1},A_{22}^{-1})  A\mathbf{\beta}=
(\mathbf{\beta}_{1}+A_{11}^{-1}A_{12}\mathbf{\beta}_{2}, \mathbf{\beta}%
_{2}+A_{22}^{-1}A_{21}\mathbf{\beta}_{1})^{\prime}%
\]
and covariance matrix%
\begin{equation}
\Cov(\widetilde{\boldsymbol{\beta}})  =
\sigma^{2}\diag(A_{11}^{-1},A_{22}^{-1}) \,A\,\diag(A_{11}^{-1},A_{22}^{-1}).
\label{eq2}%
\end{equation}

\noindent\textbf{Example 1}. Consider the linear regression model $y_{i}=\beta_{1}%
x_{i1}+\beta_{2}x_{i2}+\varepsilon_{i}$ with $\varepsilon_{i}\thicksim
N(  0,\sigma^{2})  $, $ i=1,\dots,n $. To simplify the notation we assume that the
explanatory variables have been centered and scaled so that
$(1/n)  \sum_{i=1}^{n}x_{i1}^{2}=(  1/n)  \sum_{i=1}^{n}%
x_{i2}^{2}=1$ and let us denote $(1/n)  \sum_{i=1}^{n}%
x_{i1}x_{i2}=r.$ \medskip

\noindent Suppose that $X_{1}$ is the main
effect and $X_{2}$ is a control variable. Hence, we are mainly interested in
estimating $\mathbf{a}^{\prime}\boldsymbol{\beta}=\beta_{1}$. Let us compare the
LS estimator $\widehat{\beta}_{1}\thicksim N(  \beta_{1},\sigma
^{2}[  (  1-r^{2})  n]  ^{-1})  $ with the
SPLIT estimator $\widetilde{\beta}_{1}=(  1/n)  \sum x_{i1}%
y_{i}\thicksim N(  \beta_{1}+r\beta_{2},\sigma^{2}n^{-1})$. The
MSE for $\widehat{\beta}_{1}$ and $\widetilde{\beta}_{1}$ is $\sigma
^{2}[  (  1-r^{2})  n]  ^{-1}$ and $r^{2}\beta_{2}%
^{2}+\sigma^{2}n^{-1}$, respectively. Therefore, $\widetilde{\beta}_{1}$ is
preferred if and only if $\beta_{2}^{2}<\sigma^{2}[  (
1-r^{2})  n]  ^{-1}.$ \ Hence, SPLIT becomes more attractive when
there is a larger error variance and/or a larger correlation between the
explanatory variables (multicolinearity) and/or a smaller sample size. For
example, if $\sigma=1,$ $r=0.7$ and $n=10$, then the SPLIT estimator is
preferred if $\vert \beta_{2}\vert <0.443.$ \medskip

\section{Variance Inequalities}
To compare the variability of the estimators $\widehat{\boldsymbol{\beta}}$, $\widehat{\boldsymbol{\beta}}_{\boldsymbol{\omega}}$
and $\widetilde{\boldsymbol{\beta}}$ we use the following expression
for $A^{-1}$. Let  $B_{11}=(  A_{11}-A_{12}A_{22}^{-1}%
A_{21})  ^{-1}$ and $B_{22}=(  A_{22}-A_{21}A_{11}^{-1}%
A_{12})  ^{-1}$, then we have
\begin{equation}
A^{-1}=\diag(  B_{11},B_{22})  \text{ }\overline{A}\text{
}\diag(  A_{11}^{-1},A_{22}^{-1})  . \label{eq2b}%
\end{equation}
\vskip 1ex
\noindent\textbf{Proposition 1 (Generalized Variance Inequality). }
Consider
the three estimators above, then we have the following inequalities for the generalized
variance
\begin{align*}
(a) \qquad \det(\Cov(\widetilde{\boldsymbol{\beta}}))  &\leq \det(\Cov(\widehat{\boldsymbol{\beta}})), \\
(b) \quad\ \det(\Cov(\widehat{\boldsymbol{\beta}}_{\boldsymbol{\omega}}))  & \leq \det(\Cov(\widehat{\boldsymbol{\beta}})).
\end{align*}

\noindent\textbf{Proof. \ }To prove (a) we notice that
\begin{equation}
\det(\Cov(\widetilde{\boldsymbol{\beta}}))
=\sigma^{2d}[  \det(A_{11}^{-1})  \det(  A_{22}^{-1})  ]  ^{2}\det(A)  \label{eq5}%
\end{equation}
Similarly using that $\det(A)=\det(\overline{A})
$ we have
\begin{equation}
\det(\Cov(\widehat{\boldsymbol{\beta}}))
=\sigma^{2d}[  \det(  A_{11}^{-1})  \det(  A_{22}%
^{-1})  ]  [  \det(  B_{11})  \det(
B_{22})  ]  \det(  A)  \label{eq4}%
\end{equation}
From (\ref{eq5}) and (\ref{eq4}) \ it suffices to show that
\[
\det(  A_{11}^{-1})  \det(  A_{22}^{-1})
\leq
\det(  B_{11})  \det(  B_{22})
\]
Or equivalently,%
\[
\det(A_{11})  \det(  A_{22})  \geq\det(
A_{11}-A_{12}A_{22}^{-1}A_{21})  \det(  A_{22}-A_{21}A_{11}%
^{-1}A_{12})
\]
This holds because $A_{11}\succeq A_{11}-A_{12}A_{22}^{-1}A_{21}$ \ \ and
$\ A_{22}\succeq A_{22}-A_{12}A_{11}^{-1}A_{21}$ \ implies $\det(
A_{11})  \geq\det(  A_{11}-A_{12}A_{22}^{-1}A_{21})  $ \ and
\ $\det(  A_{22})  \geq\det(  A_{22}-A_{21}A_{11}^{-1}%
A_{12})  ,$ respectively.

Part (b) follows because
\[
\det(\Cov(\widehat{\boldsymbol{\beta}}_{\boldsymbol{\omega}}))=
\sigma^{2d}\det(  D)  ^{2}\det(
A^{-1})  \leq\sigma^{2d}\det(  A^{-1})  =\det(\Cov(\widehat{\boldsymbol{\beta}})).
\]

\noindent\textbf{Proposition 2 (Total Variance Inequality).} Consider again
the three estimators $\widehat{\boldsymbol{\beta}}$
, $\widetilde{\boldsymbol{\beta}}$, $\widehat{\boldsymbol{\beta}}%
_{\boldsymbol{\omega}}$. Then we have the following inequalities for the total
variance:
\begin{align*}
(a)\quad\ \ \tr(\Cov(\widetilde{\boldsymbol{\beta}})) & \leq \tr(\Cov(\widehat{\boldsymbol{\beta}})),\\
(b)\quad \tr(\Cov(\widehat{\boldsymbol{\beta}}_{\boldsymbol{\omega}})) &\leq \tr(\Cov(\widehat{\boldsymbol{\beta}})).
\end{align*}
\noindent\textbf{Proof.} \ From (\ref{eq2}) it follows that%

\[
\tr(\Cov(\widetilde{\boldsymbol{\beta}})) =\sigma^{2}[
\tr(  A_{11}^{-1})  +\tr(  A_{22}^{-1})  ]  .
\]
On the other hand, using equation (\ref{eq2b}) we obtain that
\[
\tr(\Cov(\widehat{\boldsymbol{\beta}})) =\sigma
^{2}\tr(  A^{-1})  =\sigma^{2}[\tr(  A_{11}-A_{12}%
A_{22}^{-1}A_{21})  ^{-1}+\tr(  A_{22}-A_{21}A_{11}^{-1}%
A_{12})  ^{-1}]  .
\]
Now, notice that
\begin{align*}
A_{11}  &  \succeq A_{11}-A_{12}A_{22}^{-1}A_{21} \iff A_{11}
^{-1}\succeq(  A_{11}-A_{12}A_{22}^{-1}A_{21})  ^{-1},\\
A_{22}  &  \succeq A_{22}-A_{21}A_{11}^{-1}A_{12} \iff A_{22}
^{-1}\preceq(  A_{22}-A_{21}A_{11}^{-1}A_{12})  ^{-1},
\end{align*}
which implies
\begin{align*}
\tr(  A_{11}^{-1})   &  \leq \tr(  (  A_{11}-A_{12}%
A_{22}^{-1}A_{21})  ^{-1}), \\
\tr(  A_{22}^{-1})   &  \leq \tr(  (  A_{22}-A_{21}%
A_{11}^{-1}A_{12})  ^{-1}).
\end{align*}
To prove (b) let us set $\ B=A^{-1}$, then we have that
\[
\tr(\Cov(\widehat{\boldsymbol{\beta}}_{\boldsymbol{\omega}})) \leq
\sigma^{2}tr(  D^{2}B)  =
\sigma^{2} \sum_{i=1}^{d}w_{i}^{2}b_{ii}\leq\sigma^{2}\sum_{i=1}^{d}b_{ii}=
\tr(\Cov(\widehat{\boldsymbol{\beta}}))
\]

To show that split LS estimators improve the total and generalized variance of the least squares estimator more generally,
we now consider nested partitions of the set $\{  1,...,d\}$. A partition
$\mathcal{P}=\{  I_{1},...,I_{k}\}  $ \ is nested in the partition
$\mathcal{Q}=\{  J_{1},...,J_{m}\}  $ \ if for all $l\in
\{1,...,m\}$ there is a subset $M_{l}\subset\{  1,...,k\}  $ such
that
\[
J_{l}=U_{\alpha\in M_{l}}I_{\alpha}.
\]
For simplicity, we will consider the partitions $\mathcal{Q=}\{
J_{1},J_{2}\}  $ \ and $\mathcal{P}=\{  I_{1},I_{2},I_{3}\}
$ with $J_{1}=I_{1}\cup I_{2}$ and $J_{2}=I_{3}.$ We will also assume (w.l.g.)
that $I_{1}=\{  1,...,d_{1}\}  ,$ $I_{2}=\{  d_{1}%
+1,...,d_{2}\}  $ and $I_{3}=\{  d_{2}+1,...,d\}  .$ \ The
corresponding partitions of the matrix $A=X^{\prime}X$ \ are
\[
A=
\begin{pmatrix}
A_{11} & A_{12} & A_{13}\\
A_{21} & A_{22} & A_{23}\\
A_{31} & A_{32} & A_{33}
\end{pmatrix}
\text{ \ , \ }A=
\begin{pmatrix}
C_{11} & C_{12}\\
C_{21} & C_{22}
\end{pmatrix}
\]
with
\[
C_{11}=
\begin{pmatrix}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{pmatrix},
\text{ }C_{12}=
\begin{pmatrix}
A_{13}\\
A_{23}
\end{pmatrix} \text{ \ and \ \ }C_{22}=A_{33}.%
\]
We now compare the split estimators
$\widetilde{\boldsymbol{\beta}}=\diag(  C_{11}^{-1},C_{22}%
^{-1})  X^{\prime}\mathbf{y}$ and $\widetilde{\boldsymbol{\alpha}}=\diag(  A_{11}^{-1},A_{22}^{-1}%
,A_{33}^{-1})  X^{\prime}\mathbf{y}$.\\

\noindent\textbf{Proposition 3} \textbf{(Nested Partitions Inequality).}
\begin{align*}
(a)\ \det(\Cov(\widetilde{\boldsymbol{\alpha}}))  &\leq \det(\Cov(\widetilde{\boldsymbol{\beta}})), \\
(b) \quad \tr(\Cov(\widetilde{\boldsymbol{\alpha}}))  &\leq \tr(\Cov(\widetilde{\boldsymbol{\beta}})).
\end{align*}
\noindent\textbf{Proof.} To simplify the
notation, assume that $\sigma^{2}=1,$. We have
\[
\Cov(\widetilde{\boldsymbol{\beta}})=\diag(  C_{11}%
^{-1},C_{22}^{-1})\,A\,\diag(  C_{11}^{-1},C_{22}%
^{-1})
\]
and
\[
\Cov(\widetilde{\boldsymbol{\alpha}}) =\diag(  A_{11}%
^{-1},A_{22}^{-1},A_{33}^{-1})\,A\,\diag(  A_{11}%
^{-1},A_{22}^{-1},A_{33}^{-1})
\]
Therefore, to prove (a) write%
\begin{align*}
\det(\Cov(\widetilde{\boldsymbol{\beta}})) &
=[  \det(  C_{11}^{-1})  \det(  C_{22}^{-1})
]  ^{2}\det(  A) \\
 \det(\Cov(\widetilde{\boldsymbol{\alpha}})) &
=[  \det(  A_{11}^{-1})  \det(  A_{22}^{-1})
\det(  A_{33}^{-1})  ]  ^{2}\det(  A)
\end{align*}
Since $A_{33}^{-1}=C_{22}^{-1},$ we just need to show that $\det(
C_{11}^{-1})  \geq\det(A_{11}^{-1})  \det(A_{22}^{-1})$.
But this has been shown in Proposition 1(a) with
$C_{11}$ playing the role of $A$. \\
Similarly, to prove (b) we write
\begin{align*}
\tr(\Cov(\widetilde{\boldsymbol{\beta}})) &
=\tr(  C_{11}^{-1})  +\tr(  A_{33}^{-1}) \\
\tr(\Cov(\widetilde{\boldsymbol{\alpha}}))  &
=\tr(  A_{11}^{-1})  +\tr(  A_{22}^{-1})  +\tr(A_{33}^{-1})
\end{align*}
Hence, we just need to show that $\tr(C_{11}^{-1})  \geq \tr(
A_{11}^{-1})  +\tr(  A_{22}^{-1})  ,$ but this has already
been shown in Proposition 2(a) with $C_{11}$ playing the role of $A$.

\bigskip

\begin{remark}
{ From Theorem 2 and 3 it follows that
\[\det(\Cov(\widetilde{\boldsymbol{\alpha}}))  \leq \det(\Cov(\widetilde{\boldsymbol{\beta}}))
 \leq\det(\Cov(\widehat{\boldsymbol{\beta}})),  \]
and also
\[\tr(\Cov(\widetilde{\boldsymbol{\alpha}}))  \leq \tr(\Cov(\widetilde{\boldsymbol{\beta}})) \leq
\tr(\Cov(\widehat{\boldsymbol{\beta}})).
\]}
\end{remark}
\section{Potential benefit of split regression estimators}

 Our aim is to illustrate the potential benefits of SPLIT for the prediction of new outcomes. We use the model and notation introduced in Example 1.
The prediction of a
new outcome corresponds to the choice  $\mathbf{a=x}_{0}\ $and the prediction performance of an estimator $\widehat{\boldsymbol{\theta}}$ is measured by averaging $\MSE(\mathbf{x}_{0}^{\prime}\widehat{\boldsymbol{\theta}})$
over the values of $\mathbf{x}_{0}$, that is we calculate $\MSPE(\widehat{\boldsymbol{\theta}})=
E_{\mathbf{x}_{0}}\{\MSE(\mathbf{x}_{0}^{\prime}\widehat{\boldsymbol{\theta}})\}$.
To calculate the expectation we assume that \textbf{\ }$\mathbf{x}_{0}\thicksim N(  \mathbf{0}, %
\Gamma)$ where $\Gamma$ is a correlation matrix with parameter
$\rho.$ The empirical correlation between the predictor variables is fixed and given by the parameter $ r $. It can be shown that for LS prediction
\[
\MSPE(\widehat{\boldsymbol{\beta}})  =2(\sigma^{2}/n)
(1-r\rho )  /(  1-r^{2}).
\]
For garrote prediction with weights $\boldsymbol{\omega}$, we obtain that
\begin{align*}
\MSPE(\widehat{\boldsymbol{\beta}}_{\mathbf{\omega}}) = & (  w_{1}-1)
^{2}\beta_{1}^{2}+(  w_{2}-1)  ^{2}\beta_{2}^{2}+2\rho(
w_{1}-1)  (  w_{2}-1)  \beta_{1}\beta_{2}\\
& +\frac{\sigma^{2}}{n(  1-r^{2})  }(w_{1}^{2}+w_{2}^{2}-2rw_{1}w_{2}\rho).
\end{align*}
For ridge regression prediction with penalty $\lambda$, we have
\[
\MSPE(\widehat{\boldsymbol{\beta}}_{\lambda})  =(  (  1+\lambda/n)
^{2}-r^{2})  ^{-2}(  (  \lambda/n)  ^{2}%
\boldsymbol{\beta}^{\prime}B\Gamma B\boldsymbol{\beta}+
\sigma^{2}\tr(\Gamma V)  /n)
\]
where $B=(  b_{ij})  $, with $b_{11}=b_{22}=1+\lambda/n$,
$b_{12}=b_{21}=-r $ and $V=(  v_{ij})  $  with $v_{11}%
=v_{22}=$ $(  1+\lambda/n)  ^{2}-r^{2}(  1+2\lambda/n)
$, $v_{12}=v_{21}=r(  (  \lambda/n)  ^{2}-1+r^{2})$.
We also consider prediction based on the Lasso and elastic net estimators, denoted by $\widetilde{\boldsymbol{\beta}}_{\lambda}$ and $\widehat{\boldsymbol{\beta}}_{\alpha}$, respectively.
However, in this case we do not have a closed form expression for $\MSPE(\widetilde{\boldsymbol{\beta}}_{\lambda})$, so
we calculate it numerically.
We determine the optimal prediction performance of ridge regression, Lasso, elastic net and garrote corresponding
to values of the shrinkage parameters $\lambda$, $ \alpha $ or $\boldsymbol{\omega}$ which minimizes the MSPE for each respective method.

To make SPLIT more flexible, similarly as for the garrote, we add a shrinkage parameter $\mathbf{w}=(w_1,w_2)$ to the split coefficient estimates $\widetilde{\boldsymbol{\beta}}$.
Moreover, since splitting is not always better than joint estimation as illustrated in Example 1, we make SPLIT adaptive by allowing it to choose whether to use the joint LS estimates or to use the split LS estimates. The decision to split or not to split is made by comparing the performance of the best shrunken SPLIT estimator $\widetilde{\boldsymbol{\beta}}_{\mathbf{w}}$ to the best shrunken joint LS estimator, i.e. the simplified garrote estimator $\widehat{\boldsymbol{\beta}}_{\mathbf{\omega}}$.
The MSPE of the shrunken SPLIT estimator is given by
\[
\MSPE(\widetilde{\boldsymbol{\beta}}_{\mathbf{w}})  =\boldsymbol{\beta}^{\prime
}T^{\prime}\Gamma T\boldsymbol{\beta}+(\sigma^{2}/n)  \tr(\Gamma_{\rho}W\Gamma_{r}W),
\]
with $W=\diag(w_{1},w_{2})$ and where
$T=(  t_{i,j})  $ is a $2\times2$ matrix with $t_{11}=1-w_{1},$
$t_{22}=1-w_{2},$ $t_{12}=-w_{1}r$ \ and $t_{21}=-w_{2}r$.

 In Figures~\ref{MSPEcompare21}-\ref{MSPEcompare23}, we compare the minimal attainable MSPE via theoretical MSPE curves for the LS, ridge regression, Lasso, elastic net, simplified garrote, and adaptive SPLIT methods as a function of $\beta_{2}$ when $\beta_{1}=1$. The sample size is $n=10$ and the error variance $\sigma^{2}$ was chosen such that the signal-to-noise ratio (SNR) equals 1, 3 or 5. The correlation between the predictors in the training sample is given by $r=0.1, 0.5$ or $0.9$ and we consider the testing sample correlations $\rho=0.1, 0.5$ or $ 0.9 $. Since we use a small training sample size of $ n=10 $, the probability that the empirical correlation $ r $ between the predictor variables is greater than the underlying correlation $ \rho $ is high. For this reason, we only consider the cases where $ r \geq \rho $.

From these plots, we can see that all methods can improve on LS as expected. Moreover, the SNR has little effect on the order of performance. The elastic net and ridge regression perform identically across all cases considered and for all values of $ \beta_2 $. Ridge regression has a lower minimum attainable MSPE than Lasso, across all cases considered. The garrote performs better than the elastic when $ r=0.1 $ and $ r=0.5 $. When $ r=0.1 $ and $ r=0.5 $, the minimum attainable MSPE for the garrote is often lower than the elastic net, particularly when $ |\beta_2| $ is further away from zero. Otherwise, when $ r=0.9 $, the elastic net is generally the second best method.

The adaptive SPLIT consistently yields the best performance. The relative improvement of adaptive SPLIT over the competitors is greater when there is a larger the difference between the training data empirical correlation $ r $ and $ \rho $. Interestingly, the minimum attainable MSPE by the adaptive SPLIT does not appear to be affected highly by $ r $ and $ \rho $, as the highest theoretical MSPEs reached by each of the adaptive SPLIT curves are very similar for a fixed SNR.

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{two_vars_1.png}
	\caption{Theoretical MSPE curves for two variables and SNR = 1.}
	\label{MSPEcompare21}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{two_vars_3.png}
	\caption{Theoretical MSPE curves for two variables and SNR = 3.}
	\label{MSPEcompare23}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{two_vars_5.png}
	\caption{Theoretical MSPE curves for two variables and SNR = 5.}
	\label{MSPEcompare25}
\end{figure}

\section{Theoretical MSPE curves via simulation}

Extending the results for theoretical MSPE curves for more than two variables is most easily done via simulation. The ultimate goal of the following simulation is to provide evidence, on a theoretical level,
that splitting variables is optimal under certain conditions, motivating research on methods
that can find optimal splits for linear models in high-dimensional settings while being computationally feasible. We extend the theoretical MSPE curves from Section 4 to six variables.

\subsection{Data Generation}

We first give the details of the data generating mechanism
for the simulation experiment, in which the simulated training data has a specified target empirical covariance matrix. We assume a sample size $ n $, a population covariance $ \Gamma_{\rho}, $ and a target sample covariance $ \Gamma_r $. The procedure to generate the data with target empirical covariance is as follows.
\begin{enumerate}
	\item Generate the independent training observations\[
	\widetilde{\mathbf{X}}_{i}=(  \widetilde{X}_{i1},\dots,\widetilde{X}%
	_{i6})  \thicksim N(  \mathbf{0}, \Gamma_{\rho})  ,\text{
		\ \ }i=1, \dots, n,
	\]
	\[
	\widetilde{X}=\left(
	\begin{tabular}
	[c]{c}%
	$\widetilde{\mathbf{X}}_{1}$\\
	$\widetilde{\mathbf{X}}_{2}$\\
	$\vdots$\\
	$\widetilde{\mathbf{X}}_{n}$%
	\end{tabular}
	\ \right) ,\text{ \ \ }n\times6\text{ \ matrix.}
	\]

	\item Let $X$ be the scaled matrix $\widetilde{X}$, such that the columns have
	mean zero and variance 1, and set
	\[
	S=\Cov(  X).
	\]

	\item Get the spectral decomposition of of the matrix $ S $
	\[
	S=\sum_{j=1}^{6}\lambda_{j}\mathbf{a}_{j}\mathbf{a}_{j}^{\prime},\text{ }%
	\]
	and set
	\[
	\widetilde{Z}=XP,
	\]
	with%
	\[
	P=(  \mathbf{a}_{1},\dots,\mathbf{a}_{6})  .
	\]

	\item Denote by $Z$ the transformed matrix $\widetilde{Z}$ after scaling its
	columns so that the columns of $Z$  have mean zero and variance 1.

	\item We construct the data matrix $Y$ with sample covariance $\Gamma_{r}$. Let $ Y^j $ and $ Z^j $ be the $ j^{th} $ columns of the matrices $ Y $ and $ Z $ respectively for $ j=1, \dots, 6 $.
	The construction is done column by column, as follows. The first column is given by
	\[
	Y^1=Z^1.
	\]

	The second column is given by
	\[
	Y^2=\frac{a_{0}}{\sqrt{1+a_{0}^{2}}}Z^1+\frac{1}{\sqrt{1+a_{0}^{2}}%
	}Z^2,
	\]
	where the requirement $ \Cov(Y^1, Y^2) = [\Gamma_r]_{12}^{2} $ is satisfied by setting
	\[
	a_{0}^{2}=\frac{[\Gamma_r]_{12}^{2}}{1-[\Gamma_r]_{12}^{2}}.%
	\]

	Likewise, the third column is given by
	\[
	Y^3=\frac{a_{1}}{\sqrt{1+a_{1}^{2}+a_{2}^{2}}}Z^1+\frac{a_{2}}%
	{\sqrt{1+a_{1}^{2}+a_{2}^{2}}}Z^2+\frac{1}{\sqrt{1+a_{1}^{2}+a_{2}^{2}}%
	}Z^3%
	\]
	The values of $a_{1}$ and $a_{2}$ are determined by the conditions
	\begin{align*}
	\Cov(Y^{1},Y^{3}) &= [\Gamma_r]_{13}^{2}
	\end{align*}
	and
	\begin{align*}
	\Cov(Y^{2},Y^{3}) &= [\Gamma_r]_{23}^{2}.
	\end{align*}
	This process is continued until all six columns of the matrix $ Y $ have been computed.

\end{enumerate}

An \texttt{R} package \texttt{mvnTargetCor} was published on \texttt{CRAN} to implement the methodology described above to simulate data with a target empirical covariance.

The properties of data with pre-specified empirical correlation is considered further research.

\subsection{Split Combinations} \label{sec:SplitCombs}
We now address the problem of determining the total possible number of splits of $ p $ variables into $ G $ groups, for $ p \geq G $.

\paragraph{}
Let $ a(p,G) $ be the number of splits of $ p $ variables into $ G $ groups. Let $p_{1},p_{2},...,p_{G}>0$ be the number of elements in groups $1,2,\dots,G,$
respectively. We first assume that no variables are left out,
\[p_{1}+p_{2}%
+\cdots+p_{G}=p.\]
Let $h_i(p_{1},p_{2},...,p_{G})$ be the number of elements in the sequence $p_{1},p_{2},\dots,p_{G}$ that are equal to $ i $, $ i=1,\dots,\big\lfloor\frac{p-(G-2)}{2}\big\rfloor$. The number of possible splits of $p$ features into $G$ is given by
\[
a(p,G)  =\sum_{p_{1}\leq p_{2}\leq\cdots\leq p_{G}}\Bigg[\frac{p!}{p_{1}!p_{2}! \dots p_{G}!} \prod_{i=1}^{\big\lfloor\frac{p-(G-2)}{2}\big\rfloor}\frac{1}{h_i(p_{1},p_{2},...,p_{G})!}\Bigg]
\]

So the total number of possible splits of $ p=6 $ variables in $ G=2 $ or $ G=3 $ groups is $ a(6,2) + a(6,3) = 31 + 90 = 121 $. For the theoretical MSPE curves in Section 5, we consider any split of the six variables into two or three groups, as well as all variables in a single group. So there are a total of 122 possible splits considered for the adaptive SPLIT estimator.

Note that the number of possible splits of $ p=15 $ features into $ G=3 $ groups is 6,137,951. So for high-dimensional problems, the issue of computational infeasibility for an exhaustive search becomes apparent.

We could also consider the total number of possible splits by allowing some variables to be left out, $ b(p,G) $, given by
\[
b(  p,G)  = \sum_{j=G}^{p} {p \choose j} \ a(j,G)%
\]
The number of possible splits is even larger if we allow the variables to be shared by the different groups.

An \texttt{R} package to output all possible splits of $p$ variables into $G$ groups has been published on \texttt{CRAN}, under the name \texttt{SplitCombinations}.

\subsection{Theoretical MSPE Computation}

We now describe the steps to approximate the theoretical MSPE curves for the methods of interest. We give a full description of the procedure for SPLIT as it contains the most parameters. The steps can easily be adapted for the approximations of the theoretical MSPE curves of the other methods. Least Squares theoretical MSPE can be computed directly, so we forego providing any details.

Note that the essence of the steps given below is the following: we wish to estimate the theoretical MSPE of an estimator for the optimal parameters of this estimator, under a particular data simulation scheme.

Let $\boldsymbol{\beta }\ \ $be given and let $D=( X,
\mathbf{y}) $ be a data set such that \ \  \
\begin{equation}
\mathbf{y=}X\boldsymbol{\beta +\varepsilon}, \text{ \ } \boldsymbol{\beta} = \left(
\begin{tabular}
[c]{c}%
$ \beta_1 $ \\
$ \beta_1 $ \\
$ \beta_2 $ \\
$\vdots$\\
$\beta_2$\\
\end{tabular}
\ \right), \text{ \ }\boldsymbol{%
	\varepsilon }\thicksim N( \mathbf{0},\sigma ^{2}I) ,\text{ \ }%
\widehat{\Cov}( X) =\Gamma _{r}.  \label{eq-6}
\end{equation}

The test data, independent from $ D $, is given by
\begin{equation}
y_{0}=\mathbf{x}_{0}^{\prime }\boldsymbol{\beta +\varepsilon }_{0},\text{
	with \ }\boldsymbol{\varepsilon }_{0}\thicksim N( 0,\sigma
^{2}) \text{ and }\mathbf{x}_{0}\thicksim N( \mathbf{0},\Gamma
_{\rho }).  \label{eq-7}
\end{equation}

Let $P$ be a particular partition of the six variables into $G$ groups ($%
G=1,2,3)$ and let $\mathbf{w}=( w_{1},\dots,w_{p}) $ \ be some arbitrary set
of weights. We wish to compute
\begin{align*}
g( \boldsymbol{\widetilde{\beta}_{\mathbf{w}}}) =\min_{\mathbf{w}, P\epsilon \mathcal{P}}E[ ( \widetilde{y}_{\mathbf{w}}(D,\mathbf{x}_{0};P,\mathbf{w}) -y_{0})^{2}|\widehat{\Cov}( X) =\Gamma _{r}],
\end{align*}
where $ \widetilde{y}_{\mathbf{w}}(D,\mathbf{x}_{0};P,\mathbf{w}) $ is the SPLIT predicted value for test data $ \mathbf{x}_{0} $ using training data $ D $, fixed partition $ P $ and fixed weights $ \mathbf{w} $. To estimate $ g(\boldsymbol{\widetilde{\beta}_{\mathbf{w}}}) $, we proceed as follows:

\begin{enumerate}
	\item Genarate $D_{1},\dots,D_{N},$ independent data sets,
	each of them satisfying (\ref{eq-6}). \

	\item Generate $( y_{01},\mathbf{x}_{01}) ,\dots,( y_{0M},%
	\mathbf{x}_{0M}) $ independent satisfying  (\ref{eq-7}).

	\item Fix $P\epsilon \mathcal{P}$, and a
	conforming $\mathbf{w}$. We define the approximation of $g( \boldsymbol{\widetilde{\beta}_{\mathbf{w}}}) $ as
	\begin{align*}
	\widehat{g}(\boldsymbol{\widetilde{\beta}_{\mathbf{w}}}) = \min_{\mathbf{w}, P \epsilon \mathcal{P}} \frac{1}{N}\frac{1}{M}\sum_{i=1}^{N}%
	\sum_{j=1}^{M}( \widetilde{y}_{\mathbf{w}}( D_{i},\mathbf{x}%
	_{0j};P,\mathbf{w}) -y_{0j}) ^{2}
	\end{align*}
\end{enumerate}

The minimum attainable MSPE for ridge regression $ g(\widehat{\boldsymbol{\beta}}_{\lambda}) $, Lasso $ g(\widetilde{\boldsymbol{\beta}}_{\lambda}) $, elastic net $ g(\widehat{\boldsymbol{\beta}}_{\alpha}) $ and garrote $ g(\widetilde{\boldsymbol{\beta}}_{\boldsymbol{\omega}}) $ are approximated similarly using the prediction functions $ \widehat{y}_{\lambda}( D_{i},\mathbf{x}%
_{0j};\lambda) $, $ \widetilde{y}_{\lambda}( D_{i},\mathbf{x}%
_{0j};\lambda) $, $ \widehat{y}_{\alpha}( D_{i},\mathbf{x}%
_{0j};\alpha,\lambda) $, $ \widehat{y}_{\boldsymbol{\omega}}( D_{i},\mathbf{x}%
_{0j};\boldsymbol{\omega}) $ respectively and minimizing over their tuning parameters in the last step.

\subsection{Results}

As in the case of two variables in Section 4, all methods improve on LS and the SNR has a limited effect on the order of performance. Ridge regression and elastic net have an identical performance, outperforming the Lasso across all scenarios. This confirms the conjecture of Section 4 that on a theoretical level, it is optimal to forego any $ L_1 $ penalization ($ \alpha=0 $ is always optimal for the elastic net). With the exception of the case of low correlation in the training data with $ r=0.1 $ and test data with $ \rho=0.1 $, the elastic net has a lower minimum attainable MSPE than the garrote, making it the best competitor against SPLIT.

The relative improvement of adaptive SPLIT over the other methods is greater for six variables than for the case of two variables. The competitors break down when there is spurious correlation in the training data, whereas the minimum attainable MSPE of adaptive SPLIT is generally constant across all cases considered.

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{six_vars_1.png}
	\caption{Theoretical MSPE curves for six variables and SNR = 1.}
	\label{MSPEcompare61}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{six_vars_3.png}
	\caption{Theoretical MSPE curves for six variables and SNR = 3.}
	\label{MSPEcompare63}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{six_vars_5.png}
	\caption{Theoretical MSPE curves for six variables and SNR = 5.}
	\label{MSPEcompare65}
\end{figure}

\section{Relaxation of SPLIT estimator}

The number of possible splits of $ p $ variables into $ G $ groups grows very large as the number of variables increase, see Section \ref{sec:SplitCombs}. Consequentially, an exhaustive search over all possible splits is not a reasonable approach for high-dimensional data. In the same way the Lasso is a relaxation of the problem of best subset selection, a relaxation to the problem of best split selection, SPLIT as it is referred in this article, is even more desirable.

\subsection{SplitReg estimator}

A method proposed by {\citet{Christidis19}} to fit linear regression models that split the set of variables into groups, SplitReg, is to minimize an objective function that encourages sparsity within each group and diversity among them. The estimated coefficients are then pooled together to form the final fit. The procedure works on top of a given penalized linear regression estimator (e.g., Lasso, elastic net) by fitting it to possibly overlapping groups of features, encouraging diversity among these groups to reduce the correlation of the corresponding predictions. An extensive simulation study and real-data applications show that in general the proposed method improves the prediction accuracy of the base estimator used in the procedure.

Suppose we have training data $\left( \mathbf{y}, \mathbf{X}\right)$, where $\mathbf{y}\in \mathbb{R}^{n}$ is a  vector of response variables
and $\mathbf{X}\in \mathbb{R}^{n\times p}$ is a design  matrix comprising  $n$  measurements  on $p$
features.  We consider the linear regression model  $ \mathbf{y} = \mathbf{X}
\boldsymbol{\beta }+\boldsymbol{\varepsilon}$, where $\boldsymbol{\beta}\in \mathbb{R}^{p}$  is the vector of regression coefficients and $\boldsymbol{\varepsilon} \in \mathbb{R}^{n}$ is a vector of independent regression errors.

SplitReg is a minimizer $\hat{\boldsymbol{\beta}} =\left( \hat{\boldsymbol{\beta}}^1,\cdots,\hat{\boldsymbol{\beta}}^G \right)$ of an objective function  of the form
\begin{equation}
O(\mathbf{y}, \mathbf{X}, \boldsymbol{\beta}^1, \cdots,  \boldsymbol{\beta}^G) = \sum_{g=1}^{G}  \left\{ \frac{1}{2n}\Vert \mathbf{y} - \mathbf{X}
\boldsymbol{\beta}^g \Vert^{2}_2 + \lambda_{s}P_s\left(\boldsymbol{\beta}^g\right) +\frac{\lambda_{d}}{2} \sum_{h\neq g}^{G}P_d\left(\boldsymbol{\beta}^h,\boldsymbol{\beta}^g\right)\right\},
\label{eq:step_reg}
\end{equation}
where $G$ is the number of groups, $ P_s(\boldsymbol{\beta })$ is a penalty function that encourages shrinkage and  $P_d\left(\boldsymbol{\beta}^h,\boldsymbol{\beta}^g\right)$ is a penalty function that encourages {\it diversity}. The constants $\lambda_s, \lambda_d\geq0 $   determine the amount of shrinkage and diversity, respectively. The diversity penalty function for SplitReg is
\begin{equation*}
P_d\left(\boldsymbol{\beta}^h,\boldsymbol{\beta}^g\right) = \sum_{j=1}^p |\beta_j^g||\beta_j^h|.
\label{eq:div_fun}
\end{equation*}
The sparsity penalty $P_{s}$ is taken to be the elastic net penalty
\begin{equation*} \label{eq:elastic-net}
P_{s}(\boldsymbol{\beta }^{g})=\left( \frac{(1-\alpha
	)}{2}\Vert \boldsymbol{\beta }^{g}\Vert _{2}^{2}+\alpha \Vert \boldsymbol{%
	\beta}^{g}\Vert_{1}\right) ,
\end{equation*}%
where $\alpha \in \lbrack 0,1]$.

 The problem of minimizing \eqref{eq:step_reg} can be posed as an `artificial'
multivariate linear regression problem. Let $\mathbf{Y}\in\mathbb{R}^{n\times
	G}$ be the matrix with the vector $\mathbf{y}$ repeated $G$ times as columns.
$$
O(\mathbf{y}, \mathbf{X}, \boldsymbol{\beta}) = \frac{1}{2n}\Vert \mathbf{Y} - \mathbf{X}
\boldsymbol{\beta}\Vert_{F}^{2} + \lambda_{s} \left( \frac{(1-\alpha)}{2}\Vert
\boldsymbol{\beta}\Vert_{F}^{2} + \alpha \Vert \boldsymbol{\beta} \Vert_1\right)+ \frac{\lambda_{d}}{2}
\left( \Vert  \vert\boldsymbol{\beta}\vert^{\prime} \vert\boldsymbol{\beta}\vert\Vert_1 -
\Vert\boldsymbol{\beta}\Vert_{F}^{2} \right),
$$
where $\Vert \cdot\Vert_{F}$ is the Frobenius norm, $\vert \boldsymbol{\beta}\vert$ stands
for taking the absolute value coordinate-wise and $\Vert \cdot \Vert_{1}$ is the
sum of the absolute values of the entries of the matrix. It is seen that the
diversity penalty term in a sense penalizes correlations between the different
models.

Once the  estimates $\hat{\boldsymbol{\beta}}^{1}, \dots,
\hat{\boldsymbol{\beta}}^{G}$ have been obtained by minimizing (\ref{eq:step_reg}), they must aggregated them to form an overall fit or  prediction by averaging these models: if
$\mathbf{x}_0$ is a training point (new
observation),  the fitted value (prediction) is given by
\begin{equation*}
\frac{1}{G}\sum\limits_{g=1}^{G} \mathbf{x}_0%
^{\prime} \hat{\boldsymbol{\beta}}^{g}= \mathbf{x}_0^{\prime} \left( \frac{1}{G}%
\sum\limits_{g=1}^{G} \hat{\boldsymbol{\beta}}^{g}\right) = \mathbf{x}_0^{\prime} \hat{\boldsymbol{\beta}}_{*}.
\label{eq:avg_model}
\end{equation*}

\cite{Breiman1996} proposes to aggregate predictors by averaging them
according to weights determined by solving a constrained least squares problem
and calls the method \emph{stacking}.
In detail, given predictors $v_{k}(\mathbf{x}), k=1,\dots,K$, define their
leave-one-out versions as $v_{k}^{-i}(\mathbf{x}), i=1,\dots,n$. Let
$z_{k,i}=v_{k}^{-i}(\mathbf{x}_i)$. Then the weights used to form the final
predictor are defined as the non-negative constants
$\delta_{1},\dots,\delta_{K}$ that minimize
$$
\sum\limits_{i=1}^{n}\left(y_{i} - \sum\limits_{k=1}^{K}\delta_{k}z_{k,i}
\right)^{2}.
$$.

The theoretical properties of combining prediction procedures are discussed in
\cite{Yang2004} and references therein.

\subsection{Theoretical MSPE curves for SplitReg}

We now proceed to plot the theoretical MSPE curves for SplitReg and compare them with that of SPLIT. The minimum attainable MSPE for SplitReg $ g(\boldsymbol{\widehat{\beta}}_{\lambda_d}) $ may be approximated as described as in Section 5.2 using in the last step
\begin{align*}
\widehat{g}(\boldsymbol{\widehat{\beta}}_{\lambda_d}) = \min_{\lambda_d, \lambda_s, \alpha} \frac{1}{N}\frac{1}{M}\sum_{i=1}^{N}%
\sum_{j=1}^{M}( \widehat{y}_{\lambda_d}( D_{i},\mathbf{x}%
_{0j};\lambda_d, \lambda_s, \alpha) -y_{0j}) ^{2}
\end{align*}
where $ \widehat{y}_{\lambda_d}( D_{i},\mathbf{x}_{0j};\lambda_s, \alpha, \lambda_d) $ is the prediction function for SplitReg for fixed parameters $ \lambda_s, \alpha, \lambda_d $. It is clear from \ref{eq:step_reg} that if $ \lambda_d  $= 0, then the SplitReg estimator is simply the elastic net. The minimum attainable MSPE for SplitReg is therefore never larger than that of the elastic net.

We also consider the SplitReg estimator while optimizing over the weights $ \boldsymbol{\delta} = (\delta_1, \dots, \delta_G) $ in the aggregation of the models
\begin{equation*}
\hat{\boldsymbol{\beta}}_{*} = \sum\limits_{g=1}^{G} \delta_g \hat{\boldsymbol{\beta}}^{g}.
\label{eq:avg_model_weights}
\end{equation*}
In this case the minimum attainable MSPE for SplitReg $ g(\boldsymbol{\widetilde{\beta}}_{\lambda_d}) $ is achieved in the minimization
\begin{align*}
\widehat{g}(\boldsymbol{\widetilde{\beta}}_{\lambda_d}) = \min_{\lambda_d, \lambda_s, \alpha, \boldsymbol{\delta}} \frac{1}{N}\frac{1}{M}\sum_{i=1}^{N}%
\sum_{j=1}^{M}( \widetilde{y}_{\lambda_d}( D_{i},\mathbf{x}%
_{0j};\lambda_d, \lambda_s, \alpha, \boldsymbol{\delta}) -y_{0j}) ^{2}
\end{align*}
where $ \widetilde{y}_{\lambda_d}( D_{i},\mathbf{x}_{0j};\lambda_d, \lambda_s, \alpha, \boldsymbol{\delta}) $ is the prediction function for SplitReg for fixed parameters $ \lambda_d, \lambda_s, \alpha $ and model aggregation weights $ \boldsymbol{\delta} $.

The results are presented in Figures 7-9. Both SplitReg curves are closer to the SPLIT curves compared to the competitors in Section 5. There is a noticeable difference between SplitReg and SPLIT when the correlation is low in the training set is low, which is expected as SplitReg behaves like the elastic net in this case. This is confirmed by our experiment as the estimator was optimal when $ \lambda_d $ was close to 0. When the correlation in the training set was high, SplitReg replicates SPLIT closely. In some cases, the SplitReg estimator with optimal aggregation weights actually had a lower attainable MSPE than SPLIT. This is observed across all SNR values considered. We expect that for higher-dimensional problems, SplitReg is an even more powerful relaxation of SPLIT as the estimator allows for overlap between the groups.

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{optimal_1.png}
	\caption{Theoretical MSPE curves for SplitReg with six variables and SNR = 1.}
	\label{MSPEsplitreg1}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{optimal_3.png}
	\caption{Theoretical MSPE curves for SplitReg with six variables and SNR = 3.}
	\label{MSPEsplitreg3}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[height=18cm]{optimal_5.png}
	\caption{Theoretical MSPE curves for SplitReg with six variables and SNR = 5.}
	\label{MSPEsplitreg5}
\end{figure}

\section{Conclusions}

We have shown that in addition to regularization, splitting variables can be used to reduce the variability of LS. Both the generalized variance and the total variance of linear functions of the SPLIT coefficients are smaller than those based on LS coefficients. This smaller variability can result in a lower MSPE as illustrated in a simple setting by comparing the minimum attainable MSPE of an adaptive SPLIT method with ridge regression, Lasso, elastic net and simplified garrote. We expect that the benefits are higher in high-dimensional regression models where spurious correlations frequently occur, as demonstrated in Section 5 by increasing the number of predictors. However, determining the optimal splits is a difficult task in practice. An exhaustive search over all possible splits is not feasible, so approximate methods need to be developed. Promising results have been achieved by SplitReg, a method which minimizes loss function that penalizes for correlation between the models. We have shown that SplitReg is a promising relaxation of an exhaustive search over all possible splits and can potentially improve on SPLIT as the former allows for variables to be split in more than one model.

redpajama_set_name: RedPajamaArXiv


Phoenix News Bulletin: Phoenix Software International Acquires LifeTime Software Technologies, Inc.
Phoenix Software International Acquires LifeTime Software Technologies, Inc.
We are pleased to announce that Phoenix Software International, Inc. (PSI) and LifeTime Software Technologies, Inc. (LifeTime) signed an agreement Monday, January 22, 2001, wherein PSI acquired LifeTime. PSI will support customers of LifeTime's data entry product, LifeWorks from its Viking Software Solutions Division located in Tulsa, Oklahoma. LifeTime's software support personnel will provide training and transition assistance. LifeTime's customers can be assured a smooth transition, and expect the same high quality software support they have grown to expect.
Beginning January 29th, customers can reach the technical support hotline by dialing (918) 491-6144.